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2024-09-04T02:54:58.109592

2003.03638

arxiv-papers

Minimizing the number of optimizations for efficient community dynamic flux balance analysis. James D. Brunner1†*,Nicholas Chia1† 1Department of Surgery, Center for Individualized Medicine Microbiome Program, Mayo Clinic, Rochester, MN, USA †Current Address: Mayo Clinic, 200 First St. SW, Rochester, MN, USA *<EMAIL_ADDRESS> ## Abstract Dynamic flux balance analysis uses a quasi-steady state assumption to calculate an organism’s metabolic activity at each time-step of a dynamic simulation, using the well-known technique of flux balance analysis. For microbial communities, this calculation is especially costly and involves solving a linear constrained optimization problem for each member of the community at each time step. However, this is unnecessary and inefficient, as prior solutions can be used to inform future time steps. Here, we show that a basis for the space of internal fluxes can be chosen for each microbe in a community and this basis can be used to simulate forward by solving a relatively inexpensive system of linear equations at most time steps. We can use this solution as long as the resulting metabolic activity remains within the optimization problem’s constraints (i.e. the solution to the linear system of equations remains a feasible to the linear program). As the solution becomes infeasible, it first becomes a feasible but degenerate solution to the optimization problem, and we can solve a different but related optimization problem to choose an appropriate basis to continue forward simulation. We demonstrate the efficiency and robustness of our method by comparing with currently used methods on a four species community, and show that our method requires at least $91\%$ fewer optimizations to be solved. For reproducibility, we prototyped the method using Python. Source code is available at `https://github.com/jdbrunner/surfin_fba`. ## Author summary. The standard methods in the field for dynamic flux balance analysis (FBA) carries a prohibitively high computational cost because it requires solving a linear optimization problem at each time-step. We have developed a novel method for producing solutions to this dynamical system which greatly reduces the number of optimization problems that must be solved. We prove mathematically that we can solve the optimization problem once and simulate the system forward as an ordinary differential equation (ODE) for some time interval, and solutions to this ODE provide solutions to the optimization problem. Eventually, the system reaches an easily check-able condition which implies that another optimization problem must be solved. We compare our method against typically used methods for dynamic FBA to validate that it provides equivalent solutions while requiring fewer linear-program solutions. ## Introduction. ### Microbial communities and human health. The makeup of microbial communities is often complex, dynamic, and hard to predict. However, microbial community structure has a profound effect on human health and disease [1, 2, 3, 4, 5, 6, 7]. These two facts have lead to significant interest in mathematical models which can predict relative abundances among microbes in a community. Various dynamical models have been proposed to explain and predict microbial community population dynamics [8, 9, 10, 11, 12]. Among these are models which propose that interactions between species are mediated by the metabolites that each species produces and consumes [13, 14], and there is significant evidence that these models perform better than models which depend on direct interaction between species [15, 16]. Recently, advances in genetic sequencing have allowed the creation of genome- scale models (GEMs) that reflect the internal network of cellular metabolism, and can therefore be used to predict metabolite use and production [17, 18, 19]. This technique can be extended to microbial community modeling by combining GEMs of different species. There has been significant interest in using GEMs to predict relative populations of stable microbial communities [20, 21, 22, 23, 24, 25, 26]. Community metabolic modeling can not only predict relative populations, but also holds the potential to predict and explain the community metabolite yield, which can have a profound effect on health [4]. Furthermore, model repositories such as the online bacterial bioinformatics resource _PATRIC_ [27] or the _BiGG model database_ [28] make it possible to build community models using information from individual species investigations. GEMs can be used to predict microbial growth rates as well as metabolite consumption and production rates using a process called _flux balance analysis_ (FBA). Because these predictions appear in the form of rates of change, they can be used to define a metabolite mediated dynamical model, simply by taking as a vector field the rates of change predicted by FBA. We can therefore combine the techniques of metabolite mediated dynamic modeling and community metabolic modeling to produce dynamic predictions of microbial community population size and metabolite yield. This strategy is called _dynamic FBA_ [29, 30, 31], and has recently been used to model microbial communities [32, 33, 34]. Dynamic FBA, when implemented naïvely, requires a linear optimization problem to be repeatedly solved, and carries a high computational cost for even small communities. Furthermore, _in silico_ experiments may need to be repeated many times over various environmental conditions or using various parameter choices in order to make robust conclusions or to accurately fit model parameters. As a result, implementations of dynamic FBA which depend on optimization at every time-step carry a prohibitively high computational cost when used to simulate larger microbial communities. The implementation of dynamic FBA in the popular COBRA toolbox software package [17] is done in this way, and essentially all more efficient available tools for simulating dynamic FBA fundamentally use an ODE solver approach with optimization at each time-step [31, 35, 36, 24, 37, 38]. Dynamic FBA can be improved by taking advantage of the linear structure of the optimization problem which provides a choice of basis for an optimal solution that may be reused at future time-steps [39, 40]. However, the optimizations that are required by this strategy involve solutions with non- unique bases. This means that a basis chosen at random may not provide an optimal solution to the linear program at future time-steps because it provides a solution that is non-optimal or infeasible. In order to implement dynamic FBA without optimizing at each time step, we use an optimal basic set for the FBA linear optimization problem to create a system of linear equations whose solutions at future time-steps coincide with the solutions to the FBA optimization problem. To solve the problem of non- uniqueness among bases, we prove that there exists a choice of basis that allows forward simulation for a given optimal flux solution and provide a method to choose this basis. Note that this method does not choose among a set of non-unique optimal flux solutions, but instead chooses a basis for a single given optimum. To choose among multiple optimal flux solutions, biological, rather than mathematical, considerations should be used. In this manuscript, we detail how dynamic FBA can be simulated forward without re-optimization for some time interval, and give a method for doing so. We propose conditions on an optimal basic set for the FBA linear optimization problem which allows for forward simulation, and we prove that such a choice exists. We then detail how to choose this basis set, and finally give examples of simulations which demonstrate the power of our method. For reproducibility, we make a prototype implementation of our method in the Python language available at `https://github.com/jdbrunner/surfin_fba`. ## Background ### Flux balance analysis. With the advent of genetic sequencing and the resulting genome scale reconstruction of metabolic pathways, methods have been developed to analyze and draw insight from such large scale models [18]. To enable computation of relevant model outcomes, constraint based reconstruction and analysis (COBRA) is used to model steady state fluxes $v_{i}$ through a microorganism’s internal metabolic reactions under physically relevant constraints [18]. One of the most basic COBRA methods, called _flux balance analysis_ (FBA) optimizes some combination of reaction fluxes $\sum\gamma_{i}v_{i}$ which correspond to increased cellular biomass, subject to the constraint that the cell’s internal metabolism is at equilibrium: $\Gamma\bm{v}=0$ (1) where $\Gamma$ is the _stoichiometric matrix_ , a matrix describing the stoichiometry of the metabolic model. This optimization is chosen because it reflects the optimization carried out by nature through evolution [18]. The vector $\bm{\gamma}=(\gamma_{1},\gamma_{2},...,\gamma_{d})$ is an encoding of cellular objectives, reflecting the belief that the cell will be optimized to carry out these objectives. The constraint Eq. 1 means that any optimal set of fluxes found by FBA corresponds to a steady state of the classical model of chemical reaction networks [41]. This reflects the assumption that the cell will approach an internal chemical equilibrium. The optimization is done over a polytope of feasible solutions defined by the inequalities $v_{i,min}\leq v_{i}\leq v_{i,max}$, or possibly more complicated linear constraints. See Fig. 1 for a geometric representation of an example of the type of linear optimization problem that is carried out. By convention, forward and reverse reactions are not separated and so negative flux is allowed. Linear optimization problems like FBA often give rise to an infinite set of optimal flux vectors $\bm{v}=(v_{1},v_{2},...,v_{d})$. Geometrically, this set will correspond to some face of the polytope of feasible solutions. To draw conclusions despite this limitation, many methods have been developed to either characterize the set of optimal solutions, as with flux variability analysis (FVA), or enforce more constraints on the network to reduce the size of this set, as with loopless FVA [18]. ### Dynamic FBA. FBA provides a rate of increase of biomass which can be interpreted as a growth rate for a cell. Furthermore, a subset of the reactions of a GEM represent metabolite exchange between the cell and its environment. By interpreting constraints on nutrient exchange reactions within the metabolic network as functions of the available external metabolites and fluxes of exchange reactions as metabolite exchange rates between the cell and its environment, the coupled system can be modeled. The simplest way to do this is to use an Euler method, as in [30]. In addition to Euler’s method, more sophisticated ODE solvers may be used in the so-called “direct” method of simply recomputing the FBA optimization at every time-step. This can provide better solution accuracy and potentially larger time-steps, but may also require more than one FBA optimization at each time-step. For instance, the Runge-Kutta fourth order method [42] requires four FBA solutions at each time step. Direct methods are implemented in the COBRA toolbox [17] and are the central algorithm in many modern tools, including those of Zhuang et al. [31, 35], Harcombe et al. [36], Zomorrodi et al. [24], Louca and Doebeli [37], and Popp and Centler [38]. Notably, any direct method requires at least one complete recalculation of the network fluxes _at each time-step_. However, resolving the system at each time step is not necessary, as the solution the optimization problem at some initial time can actually be used to compute future optimal solutions. Höffner et al., [40], used this observation to introduce a variable step-size method for dynamic FBA. In that method a basic index set is chosen by adding biological constraints to the optimization problem hierarchically until a unique optimal flux vector is found. The challenge of such an approach is in choosing the basis for the optimal solution, as the optimal basis is not guaranteed to be unique even for a unique optimal flux solution. In fact, due to the nature of the method of Höffner et al. and of our method, any optimization past the initial solution that must be carried out is guaranteed to have a solution with a non-unique basis. Furthermore, many choices of optimal basis will not provide a solution for future time-steps, so that choosing among these bases must be done intelligently. Unfortunately, Höffner et al. [40] do not provide a method for choosing among non-unique bases for a single linear program solution. Our method seeks to solve this problem by choosing a basis from among the possibilities provided from an FBA solution which is most likely to remain optimal as simulation proceeds forward. We therefore prioritize reducing the number of times the linear program must be solved, choosing our basis based on the mathematical properties of the system which gives the best chance of providing a solution at future time-steps. Additionally, a method described as the “dynamic optimization approach” was introduced in Mahadevan et al., [29], however this method is computationally expensive. In particular, the method given in [29] involves optimizing over the entire time-course simulated, and so is formulated as a non-linear program which only needs to be solved once. While this method requires only one optimization, this optimization is itself prohibitively difficult due to the dimensionality of the problem growing with the fineness of time- discretization. ### The dynamic FBA model for communities. We can write a metabolite mediated model for the population dynamics of a community of organisms $\bm{x}=(x_{1},...,x_{p})$ on a medium composed of nutrients $\bm{y}=(y_{1},...,y_{m})$: $\displaystyle\dot{x}_{i}$ $\displaystyle=g_{i}(\bm{\psi}_{i}(\bm{y}))x_{i}$ (2) $\displaystyle\dot{y}_{j}$ $\displaystyle=-\sum_{i=1}^{p}\psi_{ij}(\bm{y})x_{i}$ (3) where $\bm{\psi}_{i}$ is a vector of the fluxes of nutrient exchange reactions for organism $x_{i}$ as determined by FBA. Using FBA to determine $\bm{\psi}_{i}$ is therefore a quasi-steady state assumption on the internal metabolism of the organisms $x_{i}$[43, 44, 45]. Recall that the basic assumption of flux balance analysis is that, given a matrix $\Gamma_{i}$ that gives the stoichiometry of the network of reactions in a cell of organism $x_{i}$ that growth $g_{i}(\bm{y})$ is the maximum determined by solving the following linear program [18]: $\left\\{\begin{array}[]{r}\max(\bm{v}_{i}\cdot\bm{\gamma}_{i})\\\ \Gamma_{i}\bm{v}_{i}=0\\\ \bm{c}^{1}_{i}\leq\bm{v}\leq\bm{c}^{2}_{i}(\bm{y})\end{array}\right\\}$ (4) where $\bm{c}^{1}_{i}$ is some vector of lower flux bounds while $\bm{c}^{2}_{i}(\bm{y})$ is some vector-valued function of the available metabolites which represents upper flux bounds. The key observation allowing dynamic FBA is that the optimal solution to this problem also determines $\bm{\psi}_{i}$ simply by taking $\psi_{ij}$ to be the value of the flux $v_{ij}$ of the appropriate metabolite exchange reaction. For clarity, we will relabel the elements of $\bm{v}_{i}$ so that $\psi_{ik}=v_{ij}$ if $v_{ij}$ is the $k^{th}$ exchange flux, and $\phi_{ik}=v_{ij}$ if $v_{ij}$ is the $k^{th}$ internal flux. The objective vector $\bm{\gamma}_{i}$ indicates which reactions within the cell contribute directly to cellular biomass, and so is non-zero only in elements corresponding to internal fluxes. We can therefore rewrite this vector to include only elements corresponding to internal fluxes, so that the objective of the optimization is to maximize $\bm{\gamma}_{i}\cdot\bm{\phi}_{i}$. The stoichiometry of metabolite exchange reactions is represented by standard basis vectors [18]. Therefore, we can partition $\Gamma_{i}$ as $\Gamma_{i}=\begin{bmatrix}I&-\Gamma_{i}^{*}\\\ 0&\Gamma_{i}^{\dagger}\end{bmatrix}$ (5) where $I$ is the identity matrix of appropriate size, and $\Gamma_{i}^{*}$ and $\Gamma_{i}^{\dagger}$ contain the stoichiometry of the internal reactions [18, 46, 47]. Making this change in notation allows us to see that the optimization problem of flux balance analysis is essentially internal to the cell, with external reactions providing constraints. We can see from Eq. 5 that $\ker(\Gamma_{i})$ is isomorphic to $\ker(\Gamma^{\dagger}_{i})$, and so we can maximize over this kernel. Then, the exchange reaction fluxes are determined by the internal fluxes according to the linear mapping $\bm{\psi}_{i}=\Gamma^{*}_{i}\bm{\phi}_{i}$ . The maximization of FBA becomes a maximization problem over the internal fluxes111In fact, we can project onto the kernel of the matrix $\Gamma^{\dagger}_{i}$ and so reduce the dimensionality of the problem. However, in practice this projection is not numerically stable.. We rewrite Eq. 4 using Eq. 5 and combine with Eqs. 2 and 3 to form the differential algebraic system $\displaystyle\frac{dx_{i}}{dt}=x_{i}(\bm{\gamma}_{i}\cdot\bm{\phi}_{i})$ (6) $\displaystyle\frac{d\bm{y}}{dt}=-\sum_{i}x_{i}\Gamma^{*}_{i}\bm{\phi}_{i}$ (7) $\displaystyle\left\\{\begin{array}[]{r}\max(\bm{\phi}_{i}\cdot\bm{\gamma}_{i})\\\ \Gamma^{\dagger}_{i}\bm{\phi}_{i}=0\\\ \bm{c}^{1}_{i}\leq\begin{bmatrix}\Gamma^{*}_{i}\\\ I\end{bmatrix}\bm{\phi}_{i}\leq\bm{c}^{2}_{i}(\bm{y})\end{array}\right\\}$ (11) where each $\bm{\phi}_{i}$ is determined by the optimization Eq. 11, all carried out separately. Note that this is a metabolite mediated model of community growth as defined in [15]. That is, the coupling of the growth of the separate microbes is due to the shared pool of metabolites $\bm{y}$. Each separate optimization which determines $\bm{\phi}_{i}$ at a single time-step depends on $\bm{y}$, and each $\bm{\phi}_{i}$ determines some change in $\bm{y}$. Furthermore, each optimization is carried out in a manner that depends only the status of the metabolite pool and is independent from the optimizations of other organisms. There is therefore no shared “community objective”. Instead, each organism optimizes according to only its own internal objective. We write, for full generality, upper and lower dynamic bounds on internal and exchange reactions, and assume that each function $c_{ij}(\bm{y})\in C^{\infty}$. We let $A_{i}=\begin{bmatrix}(\Gamma_{i}^{*})^{T},-(\Gamma_{i}^{*})^{T},I,-I,\end{bmatrix}^{T}$ (12) so that we can rewrite the optimization problem Eq. 11 as $\left\\{\begin{array}[]{r}\max(\bm{\phi}_{i}\cdot\bm{\gamma}_{i})\\\ A_{i}\bm{\phi}_{i}\leq\bm{c}_{i}(\bm{y},t)\\\ \Gamma^{\dagger}_{i}\bm{\phi}_{i}=\bm{0}\end{array}\right\\}$ (13) for ease of notation. We now hope to select a basic index set $\mathcal{I}_{i}$ for Eq. 13 for each organism $x_{i}$ so that each $\bm{\phi}_{i}(t)$ is a solution to the resulting linear system of equations. ## Methods. ### Linear optimization preliminaries. In this manuscript, we will rewrite the FBA optimization problem in the form $\left\\{\begin{array}[]{c}\max(\bm{\phi}\cdot\bm{\gamma})\\\ A\bm{\phi}\leq\bm{c}\\\ \Gamma^{\dagger}\bm{\phi}=0\end{array}\right\\}$ (14) where the matrices $A$ and $\Gamma^{\dagger}$ are derived from the stoichiometric matrix and flux constraints. Such a problem is often referred to as a _linear program_ (LP). We now recall some well known results from the study of linear programming (see, for example [48, 40]). First, we note that Eq. 14 can be rewritten in the so-called _standard form_ with the addition of _slack variables_ $\bm{s}=(s_{1},...,s_{n})$ which represent the distance each of the $n$ constraints is from its bound as follows: $\left\\{\begin{array}[]{c}\max(\bm{\tilde{\phi}}\cdot\bm{\tilde{\gamma}})\\\ \begin{bmatrix}\tilde{A}&I\end{bmatrix}\begin{bmatrix}\bm{\tilde{\phi}}\\\ \bm{s}\end{bmatrix}=\bm{c}\\\ \tilde{\phi}_{i}\geq 0,s_{i}\geq 0\end{array}\right\\}.$ (15) Standard form requires that we rewrite $\phi_{i}=\phi_{i}^{+}-\phi_{i}^{-}$ and then define $\bm{\tilde{\phi}}=(\phi_{1}^{+},\phi_{2}^{+},...,\phi_{d}^{+},\phi_{1}^{-},\phi_{2}^{-},...,\phi_{d}^{-})$ so that we require non-negativity of each variable, and the matrix $\tilde{A}=\left[A\;B\right]$, $B=-A$. We rewrite the problem in this form to make use of established results, and for ease of notation will write $\bm{\phi}$ instead of $\bm{\tilde{\phi}}$ when it is clear which form of the problem we are discussing. We will make use of the well-known result that there exists an _optimal basis_ or _basic set_ for a bounded linear program [49]. To state this result, we first define the notation $B_{\mathcal{J}}$ to be the matrix with columns of $[\tilde{A}\;I]$ corresponding to some index set $\\{k_{1},k_{2},...,k_{n}\\}=\mathcal{J}$, and if $B_{\mathcal{J}}$ is invertible we define the notation $\bm{w}_{\mathcal{J}}(\bm{a})$ so that $(\bm{w}_{\mathcal{J}}(\bm{a}))_{l}=\left\\{\begin{array}[]{cc}(B^{-1}_{\mathcal{I}}\bm{a})_{j}&l=k_{j}\in\mathcal{J}\\\ 0&l\not\in\mathcal{J}\end{array}\right.$ (16) for any $\bm{a}\in\mathbb{R}^{n}$. We may now define an _optimal basis_ and _optimal basic set_. ###### Definition 1. A _basic optimal solution_ to a linear program is an optimal solution along with some index set $\\{k_{1},k_{2},...,k_{n}\\}=\mathcal{I}$ such that $\bm{w}=\bm{w}_{\mathcal{I}}(\bm{c})$, where $\bm{c}$ is the vector of constraints as in Eq. 15. The variables $\\{\bm{w}_{i}|i\in\mathcal{I}\\}$ are referred to as _basic variables_ , and the index set $\mathcal{I}$ is referred to as the _basic index set_. Finally, if there exists a bounded, optimal solution to Eq. 15, then there exists a basic optimal solution and corresponding basic index set. For a given basic optimal solution vector $\bm{w}$, there may be more than one basic index set $\mathcal{I}$ such that $\bm{w}=\bm{w}_{\mathcal{I}}(\bm{b})$. Such a solution is called _degenerate_. Clearly a necessary condition for such non-uniqueness is that there exists some $k\in\mathcal{I}$ such that $w_{k}=0$. This is also a sufficient condition as long as there is some column of $[\tilde{A}\,I]$ which is not in the column space of $B_{\mathcal{I}\setminus\\{k\\}}$. ### Forward simulation without re-solving. Consider again Eq. 13, the linear program that must be solved at each time point of the dynamical system for each microbial population. Information from prior solutions can inform future time-steps as long as the region of feasible solutions has not qualitatively changed. Thus, we may only need to solve the optimization problem a few times over the course of a simulation. The key observation making this possible is that the simplex method of solving a linear program provides an optimal basis for the solution. We may often re-use this basis within some time interval, and therefore find optimal solutions without re-solving the linear program. In order to do this, we need to find a form of the solution which may be evolved in time. Thus, we turn the system of linear inequalities given in the linear program into a system of linear equations. Then, if this system has a unique solution we have reduced the task to solving a system of equations rather than optimizing over a system of inequalities. We can find such a system of equations by solving the linear program once, and using this solution to create a system of equations whose solution provides the optimal flux $\bm{\phi}_{i}$, as described above. We then use this same system to simulate forward without the need to re-solve the solution to the system of equations until there is no longer a feasible solution to the linear program. First, the linear program Eq. 13 is transformed into standard form (Eq. 15). Then, a basic optimal solution is found with corresponding basic index set $\mathcal{I}_{i}$. The dynamical system Eqs. 6, 7 and 13 can then be evolved in time using Eq. 16. This evolution is accurate until some $w_{ij}$ becomes negative (meaning that the solution is no longer a feasible solution to the linear program). At this point, a new basis must be chosen. That is, until $\bm{w}_{\mathcal{I}_{i}}(\bm{c}(t))$ becomes infeasible, we let $(\phi_{j_{1}}(\bm{c}_{i}(t)),...,\phi_{j_{m}}(\bm{c}_{i}(t)),s_{1}(\bm{c}_{i}(t)),...,s_{n}(\bm{c}_{i}(t)))=\bm{w}_{\mathcal{I}_{i}}(\bm{c}_{i}(t))$ and replace Eqs. 6, 7 and 13 with $\displaystyle\frac{dx_{i}}{dt}$ $\displaystyle=x_{i}(\bm{\gamma}_{i}\cdot\bm{\phi}_{i}(\bm{c}_{i}(t)))$ (17) $\displaystyle\frac{d\bm{y}}{dt}$ $\displaystyle=-\sum_{i}x_{i}\Gamma^{*}_{i}\bm{\phi}_{i}(\bm{c}_{i}(t))$ (18) One major difficulty in this technique is that a unique $\bm{w}_{i}$ does not guarantee a unique basis set $\mathcal{I}_{i}$. If we have some $(w_{\mathcal{I}_{i}})_{j}=0$ for $j\in\mathcal{I}_{i}$, then there exists some alternate set $\hat{\mathcal{I}}_{i}$ such that $\bm{{w}}_{\hat{\mathcal{I}}_{i}}=\bm{{w}}_{\mathcal{I}_{i}}$. Such a solution $\bm{{w}}_{\mathcal{I}_{i}}$ is called _degenerate_. In a static implementation of a linear program, the choice of basis of a degenerate solution is not important, as one is interested in the optimal vector and optimal value. However, as we will demonstrate with Example 1, the choice of basis of a degenerate solution is important in a dynamic problem. In fact, if the system given in Eqs. 17 and 18 is evolved forward until $\bm{w}_{\mathcal{I}_{i}}(\bm{c}_{i}(t))$ becomes infeasible, the time at which the system becomes infeasible is the time at which we have some $(w_{\mathcal{I}_{i}})_{j}=0$ for $j\in\mathcal{I}_{i}$. Thus, we need to resolve Eq. 13 whenever $\bm{w}_{\mathcal{I}_{i}}(\bm{c}_{i}(t))$ becomes degenerate, which will be the final time-point at which the $\bm{w}_{\mathcal{I}_{i}}(\bm{c}_{i}(t))$ is feasible. ###### Example 1. Consider the dynamic linear program $\left\\{\begin{array}[]{c}\max((1,1)\cdot\bm{v})\\\ \begin{bmatrix}1&0\\\ 0&1\\\ 1&2\end{bmatrix}\bm{v}\leq\begin{bmatrix}10\\\ 10\\\ 30-t\end{bmatrix}\\\ v_{i}\geq 0\end{array}\right\\}$ (19) In standard form at $t=0$, this linear program becomes $\left\\{\begin{array}[]{c}\max((1,1)\cdot\bm{v})\\\ \begin{bmatrix}1&0&1&0&0\\\ 0&1&0&1&0\\\ 1&2&0&0&1\end{bmatrix}\begin{bmatrix}\bm{v}\\\ \bm{s}\end{bmatrix}=\begin{bmatrix}10\\\ 10\\\ 30\end{bmatrix}\\\ v_{i},s_{i}\geq 0\end{array}\right\\}$ (20) which has the unique solution $\bm{w}=(10,10,0,0,0)$. There are three choices of basic index sets: $\mathcal{I}_{1}=\\{1,2,3\\}$, $\mathcal{I}_{2}=\\{1,2,4\\}$, and $\mathcal{I}_{3}=\\{1,2,5\\}$. The resulting bases are $\ B_{\mathcal{I}_{1}}=\begin{bmatrix}1&0&1\\\ 0&1&0\\\ 1&2&0\end{bmatrix}\quad B_{\mathcal{I}_{2}}=\begin{bmatrix}1&0&0\\\ 0&1&1\\\ 1&2&0\end{bmatrix}\quad B_{\mathcal{I}_{3}}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 1&2&1\end{bmatrix}$ Computing Eq. 16 at $t>0$ for each, we have that $B_{\mathcal{I}_{1}}$ yields $\bm{w}_{\mathcal{I}_{1}}(\bm{c}(t))=(10-t,10,t,0,0)$, $B_{\mathcal{I}_{2}}$ yields $\bm{w}_{\mathcal{I}_{2}}(\bm{c}(t))=(10,10-\nicefrac{{t}}{{2}},0,\nicefrac{{t}}{{2}},0)$, and $B_{\mathcal{I}_{3}}$ yields $\bm{w}_{\mathcal{I}_{3}}(\bm{c}(t))=(10,10,0,0,-t)$, shown in Fig. 1 for $t>0$. Thus, only $\bm{w}_{\mathcal{I}_{2}}(\bm{c}(t))$ solves the dynamic problem because $\bm{w}_{\mathcal{I}_{1}}(\bm{c}(t))$ is not optimal and $\bm{w}_{\mathcal{I}_{3}}(\bm{c}(t))$ is not feasible for $t>0$. We may follow $\bm{w}_{\mathcal{I}_{2}}$ and be insured of remaining at an optimal solution to the linear program until $t=20+\varepsilon$, at which point $\bm{w}_{\mathcal{I}_{2}}=(10,-\varepsilon/2,0,10,0)$, which is not a feasible solution to the linear program. At time $t=20$, a re-optimization is required to choose a new basis. Notice that the correct choice of basis fundamentally depends on the time- varying bound function $\bm{c}(t)=(10,10,30-t)$. To see this, consider other possible time-varying bounds $\bm{c}(t)$ which have $\bm{c}(0)=(10,10,30)$. For example, if $\bm{c}(t)=(10-t,10-t,30)$, then only $B_{\mathcal{I}_{3}}$ would give the correct $\bm{w}(\bm{c}(t))$ for $t>0$. Fig 1: Geometric representation of Example 1 for $t_{3}>t_{2}>t_{1}>0$, showing the three options for bases which are equivalent at $t=0$. Note that the best choice depends on the function $\bm{c}(t)=(10,10,30-t)$ and cannot be chosen using the static problem alone. The feasible region of the optimization problem is shown in gray. ### A basis for the flux vector. We now provide a method to choose a basis $\mathcal{I}_{i}$ for each organism $x_{i}$ in the case of a degenerate solution. Consider an optimal solution $\bm{w}_{i}$ to the linear program Eq. 15. To simulate forward according to Eqs. 17 and 18, we need for each organism $x_{i}$ a basic index set $\mathcal{I}_{i}$ such that $\left\\{\begin{array}[]{c}\bm{\dot{w}_{i}}=\bm{w}_{\mathcal{I}_{i}}\left(\frac{d}{dt}\bm{c}_{i}\right)\\\ \begin{bmatrix}\tilde{A}&I\end{bmatrix}\bm{\dot{w}}=\frac{d}{dt}\bm{c}_{i}\\\ (\bm{w}_{\mathcal{I}_{i}})_{j}=0\Rightarrow\dot{w}_{ij}\geq 0\end{array}\right\\}$ (21) so that the solution remains feasible, and furthermore that $\bm{\dot{w}}_{i}$ is optimal over the possible choice of basic index sets for $\bm{w}_{i}$. This is obviously a necessary condition for forward simulation within some non- empty time interval, and can be made sufficient (although no longer necessary) by making the inequality $(\bm{w}_{\mathcal{I}_{i}})_{j}=0\Rightarrow\dot{w}_{ij}\geq 0$ strict. We use the relaxed condition for more practical applicability. In order to develop a method based on the above observation (i.e., Eq. 21), we must know that Eq. 15 has such a solution. We therefore require the following lemma, which is proved in Appendix A: ###### Lemma 1. For a linear program with the form given in Eq. 15 with a basic optimal solution $\bm{w}$, there exists a basic index set $\mathcal{I}$ such that Eq. 21 holds and $\bm{\dot{w}}$ is optimal over the possible choice of basic index sets for $\bm{w}$. If Eq. 15 has only a non-degenerate solution, the unique basis will satisfy this requirement. The challenge remains to choose from among the possible bases of a degenerate solution. To do this, we form a second linear program analogous to Eq. 21 in the following way. We first find all constraints $\bm{a}_{j}$ (i.e. rows of $A_{i}$ or $\Gamma^{\dagger}_{i}$) such that $\bm{a}_{ij}\cdot\bm{\phi}_{i}=c_{ij}(t)$, calling this set $\mathcal{S}_{i}$. Note that this set contains all the rows of $\Gamma^{\dagger}_{i}$, for which we regard $c_{ij}(t)=0$ for all $t>0$. Note that if the solution given is a basic optimal solution, the rank of the matrix whose rows are $\bm{a}_{ij}$ for $\bm{a}_{ij}\in\mathcal{S}_{i}$ is $d$, where again $d$ is the number of internal fluxes. This is true because we include constraints of the type $a<\phi_{ij}<b$ as rows of $A_{i}$. Then, we solve the linear program $\left\\{\begin{array}[]{c}\max(\bm{\dot{w}}_{i}\cdot\bm{\gamma}_{i})\\\ \bm{a}_{j}\cdot\bm{\dot{\phi}}_{i}\leq\frac{dc_{ij}}{dt},\;\;\bm{a}_{j}\in\mathcal{S}_{i}\end{array}\right\\}$ (22) We may then use any basis $B_{\mathcal{I}}^{i}$ which solves Eq. 22 as long as it has exactly $d$ non-basic slack variables. Lemma 1 tells us that such a choice exists, although it may be necessary to manually pivot non-slack variables into the basis set given by the numerical solver222In testing the algorithm, this was necessary when using IBM ILOG CPLEX Optimization Studio to solve, but not when using The Gurobi Optimizer.. Note that we do not need the entire basis $B_{\mathcal{I}}^{i}$, but instead only need the $d\times d$ submatrix formed by rows of $A_{i}$ or $\Gamma_{i}^{\dagger}$ which correspond to non-basic slack variables in the solution to Eq. 22. These appear as rows $(\bm{a}_{i},\bm{0})$ in $B_{\mathcal{I}}^{i}$, and so this sub-matrix uniquely determines $\bm{\phi}_{i}$. We call this smaller matrix $B_{i}$, and label the set of row indices as $\mathcal{J}$. The chosen basis $\mathcal{J}$ and corresponding constraints is used to simulate forward until that particular solution becomes infeasible. At that time, we have an optimal solution to Eq. 13 simply by continuity. We therefore do not need to resolve Eq. 13 but instead re-form and solve Eq. 22. ### Pseudo-Code of the method. Below, we present as pseudo-code an outline of the method. A practical implication may need to adaptively adjust the time-step $\Delta t$ to insure that no resource is artificially over-depleted past $0$. Input: Final time $T$, initial microbial biomasses $x_{i}(0)$, initial nutrient concentrations $y_{j}(0)$, maximum inflow rates of nutrients $\alpha_{i}$, stoichiometric matrices $\Gamma_{i}$ Output: Timecourse simulation of biomass and nutrient concentrations 1 for _each microbial population $i$_ do 2 Set $\bm{w}_{i}(0)$ to be solution to eq. (13) which lies on a vertex of the feasible polytope.; 3 Solve eq. (21) to find initial basis $B_{i}$ 4 end for 5 6while _$t <T$_ do 7 Integrate eqs. (14) and (15) from $t$ to $t+\Delta t$ with $\bm{\phi}_{i}=B_{i}^{-1}\bm{c}_{\mathcal{J}}(\bm{y}(t),t)$; 8 if _$B_{i}^{-1}\bm{c}_{\mathcal{J}}(\bm{y}(t+\Delta t),t+\Delta t)$ is not a feasible solution_ then 9 reset $x_{i}=x_{i}(t)$, $y_{j}=y_{j}(t)$; 10 Solve eq. (21) to find new basis $B_{i}$, with additional constraints representing bounds violated by $B_{i}^{-1}\bm{c}_{\mathcal{J}}(\bm{y}(t),t)$. 11 end if 12 13 end while Algorithm 1 Dynamic FBA algorithm following Lemma 1. Note that for numerical stability and speed, we may store the matrices $Q_{i},R_{i}$ such that $Q_{i}R_{i}=B_{i}$ is the QR-factorization of $B_{i}$ rather than either storing $B^{-1}_{i}$ or solving completely during each time step of numerical integration. ## Results. ### Number of optimizations. We can compare the efficiency of Algorithm 1 with modern dynamic FBA methods by counting the number of times a large linear program must be carried out over the course of a simulation. At their core, state-of-art dynamic FBA tools such as _d-OptCom_ [24] and _COMETS_ [36] employ the direct method of calling an ODE-solving method with the linear program set as the right-hand-side. In the case of Euler’s method, the resulting ODE can be integrated by hand between time-steps. This last strategy is often referred to as the “static optimization approach” [40]. We compared simulation of various combinations of the organisms _Escherichia coli str. K-12 substr. MG1655_ (model iJR904), _Saccharomyces cerevisiae S288C_ (model iND705), _Pseudomonas putida KT2440_ (model iJN746) and _Mycobacterium tuberculosis H37Rv_ (model iEK1008), using models from the BiGG database [28] (see S2 File. for details). We counted the optimizations required for our model, as well as for direct methods using the numerical ODE solvers _vode_ , _zvode_ , _lsoda_ , _dopri5_ , and _dop853_ from the SciPy library. All of these numerical ODE solvers use adaptive step sizes for accuracy and stability, and so represent optimized choices of time-steps. Additionally, we compared the method of Höffner et al. as implemented in the MatLab package _DFBAlab_ [39]. For our method and the direct method, we allowed exchange of every metabolite detailed in S1 File. with initial metabolite concentrations given by that same file, and with initial biomass of $0.3$ for each species. The file `sim_comm.py` in the supplementary repository S3 Software. contains complete simulation set-up. To compare with the method of Höffner et al. [40], we use the newly available Python package from the research group of Dr. David Tourigny titled _dynamic- fba_ [50] for single organisms. This package allows simulation without secondary optimizations, as our does, and so is more similar to our prototype tool for comparison. Unfortunately, this package is currently only able to simulate single organisms at the time of publishing. For microbial communities, we can compare with the MatLab package DFBAlab [39] which requires all dynamics variables to be optimized in a secondary optimization. For simulations with DFBAlab, we use only the low-concentration metabolites D-glucose, oxygen, and cob(I)alamin from the M9 medium detailed in S1 File. as dynamically varying metabolites. It is worth noting that these are the most favorable conditions we could find for the method of H”offner [40, 39] et al. which are still biologically equivalent to our other simulations. | Solution Method ---|--- Model Combination | Algorithm 1 | Höffner | vode | zvode | lsoda | dopri5 | dop853 iJR904 | 7 | 1 | 62 | 62 | 116 | 3313 | 6228 iND750 | 4 | 1 | 91 | 91 | 85 | 3508 | 6514 iJN746 | 4 | 13 | 166 | 167 | 376 | 1176 | 2249 iEK1008 | 4 | 4 | 120 | 120 | 208 | 2768 | 5148 iJR904 + iND750 | 4 | 24 | 240 | 211 | 346 | 5586 | 10469 iJR904 + iJN746 | 30 | 479 | 420 | 420 | 744 | 2695 | 5579 iJR904 + iEK1008 | 20 | 136 | 216 | 216 | 454 | 3385 | 6411 iND750 + iEK1008 | 8 | 32 | 311 | 311 | 509 | 5284 | 9888 iJR904 + iND750 + iEK1008 | 18 | 32* | 451 | 451 | 1282 | 6225 | 11961 iJR904 + iND750 + iJN746 + iEK1008 | 56 | 672 | 1122 | 1122 | 2242 | 6837 | 13529 Table 1: Number of realizations required to simulate to time $t=5$ with no cell death or metabolite flow, using M9 minimal medium. *Simulation failed at $t=3.034277$. Fig 2: Time-points of re-optimizations required in simulations using the proposed method, the method of Höffner et al. [40] and various direct methods, shown in blue. Shown in orange are times at which the direct method solver encountered an infeasible linear program due to numerical error. ### Error estimation. Our method provides much less theoretical error in dynamic FBA solutions than traditional methods. In fact, Algorithm 1 implies that a simulation of a microbial community can be divided into time intervals on which the algorithm is exact. Of course, this assumes that the linear ODE solved in these intervals is solved exactly rather than numerically. Precisely, there exits some sequence $t_{0}=0<t_{1}<\cdots<t_{n-1}<t_{n}=T$ such that if we know the optimal flux vectors $\bm{w}_{i}(t_{l})$ at time $t_{l}$, then Lemma 1 implies the existence of a set of invertible matrices $B_{i}^{l}$ such that solutions to Eqs. 17 and 18 are solutions to Eqs. 6, 7 and 13 for $t\in[t_{l},t_{l+1}]$. Therefore, if we are able to identify the $t_{l}$ exactly, then Algorithm 1 provides exact solutions to the dynamic FBA problem Eqs. 6, 7 and 13. Of course, numerical limitations imply that we will not re-optimize precisely at each $t_{l}$, and so we must investigate the impact of this error. However, once re-optimization is done, the method is again exact. The result is that we have no local truncation error for any time step taken between re-optimization after $t_{l}$ and the interval endpoint $t_{l+1}$, except for error due to numerical integration. In comparison, direct methods from some integration error at every time step. This error depends on the integration strategy used, and so for example the Euler’s method based static optimization approach carries first order local truncation error at each time step. This can easily lead to ODE overshoot and infeasible linear programs at future time-step. Assume that $t_{l-1}$ is known exactly, and $N$ is such that $t^{1}=t_{l-1}+(N-1)\Delta t\leq t_{l}<t_{l-1}+N\Delta t=t^{2}$, so that there is some possible error in the interval $[t^{1},t^{2}]$. We can estimate the accumulated error in this time interval using a power series expansion. Let $\bm{x}(t),\bm{y}(t)$ be solutions to Eqs. 6, 7 and 13 and $\bm{\tilde{x}},\bm{\tilde{y}}$ be solutions given by Algorithm 1 for $t\in[t^{1},t^{2})$. Furthermore, let $B_{i}^{l-1}$ be the invertible matrices derived by solving Eq. 13 at $t_{l-1}$ and $B_{i}^{l}$ those derived by solving at $t_{l}$. Then, $\bm{x}(t^{1})=\bm{\tilde{x}}(t^{1})$ and $\bm{y}(t^{1})=\bm{\tilde{y}}(t^{1})$. For each $x_{i}$ we expand, assuming some regularity of the functions $\bm{c}(\bm{y})$, $x_{i}(t^{2})-\tilde{x}_{i}(t^{2})=(\Delta t)x_{i}(t_{1})(\bm{\gamma}_{i}\cdot\left((B_{i}^{l-1})^{-1}-(B_{i}^{l-1})^{-1}\right)\bm{\hat{c}}_{i}(\bm{y}(t^{1}))+o(\Delta t)$ (23) and see that this method gives first order local error in time steps that require a re-optimization. The local error, while first order, only appears at time steps in which a re- optimization occurred, and so global error will scale with the number of necessary re-optimizations. This is in contrast with the classical use of Euler’s method, which gives first order local error at every time-step, or any other direct ODE method, whose error is dependent on the solver used. We may compare the solutions provided by direct methods with those provided by the method presented in Algorithm 1 and by the method of Höffner et al. [40]. The root-sum-square ($l_{2}$) difference in results are shown in Table 2. As we argue above, direct methods are less accurate in theory that the algorithm presented in Algorithm 1. Furthermore, direct simulations routinely failed to simulate to time $t=5$ without encountering an infeasible linear program. This infeasibility is the result of numerical error accumulating throughout the simulation. The comparisons in Table 2 can be summarized by three distinct characteristics. First, in the case of _S.cerevisiae_ , the direct methods agree well with the newly presented method. Secondly, in the case of _E.coli_ and _M.tuberculosis_ , error seems to begin accumulating immediately. Finally, in the case of _P.putida_ , the simulations agree well up to some time-point at which the direct method fails and either quits entirely (as in the case of the _dopri5_ solver which returns small error) or continues at a constant value. We note that discrepancies in dynamic FBA simulation may not always be due to numerical error, but instead due to non-uniqueness in optimal flux solutions. Our method provides a strategy for choosing between non-unique representations (in the form of a basis) of a single optimal flux solution. The method of Höffner et al. [40] provides a lexicographic strategy for choosing between non-unique optimal flux solutions based on biological, rather than mathematical, considerations. We note that for complete reproducibility, our method should be integrated with some biologically based strategy for choosing between non-unique optima. | vode | zvode | lsoda | dopri5 | dop853 | Hoffner et al. ---|---|---|---|---|---|--- E.coli | 5.09933 | 5.09933 | 4.61467 | 5.09928 | 5.09928 | 4.68578 M.tuberculosis | 1.45401 | 1.45401 | 1.45417 | 1.45415 | 1.45415 | 2.48691 S.cerevisiae | 0.00426 | 0.00426 | 0.00430 | 0.00429 | 0.00429 | 3.06105 P.putida | 15.29177 | 15.29177 | 0.07080 | 15.23826 | 15.26221 | 4.78751 Table 2: $l_{2}$ difference in solutions to single-organism simulations between direct methods and the method presented in Algorithm 1. Fig 3: Simulations of _E.coli_ , _S.cerevisae_ , _M.tuberculosis_ and _P.putida_ using Algorithm 1, direct solvers, and the method of Höffner et al. In simulations of _E.coli_ _M.tuberculosis_ , there is discrepancy early in the simulation. In contrast, simulations of _P.putida_ agree up to the point that an ODE solver fails. ## Examples & applications. There has been a recent surge in interest in modeling microbial communities using genome-scale metabolic models, much of which has focused on equilibrium methods [22, 21, 4, 51, 26]. In order to capture transient behavior and dynamic responses to stimuli, dynamic FBA has also been applied to microbial communities [24, 52, 34]. However, community dynamic FBA invariable leads to a large dynamical system with a high-dimensional parameter space, often with little to know knowledge of parameter values. Any parameter fitting therefore requires repeated numerical simulation of the system. Existing tools to do this are built around a direct simulation approach, requiring many linear program solutions. By drastically reducing the number of optimizations required for numerical simulation, our approach offers the promise of efficient numerical simulation of dynamic FBA which will make parameter fitting more tractable, and may even allow conclusions without well-fit parameters. Below, we demonstrate that the problem of parameter fitting is an important one by show that experimental outcome in even small communities is sensitive to changes in kinetic parameters. Precisely, the kinetic parameters governing the uptake rate of nutrients (i.e., the parameters of the functions $\bm{c}^{2}_{i}$ in Eq. 4) have a profound effect on species competition. Next, we show how repeated simulation with randomly sampled parameters can provide some insight into community structure even without a well-fit set of nutrient uptake parameters. These examples demonstrate the importance of efficient dynamic FBA to microbial community modeling. ### Prediction dependence on nutrient uptake. The set of unknown functions $\bm{c}_{i}^{2}(\bm{y})$ in Eq. 4 present a profound problem for dynamic FBA simulation. If the behavior of the system is sensitive to the functions chosen and parameters of those functions, a single simulation will be of little use in drawing biological conclusion. In order to demonstrate that such a sensitivity exists, we repeatedly simulated the same simple community with different randomly drawn parameters. While a more realistic choice of function may be saturating or sigmoidal (as with Hill or Michaelis-Menten kinetics), for the following experiment we take these functions to be linear: $c_{ij}^{2}(\bm{y})=\kappa_{ij}y_{j},$ (24) meaning that the maximum uptake rate of nutrient $y_{j}$ by organism $x_{i}$ is proportional to the concentration of $y_{j}$. This choice minimizes the number of parameters that must be chosen for our analysis of parameter sensitivity, and is in line with an assumption of simple mass action kinetics[53, 54]. The choice of $\kappa_{ij}$ may have a profound effect on the outcome of a community simulation, as it represents how well an organism can sequester a resource when this will optimize the organism’s growth. In order study this effect in a small community, we sampled a three-species community model with $\kappa_{ij}\in(0,1)$ chosen uniformly at random. We used models for _E.coli_ , _S.cerevisiae_ and _M.tuberculosis_ downloaded from the BiGG model database[28]. We simulated with no dilution of metabolites or microbes, and no replenishment of nutrients. In every simulation, some critical metabolite was eventually depleted and the organisms stopped growing. We recorded the simulated final biomass of each organism from each simulation, and the results are shown in Fig. 4. Fig 4: (Top) Histogram of the final simulated biomass of each of _E.coli_ , _S.cerevisiae_ and _M.tuberculosis_ from 95 simulations, each with different metabolite uptake rates $\kappa_{ij}$. (Bottom) Pair-wise comparison of the final simulated biomass densities using a kernel density estimation. In red is the result of uniform uptake rates $\kappa_{ij}=1$ for all $i,j$. ### Community growth effects. As we saw in previous section, community growth outcomes depend on the choice of nutrient uptake rates $\kappa_{ij}$. Using Algorithm 1, we can perform Monte-Carlo sampling in order to understand the possible effects on some microorganism growing in some community. To do this, we randomly sample the set of uptake rates $\kappa_{ij}$ and run simulations of various communities for the chosen uptake rates. Then, the correlation between communities of final simulated biomass of some organism can be interpreted as the effect of the community on the growth of that organism. A correlation less than $1$ between growth of an organism in different communities indicates that the community is having some effect. To see the direction of this effect, we can fit a simple linear regression model (best fit line) to the final simulated biomasses. Then, the slope of this line tells us if the organism benefits or is harmed by being in one community over another. We again simulated _E.coli_ , _S.cerevisiae_ and _M.tuberculosis_ downloaded from the BiGG model database [28]. Simulations were run with the M9 medium described in S1 File., with no replenishment of resources. Each organism grew to a larger final simulated biomass when alone compared to when in a trio with the other two, which is unsurprising given the finite resources. This difference was the least pronounced for _S.cerevisiae_ , suggesting that this organism is the least negatively effected by the competition. However, this can be seen as only a preliminary observation without better estimates of uptake parameters. Best-fit lines are shown in Fig. 5. Efficient dynamic FBA allows repeated simulation with randomly sampled parameters, which gives an indication of likely behavior even without accurate parameter fitting. Fig 5: Final simulated biomass of _E.coli_ , _S.cerevisiae_ and _M.tuberculosis_ when grown alone or in pairs, for randomly sampled modeled parameters. Best fit lines indicate the average effect of the community on an organism’s growth. ## Conclusion Understanding, predicting, and manipulating the make-up of microbial communities requires understanding a complex dynamic process. Genome-scale metabolic models provide an approximation to this process through the quasi- steady state assumption which leads to dynamic flux balance analysis. However, this system is large and hard to simulate numerically, let alone analyze for qualitative behaviors. As a first step towards a thorough analysis of community of organisms modeled with dynamic FBA, an efficient method of numerical simulation would provide an essential tool. However, modern tools for simulating dynamic FBA rely on repeatedly solving an optimization problem at every time step [31, 35, 36, 24, 37, 38]. Dynamic FBA simulation can be improved by considering the structure of these linear programs so that many fewer optimizations are required. As of now, the algorithm of Höffner et al. [40] is the only published method which takes advantage of this observation. However, that method does not account for the degeneracy of solutions to the relevant linear programs, meaning that it can choose a solution that cannot be carried forward in time. We present a method that chooses a basis to for forward simulation. In contrast to the method of Höffner et al., we choose this basis in such a way that increases the likelihood that this forward simulation is actually possible. Efficient dynamic FBA will allow better parameter fitting to time-longitudinal data. Furthermore, it allows for a search of parameter space which can help predict likely model outcomes or learn maps from parameter values to model outcomes. ## Supporting information. #### S1 File. M9 medium File. `m9med.csv` defines an M9 minimal medium as adapted from Monk et al. [55]. #### S2 File. List of Models Used. `modelsUsed.csv` provides name, ID, and URL for the four models used in analysis of the method. #### S3 Software. `https://github.com/jdbrunner/surfin_fba`. Available code for the algorithm described in the Python language. This code requires the popular COBRAPy package for metabolic models. ## Acknowledgments This work was supported by funding from the DeWitt and Curtiss Family Foundation, National Cancer Institute grant R01 CA179243, and the Center for Individualized Medicine, Mayo Clinic. ## References * 1. Braundmeier AG, Lenz KM, Inman KS, Chia N, Jeraldo P, Walther-António MRS, et al. Individualized medicine and the microbiome in reproductive tract. 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Modeling microbial communities from atrazine contaminated soils promotes the development of biostimulation solutions. The ISME journal. 2019;13(2):494–508. * 53. Horn F, Jackson R. General mass action kinetics. Archive for Rational Mechanics and Analysis. 1972;47. * 54. Feinberg M. Lectures on Chemical Reaction Networks; 1979. http://www.crnt.osu.edu/LecturesOnReactionNetworks. * 55. Monk JM, Charusanti P, Aziz RK, Lerman JA, Premyodhin N, Orth JD, et al. Genome-scale metabolic reconstructions of multiple Escherichia coli strains highlight strain-specific adaptations to nutritional environments. Proceedings of the National Academy of Sciences. 2013;110(50):20338–20343. ## Appendix A Existence of desired optimal basis. ###### Lemma 1. For a linear program with the form given in Eq. 15 with a basic optimal solution $\bm{w}$, there exists a basic index set $\mathcal{I}$ such that Eq. 21 holds and $\bm{\dot{w}}$ is optimal over the possible choice of basic index sets for $\bm{w}$. ###### Proof. For convenience, we now restate Eq. 15: $\left\\{\begin{array}[]{c}\max(\bm{\tilde{\phi}}\cdot\bm{\tilde{\gamma}})\\\ \begin{bmatrix}\tilde{A}&I\end{bmatrix}\begin{bmatrix}\bm{\tilde{\phi}}\\\ \bm{s}\end{bmatrix}=\bm{c}\\\ \tilde{\phi}_{i}\geq 0,s_{i}\geq 0\end{array}\right\\}$ where we write $(\bm{\tilde{\phi}},\bm{s})=\bm{w}$. We note that there is a finite number of basic index sets for $\bm{w}$, and so we need only show that there exists $\mathcal{I}$ such that Eq. 21 holds. Then, the existence of an optimal such $\mathcal{I}$ follows trivially. If $\bm{w}$ is not degenerate, then the unique choice of basic index set $\mathcal{I}$ satisfies Eq. 21. To see this, simply note that if $\bm{w}$ is non-degenerate, then for every $i\in\mathcal{I}$, $w_{i}>0$. Thus, Eq. 21 only includes non-negativity constraints on $\dot{w}_{i}$ if $i\not\in\mathcal{I}$, and for any $i\not\in\mathcal{I}$, $\dot{w}_{i}=0$. Thus, the non-negativity constraints are enforced. The equality constraints are enforced by the definition of $\bm{w}_{\mathcal{I}}(\bm{a})$ given in Eq. 16, which implies that $[\tilde{A}\;I]\bm{w}_{\mathcal{I}}(\bm{a})=\bm{a}$ for any vector $\bm{a}\in\mathbb{R}^{n}$. In the case of a degenerate solution $\bm{w}$, we use the following procedure to choose a set of basic variables. Let $\mathcal{J}\subset\\{1,...,n\\}$ be the indices of the $n_{1}$ slack variables such that $s_{j}=0$ if $j\in\mathcal{J}$ (recalling that each $s_{i}$ is a component of the vector $\bm{w}$). Then, let $\tilde{A}_{\mathcal{J}}$ be the matrix with rows $m_{j}$ of $\tilde{A}$ for $j\in\mathcal{J}$. Next, let $\mathcal{J}^{*}$ be the indices of the $n_{2}$ non-slack variables such that $\phi_{j}=0$ and $I_{\mathcal{J}^{*}}$ the corresponding rows of the identity matrix $I$. Notice that we now have that $M\bm{\tilde{\phi}}=\begin{bmatrix}\tilde{A}_{\mathcal{J}}\\\ -I_{\mathcal{J}^{*}}\end{bmatrix}\bm{\tilde{\phi}}=\begin{bmatrix}\bm{c}_{\mathcal{J}}\\\ \bm{0}\end{bmatrix}.$ (25) and that if $w_{j}=0$ then either $j\in\mathcal{J}^{*}$ or $w_{j}=s_{k}$ where $k\in\mathcal{J}$ so that $\bm{m}_{k}\cdot\bm{\tilde{\phi}}=c_{k}$ (i.e. $s_{k}$ is a slack variable and $s_{k}=0$). Notice that because Eq. 15 has a bounded solution, then we can assume without loss of generality that if $M\in\mathbb{R}^{q\times r}$, then $\mathit{rank}(M)=r$ (i.e. $M$ is full rank) because $\bm{w}$ must satisfy at least $r$ linearly independent constraints. If this is not the case, then the problem can be projected onto a lower dimensional subspace. Consider the linear program $\left\\{\begin{array}[]{c}\max(\bm{y}\cdot\bm{\gamma})\\\ \begin{bmatrix}M&I\end{bmatrix}\begin{bmatrix}\bm{y}_{\bm{\tilde{\phi}}}\\\ \bm{y}_{\bm{s}}\end{bmatrix}=\begin{bmatrix}\frac{d}{dt}\bm{c}_{\mathcal{J}}\\\ \bm{0}\end{bmatrix}\\\ y_{j}\geq 0\end{array}\right\\}.$ (26) Assume that there is some basic optimal solution to Eq. 26 with a basic index set $\hat{\mathcal{I}}$ such that exactly $r$ slack variables are non-basic, where again $r=|\bm{\phi}|$ is the rank of the matrix $M$. This implies that there are $r$ linearly independent rows of $M$ (which we index by $\mathcal{J}^{{\dagger}}$) which form an invertible matrix $\tilde{M}$ such that $\tilde{M}\bm{y}_{\bm{\tilde{\phi}}}=\begin{bmatrix}\frac{d}{dt}\bm{c}_{\mathcal{J}^{{\dagger}}}\\\ \bm{0}\end{bmatrix}$ (27) and we can then determine $\bm{y}_{\bm{s}}$ by $\bm{y}_{\bm{s}}=\begin{bmatrix}\frac{d}{dt}\bm{c}_{\mathcal{J}}\\\ \bm{0}\end{bmatrix}-M\bm{y}_{\bm{\tilde{\phi}}}$ (28) and note that each $(\bm{y}_{\bm{s}})_{i}\geq 0$. We now rewrite $\bm{\dot{w}}=(\bm{\dot{w}}_{\bm{\tilde{\phi}}},\bm{\dot{w}}_{\bm{s}})$ from Eq. 21 and define $\bm{\dot{w}}_{\bm{\tilde{\phi}}}=\bm{y}_{\bm{\tilde{\phi}}}$ and $\bm{\dot{w}}_{\bm{s}}=\frac{d}{dt}\bm{c}-M\bm{\dot{w}}_{\bm{\tilde{\phi}}}$ (29) and conclude that this satisfies the constraints of Eq. 21. Next, we take $\bm{\tilde{\phi}}$ to be the unique solution to $\tilde{M}\bm{\tilde{\phi}}=\begin{bmatrix}\bm{c}_{\mathcal{J}^{{\dagger}}}\\\ \bm{0}\end{bmatrix}$ (30) and $\bm{s}=\bm{c}-\tilde{A}\bm{\tilde{\phi}}$. Finally, we take $\mathcal{I}=(\hat{\mathcal{I}}\setminus\mathcal{J}^{*})\cup\mathcal{J}^{c}$ and note that this basis set enforces exactly the same $r$ linearly independent constraints as $\tilde{M}$333In practice, we may simply use $\tilde{M}$ to find $\tilde{\bm{\phi}}$. We now prove that there is some basic optimal solution to Eq. 26 with a basic index set $\hat{\mathcal{I}}$ such that exactly $r$ slack variables are non- basic, where $r$ is the rank of the matrix $M$. First we note that for any basic optimal solution, if there are $r^{*}>r$ slack variables which are non-basic, then there are $r^{*}$ rows of $B_{\hat{\mathcal{I}}}$ which are non-zero only in the columns of ${M}$. Therefore, $B_{\hat{\mathcal{I}}}$ is not invertible. We can conclude that the number of non-basic slack variables is at most $r$. Next, suppose $\bm{\dot{w}}^{*}$ is a basic optimal solution with basis $\mathcal{I}^{*}$ such that there are $r^{*}<r$ slack variables which are non- basic. We would like to assume that there are at least $r$ slack variables $s_{k}^{*}$ corresponding to $r$ linearly independent constraints such that $s_{k}^{*}=0$. Recall that $\tilde{A}$ was formed with repeated (negated) columns in order write the problem in standard form (the non-negativity bounds of Eq. 15 are artificial). Therefore, we can find some vector $\bm{x}$ in the kernel of the matrix formed by the rows of $\tilde{A}$ corresponding to zero slacks which also has $\bm{x}\cdot\bm{\gamma}=0$. We can therefore find a vector $\bm{y}$ in the kernel of $\begin{bmatrix}\tilde{A}_{\mathcal{J}}&I&0\\\ -I_{\mathcal{J}^{*}}&0&I\end{bmatrix}$ which has $y_{k}=0$ if $s_{k}=0$ and $y_{j}\neq 0$ if $s_{j}\neq 0$ and $s_{j}$ corresponds to a constraint that is not a linear combination of the constraints corresponding to the $s_{k}=0$. There is at least one such constraint as long as the $0$ slack variables correspond to constraints with span less than dimension $r$, and so we can take $\bm{\dot{w}}+\lambda\bm{y}$ for some $\lambda$ and so increase the number of non-zero slack variables. We can therefore assume without loss of generality that there are at least $r$ slack variables $s_{k}^{*}$ corresponding to $r$ linearly independent constraints such that $s_{k}^{*}=0$, as was desired. We can finally choose some linearly independent set of $r$ constraints which correspond to $0$ slack variables, and call the matrix whose rows are these constraint vectors $M^{*}$. Now, because there are $r^{*}<r$ non-slack basic variables, there is some non-slack, non-basic variable $v_{j}$ such that the column $m_{j}^{*}$ of $M^{*}$ (and ${m}_{j}$ of ${M}$) is linearly independent from the columns corresponding to the $r^{*}$ non-slack basic variables. We can conclude that if $B_{\mathcal{I}^{*}}\bm{\lambda}={m}_{j}$ (31) then there is some $\lambda_{k}\neq 0$ where $k$ corresponds to the index of a slack variable with $s_{k}=0$. We can remove $k$ from the basic index set and add $j$ without changing $\bm{\dot{w}}^{*}$, and therefore preserving optimality and feasibility. We have then increased the number of non-basic slack variables, and we can repeat if necessary to form $\hat{\mathcal{I}}$ with exactly $r$ non-basic slack variables. ∎

2024-09-04T02:54:58.132395

2003.03685

arxiv-papers

# Discovering contemporaneous and lagged causal relations in autocorrelated nonlinear time series datasets Jakob Runge German Aerospace Center Institute of Data Science 07745 Jena, Germany and Technische Universität Berlin 10623 Berlin, Germany ###### Abstract The paper introduces a novel conditional independence (CI) based method for linear and nonlinear, lagged and contemporaneous causal discovery from observational time series in the causally sufficient case. Existing CI-based methods such as the PC algorithm and also common methods from other frameworks suffer from low recall and partially inflated false positives for strong autocorrelation which is an ubiquitous challenge in time series. The novel method, PCMCI+, extends PCMCI [Runge et al., 2019b] to include discovery of contemporaneous links. PCMCI+ improves the reliability of CI tests by optimizing the choice of conditioning sets and even benefits from autocorrelation. The method is order-independent and consistent in the oracle case. A broad range of numerical experiments demonstrates that PCMCI+ has higher adjacency detection power and especially more contemporaneous orientation recall compared to other methods while better controlling false positives. Optimized conditioning sets also lead to much shorter runtimes than the PC algorithm. PCMCI+ can be of considerable use in many real world application scenarios where often time resolutions are too coarse to resolve time delays and strong autocorrelation is present. ## 1 INTRODUCTION A number of frameworks address the problem of causal discovery from observational data utilizing different assumptions. Next to Bayesian score- based methods [Chickering, 2002], classical Granger causality (GC) [Granger, 1969], and the more recent restricted structural causal models (SCM) framework [Peters et al., 2017, Spirtes and Zhang, 2016], conditional independence (CI) based network learning algorithms [Spirtes et al., 2000] form a main pillar. A main representative of the CI framework in the causally sufficient case (no unobserved common drivers) is the PC algorithm [Spirtes and Glymour, 1991]. Its advantages lie, firstly, in the flexibility of utilizing a wide and growing class of CI tests, from linear partial correlation (ParCorr) and non- parametric residual-based approaches [Ramsey, 2014, Runge et al., 2019b] to Kernel measures [Zhang et al., 2011], tests based on conditional mutual information [Runge, 2018b], and neural networks [Sen et al., 2017]. Secondly, the PC algorithm utilizes sparsity making it applicable also to large numbers of variables while score- and SCM-based methods are more difficult to adapt to nonlinear high-dimensional causal discovery. Causal discovery in the time series case is partially less and partially more challenging [Runge et al., 2019a]. Obviously, time-order greatly helps in identifying causal directions for lagged links (causes precede effects). This forms the basis of GC which, however, cannot deal with contemporaneous links and suffers from the curse of dimensionality [Runge et al., 2019b]. SCM-based methods such as LiNGAM [Hyvärinen et al., 2010] and also CI-based methods [Runge et al., 2019b, Entner and Hoyer, 2010, Malinsky and Spirtes, 2018] have been adapted to the time series case. In [Moneta et al., 2011] GC is augmented by the PC algorithm. However, properties such as non-stationarity and especially autocorrelation can make causal discovery much less reliable. Here I show that autocorrelation, an ubiquitous property of time series (e.g., temperature data), is especially detrimental and propose a novel CI-based method, PCMCI+, that extends the PCMCI method from [Runge et al., 2019b] to also include discovery of contemporaneous links, which requires substantial changes. PCMCI+ is based on two central ideas that deviate from the PC algorithm and the time-series adaptations of FCI in [Entner and Hoyer, 2010, Malinsky and Spirtes, 2018]: First, an edge removal phase is conducted separately for lagged and contemporaneous conditioning sets and the lagged phase uses much fewer CI tests. Secondly, and more importantly, PCMCI+ optimizes the choice of conditioning sets for the individual CI tests to make them better calibrated under autocorrelation and increase detection power by utilizing the momentary conditional independence idea [Runge et al., 2019b]. The paper is structured as follows. Section 2 briefly introduces the problem and Sect. 3 describes the method and states theoretical results. Numerical experiments in Sect. 4 show that PCMCI+ benefits from strong autocorrelation and yields much more adjacency detection power and especially more orientation recall for contemporaneous links while better controlling false positives at much shorter runtimes than the PC algorithm. A Supplementary Material (SM) contains proofs and further numerical experiments. ## 2 TIME SERIES CAUSAL DISCOVERY ### 2.1 PRELIMINARIES We are interested in discovering time series graphs (e.g., [Runge, 2018a]) that can represent the temporal dependency structure underlying complex dynamical systems. Consider an underlying discrete-time structural causal process $\mathbf{X}_{t}=(X^{1}_{t},\ldots,X^{N}_{t})$ with $\displaystyle X^{j}_{t}$ $\displaystyle=f_{j}\left(\mathcal{P}(X^{j}_{t}),\,\eta^{j}_{t}\right)$ (1) where $f_{j}$ are arbitrary measurable functions with non-trivial dependencies on their arguments and $\eta^{j}_{t}$ represents mutually ($i\neq j$) and serially ($t^{\prime}\neq t$) independent dynamical noise. The nodes in a time series graph $\mathcal{G}$ (example in Fig. 1) represent the variables $X^{j}_{t}$ at different lag-times and the set of variables that $X^{j}_{t}$ depends on defines the causal parents $\mathcal{P}(X^{j}_{t})\subset\mathbf{X}^{-}_{t+1}=(\mathbf{X}_{t},\mathbf{X}_{t-1},\ldots){\setminus}\\{X^{j}_{t}\\}$. We denote _lagged parents_ by $\mathcal{P}^{-}_{t}(X^{j}_{t})=\mathcal{P}(X^{j}_{t})\cap\mathbf{X}^{-}_{t}$. A lagged ($\tau>0$) or contemporaneous ($\tau=0$) causal link $X^{i}_{t-\tau}\to X^{j}_{t}$ exists if $X^{i}_{t-\tau}\in\mathcal{P}(X^{j}_{t})$. Throughout this work the graph $\mathcal{G}$ is assumed _acyclic_ and the causal links _stationary_ meaning that if $X^{i}_{t-\tau}\to X^{j}_{t}$ for some $t$, then $X^{i}_{t^{\prime}-\tau}\to X^{j}_{t^{\prime}}$ for all $t^{\prime}\neq t$. Then we can always fix one variable at $t$ and take $\tau\geq 0$. Note that the stationarity assumption may be relaxed. The graph is actually infinite in time, but in practice only considered up to some maximum time lag $\tau_{\max}$. We define the set of adjacencies $\mathcal{A}(X^{j}_{t})$ of a variable $X^{j}_{t}$ to include all $X^{i}_{t-\tau}$ for $\tau\geq 0$ that have a (lagged or contemporaneous) link with $X^{j}_{t}$ in $\mathcal{G}$. We define contemporaneous adjacencies as $\mathcal{A}_{t}(X^{j}_{t})=\mathcal{A}(X^{j}_{t})\cap\mathbf{X}_{t}$. A sequence of $m$ contemporaneous links is called a _directed contemporaneous path_ if for all $k\in\\{1,\ldots,m\\}$ the link $X^{i+k-1}_{t}\to X^{i+k}_{t}$ occurs. We call $X^{i}_{t}$ a _contemporaneous ancestor_ of $X^{j}_{t}$ if there is a directed contemporaneous path from $X^{i}_{t}$ to $X^{j}_{t}$ and we denote the set of all contemporaneous ancestors as $\mathcal{C}_{t}(X^{j}_{t})$ (which excludes $X^{j}_{t}$ itself). We denote separation in the graph by $\bowtie$, see [Runge, 2018a] for further notation details. ### 2.2 PC ALGORITHM The PC algorithm is the most wide-spread CI-based causal discovery algorithm for the causally sufficient case and utilizes the Markov and Faithfulness assumptions as formally defined in Sect. S1. Adapted to time series (analogously to the methods for the latent case in [Entner and Hoyer, 2010, Malinsky and Spirtes, 2018]), it consists of three phases: First, a skeleton of adjacencies is learned based on iteratively testing which pairs of variables (at different time lags) are conditionally independent at some significance level $\alpha_{\rm PC}$ (Alg. 2 with the PC option). For lagged links, time-order automatically provides orientations, while for contemporaneous links a collider phase (Alg. S2) and rule phase (Alg. S3) determine the orientation of links. CI-based discovery algorithms can identify the contemporaneous graph structure only up to a Markov equivalence class represented as a completed partially directed acyclic graph (CPDAG). We denote links for which more than one orientation occurs in the Markov equivalence class by $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$. Here we consider a modification of PC that removes an undesired dependence on the order of variables, called PC-stable [Colombo and Maathuis, 2014]. These modifications also include either the _majority_ or _conservative_ [Ramsey et al., 2006] rule for handling ambiguous triples where separating sets are inconsistent, and conflicting links where different triples in the collider or orientation phase lead to conflicting link orientations. With the _conservative_ rule the PC algorithm is consistent already under the weaker Adjacency Faithfulness condition [Ramsey et al., 2006]. Another approach for the time series case (considered in the numerical experiments) is to combine vector-autoregressive modeling to identify lagged links with the PC algorithm for the contemporaneous causal structure [Moneta et al., 2011]. ### 2.3 AUTOCORRELATION Figure 1: The curse and blessing of autocorrelation. Linear example of model (3) with ground truth links shown for the PCMCI+ case (right panel). All autodependency coefficients are $0.95$ (except $0.475$ for $X^{5,6}$) and all cross-coupling coefficients are $0.4$ ($\pm$ indicated by red/blue links). The graphs show true and false link detection rates as the link width (if $>$ 0.06) for true (color indicating ParCorr) and incorrect links (grey) for the PC algorithm, Alg. 1, and the variants PCMCI+ and PCMCI${}^{+}_{0}$ as explained in the text (detection rates based on $500$ realizations run at $\alpha_{\rm PC}=0.01$ for $T=500$). To illustrate the challenge of autocorrelation, in Fig. 1 we consider a linear example with lagged and contemporaneous ground truth links shown for the PCMCI+ case (right panel). The PC algorithm (Alg. 2 with ParCorr CI test) starts by testing all unconditional independencies ($p=0$). Here the coupled pairs $(X^{5},X^{6})$ as well as $(X^{7},X^{8})$ are independent of the other variables and removed from each others adjacency sets, which shows how PC exploits sparsity and reduces the estimation dimension compared to fitting a full model on the whole past as in the GC framework. Due to the strong autocorrelation the remaining variables, on the other hand, are almost all adjacent to each other at multiple time lags in this iteration. In the next iteration, CI for all remaining links is tested conditional on all one- dimensional ($p=1$) conditioning sets. Here the PC algorithm removes the true lagged link $X^{1}_{t-1}\to X^{0}_{t}$ (black dots) due to the incorrect CI result $X^{1}_{t-1}\perp\\!\\!\\!\perp X^{0}_{t}|X^{1}_{t-2}$ (condition marked by blue box). Later this then leads to the false positive $X^{1}_{t-2}\to X^{0}_{t}$ (grey link) since $X^{1}_{t-1}$ is not conditioned on. In a similar way the true link $X^{1}_{t-2}\to X^{3}_{t}$ is missed leading to the false positive $X^{0}_{t-1}\to X^{3}_{t}$. Further, the true contemporaneous link $X^{2}_{t}{\circ\\!{\\--}\\!\circ}X^{3}_{t}$ (similarly $X^{3}_{t}{\circ\\!{\\--}\\!\circ}X^{4}_{t}$) is removed when conditioning on $\mathcal{S}=(X^{4}_{t-1},X^{3}_{t-1})$ (blue boxes), which leads to the false positive autodependencies at lag $2$ for $X^{2}_{t},X^{4}_{t}$, while the false autodependency $X^{3}_{t-2}\to X^{3}_{t}$ is due to missing $X^{1}_{t-2}\to X^{3}_{t}$. This illustrates the pattern of a cascade of false negative errors (missing links) leading to false positives in later stages of the PC algorithm. What determines the removal of a true link in the finite sample case? Detection power depends on sample size, the significance level $\alpha_{\rm PC}$, the CI test dimension ($p+2$), and effect size, e.g., the absolute ParCorr (population) value, here denoted $I(X^{i}_{t-\tau};X^{j}_{t}|\mathcal{S})$ for some conditioning set $\mathcal{S}$. Within each $p$-iteration the sample size, $\alpha_{\rm PC}$, and the dimension are the same and a link will be removed if $I(X^{i}_{t-\tau};X^{j}_{t}|\mathcal{S})$ falls below the $\alpha_{\rm PC}$-threshold for _any_ considered $\mathcal{S}$. Hence, the overall minimum effect size $\min_{\mathcal{S}}[I(X^{i}_{t-\tau};X^{j}_{t}|\mathcal{S})]$ determines whether a link is removed. The PC algorithm will iterate through _all_ subsets of adjacencies such that this minimum can become very small. Low effect size can be understood as a low (causal) signal-to-noise ratio: Here $I(X^{1}_{t-1};X^{0}_{t}|X^{1}_{t-2})$ is small since the signal $X^{1}_{t-1}$ is reduced by conditioning on its autodependency $X^{1}_{t-2}$ and the ‘noise’ in $X^{0}_{t}$ is large due to its strong autocorrelation. But autocorrelation can also be a blessing. The contemporaneously coupled pair $(X^{7},X^{8})$ illustrates a case where autocorrelation helps to identify the orientation of the link. Without autocorrelation the output of PC would be an unoriented link to indicate the Markov equivalence class. On the other hand, the detection rate here is rather weak since, as above, the signal (link from $X^{8}_{t}$) is small compared to the noise (autocorrelation in $X^{7}$). This illustrates the curse and blessing of autocorrelation. In summary, the PC algorithm often results in false negatives (low recall) and these then lead to false positives. Another reason for false positives are ill-calibrated tests: To correctly model the null distribution, each individual CI test would need to account for autocorrelation, which is difficult in a complex multivariate and potentially nonlinear setting [Runge, 2018a]. In the experiments we will see that the PC algorithm features inflated false positives. As a side comment, the pair $(X^{5},X^{6})$ depicts a feedback cycle. These often occur in real data and the example shows that time series graphs allow to resolve time-delayed feedbacks while an aggregated _summary graph_ would contain a cyclic dependency and summary graph-based methods assuming acyclic graphs would not work. The orientation of the contemporaneous link $X^{6}_{t}\to X^{5}_{t}$ is achieved via rule R1 in the orientation phase of PC (Alg. S3). ## 3 PCMCI+ Figure 2: Schematic of PCMCI+. Note that for ease of visualization repeating edges due to stationarity are only partially shown. ### 3.1 ALGORITHM The goal of PCMCI+ is to optimize the choice of conditioning sets in CI tests in order to increase detection power and at the same time maintain well- calibrated tests. The approach is based on two central ideas, (1) separating the skeleton edge removal phase into a lagged and contemporaneous conditioning phase with much fewer CI tests and (2) utilizing the momentary conditional independence (MCI) test [Runge et al., 2019b] idea in the contemporaneous conditioning phase. Below, I explain the reasoning behind. Figure 2 illustrates the steps. First, the goal of PC’s skeleton phase is to remove all those adjacencies that are due to indirect paths and common causes by conditioning on subsets $\mathcal{S}$ of the variables’ neighboring adjacencies in each iteration. Consider a variable $X^{j}_{t}$. If we test lagged adjacencies from nodes $X^{i}_{t-\tau}\in\mathbf{X}^{-}_{t}$ conditional on the whole past, i.e., $\mathcal{S}=\mathbf{X}^{-}_{t}\setminus\\{X^{i}_{t-\tau}\\}$, the only indirect adjacencies remaining are due to paths through contemporaneous parents of $X^{j}_{t}$. This is in contrast to conditioning sets on contemporaneous adjacencies which can also open up paths $X^{j}_{t}\to X^{k}_{t}\leftarrow X^{i}_{t-\tau}$ if $X^{k}_{t}$ is conditioned on. One reason why the PC algorithm tests _all_ combinations of subsets $\mathcal{S}$ is to avoid opening up such collider paths. Therefore, one approach would be to start by $\mathcal{S}=\mathbf{X}^{-}_{t}\setminus\\{X^{i}_{t-\tau}\\}$ and then iterate through contemporaneous conditions. A similar idea lies behind the combination of GC and the PC algorithm in [Moneta et al., 2011]. However, conditioning on large-dimensional conditioning sets strongly affects detection power [Runge et al., 2019b]. To avoid this, the lagged conditioning phase of PCMCI+ (Alg. 1, see Fig. 2 left panels) tests all pairs $(X^{i}_{t-\tau},X^{j}_{t})$ for $\tau>0$ conditional on only the _strongest_ $p$ adjacencies of $X^{j}_{t}$ in each $p$-iteration without going through all $p$-dimensional subsets of adjacencies. This choice $(i)$ improves the causal signal-to-noise ratio and recall since for a given test $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S}$ the ‘noise’ in $X^{j}_{t}$ due to other lagged adjacencies is conditioned out, $(ii)$ leads to fewer CI tests further improving recall, and $(iii)$ speeds up the skeleton phase. We denote the lagged adjacency set resulting from Alg. 1 as $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$. Lemma 1 in Sect. 3.2 states that the only remaining indirect adjacencies in $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ are then due to paths passing through contemporaneous parents of $X^{j}_{t}$. In the schematic in Fig. 2 this is the link $Y_{t-1}\to X_{t}$. Secondly, in Alg. 2 (Fig. 2 center panels) the graph $\mathcal{G}$ is initialized with all contemporaneous adjacencies plus all lagged adjacencies from $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ for all $X^{j}_{t}$. Algorithm 2 tests all (unordered lagged and ordered contemporaneous) adjacent pairs $(X^{i}_{t-\tau},X^{j}_{t})$ and iterates through contemporaneous conditions $\mathcal{S}\subseteq\mathcal{A}_{t}(X^{j}_{t})$ with the MCI test $\displaystyle X^{i}_{t-\tau}$ $\displaystyle{\perp\\!\\!\\!\perp}X^{j}_{t}~{}|~{}\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau}).$ (2) In the schematic in Fig. 2 the condition on $\mathcal{S}=Y_{t}$, as part of the full conditioning set $\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$, removes the link $X_{t}{\circ\\!{\\--}\\!\circ}Z_{t}$. The condition on $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ blocks paths through lagged parents and the advantage of the additional conditioning on $\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$ is discussed in the following. We denote the variant without the condition on $\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$ as PCMCI${}^{+}_{0}$. Both versions are followed by the collider orientation phase (Alg. S2) and rule orientation phase (Alg. S3) which are deferred to the SM since they are equivalent to the PC algorithm with the modification that the additional CI tests in the collider phase for the conservative or majority rule are also based on the test (2) (Fig. 2 right panel). We now discuss PCMCI${}^{+}_{0}$ and PCMCI+ on the example in Fig. 1. Algorithm 1 tests $X^{1}_{t-1}\to X^{0}_{t}$ conditional on $\mathcal{S}=\\{X^{0}_{t-1}\\}$ for $p=1$ and $\mathcal{S}=\\{X^{0}_{t-1},X^{1}_{t-2}\\}$ for $p=2$ as the two strongest adjacencies (as determined by the test statistic value, see pseudo-code). In both of these tests the effect size $I$ (causal signal-to-noise ratio) is much larger than for the condition on $\mathcal{S}=\\{X^{1}_{t-2}\\}$ which lead to the removal of $X^{1}_{t-1}\to X^{0}_{t}$ in the PC algorithm. In Sect. 3.2 we elaborate more rigorously on effect size. In the example $\widehat{\mathcal{B}}^{-}_{t}(X^{2}_{t})$ is indicated as blue boxes in the second panel and contains lagged parents as well as adjacencies due to paths passing through contemporaneous parents of $X^{2}_{t}$. One false positive, likely due to an ill-calibrated test caused by autocorrelation, is marked by a star. Based on these lagged adjacencies, Alg. 2 with the PCMCI${}^{+}_{0}$ option then recovers all lagged links (3rd panel), but it still the misses contemporaneous adjacencies $X^{2}_{t}{\circ\\!{\\--}\\!\circ}X^{3}_{t}$ and $X^{3}_{t}{\circ\\!{\\--}\\!\circ}X^{4}_{t}$ and we also see strong lagged false positives from $X^{3}$ to $X^{2}$ and $X^{4}$. What happened here? The problem are now tests on contemporaneous links: The CI test for PCMCI${}^{+}_{0}$ in the $p=0$ loop, like the original PC algorithm, will test _ordered_ contemporaneous pairs. Hence, first $X^{2}_{t}{\circ\\!{\\--}\\!\circ}X^{3}_{t}$ conditional on $\widehat{\mathcal{B}}^{-}_{t}(X^{3}_{t})$ and, if the link is not removed, $X^{3}_{t}{\circ\\!{\\--}\\!\circ}X^{2}_{t}$ conditional on $\widehat{\mathcal{B}}^{-}_{t}(X^{2}_{t})$. Here $X^{2}_{t}{\circ\\!{\\--}\\!\circ}X^{3}_{t}$ is removed conditional on $\widehat{\mathcal{B}}^{-}_{t}(X^{3}_{t})$ (indicated by blue boxes in the panel) because $I(X^{2}_{t};X^{3}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{3}_{t}))$ falls below the significance threshold. The second central idea of PCMCI+ is to improve the effect size of CI tests for contemporaneous links by conditioning on _both_ lagged adjacencies $\widehat{\mathcal{B}}^{-}_{t}$ in the CI test (2) (see blue and red boxes in Fig. 1 right panel). At least for the initial phase $p=0$ one can prove that for non-empty $\widehat{\mathcal{B}}^{-}_{t}$ the effect size of the PCMCI+ CI test is always strictly larger than that of the PCMCI${}^{+}_{0}$ test (Thm. 4). I conjecture that this similarly holds for PCMCI+ vs. the PC algorithm. Higher effect size leads to higher recall and PCMCI+ now recovers all lagged as well as contemporaneous links and also correctly removes the lagged false positives that PCMCI${}^{+}_{0}$ obtains. Also the contemporaneous coupled pair $(X^{7},X^{8})$ is now much better detected since the MCI effect size $I(X^{7}_{t};X^{8}_{t}|X^{7}_{t-1})$ is larger than $I(X^{7}_{t};X^{8}_{t})$, one of the two PCMCI${}^{+}_{0}$ and PC algorithm effect sizes tested here. Another advantage, discussed in [Runge et al., 2019b] is that PCMCI+ CI tests are better calibrated, in contrast to PCMCI${}^{+}_{0}$ and PC algorithm tests, since the condition on both parents removes autocorrelation effects. Note that for lagged links the effect size of PCMCI+ is generally smaller than that of PCMCI${}^{+}_{0}$ since the extra condition on $\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$ can only reduce effect size (see [Runge et al., 2012]). This is the cost of avoiding inflated false positives. In summary, the central PCMCI+ idea is to increase effect size in individual CI tests to achieve higher detection power and at the same time maintain well- controlled false positives also for high autocorrelation. Correct adjacency information then leads to better orientation recall in Alg. S2, S3. The other advantage of PCMCI+ compared to the PC algorithm is a much faster and, as numerical examples show, also much less variable runtime. The full algorithm is detailed in pseudo-code Algorithms 1,2,S2,S3 with differences to PC and PCMCI${}^{+}_{0}$ indicated. Note that pairs $(X^{i}_{t-\tau},X^{j}_{t})$ in lines 5 and 6 of Alg. 2 are ordered for $\tau=0$ and unordered for $\tau>0$. One can construct (rather conservative) $p$-values for the skeleton adjacencies $(X^{i}_{t-\tau},X^{j}_{t})$ by taking the maximum $p$-value over all CI tests conducted in Alg. 2. A link strength can be defined corresponding to the test statistic value of the maximum $p$-value. Based on the PC stable variant, PCMCI+ is fully order-independent. Here shown is the majority-rule implementation of the collider phase, the version without handling ambiguous triples and for the conservative rule are detailed in Alg. S2. Note that the tests in the collider phase also use the CI tests (2). Like other CI-based methods, PCMCI+ has the free parameters $\alpha_{\rm PC}$, $\tau_{\max}$, and the choice of the CI test. $\alpha_{\rm PC}$ can be chosen based on cross-validation or an information criterion (implemented in tigramite). $\tau_{\max}$ should be larger or equal to the maximum true time lag of any parent and can in practice also be chosen based on model selection. However, the numerical experiments indicate that, in contrast to GC, a too large $\tau_{\max}$ does not degrade performance much and $\tau_{\max}$ can also be chosen based on the lagged dependence functions, see [Runge et al., 2019b]. PCMCI+ can flexibly be combined with different CI tests for nonlinear causal discovery, and for different variable types (discrete or continuous, univariate or multivariate). The computational complexity of PCMCI+ strongly depends on the network structure. The sparser the causal dependencies, the faster the convergence. Compared to the original PC algorithm with worst-case exponential complexity, the complexity is much reduced since Alg. 1 only has polynomial complexity [Runge et al., 2019b] and Alg. 2 only iterates through contemporaneous conditioning sets, hence the worst-case exponential complexity only applies to $N$ and not to $N\tau_{\max}$. Algorithm 1 (PCMCI+ / PCMCI${}^{+}_{0}$ lagged skeleton phase) 1:Time series dataset $\mathbf{X}=(X^{1},\,\ldots,X^{N})$, max. time lag $\tau_{\max}$, significance threshold $\alpha_{\rm PC}$, CI test ${\rm CI}(X,\,Y,\,\mathbf{Z})$ returning $p$-value and test statistic value $I$ 2:for all $X^{j}_{t}$ in $\mathbf{X}_{t}$ do 3: Initialize $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){=}\mathbf{X}^{-}_{t}{=}(\mathbf{X}_{t-1},\dots,\mathbf{X}_{t-\tau_{\max}})$ and $I^{\min}(X^{i}_{t-\tau},X^{j}_{t})=\infty~{}~{}\forall~{}X^{i}_{t-\tau}\in\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ 4: Let $p=0$ 5: while any $X^{i}_{t-\tau}\in\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ satisfies $|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}|\geq p$ do 6: for all $X^{i}_{t-\tau}$ in $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ satisfying $|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}|\geq p$ do 7: $\mathcal{S}=$ first $p$ variables in $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{X^{i}_{t-\tau}\\}$ 8: $(\text{$p$-value},\,I)\leftarrow$ CI($X^{i}_{t-\tau},\,X^{j}_{t},\,\mathcal{S}$) 9: $I^{\min}(X^{i}_{t-\tau},X^{j}_{t})=\min(|I|,I^{\min}(X^{i}_{t-\tau},X^{j}_{t}))$ 10: if $p$-value $>\alpha_{\rm PC}$ then mark $X^{i}_{t-\tau}$ for removal 11: Remove non-significant entries and sort $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ by $I^{\min}(X^{i}_{t-\tau},X^{j}_{t})$ from largest to smallest 12: Let $p=p+1$ 13:return $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ for all $X^{j}_{t}$ in $\mathbf{X}_{t}$ Algorithm 2 (PCMCI+ / PCMCI${}^{+}_{0}$ contemporaneous skeleton phase / PC full skeleton phase) 1:Time series dataset $\mathbf{X}=(X^{1},\,\ldots,X^{N})$, max. time lag $\tau_{\max}$, significance threshold $\alpha_{\rm PC}$, ${\rm CI}(X,\,Y,\,\mathbf{Z})$, PCMCI+ / PCMCI${}^{+}_{0}$: $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ for all $X^{j}_{t}$ in $\mathbf{X}_{t}$ 2:PCMCI+ / PCMCI${}^{+}_{0}$: Form time series graph $\mathcal{G}$ with lagged links from $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ for all $X^{j}_{t}$ in $\mathbf{X}_{t}$ and fully connect all contemporaneous variables, i.e., add $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ for all $X^{i}_{t}\neq X^{j}_{t}\in\mathbf{X}_{t}$ 3:PC: Form fully connected time series graph $\mathcal{G}$ with lagged and contemporaneous links 4:PCMCI+ / PCMCI${}^{+}_{0}$: Initialize contemporaneous adjacencies $\widehat{\mathcal{A}}(X^{j}_{t}):=\widehat{\mathcal{A}}_{t}(X^{j}_{t})=\\{X^{i}_{t}{\neq}X^{j}_{t}\in\mathbf{X}_{t}:X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}~{}\text{in $\mathcal{G}$}\\}$ 5:PC: Initialize full adjacencies $\widehat{\mathcal{A}}(X^{j}_{t})$ for all (lagged and contemporaneous) links in $\mathcal{G}$ 6:Initialize $I^{\min}(X^{i}_{t-\tau},X^{j}_{t})=\infty$ for all links in $\mathcal{G}$ 7:Let $p=0$ 8:while any adjacent pairs $(X^{i}_{t-\tau},X^{j}_{t})$ for $\tau\geq 0$ in $\mathcal{G}$ satisfy $|\widehat{\mathcal{A}}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}|\geq p$ do 9: Select new adjacent pair $(X^{i}_{t-\tau},X^{j}_{t})$ for $\tau\geq 0$ satisfying $|\widehat{\mathcal{A}}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}|\geq p$ 10: while $(X^{i}_{t-\tau},X^{j}_{t})$ are adjacent in $\mathcal{G}$ and not all $\mathcal{S}\subseteq\widehat{\mathcal{A}}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$ with $|\mathcal{S}|=p$ have been considered do 11: Choose new $\mathcal{S}{\subseteq}\widehat{\mathcal{A}}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$ with $|\mathcal{S}|{=}p$ 12: PCMCI+: Set $\mathbf{Z}{=}(\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau}))$ 13: PCMCI${}^{+}_{0}$: Set $\mathbf{Z}{=}(\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\})$ 14: PC: Set $\mathbf{Z}{=}\mathcal{S}$ 15: $(\text{$p$-value},\,I)\leftarrow$ CI($X^{i}_{t{-}\tau},X^{j}_{t},\mathbf{Z})$ 16: $I^{\min}(X^{i}_{t-\tau},X^{j}_{t})=\min(|I|,I^{\min}(X^{i}_{t-\tau},X^{j}_{t}))$ 17: if $p$-value $>\alpha_{\rm PC}$ then 18: Delete link $X^{i}_{t-\tau}\to X^{j}_{t}$ for $\tau>0$ (or $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ for $\tau=0$) from $\mathcal{G}$ 19: Store (unordered) ${\rm sepset}(X^{i}_{t-\tau},X^{j}_{t})=\mathcal{S}$ 20: Let $p=p+1$ 21: Re-compute $\widehat{\mathcal{A}}(X^{j}_{t})$ from $\mathcal{G}$ and sort by $I^{\min}(X^{i}_{t-\tau},X^{j}_{t})$ from largest to smallest 22:return $\mathcal{G}$, sepset ### 3.2 THEORETICAL RESULTS This section states asymptotic consistency, finite sample order-independence, and further results regarding effect size and false positive control. The consistency of network learning algorithms is separated into _soundness_ , i.e., the returned graph has correct adjacencies, and _completeness_ , i.e., the returned graph is also maximally informative (links are oriented as much as possible). We start with the following assumptions. ###### Assumptions 1 (Asymptotic case). Throughout this paper we assume Causal Sufficiency, the Causal Markov Condition, the Adjacency Faithfulness Conditions, and consistent CI tests (oracle). In the present time series context we also assume stationarity and time-order and that the maximum time lag $\tau_{\max}\geq\tau^{\mathcal{P}}_{\max}$, where $\tau^{\mathcal{P}}_{\max}$ is the maximum time lag of any parent in the SCM (1). Furthermore, we rule out _selection variables_ and _measurement error_. Definitions of these assumptions, adapted from [Spirtes et al., 2000] to the time series context, are in Sect. S1 and all proofs are in Sect. S2. We start with the following lemma. ###### Lemma 1. Under Assumptions 1 Alg. 1 returns a set that always contains the parents of $X^{j}_{t}$ and, _at most_ , the lagged parents of all contemporaneous ancestors of $X^{j}_{t}$, i.e., $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})=\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$. $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ contains _all_ lagged parents of all contemporaneous ancestors if the weaker Adjacency Faithfulness assumption is replaced by standard Faithfulness. This establishes that the conditions $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ estimated in the first phase of PCMCI+ will suffice to block all lagged confounding paths that do not go through contemporaneous links. This enables to prove the soundness of Alg. 2, even though Alg. 2 is a variant of the PC algorithm that only iterates through contemporaneous conditioning sets. ###### Theorem 1 (Soundness of PCMCI+). Algorithm 2 returns the correct adjacencies under Assumptions 1, i.e., $\widehat{\mathcal{G}^{*}}=\mathcal{G}^{*}$, where the $\mathcal{G}^{*}$ denotes the skeleton of the time series graph. To prove the completeness of PCMCI+, we start with the following observation. ###### Lemma 2. Due to time-order and the stationarity assumption, the considered triples in the collider phase (Alg. S2) and rule orientation phase (Alg. S3) can be restricted as follows: In the collider orientation phase only unshielded triples $X^{i}_{t-\tau}\to X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ (for $\tau>0$) or $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ (for $\tau=0$) in $\mathcal{G}$ where $(X^{i}_{t-\tau},X^{j}_{t})$ are not adjacent are relevant. For orientation rule R1 triples $X^{i}_{t-\tau}\to X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ where $(X^{i}_{t-\tau},X^{j}_{t})$ are not adjacent, for orientation rule R2 triples $X^{i}_{t}\to X^{k}_{t}\to X^{j}_{t}$ with $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$, and for orientation rule R3 pairs of triples $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}\to X^{j}_{t}$ and $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{l}_{t}\to X^{j}_{t}$ where $(X^{k}_{t},X^{l}_{t})$ are not adjacent and $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ are relevant. These restrictions imply that only contemporaneous parts of separating sets are relevant for the collider phase. ###### Theorem 2 (PCMCI+ is complete). PCMCI+ (Algorithms 1,2,S2,S3) when used with the conservative rule for orienting colliders in Alg. S2 returns the correct CPDAG under Assumptions 1. Under standard Faithfulness also PCMCI+ when used with the majority rule or the standard orientation rule is complete. Also the proof of order-independence follows straightforwardly from the proof in [Colombo and Maathuis, 2014]. Of course, order independence does not apply to time-order. ###### Theorem 3 (Order independence). Under Assumptions 1 PCMCI+ with the conservative or majority rule in Alg. S2 is independent of the order of variables $(X^{1},\ldots,X^{N})$. Next, we consider effect size. The toy example showed that a major problem of PCMCI${}^{+}_{0}$ (and also PC) is lack of detection power for contemporaneous links. A main factor of statistical detection power is effect size, i.e., the population value of the test statistic considered (e.g., absolute partial correlation). In the following, I will base my argument in an information- theoretic framework and consider the conditional mutual information as a general test statistic, denoted $I$. In Alg. 2 PCMCI${}^{+}_{0}$ will test a contemporaneous dependency $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ first with the test statistic $I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}))$, and, if that test was positive, secondly with $I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t})))$. If either of these tests finds (conditional) independence, the adjacency is removed. Therefore, the minimum test statistic value determines the relevant effect size. On the other hand, PCMCI+ treats both cases symmetrically since the test statistic is always $I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}),\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t}))$. ###### Theorem 4 (Effect size of MCI tests for $p=0$). Under Assumptions 1 the PCMCI+ oracle case CI tests in Alg. 2 for $p=0$ for contemporaneous true links $X^{i}_{t}\to X^{j}_{t}\in\mathcal{G}$ have an effect size that is always greater than that of the PCMCI${}^{+}_{0}$ CI tests, i.e., $I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}),\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t}))>\min(I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})),\,I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t})))$ if both $X^{i}_{t}$ and $X^{j}_{t}$ have parents that are not shared with the other. I conjecture that this result holds similarly for $p>0$ and also that PCMCI+ has greater effect sizes than the PC algorithm since the latter iterates over _all_ subsets of adjacencies and, hence, the minimum is taken generally over an even larger set leading to even smaller effect sizes. For lagged links the effect size of the PCMCI+ tests is always smaller (or equal) than that of the PCMCI${}^{+}_{0}$ tests (see [Runge et al., 2012]). Last, we discuss false positive control. While the effect size result regards detection power, in the following I give a mathematical intuition why the MCI tests are better calibrated than the PC algorithm CI tests and control false positives below the expected significance level. Lemma 1 implies that even though Alg. 1 does not aim to estimate the contemporaneous parents, it still yields a set of conditions that shields $X^{j}_{t}$ from the ‘infinite’ past $\mathbf{X}^{-}_{t}$, either by blocking the parents of $X^{j}_{t}$ or by blocking indirect contemporaneous paths through contemporaneous ancestors of $X^{j}_{t}$. Blocking paths from the infinite past, I conjecture, is key to achieve well-calibrated CI tests in Alg. 2. The authors in [Runge et al., 2019b] showed that under certain model assumptions the MCI tests reduce to CI tests among the noise terms $\eta$ from model (1) which are assumed to be i.i.d. and help to achieve well-calibrated CI tests. In the numerical experiments below we can see that the PC algorithm has inflated false positive for high autocorrelation, while PCMCI+ well controls false positives, but a formal proof of correct false positive control for this challenging nonlinear, high-dimensional setting is beyond the scope of this paper. ## 4 NUMERICAL EXPERIMENTS We consider a number of typical challenges [Runge et al., 2019a], contemporaneous and time lagged causal dependencies, strong autocorrelation, large number of variables and considered time lags, different noise distributions and nonlinearity, in the following additive variant of model (1): $\displaystyle X_{t}^{j}$ $\displaystyle=a_{j}X^{j}_{t-1}+\textstyle{\sum_{i}}c_{i}f_{i}(X^{i}_{t-\tau_{i}})+\eta^{j}_{t}$ (3) for $j\in\\{1,\ldots,N\\}$. Autocorrelations $a_{j}$ are uniformly drawn from $[\max(0,a-0.3),\,a]$ for $a$ as indicated in Fig. 3 and $\eta^{j}$ is _i.i.d._ and follows a zero-mean Gaussian $\mathcal{N}$ or Weibull $\mathcal{W}$ (scale parameter $2$) distribution (depending on setup) with standard deviation drawn from $[0.5,\,2]$. In addition to autodependency links, for each model $L=\lfloor 1.5\cdot N\rfloor$ (except for $N=2$ with $L=1$) cross-links are chosen whose functional dependencies are linear or $f_{i}(x)=f^{(2)}(x)=(1+5xe^{-x^{2}/20})x$ (depending on setup), with $f^{(2)}$ designed to yield more stationary dynamics. Coefficients $c_{i}$ are drawn uniformly from $\pm[0.1,0.5]$. 30% of the links are contemporaneous ($\tau_{i}=0$) and the remaining $\tau_{i}$ are drawn from $[1,\,5]$. Only stationary models are considered. We have an average cross-in-degree of $d=1.5$ for all network sizes (plus an auto-dependency) implying that models become sparser for larger $N$. We consider several model setups: linear Gaussian, linear mixed noise (among the $N$ variables: 50% Gaussian, 50% Weibull), and nonlinear mixed noise (50% linear, 50% $f^{(2)}(x)$; 66% Gaussian, 34% Weibull). For the linear model setups we consider the PC algorithm and PCMCI+ in the majority-rule variant with ParCorr and compare these with GCresPC [Moneta et al., 2011], a combination of GC with PC applied to residuals, and a autoregressive model version of LiNGAM [Hyvärinen et al., 2010], a representative of the SCM framework (implementation details in Sect. S4). For the LiNGAM implementation I could not find a way to set a significance level and used the LASSO option which prunes ‘non-active’ links to zero. Both GCresPC and LiNGAM assume linear dependencies and LiNGAM also non-Gaussianity. For the nonlinear setup the PC algorithm and PCMCI+ are implemented with the GPDC test [Runge et al., 2019b] that is based on Gaussian process regression and a distance correlation test on the residuals, which is suitable for a large class of nonlinear dependencies with additive noise. Performance is evaluated as follows: True (TPR) and false positive rates (FPR, shown to evaluate false positive control, not applicable to LiNGAM) for adjacencies are distinguished between lagged cross-links ($i\neq j$), contemporaneous, and autodependency links. Due to time order, lagged links (and autodependencies) are automatically oriented. Contemporaneous orientation precision is measured as the fraction of correctly oriented links (${\circ\\!{\\--}\\!\circ}$ or $\to$) among all estimated adjacencies, and recall as the fraction of correct orientations among all true contemporaneous links. Further shown is the fraction of conflicting links among all detected contemporaneous adjacencies (not applicable to LiNGAM). All metrics (and their std. errors) are computed across all estimated graphs from $500$ realizations of model (3) at time series length $T$. The average runtimes were evaluated on Intel Xeon Platinum 8260. In Fig. 3 results for the linear Gaussian setup with default model parameters $N=5,\,T=500,\,a=0.95$ and method parameters $\tau_{\max}=5$ and $\alpha=0.01$ (not applicable to LiNGAM) are shown. Each of the four panels shows results for varying one of $a,\,N,\,T,\,\tau_{\max}$. The insets show ANOVA statistics $r\pm\bar{\Delta}r$ [per unit], where $r$ is the performance metric at the leftmost parameter on the $x$-axis ($a,\,N,\,T,\,\tau_{\max}$, respectively) and $\bar{\Delta}r$ denotes the average change per parameter unit. In the adjacency subplots the statistics refer to lagged links. Figure 3: Numerical experiments with linear Gaussian setup for varying (A) autocorrelation strength $a$ (B) number of variables $N$ (C) sample size $T$ and (D) maximum time lag $\tau_{\max}$. All remaining setup parameters indicated in the top right. Errorbars show std. errors or the 90% range (for runtime). The insets show ANOVA statistics. Figure 3A demonstrates that the TPR of PCMCI+ and GCresPC for contemporaneous links is stable even under high autocorrelation while PC and LiNGAM show strong declines. Since LiNGAM has no $\alpha_{\rm PC}$ for FPR-control we focus on its relative changes rather than absolute performance. Lagged TPR decreases strongly for PC while the other methods are more robust. FPR is well-controlled for PCMCI+ while PC and slightly also GCresPC show inflated lagged FPR for high autocorrelation. LiNGAM features a strong increase of lagged FPR. These adjacency results translate into higher contemporaneous orientation recall for PCMCI+ which increases with autocorrelation, while it decreases for all other methods. GCresPC has steady low recall since it does not use lagged links in the orientation phase. Except for GCresPC, all methods have increasing precision with PCMCI+ and PC outperforming LiNGAM. PCMCI+ shows almost no conflicts while PC’s conflicts increase with autocorrelation until low power reduces them again. Finally, runtimes are almost constant for GCresPC and LiNGAM, while they increase for PCMCI+ and much stronger for PC. Figure 3B shows that PCMCI+ and GCresPC have the highest TPR for increasing number of variables $N$, especially for contemporaneous links. FPR is well controlled only for PCMCI+ while PC has false positives for small $N$ where model connectivity is denser and false negatives are more likely leading to false positives. For high $N$ PC has false positives only regarding autodependencies while inflated FPR appears for GCresPC. PCMCI+ has more than twice as much contemporaneous recall compared to the other methods and is almost not affected by higher $N$. Orientation precision is decreasing for all methods (except PC) with a higher decrease for PCMCI+. Runtime is increasing at a much smaller rate for PCMCI+ compared to PC, which also has a very high runtime variability across the different model realizations. LiNGAM and especially GCresPC are fastest. PCMCI+, GCresPC, and LiNGAM benefit similarly and PC less so for increasing sample size regarding TPR (Fig. 3C). FPR is still not controlled for PC for large sample sizes, lagged FPR increases for GCresPC. PCMCI+ shows the highest increases in contemporaneous recall and precision. Runtime increases are moderate compared to PC, conflicts decrease. Last, Fig. 3D shows that all methods are relatively robust to large maximum time lags $\tau_{\max}$ (beyond the true max. time lag $5$) for the considered sample size $T=500$. Contemporaneous FPR and runtime increase for PC. In the SM further results are shown. For too large $N\tau_{\max}$ (relative to $T$) GCresPC and LiNGAM (despite LASSO-regularization) sharply drop in performance. For the linear mixed noise setup (Fig. S2) results are almost unchanged for all methods except for LiNGAM for which recall and precision rise, as expected. Recall is then higher than PCMCI+ for low autocorrelation, but still much lower for high autocorrelation and large $N$ or $\tau_{\max}$, at similar precision. In the nonlinear mixed noise setup (Fig. S3), the difference between PC and PCMCI+ is similar. We observe slight FPR inflation for high autocorrelation. GPDC seems to not work well in high-dimensional, highly autocorrelated settings. Runtime for GPDC compared to ParCorr is orders of magnitude longer, especially for PC. Further figures in the SM show many combinations of $a,\,N,\,T,\,\tau_{\max}$ and $\alpha_{\rm PC}$ for the model setups and demonstrate that the above findings are robust. ## 5 CONCLUSIONS PCMCI+ improves the reliability of CI tests by optimizing the choice of conditioning sets and yields much higher recall, well-controlled false positives, and faster runtime than the original PC algorithm for highly autocorrelated time series, while maintaining similar performance for low autocorrelation. The algorithm well exploits sparsity in high-dimensional settings and can flexibly be combined with different CI tests for nonlinear causal discovery, and for different variable types (discrete or continuous, univariate or multivariate). Autocorrelation is actually key to increase contemporaneous orientation recall since it creates triples $X^{i}_{t-1}\to X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ that can often be oriented while an isolated link $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ stays undirected in the Markov equivalence class, a drawback of CI-based methods. If the data is at least non-Gaussian, a SCM method like LiNGAM can exploit this property and recover directionality in such cases. Still, we saw that LiNGAM suffers from large autocorrelation. PCMCI+ is available as part of the _tigramite_ Python package at https://github.com/jakobrunge/tigramite. A next step will be to extend the present ideas to an algorithm accounting for latent confounders and to explore combinations between SCM-based methods and PCMCI+. The numerical results will be contributed to the causality benchmark platform `www.causeme.net` [Runge et al., 2019a] to facilitate a further expanded method evaluation. #### Acknowledgments DKRZ provided computational resources (grant no. 1083). I thank Andreas Gerhardus for helpful comments. ## References * [Chickering, 2002] Chickering, D. M. (2002). Learning Equivalence Classes of Bayesian-Network Structures. J. Mach. Learn. Res., 2:445–498. * [Colombo and Maathuis, 2014] Colombo, D. and Maathuis, M. H. (2014). Order-Independent Constraint-Based Causal Structure Learning. J. Mach. Learn. Res., 15:3921–3962. * [Entner and Hoyer, 2010] Entner, D. and Hoyer, P. O. (2010). On causal discovery from time series data using FCI. In Proc. Fifth Eur. Work. Probabilistic Graph. Model., pages 121–128. * [Granger, 1969] Granger, C. W. J. (1969). Investigating causal relations by econometric models and cross-spectral methods. 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Causation, Prediction, and Search. MIT Press, Boston, MA. * [Spirtes and Zhang, 2016] Spirtes, P. and Zhang, K. (2016). Causal discovery and inference: concepts and recent methodological advances. Appl. Informatics, 3(1):3. * [Zhang et al., 2011] Zhang, K., Peters, J., Janzing, D., and Schölkopf, B. (2011). Kernel-based Conditional Independence Test and Application in Causal Discovery. In Proc. 27th Conf. Uncertain. Artif. Intell., pages 804–813. ## Appendix S1 Definitions The following definitions are adaptations of the standard assumptions of causal discovery to the time series case. Here we consider the causally sufficient case and assume that all variables $\mathbf{X}=(X^{1},\ldots,X^{N})$ of the underlying SCM (1) are observed. Additionally, we assume that the maximum PCMCI+ time lag $\tau_{\max}\geq\tau^{\mathcal{P}}_{\max}$, where $\tau^{\mathcal{P}}_{\max}$ is the maximum time lag of any parent in the SCM (1). ###### Definition S1 (Causal Markov Condition). The joint distribution of a process $\mathbf{X}$ whose causal structure can be represented in a time series graph $\mathcal{G}$ fulfills the Causal Markov Condition iff for all $X^{j}_{t}\in\mathbf{X}_{t}$ every non-descendent of $X^{j}_{t}$ in $\mathcal{G}$ is independent of $X^{j}_{t}$ given the parents $\mathcal{P}(X^{j}_{t})$. In particular, $\mathbf{X}_{t}^{-}{\setminus}\mathcal{P}(X^{j}_{t})\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{P}(X^{j}_{t})$ since all variables in $\mathbf{X}_{t}^{-}$ are non-descendants of $X^{j}_{t}$ by time-order. Note that for the SCM (1) with independent noise terms the Causal Markov Condition is automatically fulfilled. ###### Definition S2 (Adjacency and standard faithfulness Condition). The joint distribution of a process $\mathbf{X}$ whose causal structure can be represented in a time series graph $\mathcal{G}$ fulfills the Adjacency Faithfulness Condition iff for all disjoint $X^{i}_{t-\tau},X^{j}_{t},\mathcal{S}\in\mathbf{X}^{-}_{t+1}$ with $\tau>0$ $\displaystyle X^{i}_{t-\tau}$ $\displaystyle\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S}~{}\Rightarrow~{}X^{i}_{t-\tau}\to X^{j}_{t}\notin\mathcal{G}$ $\displaystyle X^{i}_{t-\tau}$ $\displaystyle\to X^{j}_{t}\in\mathcal{G}~{}\Rightarrow~{}X^{i}_{t-\tau}\cancel{\perp\\!\\!\\!\perp}X^{j}_{t}~{}|~{}\mathcal{S}~{}~{}\text{(contrapositive)}$ and with $\tau=0$ $\displaystyle X^{i}_{t}$ $\displaystyle\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S}~{}\Rightarrow~{}X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}\notin\mathcal{G}$ $\displaystyle X^{i}_{t}$ $\displaystyle{\circ\\!{\\--}\\!\circ}X^{j}_{t}\in\mathcal{G}~{}\Rightarrow~{}X^{i}_{t}\cancel{\perp\\!\\!\\!\perp}X^{j}_{t}~{}|~{}\mathcal{S}~{}~{}\text{(contrapositive)}\,.$ Furthermore, the variables fulfill the (standard) Faithfulness Condition iff for $\tau\geq 0$ $\displaystyle X^{i}_{t-\tau}$ $\displaystyle\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S}~{}\Rightarrow~{}X^{i}_{t-\tau}\bowtie X^{j}_{t}~{}|~{}\mathcal{S}$ $\displaystyle X^{i}_{t-\tau}$ $\displaystyle\cancel{\bowtie}X^{j}_{t}~{}|~{}\mathcal{S}~{}\Rightarrow~{}X^{i}_{t-\tau}\cancel{\perp\\!\\!\\!\perp}X^{j}_{t}~{}|~{}\mathcal{S}~{}~{}\text{(contrapositive)}\,.$ ## Appendix S2 Proofs ### S2.1 Proof of Lemma 1 We first consider the following Lemma: ###### Lemma S1. Algorithm 1 returns a superset of lagged parents under Assumptions 1, i.e., $\mathcal{P}^{-}_{t}(X^{j}_{t})\subseteq\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ for all $X^{j}_{t}$ in $\mathbf{X}_{t}$. ###### Proof. We need to show that for arbitrary $(X^{i}_{t-\tau},X^{j}_{t})$ with $\tau>0$ we have $X^{i}_{t-\tau}\notin\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})~{}\Rightarrow~{}X^{i}_{t-\tau}\notin\mathcal{P}^{-}_{t}(X^{j}_{t})$. Algorithm 1 removes $X^{i}_{t-\tau}$ from $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ iff $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S}$ for some $\mathcal{S}\subseteq\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\}$ in the iterative CI tests. Then Adjacency Faithfulness directly implies that $X^{i}_{t-\tau}$ is not adjacent to $X^{j}_{t}$ and in particular $X^{i}_{t-\tau}\notin\mathcal{P}^{-}_{t}(X^{j}_{t})$. ∎ With this step we can prove Lemma 1. ###### Proof. The lemma states that under Assumptions 1 with Adjacency Faithfulness replaced by standard Faithfulness Alg. 1 for all $X^{j}_{t}\in\mathbf{X}_{t}$ returns $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})=\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$ where $\mathcal{C}_{t}(X^{j}_{t})$ denotes the contemporaneous ancestors of $X^{j}_{t}$. We need to show that for arbitrary $X^{i}_{t-\tau},X^{j}_{t}\in\mathbf{X}^{-}_{t+1}$ with $\tau>0$: (1) $X^{i}_{t-\tau}\notin\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})~{}\Rightarrow~{}X^{i}_{t-\tau}\notin\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$ and (2) $X^{i}_{t-\tau}\in\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})~{}\Rightarrow~{}X^{i}_{t-\tau}\in\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$. Ad 1) Algorithm 1 removes $X^{i}_{t-\tau}$ from $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ iff $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S}$ for some $\mathcal{S}\subseteq\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$ in the iterative CI tests. Then standard Faithfulness implies that $X^{i}_{t-\tau}\bowtie X^{j}_{t}~{}|~{}\mathcal{S}$ and in particular $X^{i}_{t-\tau}\notin\mathcal{P}^{-}_{t}(X^{j}_{t})$, as proven already in Lemma S1 under the weaker Adjacency Faithfulness Condition. To show that $X^{i}_{t-\tau}\notin\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$ we note that $\mathcal{S}\subseteq\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$ does not include any contemporaneous conditions and, hence, all contemporaneous directed paths from contemporaneous ancestors of $X^{j}_{t}$ are open and also paths from parents of those ancestors are open. If $X^{i}_{t-\tau}\in\bigcup_{X^{i}_{t}\in\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$, by the contraposition of standard Faithfulness we should observe $X^{i}_{t-\tau}\cancel{\perp\\!\\!\\!\perp}X^{j}_{t}~{}|~{}\mathcal{S}$. Then the fact that on the contrary we observe $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S}$ implies that $X^{i}_{t-\tau}\notin\bigcup_{X^{i}_{t}\in\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$. Ad 2) Now we have $X^{i}_{t-\tau}\in\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ which implies that $X^{i}_{t-\tau}\cancel{\perp\\!\\!\\!\perp}X^{j}_{t}~{}|~{}\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$ in the last iteration step of Alg. 1. By (1), $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ is a superset of $\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$. Define the lagged extra conditions as $W^{-}_{t}=\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t}),X^{i}_{t-\tau}\\}$. Since $W^{-}_{t}$ is lagged, it is a non-descendant of $X^{j}_{t}$ or any $X^{k}_{t}\in\mathcal{C}_{t}(X^{j}_{t})$. We now proceed by a proof by contradiction. Suppose to the contrary that $X^{i}_{t-\tau}\notin\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$. The Causal Markov Condition applies to both $X^{i}_{t-\tau}$ and $W^{-}_{t}$ and implies that $(X^{i}_{t-\tau},W^{-}_{t})\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$. From the weak union property of conditional independence we get $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t}),W^{-}_{t}$ which is equivalent to $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{X^{i}_{t-\tau}\\}$, contrary to the assumption, hence $X^{i}_{t-\tau}\in\bigcup_{X^{i}_{t}\in\\{X^{j}_{t}\\}\cup\mathcal{C}_{t}(X^{j}_{t})}\mathcal{P}^{-}_{t}(X^{i}_{t})$. ∎ ### S2.2 Proof of Theorem 1 ###### Proof. The theorem states that under Assumptions 1 $\widehat{\mathcal{G}^{*}}=\mathcal{G}^{*}$, where the $\mathcal{G}^{*}$ denotes the skeleton of the time series graph. We denote the two types of skeleton links $\to$ and ${\circ\\!{\\--}\\!\circ}$ here generically as ${\star\\!{\\--}\\!\star}$ and can assume $\tau_{\max}\geq\tau\geq 0$. We need to show that for arbitrary $X^{i}_{t-\tau},X^{j}_{t}\in\mathbf{X}^{-}_{t+1}$: (1) $X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}\notin\widehat{\mathcal{G}^{*}}~{}\Rightarrow~{}X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}\notin\mathcal{G}^{*}$ and (2) $X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}\notin\mathcal{G}^{*}~{}\Rightarrow~{}X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}\notin\widehat{\mathcal{G}^{*}}$. Ad (1): Algorithm 2 deletes a link $X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}$ from $\widehat{\mathcal{G}^{*}}$ iff $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$ for some $\mathcal{S}\subseteq\widehat{\mathcal{A}}_{t}(X^{j}_{t})$ in the iterative CI tests with $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ estimated in Alg. 1. $\widehat{\mathcal{A}}_{t}(X^{j}_{t})$ denotes the contemporaneous adjacencies. Then Adjacency Faithfulness directly implies that $X^{i}_{t-\tau}$ is not adjacent to $X^{j}_{t}$: $X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}\notin\mathcal{G}^{*}$. Ad (2): By Lemma 1 we know that $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ is a superset of the lagged parents of $X^{j}_{t}$. Denote the lagged, extra conditions occurring in the CI tests of Alg. 2 as $W^{-}_{t}=(\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{X^{i}_{t-\tau}\\},\widehat{\mathcal{B}}^{-}_{t-\tau}(X^{i}_{t-\tau}))\setminus\mathcal{P}(X^{j}_{t})$. $W^{-}_{t}$ does not contain parents of $X^{j}_{t}$ and by the assumption also $X^{i}_{t-\tau}$ is not a parent of $X^{j}_{t}$. We further assume that for $\tau=0$ $X^{i}_{t}$ is also not a descendant of $X^{j}_{t}$ since that case is covered if we exchange $X^{i}_{t}$ and $X^{j}_{t}$. Then the Causal Markov Condition implies $(X^{i}_{t-\tau},W^{-}_{t})\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{P}(X^{j}_{t})$. By the weak union property of conditional independence this leads to $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{P}(X^{j}_{t}),W^{-}_{t}$ which is equivalent to $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{P}(X^{j}_{t}),\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{X^{i}_{t-\tau}\\},\widehat{\mathcal{B}}^{-}_{t-\tau}(X^{i}_{t-\tau})$. Now Alg. 2 iteratively tests $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{X^{i}_{t-\tau}\\},\widehat{\mathcal{B}}^{-}_{t-\tau}(X^{i}_{t-\tau})$ for all $\mathcal{S}\subseteq\widehat{\mathcal{A}_{t}}(X^{j}_{t})$. By the first part of this proof, the estimated contemporaneous adjacencies are always a superset of the true contemporaneous adjacencies, i.e., $\mathcal{A}_{t}(X^{j}_{t})\subseteq\widehat{\mathcal{A}_{t}}(X^{j}_{t})$, and by Lemma 1 $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ is a superset of the lagged parents. Hence, at some iteration step $\mathcal{S}=\mathcal{P}_{t}(X^{j}_{t})$ and Alg. 2 will find $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{P}(X^{j}_{t}),\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\\{X^{i}_{t-\tau}\\},\widehat{\mathcal{B}}^{-}_{t-\tau}(X^{i}_{t-\tau})$ and remove $X^{i}_{t-\tau}{\star\\!{\\--}\\!\star}X^{j}_{t}$ from $\widehat{\mathcal{G}^{*}}$. ∎ For empty conditioning sets $\mathcal{S}$ ($p=0$), Alg. 2 is equivalent to the MCI algorithm [Runge et al., 2019b] with the slight change that the latter is initialized with a fully connected (lagged) graph, which has no effect asymptotically. In [Runge et al., 2019b] the authors prove the consistency of PCMCI assuming no contemporaneous causal links under the standard Faithfulness Condition. The proof above implies that PCMCI is already consistent under the weaker Adjacency Faithfulness Condition. ### S2.3 Proof of Lemma 2 ###### Proof. Time order and stationarity can be used to constrain the four cases as follows. Let us first consider a generic triple $X^{i}_{t_{i}}{\star\\!{\\--}\\!\star}X^{k}_{t_{k}}{\star\\!{\\--}\\!\star}X^{j}_{t_{j}}$. By stationarity we can fix $t=t_{j}$. We only need to consider cases with $t_{i},t_{k}\leq t$. If $t_{k}>t_{j}$, the triple is oriented already by time order and the case $t_{i}>t_{j}$ is symmetric. The possible triples in the collider phase of the original PC algorithm are $X^{i}_{t_{i}}{\star\\!{\\--}\\!\star}X^{k}_{t_{k}}{\star\\!{\\--}\\!\star}X^{j}_{t}$ where $(X^{i}_{t_{i}},X^{j}_{t})$ are not adjacent. For $t_{k}<t$ the time- order constraint automatically orients $X^{k}_{t_{k}}\to X^{j}_{t}$ and hence $X^{k}_{t_{k}}$ is a parent of $X^{j}_{t}$ and must always be in the separating set that makes $X^{i}_{t_{i}}$ and $X^{j}_{t}$ independent. Hence we only need to consider $t_{k}=t$ and can set $\tau=t-t_{i}$ ($\tau_{\max}\geq\tau\geq 0$), leaving the two cases of unshielded triples $X^{i}_{t-\tau}\to X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ (for $\tau>0$) or $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ (for $\tau=0$) in $\mathcal{G}$ where $(X^{i}_{t-\tau},X^{j}_{t})$ are not adjacent. Since $X^{k}_{t}$ is contemporaneous to $X^{j}_{t}$, this restriction implies that only contemporaneous parts of separating sets are relevant for the collider orientation phase. For rule R1 in the orientation phase the original PC algorithm considers the remaining triples with $X^{i}_{t-\tau}\to X^{k}_{t}$ that were not oriented by the collider phase (or by time order). This leaves $X^{i}_{t-\tau}\to X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ where $\tau_{\max}\geq\tau\geq 0$. For rule R2 the original PC algorithm considers $X^{i}_{t_{i}}\to X^{k}_{t_{k}}\to X^{j}_{t}$ with $X^{i}_{t_{i}}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$. The latter type of link leads to $t_{i}=t$ and time order restricts the triples to $X^{i}_{t}\to X^{k}_{t}\to X^{j}_{t}$ with $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$. For rule R3 the original PC algorithm considers $X^{i}_{t_{i}}{\circ\\!{\\--}\\!\circ}X^{k}_{t_{k}}\to X^{j}_{t}$ and $X^{i}_{t_{i}}{\circ\\!{\\--}\\!\circ}X^{l}_{t_{l}}\to X^{j}_{t}$ where $(X^{k}_{t_{k}},X^{l}_{t_{l}})$ are not adjacent and $X^{i}_{t_{i}}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$. The latter constraint leads to $t_{i}=t$ and $X^{i}_{t_{i}}{\circ\\!{\\--}\\!\circ}X^{k}_{t_{k}}$ and $X^{i}_{t_{i}}{\circ\\!{\\--}\\!\circ}X^{k}_{t_{l}}$ imply $t_{k}=t_{l}=t$. Hence we only need to check triples $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}\to X^{j}_{t}$ and $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{l}_{t}\to X^{j}_{t}$ where $(X^{k}_{t},X^{l}_{t})$ are not adjacent and $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$. ∎ ### S2.4 Proof of Theorem 2 ###### Proof. We first consider the case under Assumptions 1 with Adjacency Faithfulness and PCMCI+ in conjunction with the conservative collider orientation rule in Alg. S2. We need to show that all separating sets estimated in Alg. S2 during the conservative orientation rule are correct. From the soundness (Theorem 1) and correctness of the separating sets follows the correctness of the collider orientation phase and the rule orientation phase which implies the completeness. By Lemma 2 we only need to prove that in Alg. S2 for unshielded triples $X^{i}_{t-\tau}\to X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ (for $\tau>0$) or $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ (for $\tau=0$) the separating sets among subsets of contemporaneous neighbors of $X^{j}_{t}$ and, if $\tau=0$, of $X^{i}_{t}$, are correct. Algorithm S2 tests $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$ for all $\mathcal{S}{\subseteq}\widehat{\mathcal{A}}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$ and for all $\mathcal{S}{\subseteq}\widehat{\mathcal{A}}_{t}(X^{i}_{t}){\setminus}\\{X^{j}_{t}\\}$ (if $\tau=0$). Since PCMCI+ is sound, all adjacency information is correct and since all CI tests are assumed correct, all information on separating sets is correct. Furthermore, with the conservative rule those triples where only Adjacency Faithfulness, but not standard Faithfulness, holds will be correctly marked as ambiguous triples. Under standard Faithfulness the completeness requires to prove that PCMCI+ without the conservative orientation rule yields correct separating set information. By Lemma 2 also here we need to consider only separating sets among subsets of contemporaneous neighbors of $X^{j}_{t}$. Algorithm 2 tests $X^{i}_{t-\tau}\perp\\!\\!\\!\perp X^{j}_{t}~{}|~{}\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau})$ for all $\mathcal{S}{\subseteq}\widehat{\mathcal{A}}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$. And again, since PCMCI+ is sound, all adjacency information is correct and since all CI tests are assumed correct, all information on separating sets is correct, from which the completeness for this case follows. ∎ ### S2.5 Proof of Theorem 3 ###### Proof. Order-independence follows straightforwardly from sticking to the PC algorithm version in [Colombo and Maathuis, 2014]. In particular, Alg. 1 and Alg. 2 are order-independent since they are based on PC stable where adjacencies are removed only after each loop over conditions of cardinality $p$. Furthermore, the collider phase (Alg. S2) and rule orientation phase (Alg. S3) are order- independent by marking triples with inconsistent separating sets as ambiguous and consistently marking conflicting link orientations by ${x\\!{\\--}\\!x}$. ∎ ### S2.6 Proof of Theorem 4 ###### Proof. The theorem states that under Assumptions 1 the effect size for the PCMCI+ oracle case CI tests in Alg. 2 for $p=0$ for contemporaneous true links $X^{i}_{t}\to X^{j}_{t}\in\mathcal{G}$ is greater than that of PCMCI${}^{+}_{0}$: $I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}),\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t}))>\min(I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})),\,I(X^{i}_{t};X^{j}_{t}|\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t})))$ if both $X^{i}_{t}$ and $X^{j}_{t}$ have parents that are not shared with the other. We will use an information-theoretic framework here and consider the conditional mutual information. To prove this statement, we denote by $\mathcal{B}_{i}=\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t})\setminus\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ the lagged conditions of $X^{i}_{t}$ that are not already contained in those of $X^{j}_{t}$ and, correspondingly, $\mathcal{B}_{j}=\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\setminus\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t})$. Since both $X^{i}_{t}$ and $X^{j}_{t}$ have parents that are not shared with the other and we assume the oracle case, both these sets are non-empty. Further, we denote the common lagged conditions as $\mathcal{B}_{ij}=\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})\cap\widehat{\mathcal{B}}^{-}_{t}(X^{i}_{t})$ and make use of the following conditional independencies, which hold by the Markov assumption: (1) $\mathcal{B}_{i}\perp\\!\\!\\!\perp X^{j}_{t}|\mathcal{B}_{j},\mathcal{B}_{ij},X^{i}_{t}$ and (2) $\mathcal{B}_{j}\perp\\!\\!\\!\perp X^{i}_{t}|\mathcal{B}_{i},\mathcal{B}_{ij}$. We first prove that, given a contemporaneous true link $X^{i}_{t}\to X^{j}_{t}\in\mathcal{G}$, $I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{j})>I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{i})$ by using the following two ways to apply the chain rule of conditional mutual information: $\displaystyle I(X^{i}_{t},\mathcal{B}_{i};X^{j}_{t},\mathcal{B}_{j}|\mathcal{B}_{ij})=$ $\displaystyle=I(X^{i}_{t},\mathcal{B}_{i};\mathcal{B}_{j}|\mathcal{B}_{ij})+I(X^{i}_{t},\mathcal{B}_{i};X^{j}_{t}|\mathcal{B}_{ij}\mathcal{B}_{j})$ $\displaystyle=I(\mathcal{B}_{i};\mathcal{B}_{j}|\mathcal{B}_{ij})+\underbrace{I(X^{i}_{t};\mathcal{B}_{j}|\mathcal{B}_{ij},\mathcal{B}_{i})}_{=0~{}~{}\text{(Markov)}}$ $\displaystyle\phantom{=}+I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{j})+\underbrace{I(\mathcal{B}_{i};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{j},X^{i}_{t})}_{=0~{}~{}\text{(Markov)}}$ (S1) and $\displaystyle I(X^{i}_{t},\mathcal{B}_{i};X^{j}_{t},\mathcal{B}_{j}|\mathcal{B}_{ij})=$ $\displaystyle=I(\mathcal{B}_{i};X^{j}_{t},\mathcal{B}_{j}|\mathcal{B}_{ij})+I(X^{i}_{t};X^{j}_{t},\mathcal{B}_{j}|\mathcal{B}_{ij}\mathcal{B}_{i})$ $\displaystyle=I(\mathcal{B}_{i};\mathcal{B}_{j}|\mathcal{B}_{ij})+\underbrace{I(\mathcal{B}_{i};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{j})}_{>0~{}~{}\text{since $X^{i}_{t}\to X^{j}_{t}$}}$ $\displaystyle\phantom{=}+I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{i})+\underbrace{I(X^{i}_{t};\mathcal{B}_{j}|\mathcal{B}_{ij},\mathcal{B}_{i},X^{j}_{t})}_{>0~{}~{}\text{since $X^{i}_{t}\to X^{j}_{t}$}}$ (S2) where (S1) and (S2) denote two different applications of the chain rule. From this is follows that $I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{j})>I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{i})$. Hence, it remains to prove that $I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{j},\mathcal{B}_{i})>I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{i})$, which we also do by the chain rule: $\displaystyle I(X^{i}_{t};X^{j}_{t},\mathcal{B}_{j}|\mathcal{B}_{ij},\mathcal{B}_{i})=$ $\displaystyle=I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{i})+\underbrace{I(X^{i}_{t};\mathcal{B}_{j}|\mathcal{B}_{ij},\mathcal{B}_{i},X^{j}_{t})}_{>0~{}~{}\text{since $X^{i}_{t}\to X^{j}_{t}$}}$ (S3) $\displaystyle=\underbrace{I(X^{i}_{t};\mathcal{B}_{j}|\mathcal{B}_{ij},\mathcal{B}_{i})}_{=0~{}~{}\text{(Markov)}}+I(X^{i}_{t};X^{j}_{t}|\mathcal{B}_{ij},\mathcal{B}_{i},\mathcal{B}_{j})$ (S4) ∎ ## Appendix S3 Further pseudo code Algorithms S2 and S3 detail the pseudo-code for the PCMCI+ / PCMCI${}^{+}_{0}$ / PC collider phase with different collider rules and the orientation phase. Algorithm S2 (Detailed PCMCI+ / PCMCI${}^{+}_{0}$ / PC collider phase with different collider rules) 1:$\mathcal{G}$ and sepset from Alg. 2, rule $=\\{$’none’, ’conservative’, ’majority’$\\}$, time series dataset $\mathbf{X}=(X^{1},\,\ldots,X^{N})$, significance threshold $\alpha_{\rm PC}$, ${\rm CI}(X,\,Y,\,\mathbf{Z})$, PCMCI+ / PCMCI${}^{+}_{0}$: $\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t})$ for all $X^{j}_{t}$ in $\mathbf{X}_{t}$ 2:for all unshielded triples $X^{i}_{t-\tau}\to X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ ($\tau>0$) or $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ ($\tau=0$) in $\mathcal{G}$ where $(X^{i}_{t-\tau},X^{j}_{t})$ are not adjacent do 3: if rule $=$ ’none’ then 4: if $X^{k}_{t}$ is not in sepset$(X^{i}_{t-\tau},X^{j}_{t})$ then 5: Orient $X^{i}_{t-\tau}\to X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ ($\tau>0$) or $X^{i}_{t-\tau}{\circ\\!{\\--}\\!\circ}X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ ($\tau=0$) as $X^{i}_{t-\tau}\to X^{k}_{t}\leftarrow X^{j}_{t}$ 6: else 7: PCMCI+ / PCMCI${}^{+}_{0}$: Define contemporaneous adjacencies $\widehat{\mathcal{A}}(X^{j}_{t})=\widehat{\mathcal{A}}_{t}(X^{j}_{t})=\\{X^{i}_{t}{\neq}X^{j}_{t}\in\mathbf{X}_{t}:X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}~{}\text{in $\mathcal{G}$}\\}$ 8: PC: Define full adjacencies $\widehat{\mathcal{A}}(X^{j}_{t})$ for all (lagged and contemporaneous) links in $\mathcal{G}$ 9: for all for all $\mathcal{S}{\subseteq}\widehat{\mathcal{A}}(X^{j}_{t}){\setminus}\\{X^{i}_{t-\tau}\\}$ and for all $\mathcal{S}{\subseteq}\widehat{\mathcal{A}}(X^{i}_{t}){\setminus}\\{X^{j}_{t}\\}$ (if $\tau=0$) do 10: Evaluate CI($X^{i}_{t{-}\tau},X^{j}_{t},\mathbf{Z})$ with 11: PCMCI+: $\mathbf{Z}{=}(\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\},\widehat{\mathcal{B}}^{-}_{t{-}\tau}(X^{i}_{t{-}\tau}))$ 12: PCMCI${}^{+}_{0}$: $\mathbf{Z}{=}(\mathcal{S},\widehat{\mathcal{B}}^{-}_{t}(X^{j}_{t}){\setminus}\\{X^{i}_{t{-}\tau}\\})$ 13: PC: $\mathbf{Z}{=}\mathcal{S}$ 14: Store all subsets $\mathcal{S}$ with $p$-value $>\alpha_{\rm PC}$ as separating subsets 15: if no separating subsets are found then 16: Mark triple as ambiguous 17: else 18: Compute fraction $n_{k}$ of separating subsets that contain $X^{k}_{t}$ 19: if rule $=$ ’conservative’ then 20: Orient triple as collider if $n_{k}{=}0$, leave unoriented if $n_{k}{=}1$, and mark as ambiguous if $0{<}n_{k}{<}1$ 21: else if rule $=$ ’majority’ then 22: Orient triple as collider if $n_{k}{<}0.5$, leave unoriented if $n_{k}{>}0.5$, and mark as ambiguous if $n_{k}{=}0.5$ 23: Mark links in $\mathcal{G}$ with conflicting orientations as ${x\\!{\\--}\\!x}$ 24:return $\mathcal{G}$, sepset, ambiguous triples, conflicting links Algorithm S3 (Detailed PCMCI+ / PCMCI${}^{+}_{0}$ / PC rule orientation phase) 1:$\mathcal{G}$, ambiguous triples, conflicting links 2:while any unambiguous triples suitable for rules R1-R3 are remaining do 3: Apply rule R1 (orient unshielded triples that are not colliders): 4: for all unambiguous triples $X^{i}_{t-\tau}\to X^{k}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ where $(X^{i}_{t-\tau},X^{j}_{t})$ are not adjacent do 5: Orient as $X^{i}_{t-\tau}\to X^{k}_{t}\to X^{j}_{t}$ 6: Mark links with conflicting orientations as ${x\\!{\\--}\\!x}$ 7: Apply rule R2 (avoid cycles): 8: for all unambiguous triples $X^{i}_{t}\to X^{k}_{t}\to X^{j}_{t}$ with $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ do 9: Orient as $X^{i}_{t}\to X^{j}_{t}$ 10: Mark links with conflicting orientations as ${x\\!{\\--}\\!x}$ 11: Apply rule R3 (orient unshielded triples that are not colliders and avoid cycles): 12: for all pairs of unambiguous triples $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{k}_{t}\to X^{j}_{t}$ and $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{l}_{t}\to X^{j}_{t}$ where $(X^{k}_{t},X^{l}_{t})$ are not adjacent and $X^{i}_{t}{\circ\\!{\\--}\\!\circ}X^{j}_{t}$ do 13: Orient as $X^{i}_{t}\to X^{j}_{t}$ 14: Mark links with conflicting orientations as ${x\\!{\\--}\\!x}$ 15:return $\mathcal{G}$, conflicting links ## Appendix S4 Implementation details In the linear and nonlinear numerical experiments PCMCI+ is compared with the PC algorithm, both implemented with the appropriate CI test (ParCorr for the linear case, GPDC for the nonlinear case). For the linear numerical experiments we additionally consider representatives from two further frameworks: GCresPC, a combination of GC with PC applied to residuals, and an autoregressive model version of LiNGAM [Hyvärinen et al., 2010], a representative of the SCM framework. Their implementations are as follows. ### S4.1 LiNGAM For LiNGAM the code was taken from https://github.com/cdt15/lingam which provides a class VARLiNGAM. The method was called follows: Input: data, tau_max model = lingam.VARLiNGAM(lags=tau_max, criterion=None, prune=True) model.fit(data) val_matrix = model.adjacency_matrices_.transpose(2,1,0) graph = (val_matrix != 0.).astype(’int’) Output: graph The causal graph `graph` encodes the causal relations in an array of shape `(N, N, tau_max + 1)`. The option `criterion=None` just ignores the optional automatic selection of `lags`, which is here set to the same `tau_max` for all methods. I could not find a way to obtain p-values in the VARLiNGAM implementation, but with the parameter setting `prune=True` the resulting adjacency matrices are regularized with an adaptive LASSO approach using the BIC criterion to find the optimal regularization hyper-parameter (`sklearn.LassoLarsIC(criterion=’bic’)`). Non-zero adjacencies were then evaluated as causal links. Note that all other methods can be intercompared at different $\alpha_{\rm PC}$ levels while for comparison against LiNGAM we focus on its relative changes rather than absolute performance. ### S4.2 GCresPC There was no code available for the method proposed in [Moneta et al., 2011]. The present implementation first fits a VAR model up to $\tau_{\max}$ and applies the PC algorithm on the residuals. To remove spurious lagged links (due to contemporaneous paths), the PC algorithm was additionally run on significant lagged and contemporaneous links, but the orientation phase was restricted to contemporaneous links, as proposed in [Moneta et al., 2011]. The following Python pseudo-code utilizes functionality from the tigramite package, numpy, and statsmodels: Input: data, tau_max, alpha import functions/classes ParCorr, PCMCI, DataFrame from tigramite graph = np.zeros((N, N, tau_max + 1)) # 1. Estimate lagged adjacencies (to be updated in step 3.) tsamodel = tsa.var.var_model.VAR(data) results = tsamodel.fit(maxlags=tau_max, trend=’nc’) pvalues = results.pvalues values = results.coefs residuals = results.resid lagged_parents = significant lagged links at alpha # 2. Run PC algorithm on residuals (with tau_max=0) pcmci = PCMCI(dataframe=DataFrame(residuals), cond_ind_test=ParCorr()) pcmcires = pcmci.run_pcalg(pc_alpha=alpha, tau_min=0, tau_max=0) # Update contemporaneous graph graph[:,:,0] = pcmcires[’graph’][:,:,0] # 3. Run PC algorithm on significant lagged and contemporaneous adjacencies # to remove spurious lagged links due to contemporaneous parents selected_links = lagged_parents + significant contemporaneous adjacencies pcmci = PCMCI(dataframe=DataFrame(data), cond_ind_test=ParCorr()) pcmcires = pcmci.run_pcalg(selected_links=selected_links, pc_alpha=alpha, tau_min=0, tau_max=tau_max) # Update lagged part of graph graph[:,:,1:] = pcmcires[’graph’][:,:,1:] Output: graph Note that the contemporaneous graph structure in `graph` comes only from applying the PC algorithm to the residuals and, hence, does not utilize triples containing lagged adjacencies. Step 3 is necessary to remove spurious lagged links due to contemporaneous parents. The output of GCresPC depends on $\alpha_{\rm PC}$ as for PCMCI+ and the PC algorithm. ## Appendix S5 Further numerical experiments Next to repeating the overview figure for the linear Gaussian model setup from the main text in Fig. S1, in Fig. S2 we show the linear mixed noise setup, and in Fig. S3 the nonlinear mixed noise setup. The remaining pages contain results of further numerical experiments that evaluate different $a,\,N,\,T,\,\tau_{\max}$ and $\alpha_{\rm PC}$ for the linear model setups. All results and more will be contributed to the causality benchmark platform `www.causeme.net` [Runge et al., 2019a] to facilitate a further expanded method evaluation. Figure S1: Numerical experiments with linear Gaussian setup for varying (A) autocorrelation strength $a$ (B) number of variables $N$ (C) sample size $T$ and (D) maximum time lag $\tau_{\max}$. All remaining setup parameters indicated in the top right. Errorbars show std. errors or the 90% range (for runtime). The insets show ANOVA statistics. Figure S2: Numerical experiments with linear mixed noise setup for varying (A) autocorrelation strength $a$ (B) number of variables $N$ (C) sample size $T$ and (D) maximum time lag $\tau_{\max}$. All remaining setup parameters indicated in the top right. Errorbars show std. errors or the 90% range (for runtime). The insets show ANOVA statistics. Figure S3: Numerical experiments with nonlinear mixed noise setup for varying (A) autocorrelation strength $a$ (B) number of variables $N$ (C) sample size $T$ and (D) maximum time lag $\tau_{\max}$. All remaining setup parameters indicated in the top right. Errorbars show std. errors or the 90% range (for runtime). The insets show ANOVA statistics. Figure S4: Numerical experiments with linear Gaussian setup for varying autocorrelation $a$ and $T=200$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for $N=2,\,3,\,5,\,10$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S5: Numerical experiments with linear Gaussian setup for varying autocorrelation $a$ and $T=500$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for $N=2,\,3,\,5,\,10$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S6: Numerical experiments with linear Gaussian setup for varying autocorrelation $a$ and $T=1000$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for $N=2,\,3,\,5,\,10$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S7: Numerical experiments with linear Gaussian setup for varying number of variables $N$ and $T=200$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S8: Numerical experiments with linear Gaussian setup for varying number of variables $N$ and $T=500$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S9: Numerical experiments with linear Gaussian setup for varying number of variables $N$ and $T=1000$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S10: Numerical experiments with linear Gaussian setup for varying sample size $T$ for $N=5$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S11: Numerical experiments with linear Gaussian setup for varying sample size $T$ for $N=10$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S12: Numerical experiments with linear Gaussian setup for varying sample size $T$ for $N=20$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S13: Numerical experiments with linear Gaussian setup for varying maximum time lag $\tau_{\max}$ and $T=200$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S14: Numerical experiments with linear Gaussian setup for varying maximum time lag $\tau_{\max}$ and $T=500$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S15: Numerical experiments with linear Gaussian setup for varying maximum time lag $\tau_{\max}$ and $T=1000$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S16: Numerical experiments with linear mixed noise setup for varying autocorrelation $a$ and $T=200$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for $N=2,\,3,\,5,\,10$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S17: Numerical experiments with linear mixed noise setup for varying autocorrelation $a$ and $T=500$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for $N=2,\,3,\,5,\,10$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S18: Numerical experiments with linear mixed noise setup for varying autocorrelation $a$ and $T=1000$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for $N=2,\,3,\,5,\,10$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S19: Numerical experiments with linear mixed noise setup for varying number of variables $N$ and $T=200$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S20: Numerical experiments with linear mixed noise setup for varying number of variables $N$ and $T=500$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S21: Numerical experiments with linear mixed noise setup for varying number of variables $N$ and $T=1000$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S22: Numerical experiments with linear mixed noise setup for varying sample size $T$ for $N=5$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S23: Numerical experiments with linear mixed noise setup for varying sample size $T$ for $N=10$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S24: Numerical experiments with linear mixed noise setup for varying sample size $T$ for $N=20$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S25: Numerical experiments with linear mixed noise setup for varying maximum time lag $\tau_{\max}$ and $T=200$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S26: Numerical experiments with linear mixed noise setup for varying maximum time lag $\tau_{\max}$ and $T=500$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel. Figure S27: Numerical experiments with linear mixed noise setup for varying maximum time lag $\tau_{\max}$ and $T=1000$ . The left (right) column shows results for significance level $\alpha=0.01$ ($\alpha=0.05$). The rows depict results for increasing autocorrelations $a$ (top to bottom). All model and method parameters are indicated in the upper right of each panel.

2024-09-04T02:54:58.146594

2003.03694

arxiv-papers

††thanks: These authors contributed equally††thanks: These authors contributed equally # Bandgap Control in Two-Dimensional Semiconductors via Coherent Doping of Plasmonic Hot Electrons Yu-Hui Chen School of Physics, Beijing Institute of Technology, Beijing 10081, China Ronnie R. Tamming MacDiarmid Institute for Advanced Materials and Nanotechnology, Dodd-Walls Centre for Photonic and Quantum Technologies, School of Chemical and Physical Sciences, Victoria University of Wellington, Wellington 6012, New Zealand Kai Chen MacDiarmid Institute for Advanced Materials and Nanotechnology, Dodd-Walls Centre for Photonic and Quantum Technologies, School of Chemical and Physical Sciences, Victoria University of Wellington, Wellington 6012, New Zealand Zhepeng Zhang Department of Materials Science and Engineering, College of Engineering, Center for Nanochemistry (CNC), College of Chemistry and Molecular Engineering, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China Yanfeng Zhang Department of Materials Science and Engineering, College of Engineering, Center for Nanochemistry (CNC), College of Chemistry and Molecular Engineering, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China Justin M. Hodgkiss MacDiarmid Institute for Advanced Materials and Nanotechnology, Dodd-Walls Centre for Photonic and Quantum Technologies, School of Chemical and Physical Sciences, Victoria University of Wellington, Wellington 6012, New Zealand Richard J. Blaikie MacDiarmid Institute for Advanced Materials and Nanotechnology, Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, University of Otago, PO Box 56, Dunedin 9016, New Zealand Boyang Ding <EMAIL_ADDRESS>MacDiarmid Institute for Advanced Materials and Nanotechnology, Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, University of Otago, PO Box 56, Dunedin 9016, New Zealand Min Qiu<EMAIL_ADDRESS>Key Laboratory of 3D Micro/Nano Fabrication and Characterization of Zhejiang Province, School of Engineering, Westlake University, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China Institute of Advanced Technology, Westlake Institute for Advanced Study, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China ###### Abstract Bandgap control is of central importance for semiconductor technologies. The traditional means of control is to dope the lattice chemically, electrically or optically with charge carriers. Here, we demonstrate for the first time a widely tunable bandgap (renormalisation up to 650 meV at room-temperature) in two-dimensional (2D) semiconductors by coherently doping the lattice with plasmonic hot electrons. In particular, we integrate tungsten-disulfide (WS2) monolayers into a self-assembled plasmonic crystal, which enables coherent coupling between semiconductor excitons and plasmon resonances. Accompanying this process, the plasmon-induced hot electrons can repeatedly fill the WS2 conduction band, leading to population inversion and a significant reconstruction in band structures and exciton relaxations. Our findings provide an innovative and effective measure to engineer optical responses of 2D semiconductors, allowing a great flexiblity in design and optimisation of photonic and optoelectronic devices. Two-dimensional (2D) semiconductors, such as transition metal dichalcogenides (TMDCs)Mak _et al._ (2010); Splendiani _et al._ (2010), have direct bandgap at their monolayer limit, exhibiting tremendous potential in development of next-generation nanoscale devices. Like in their bulk counterparts, bandgap control plays a vital role in 2D semiconductor technoglogies, since it enables the creation of desirable optoelectronic properties that are required in numerous applications, ranging from lasersYe _et al._ (2015) to modulatorsMak and Shan (2016), photodetectorsLopez-Sanchez _et al._ (2013) and photocatalysisVoiry _et al._ (2013). The traditional means of control is to dope the lattice chemicallyKim _et al._ (2015), electricallyChernikov _et al._ (2015a) or opticallyChernikov _et al._ (2015b) with charge carriers, the practicality of which is, however, limited by many factors, e.g. the irreversible bandgap modification, contact-type control and requirement of ultrastrong pump. Here we report that one can flexibly and effectively modify the electronic band structures of 2D semiconductors by establishing coherent strong coupling between the semiconductor excitons and a plasmonic resonatorEbbesen _et al._ (1998); Liu and Lalanne (2008). In particular, plasmonic resonators are metallic nanostructures that support collective oscillation of electrons, known as plasmons. The excitation of plasmons can produce hot electrons, i.e. highly energetic electrons with non-equilibrium thermal distributionsClavero (2014); Brongersma _et al._ (2015), which, in the strong coupling regime, can repeatedly dope the lattice along with the coherent plasmon-exciton energy exchange. As a result, the bandgap of 2D semiconductors is significantly renormalised and the renormalisation can be easily altered through changing the detuning between plasmons and excitons. The schematic of our sample in Fig.1a demonstrates a WS2 monolayer (ML) deposited onto a plasmonic crystal (PC)Ding _et al._ (2013, 2019), which comprises of a periodic array of silver capped silica nanospheres that are coated with an ultrathin Al2O3 spacer. This metal-insulator-semiconductor configuration constitutes PC-WS2 hybrid systems, supporting plasmon lattice modes propagating on the PC-WS2 interface. Here the top WS2 MLs belong to the family of atomically thin TMDCs, having been extensively studiedYe _et al._ (2014); Sie _et al._ (2017); Ruppert _et al._ (2017); Cunningham _et al._ (2017); Steinhoff _et al._ (2017) for their unusual exciton-dominated optical responses, such as high absorption and emission efficiency. These properties make the PC-WS2 systems a suitable platform to study plasmon-exciton interactionsDing _et al._ (2019). The PC geometries were chosen to excite plasmon lattice modesEbbesen _et al._ (1998); Liu and Lalanne (2008); Ding _et al._ (2013, 2019) that can match the frequency of exciton A in WS${}_{\text{2}}$ MLs at certain incident angles $\theta$. The plasmon modes show red-shift dispersion at higher $\theta$ (yellow curve in Fig.1b), matching the frequency of exciton A ($E=2.061$ eV) at $\theta=22^{\circ}$. In this case, plasmon modes can coherently couple with excitons, leading to the formation of plasmon-exciton polaritons, i.e. half- light half-matter quasiparticles that inherit properties from both the plasmonic and excitonic components. As a result, the transmission maxima exhibit pronounced splitting features that follow the dispersions of upper polariton (UP) and lower polariton (LP), indicating the establishment of strong coupling between plasmons and excitons. When the frequency of the plasmon mode is tuned in resonance with exciton A ($\theta=22^{\circ}$), the hybrid system is characterised by a vacuum Rabi splitting of $\hbar\cdot\Omega_{\text{R}}\approx 136$ meV. More detailed analysis of strong plasmon-exciton coupling in equilibrium states can be found in a previous workDing _et al._ (2019) and Fig.S1 in the Supplementary Information (SI). Upon photoexcitation, the transient optical responses of PC-WS2 samples can be characterised using femtosecond transient absorption (TA) spectroscopy (Fig.2a and Methods), which enables incident angle-resolved probes of the optical properties and dynamics of WS2 MLs that are strongly coupled with plasmon resonancesDarby _et al._ (2016). Fig.2b shows the transient transmission spectra ($\Delta\text{T}/\text{T}$) with a pump fluence of $12\mu\text{J/cm}^{2}$ as a function of time delay and energy at the tuned state ($\theta=22^{\circ}$), which displays two split relaxation traces flanking the spectral position of exciton A ($E=2.061$ eV), corresponding to UP and LP. This sharply contrasts with the single-trace relaxations of exciton B ($E=2.471$ eV, Fig.2b) and uncoupled exciton A in bare WS2 MLs (Fig.S2 in SI). When the PC is detuned from exciton A, e.g. at $\theta=30^{\circ}$ (Fig.2c), a single relaxation trace appears, highly resembling the trace of bare exciton A. These ultrashort timescale results confirm again the strong coupling nature of our PC-WS2 systems. It is worth noting that the photoinduced absorption minimum associated with tuned polaritons appears at the 1 to 10 ps range (blue area centred at $E=1.946$ eV in Fig. 2b and the corresponding $\Delta\text{T}/\text{T}$ transient with negative magnitudes in Fig.2f), obviously delayed compared to the minimum near exciton B (Fig.2b) and its counterpart in the detuned polaritons (Fig. 2c), which all emerge simultaneously after the arrival of the pump pulse. Similar postponed minima have been found in transient spectra of bare TMDC MLs, which typically arise from enhanced exciton-exciton and/or exciton-electron interactions under high-power pump that can populate high- density carriers in the latticeCeballos _et al._ (2016); Ruppert _et al._ (2017); Cunningham _et al._ (2017); Sie _et al._ (2017) (see Section 2 in SI for detailed discussions). What is different is that, in our hybrid systems, the delayed minima appear under much lower pump intensity than that in the reference experiments for bare WS2 MLs and are only associated with tuned polaritons. More importantly, it is noted that a $\Delta\text{T}/\text{T}$ maximum lasting for $\sim 1$ ps in $E=1.6$ to $1.8$ eV arise in the tuned polariton spectra (Fig.2b), which, in contrast, is remarkably weaker in the detuned state (Fig.2c) and is completely absent in bare WS2 MLs (Fig.S2 in SI). The integrated $\Delta\text{T}/\text{T}$ spectrum near zero probe delay (Fig.2d) shows that the broad maximum has positive magnitudes, which indicates negative optical absorption or positive gain, being a clear evidence of bandgap renormalisation accompanied by population inversion.Chernikov _et al._ (2015b) Such phenomena are typically induced by the population of high-density carriers in 2D semiconductor latticeMeckbach _et al._ (2018), which leads to the non-equilibrium occupation of electron and/or hole states that can induce the formation of new quasiparticle bandgaps. This process can be decribed byPeyghambarian _et al._ (1993): $\Delta E_{\text{g}}=-\underset{q\neq 0}{\sum}V_{\text{s}}(q)\,[f_{\text{e}}(q)+f_{\text{h}}(q)]-\underset{q\neq 0}{\sum}[V_{\text{s}}(q)-V(q)]$ (1) where $V_{\text{s}}(q)$ and $V(q)$ represent fourier transforms of screened and unscreened Coulomb potentials, while $f_{\text{e}}(q)$ and $f_{\text{h}}(q)$ are occupation probabilities of electron and hole with momentum $q$. The onset of the new bandgap can be extracted from the low- energy end of the broad maximum. It means that in our experiments, the renormalised bandgap starts at $E_{\text{g}}\approx 1.60$ eV, lying $\sim 400$ meV below LP and $\sim 650$ meV below the initial bandgap of WS2 MLs (given that the binding energy of exciton A is $\sim 200$ meVCunningham _et al._ (2017)). This is, to the best of our knowledge, the largest bandgap renormalisation in 2D semiconductors under such a low pump intensity (12$\mu$J$/$cm2) to date, which, in the meanwhile, results in the inversion of carrier population near the newly formed band edgeChernikov _et al._ (2015b); Meckbach _et al._ (2018), presenting as optical gains, i.e. the broad maximum in Fig.2b and 2d. These unusual spectral and transient features are broadly understood as the presence of high-density carriers, which, in our tuned PC-WS2 systems, surprisingly have been achieved under room temperature and extremely low pump intensity ($12\mu$J$/$cm2). This sharply contrasts with similar observationsChernikov _et al._ (2015b) in bare WS2 single/bi-layers with ultrastrong photoexcitation ($840\mu$J$/$cm2 at 70 K or $3400\mu$J$/$cm2 at room temperature). In their study, the population of high-density carriers are a result of Mott-transition, which are induced by enhanced exciton-exciton interactions under high-power pump, reducing exciton binding energy, finally breaking excitons into unbound electron-hole plasmaChernikov _et al._ (2015b); Steinhoff _et al._ (2017). In our experiments, the pump power is too low to develop a Mott-transition, suggesting that there must be other sources that can provide large numbers of additional carriers. To understand the origin of these carriers, we turn to discuss one unique property of plasmon-exciton polaritons, i.e. the generation of hot electrons that are inherited from the polaritons’ plasmon root. In particular, hot electrons are electrons with non-equilibrium thermal distributions, generated by plasmon dephasing from wave-like states through non-radiative decayBrongersma _et al._ (2015), which can electrically dope adjacent semiconductorsFang _et al._ (2012), modifying their photovoltaic and photocatalytic performanceClavero (2014). When plasmons are coupled to exciton-like resonances in semiconductors, the hot electron density can be highly enhanced in the lattice through direct electron tunnelingGarcía De Arquer _et al._ (2013) or dipole-dipole interactionCushing _et al._ (2012). Therefore it is very likely that the high-density carriers in tuned PC-WS2 systems are the hot electrons introduced during strong coupling process. (See Section 3 in SI for detailed discussions) The analyses of relaxation dynamics of tuned and detuned polaritons support the hot electron model. We note that both the UP and LP in Fig.2f demonstrate slower decays than that of detuned states in Fig.2g (Table S2 in SI for fitting parameters). This observation coincides with a previous studyBoulesbaa _et al._ (2016), clearly indicating the involvement of plasmonic hot electrons in the strong plasmon-exciton coupling process. Specifically, as the system sits in the strong coupilng regime, after photoexcitation, excitons and plasmons coherently exchange energy at the Rabi frequency ($\sim 136$ meV)Vasa _et al._ (2013), while the plasmon-to-exciton process is accompanied by hot electron population in the lattice. Such a charge population runs at an ultrashort period of $\sim 30$ fs ($T_{\text{R}}=2\pi/\Omega_{\text{R}}$), which is too short to be caught by our equipment, also greatly shorter than the exciton formation ($<1$ ps)Ceballos _et al._ (2016), the non-radiative decay (at scales of $10$ ps) and the radiative decay process (up to few- hundred ps) in WS2 MLsRuppert _et al._ (2017); Sie _et al._ (2017). This means that during exciton relaxation, there is frequent tunneling/generation of hot electrons that can repeatedly fill the unoccupied states in conduction band of WS2 monolayers, which slow down the exciton bleaching via Pauli blocking and lead to the extended lifetimes (Section 4 in SI for more details). Given that there is little evidence for other possible carrier sources, e.g. polariton condensates Byrnes _et al._ (2014), we conclude that coherent doping of plasmonic hot electrons is the origin of the spectral and transient features that require high-density population. In particular, the hot electron population repeatedly takes place throughout the whole relaxation process, while the Al2O3 spacer can form a Schottky-like barrier that prevents charges from returning back to the metalsCushing _et al._ (2012); García De Arquer _et al._ (2013). As a result, hot electrons can be accumulated in the lattice before they decay (within 1 psBrongersma _et al._ (2015)), which simultaneously competes with rapid exciton relaxations, transiently converting the intrinsic WS2 monolayers to ”n-doped” ones. This leads to the giant bandgap renormalisation with population inversion that peak at few-hundred femtoseconds (Fig.S10 in SI), and also induces the delayed absorption minima in Fig.2b and 2f (Section 5 in SI). To confirm our observations, we have performed meaurements under $\sim 10$ times higher pump fluence ($100\mu$J$/$cm2) (Fig. 3a). Apart from the pronounced broad maxima at low energies, we can see large spectral shift as well as remarkably delayed occurance of UP and LP maxima, revealing that the accumulation of hot electrons competes with the relaxation dynamics, which significantly enhances the systems’ nonlinear responses on ultrashort timescales. (Detailed discussions in Section 6 of SI). Similar to the low- power case, the transient variation of the broad maximum (Fig. 3c) under intense photoexcitation takes $\sim 1.5$ ps from initial excitation to fading. Fig. 3d shows the evolution of population inversion, where the magnitude and width of the maximum is highly dependent on pump intensity. Under $100\mu$J$/$cm2 pump fluence, the full-width at half-maximum can reach at $\sim 200$ meV with highly enhanced magnitudes as compared to the maximum under $5\mu$J$/$cm2 pump, also contrasting the unchanged flat spectral features in bare WS2 MLs. But, even in this case, the pump fluence is still sigfinicantly lower than that in Ref.Chernikov _et al._ (2015b). As discussed above, the strong plasmon-exction coupling dramatically modifies the electronic band structures of WS2 monolayers, which are induced, to a large degree, by plasmonic hot electron doping via strong coupling. This effect is extremely hard to observe in traditional exciton-polaritonsByrnes _et al._ (2014), being a non-trivial factor that has to be considered when studying light-matter interactions using plasmonic resonators, which, on the other hand, provides new and effective measures to engineer bandgap of 2D semiconductors. ## Acknowledgments The authors acknowledge the New Idea Research Funding 2018 (Dodd-Walls Centre for photonic and quantum technologies), the Marsden Fast-start Fund by Royal Society of New Zealand through contract MFP-UOO1827 and the Smart Ideas Fund by Ministry of Business, Innovation and Employment, New Zealand through contract UOOX1802. In addition, this work was supported in part by the National Key Research and Development Program of China (no. 2017YFA0205700) and the National Natural Science Foundation of China (nos. 61425023, 61235007, 61575177 and 51861135201). The authors also acknowledge the visiting Fellowship awarded by New Zealand Centre at Peking University. We thank Dr. M. Yan and Dr. F. Hong for their help with thin-film deposition, AFM, and SEM measurements. ## Author Contributions B.D. and Y.-H.C. conceived the project; Z.Z, and B.D. prepared the samples; R.T., K.C., Y.-H.C. and B.D. carried out the optical and other characterization; Y.-H.C. and B.D. performed the simulation; Y.Z., M.Q., R.J.B., and B.D. supervised the projects; Y.-H.C. and B.D. prepared the manuscript; all authors discussed and analyzed the results. ## References * Mak _et al._ (2010) K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev. Lett. 105, 136805 (2010), arXiv:1004.0546 . * Splendiani _et al._ (2010) A. 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Phys. 10, 803 (2014). Figure 1: Structures of a PC-WS${}_{\text{2}}$ sample and steady-state optical properties. a, schematic of polariton formation in a WS${}_{\text{2}}$ ML that is supported on a self-assembled plasmonic crystal. The Al2O3 spacer is not depicted for similicity. right insets: side and top-view scanning electron microscope (SEM) images; b, angle-resolved transmission spectra under p-polarised illumination and their projection (top x-y plane), in which the spectral positions of exciton A (X${}_{\text{A}}$) and B (X${}_{\text{B}}$), calculated dispersions of plasmon lattice modes (yellow curve), and upper and lower branches of polaritons (orange curves) are indicated. The tuned angle ($\theta=22^{\circ}$) is marked with a blacked dahsed line. Refer to Section 1 in the SI for detailed discussion of the strong plasmon-exciton coupling and its dispersion. Figure 2: Transient optical responses. a, schematic of angle- resolved ultrafast pump-probe spectroscopy; b, d and f refer to normalised differential transmission spectra ($\Delta\text{T}/\text{T}$) at the tuned angle ($\theta=22^{\circ}$), while c, e and g refer to $\Delta\text{T}/\text{T}$ at the detuned angle ($\theta=30^{\circ}$); b and c are intensity plots of $\Delta\text{T}/\text{T}$ as function of time delay and probe photon energy, using the same colour bar (which is also used by Fig.3a); d and e are $\Delta\text{T}/\text{T}$ spectra averaged within the time span from 0.1 to 0.7 ps after pump; f and g are $\Delta\text{T}/\text{T}$ transient at specific energies (labelled with different colours), in which scatter symbols and solid curves represent measured and fitted data, respectively. Dashed frames in panel b, d and e mark the spectral region of the broad maxima (see main text). All measurements were carried out using 400 nm ($E=3.1$ eV) pump pulses that have 100 fs duration and pump fluence of 12 $\mu$J/cm2 at room temperature. The instrument-response-function is shown as the grey area in panel g Figure 3: Bandgap renormalisation and evloution of population inversion. a, intensity plot of $\Delta\text{T}/\text{T}$ spectra of PC-WS2 under $100\mu$J$/$cm2 pump fluence at $\theta=22^{\circ}$, where orange (blue) colour represents the maximum (minimum) value. b, delay time dependent spectra ($\Delta\text{T}/\text{T}$) at energies of UP, LP and exciton B extracted from panel a. Solid curves are plotted only for visual guidance c, $\Delta\text{T}/\text{T}$ spectra at different delay times, extracted from the white dashed frame in panel a; red dashed vertical line indicates the onset of renormalised bandgap. d, comparison of $\Delta\text{T}/\text{T}$ spectra at delay of 0.96 ps between PC-WS2 (left) and WS2 MLs (right) under gradually increasing pump fluence.

2024-09-04T02:54:58.158122

2003.03734

arxiv-papers

11institutetext: National Key Laboratory for Novel Software Technology, Nanjing University, China 11email<EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext: Samsung Research America, Mountain View CA, USA 22email<EMAIL_ADDRESS> # ESBM: An Entity Summarization BenchMark Qingxia Liu 11 Gong Cheng 11 Kalpa Gunaratna 22 Yuzhong Qu 11 ###### Abstract Entity summarization is the problem of computing an optimal compact summary for an entity by selecting a size-constrained subset of triples from RDF data. Entity summarization supports a multiplicity of applications and has led to fruitful research. However, there is a lack of evaluation efforts that cover the broad spectrum of existing systems. One reason is a lack of benchmarks for evaluation. Some benchmarks are no longer available, while others are small and have limitations. In this paper, we create an Entity Summarization BenchMark (ESBM) which overcomes the limitations of existing benchmarks and meets standard desiderata for a benchmark. Using this largest available benchmark for evaluating general-purpose entity summarizers, we perform the most extensive experiment to date where 9 existing systems are compared. Considering that all of these systems are unsupervised, we also implement and evaluate a supervised learning based system for reference. ###### Keywords: Entity summarization Triple ranking Benchmarking. ## 1 Introduction RDF data describes entities with triples representing property values. In an RDF dataset, the description of an entity comprises all the RDF triples where the entity appears as the subject or the object. An example entity description is shown in Fig. 1. Entity descriptions can be large. An entity may be described in dozens or hundreds of triples, exceeding the capacity of a typical user interface. A user served with all of those triples may suffer information overload and find it difficult to quickly identify the small set of triples that are truly needed. To solve the problem, an established research topic is _entity summarization_ [15], which aims to compute an optimal compact summary for the entity by selecting a size-constrained subset of triples. An example entity summary under the size constraint of 5 triples is shown in the bottom right corner of Fig. 1. Figure 1: Description of entity Tim Berners-Lee and a summary thereof. Entity summarization supports a multiplicity of applications [6, 21]. Entity summaries constitute entity cards displayed in search engines [9], provide background knowledge for enriching documents [26], and facilitate research activities with humans in the loop [3, 4]. This far-reaching application has led to fruitful research as reviewed in our recent survey paper [15]. Many entity summarizers have been developed, most of which generate summaries for general purposes. Research Challenges. However, two challenges face the research community. First, there is a _lack of benchmarks_ for evaluating entity summarizers. As shown in Table 1, some benchmarks are no longer available. Others are available [22, 7, 8] but they are small and have limitations. Specifically, [22] has a task-specific nature, and [7, 8] exclude classes and/or literals. These benchmarks could not support a comprehensive evaluation of general- purpose entity summarizers. Second, there is a _lack of evaluation efforts_ that cover the broad spectrum of existing systems to compare their performance and assist practitioners in choosing solutions appropriate to their applications. Contributions. We address the challenges with two contributions. First, we create an Entity Summarization BenchMark (ESBM) which overcomes the limitations of existing benchmarks and meets the desiderata for a successful benchmark [18]. ESBM has been published on GitHub with extended documentation and a permanent identifier on w3id.org111https://w3id.org/esbm under the ODC- By license. As the largest available benchmark for evaluating general-purpose entity summarizers, ESBM contains 175 heterogeneous entities sampled from two datasets, for which 30 human experts create 2,100 general-purpose ground-truth summaries under two size constraints. Second, using ESBM, we evaluate 9 existing general-purpose entity summarizers. It represents the most extensive evaluation effort to date. Considering that existing systems are unsupervised, we also implement and evaluate a supervised learning based entity summarizer for reference. In this paper, for the first time we comprehensively describe the creation and use of ESBM. We report ESBM v1.2—the latest version, while early versions have successfully supported the entity summarization shared task at the EYRE 2018 workshop222https://sites.google.com/view/eyre18/sharedtasks and the EYRE 2019 workshop.333https://sites.google.com/view/eyre19/sharedtasks We will also educate on the use of ESBM at an ESWC 2020 tutorial on entity summarization444https://sites.google.com/view/entity-summarization- tutorials/eswc2020. The remainder of the paper is organized as follows. Section 2 reviews related work and limitations of existing benchmarks. Section 3 describes the creation of ESBM, which is analyzed in Section 4. Section 5 presents our evaluation. In Section 6 we discuss limitations of our study and perspectives for future work. Table 1: Existing benchmarks for evaluating entity summarization. | Dataset | Number of entities | Availability ---|---|---|--- WhoKnows?Movies! [22] | Freebase | 60 | Available111http://yovisto.com/labs/iswc2012 Langer et al. [13] | DBpedia | 14 | Unavailable FRanCo [1] | DBpedia | 265 | Unavailable Benchmark for evaluating RELIN [2] | DBpedia | 149 | Unavailable Benchmark for evaluating DIVERSUM [20] | IMDb | 20 | Unavailable Benchmark for evaluating FACES [7] | DBpedia | 50 | Available222http://wiki.knoesis.org/index.php/FACES Benchmark for evaluating FACES-E [8] | DBpedia | 80 | Available222http://wiki.knoesis.org/index.php/FACES ## 2 Related Work We review methods and evaluation efforts for entity summarization. Methods for Entity Summarization. In a recent survey [15] we have categorized the broad spectrum of research on entity summarization. Below we briefly review _general-purpose_ entity summarizers which mainly rely on generic technical features that can apply to a wide range of domains and applications. We will not address methods that are domain-specific (e.g., for movies [25] or timelines [5]), task-specific (e.g., for facilitating entity resolution [3] or entity linking [4]), or context-aware (e.g., contextualized by a document [26] or a query [9]). RELIN [2] uses a weighted PageRank model to rank triples according to their statistical informativeness and relatedness. DIVERSUM [20] ranks triples by property frequency and generates a summary with a strong constraint that avoids selecting triples having the same property. SUMMARUM [24] and LinkSUM [23] mainly rank triples by the PageRank scores of property values that are entities. LinkSUM also considers backlinks from values. FACES [7], and its extension FACES-E [8] which adds support for literals, cluster triples by their bag-of-words based similarity and choose top-ranked triples from as many different clusters as possible. Triples are ranked by statistical informativeness and property value frequency. CD [28] models entity summarization as a quadratic knapsack problem that maximizes the statistical informativeness of the selected triples and in the meantime minimizes the string, numerical, and logical similarity between them. In ES-LDA [17], ES- LDAext [16], and MPSUM [27], a Latent Dirichlet Allocation (LDA) model is learned where properties are treated as topics, and each property is a distribution over all the property values. Triples are ranked by the probabilities of properties and values. MPSUM further avoids selecting triples having the same property. BAFREC [12] categorizes triples into meta-level and data-level. It ranks meta-level triples by their depths in an ontology and ranks data-level triples by property and value frequency. Triples having textually similar properties are penalized to improve diversity. KAFCA [11] ranks triples by the depths of properties and values in a hierarchy constructed by performing the Formal Concept Analysis (FCA). It tends to select triples containing infrequent properties but frequent values, where frequency is computed at the word level. Limitations of Existing Benchmarks. For evaluating entity summarization, compared with task completion based _extrinsic evaluation_ , ground truth based _intrinsic evaluation_ is more popular because it is easy to perform and the results are reproducible. Its idea is to create a benchmark consisting of human-made ground-truth summaries, and then compute how much a machine- generated summary is close to a ground-truth summary. Table 1 lists known benchmarks, including dedicated benchmarks [22, 13, 1] and those created for evaluating a particular entity summarizer [2, 20, 7, 8]. It is not surprising that these benchmarks are not very large since it is expensive to manually create high-quality summaries for a large set of entities. Unfortunately, some of these benchmarks are not publicly available at this moment. Three are available [22, 7, 8] but they are relatively small and have limitations. Specifically, WhoKnows?Movies! [22] is not a set of ground-truth summaries but annotates each triple with the ratio of movie questions that were correctly answered based on that triple, as an indicator of its importance. This kind of task-specific ground truth may not be suitable for evaluating general-purpose entity summarizers. The other two available benchmarks were created for evaluating FACES/-E [7, 8]. Classes and/or literals are not included because they could not be processed by FACES/-E and hence were filtered out. Such benchmarks could not comprehensively evaluate most of the existing entity summarizers [2, 20, 28, 27, 12, 11] that can handle classes and literals. These limitations of available benchmarks motivated us to create a new ground truth consisting of _general-purpose summaries_ for a _larger set of entities_ involving _more comprehensive triples_ where property values can be entities, classes, or literals. ## 3 Creating ESBM To overcome the above-mentioned limitations of existing benchmarks, we created a new benchmark called ESBM. To date, it is the largest available benchmark for evaluating general-purpose entity summarizers. In this section, we will first specify our design goals. Then we describe the selection of entity descriptions and the creation of ground-truth summaries. We partition the data to support cross-validation for parameter fitting. Finally we summarize how our design goals are achieved and how ESBM meets standard desiderata for a benchmark. ### 3.1 Design Goals The creation of ESBM has two main design goals. First, a successful benchmark should meet seven desiderata [18]: accessibility, affordability, clarity, relevance, solvability, portability, and scalability, which we will detail in Section 3.5. Our design of ESBM aims to satisfy these basic requirements. Second, in Section 2 we discussed the limitations of available benchmarks, including task specificness, small size, and triple incomprehensiveness. Besides, all the existing benchmarks use a single dataset and hence may weaken the generalizability of evaluation results. We aim to overcome these limitations when creating ESBM. In Section 3.5 we will summarize how our design goals are achieved. ### 3.2 Entity Descriptions To choose entity descriptions to summarize, we sample entities from selected datasets and filter their triples. The process is detailed below. Datasets. We sample entities from two datasets of different kinds: an encyclopedic dataset and a domain-specific dataset. For the encyclopedic dataset we choose DBpedia [14], which has been used in other benchmarks [13, 1, 2, 7, 8]. We use the English version of DBpedia 2015-10555http://wiki.dbpedia.org/dbpedia-dataset-version-2015-10—the latest version when we started to create ESBM. For the domain-specific dataset we choose LinkedMDB [10], which is a popular movie database. The movie domain is also the focus of some existing benchmarks [22, 20] possibly because this domain is familiar to the lay audience so that it would be easy to find qualified human experts to create ground-truth summaries. We use the latest available version of LinkedMDB.666http://www.cs.toronto.edu/~oktie/linkedmdb/linkedmdb-latest- dump.zip Entities. For DBpedia we sample entities from five large classes: Agent, Event, Location, Species, and Work. They collectively contain 3,501,366 entities (60%) in the dataset. For LinkedMDB we sample from Film and Person, which contain 159,957 entities (24%) in the dataset. Entities from different classes are described by very different properties as we will see in Section 4.3, and hence help to assess the generalizability of an entity summarizer. According to the human efforts we could afford, from each class we randomly sample 25 entities. The total number of selected entities is 175. Each selected entity should be described in at least 20 triples so that summarization would not be a trivial task. This requirement follows common practice in the literature [1, 2, 20, 7] where a minimum constraint in the range of 10–20 was posed. (a) Average number of triples describing an entity. (b) Average number of distinct properties describing an entity. Figure 2: Composition of entity descriptions (the left bar in each group), top-5 ground-truth summaries (the middle bar), and top-10 ground-truth summaries (the right bar), grouped by class in DBpedia (D) and LinkedMDB (L). Triples. For DBpedia, entity descriptions comprise triples in the following dump files: _instance types_ , _instance types transitive_ , _YAGO types_ , _mappingbased literals_ , _mappingbased objects_ , _labels_ , _images_ , _homepages_ , _persondata_ , _geo coordinates mappingbased_ , and _article categories_. We do not import dump files that provide metadata about Wikipedia articles such as _page links_ and _page length_. We do not import _short abstracts_ and _long abstracts_ as they provide handcrafted textual entity summaries; it would be inappropriate to include them in a benchmark for evaluating entity summarization. For LinkedMDB we import all the triples in the dump file except sameAs links which do not express facts about entities but are of more technical nature. Finally, as shown in Fig. 2a (the left bar in each group), the mean number of triples in an entity description is in the range of 25.88–52.44 depending on the class, and the overall mean value is 37.62. ### 3.3 Ground-Truth Summaries We invite 30 researchers and students to create ground-truth summaries for entity descriptions. All the participants are familiar with RDF. Task Assignment. Each participant is assigned 35 entities consisting of 5 entities randomly selected from each of the 7 classes in ESBM. The assignment is controlled to ensure that each entity in ESBM is processed by 6 participants. A participant creates two summaries for each entity description by selecting different numbers of triples: a _top-5 summary_ containing 5 triples, and a _top-10 summary_ containing 10 triples. Therefore, we will be able to evaluate entity summarizers under different size constraints. The choice of these two numbers follows previous work [2, 7, 8]. Participants work independently and they may create different summaries for an entity. It is not feasible to ask participants to reach an agreement. It is also not reasonable to merge different summaries into a single version. So we keep different summaries and will use all of them in the evaluation. The total number of ground-truth summaries is $175\cdot 6\cdot 2=2,100$. Figure 3: User interface for creating ground-truth entity summaries. Procedure. Participants are instructed to create _general-purpose summaries_ that are not specifically created for any particular task. They read and select triples using a Web-based user interface shown in Fig. 3. All the triples in an entity description are listed in random order but those having a common property are placed together for convenient reading and comparison. For IRIs, their human-readable labels (rdfs:label) are shown if available. To help participants understand a property value that is an unfamiliar entity, a click on it will open a pop-up showing a short textual description extracted from the first paragraph of its Wikipedia/IMDb page. Any triple can be selected into the top-5 summary, the top-10 summary, or both. The top-5 summary is not required to be a subset of the top-10 summary. ### 3.4 Training, Validation, and Test Sets Some entity summarizers need to tune hyperparameters or fit models. To make their evaluation results comparable with each other, we specify a split of our data into training, validation, and test sets. We provide a partition of the 175 entities in ESBM into 5 equally sized subsets $P_{0},\ldots,P_{4}$ to support 5-fold cross-validation. Entities of each class are partitioned evenly among the subsets. For $0\leq i\leq 4$, the $i$-th fold uses $P_{i},P_{i+1\text{ mod }5},P_{i+2\text{ mod }5}$ as the training set (e.g., for model fitting), uses $P_{i+3\text{ mod }5}$ for validation (e.g., tuning hyperparameters), and retains $P_{i+4\text{ mod }5}$ as the test set. Evaluation results are averaged over the 5 folds. ### 3.5 Conclusion ESBM overcomes the limitations of available benchmarks discussed in Section 2. It contains 175 entities which is 2–3 times as large as available benchmarks [22, 7, 8]. In ESBM, property values are not filtered as in [7, 8] but can be any entity, class, or literal. Different from the task-specific nature of [22], ESBM provides general-purpose ground-truth summaries for evaluating general-purpose entity summarizers. Besides, ESBM meets the seven desiderata proposed in [18] as follows. * • Accessibility. ESBM is publicly available and has a permanent identifier on w3id.org. * • Affordability. ESBM is with an open-source program and example code for evaluation. The cost of using ESBM is minimized. * • Clarity. ESBM is documented clearly and concisely. * • Relevance. ESBM samples entities from two real datasets that have been widely used. The summarization tasks are natural and representative. * • Solvability. An entity description in ESBM has at least 20 triples and a mean number of 37.62 triples, from which 5 or 10 triples are to be selected. The summarization tasks are not trivial and not too difficult. * • Portability. ESBM can be used to evaluate any general-purpose entity summarizer that can process RDF data. * • Scalability. ESBM samples 175 entities from 7 classes. It is reasonably large and diverse to evaluate mature entity summarizers but is not too large to evaluate research prototypes. However, ESBM has its own limitations, which we will discuss in Section 6. ## 4 Analyzing ESBM In this section, we will first characterize ESBM by providing some basic statistics and analyzing the triple composition and heterogeneity of entity descriptions. Then we compute inter-rater agreement to show how much consensus exists in the ground-truth summaries given by different participants. ### 4.1 Basic Statistics The 175 entity descriptions in ESBM collectively contain 6,584 triples, of which 37.44% are selected into at least one top-5 summary and 58.15% appear in at least one top-10 summary, showing a wide selection by the participants. However, many of them are selected only by a single participant; 20.46% and 40.23% are selected by different participants into top-5 and top-10 summaries, respectively. We will further analyze inter-rater agreement in Section 4.4. We calculate the overlap between the top-5 and the top-10 summaries created by the same participant for the same entity. The mean overlap is in the range of 4.80–4.99 triples depending on the class, and the overall mean value is 4.91, showing that the top-5 summary is usually a subset of the top-10 summary. ### 4.2 Triple Composition In Fig. 2 we present the composition of entity descriptions (the left bar in each group) and their ground-truth summaries (the middle bar for top-5 and the right bar for top-10) in ESBM, in terms of the average number of triples describing an entity (Fig. 2a) and in terms of the average number of distinct properties describing an entity (Fig. 2b). Properties are divided into literal-valued, class-valued, and entity-valued. Triples are divided accordingly. In Fig. 2a, both class-valued and entity-valued triples occupy a considerable proportion of the entity descriptions in DBpedia. Entity-valued triples predominate in LinkedMDB. Literal-valued triples account for a small proportion in both datasets. However, they constitute 30% in top-5 ground- truth summaries and 25% in top-10 summaries. Entity summarizers that cannot process literals [24, 23, 7, 17] have to ignore these notable proportions, thereby significantly influencing their performance. In Fig. 2b, in terms of distinct properties, entity-valued and literal-valued triples have comparable numbers in entity descriptions since many entity- valued properties are multi-valued. Specifically, an entity is described by 13.24 distinct properties, including 5.31 literal-valued (40%) and 6.93 entity-valued (52%). Multi-valued properties appear in every entity description and they constitute 35% of the triples. However, in top-5 ground- truth summaries, the average number of distinct properties is 4.70 and is very close to 5, indicating that the participants are not inclined to select multiple values of a property. Entity summarizers that prefer diverse properties [20, 7, 8, 28, 27, 12] may exhibit good performance. Figure 4: Jaccard similarity between property sets describing different classes. Table 2: Popular properties in ground-truth summaries. In top-5 summaries | In top-10 summaries ---|--- Agent | Event | Location | Species | Work | Film | Person | Agent | Event | Location | Species | Work | Film | Person type | type | type | type | type | director | type | type | type | type | family | type | director | type birthDate | date | country | family | | type | actor | subject | subject | country | type | subject | actor | actor | | | | | | | birthDate | date | subject | order | genre | type | label | | | | | | | | label | | class | | writer | page | | | | | | | | | | genus | | producer | | | | | | | | | | | subject | | date | | | | | | | | | | | kingdom | | language | ### 4.3 Entity Heterogeneity Entities from different classes are described by different sets of properties. For each class we identify the set of properties describing at least one entity from the class. The Jaccard similarity between properties sets for each pair of classes is very low, as shown in Fig. 4. Such heterogeneous entity descriptions help to assess the generalizability of an entity summarizer. Table 2 shows popular properties that appear in at least 50% of the ground- truth summaries for each class. Some universal properties like rdf:type and dct:subject are popular for most classes. We also see class-specific properties, e.g., dbo:birthDate for Agent, dbo:family for Species. However, the results suggest that it would be unrealistic to generate good summaries by manually selecting properties for each class. For example, among 13.24 distinct properties describing an entity, only 1–2 are popular in top-5 ground-truth summaries. The importance of properties is generally contextualized by concrete entities. ### 4.4 Inter-Rater Agreement Recall that each entity in ESBM has six top-5 ground-truth summaries and six top-10 summaries created by different participants. We calculate the average overlap between these summaries in terms of the number of common triples they contain. As shown in Table 3, the results are generally comparable with those reported for other benchmarks in the literature. There is a moderate degree of agreement between the participants. Table 3: Inter-rater agreement. | ESBM | [2] | [7] | [8] ---|---|---|---|--- Overlap between top-5 summaries | 1.99 (39.8$\%$) | 2.91 (58.2$\%$) | 1.92 (38.4$\%$) | 2.12 (42.4$\%$) Overlap between top-10 summaries | 5.42 (54.2$\%$) | 7.86 (78.6$\%$) | 4.64 (46.4$\%$) | 5.44 (54.4$\%$) Ground-truth summaries per entity | 6 | 4.43 | $\geq$ 7 | $\geq$ 4 ## 5 Evaluating with ESBM We used ESBM to perform the most extensive evaluation of general-purpose entity summarizers to date. In this section, we will first describe evaluation criteria. Then we introduce the entity summarizers that we evaluate. Finally we present evaluation results. ### 5.1 Evaluation Criteria Let $S_{m}$ be a machine-generated entity summary. Let $S_{h}$ be a human-made ground-truth summary. To compare $S_{m}$ with $S_{h}$ and assess the quality of $S_{m}$ based on how much $S_{m}$ is close to $S_{h}$, it is natural to compute precision (P), recall (R), and F1. The results are in the range of 0–1: $\text{P}=\frac{|S_{m}\cap S_{h}|}{|S_{m}|}\,,\quad\text{R}=\frac{|S_{m}\cap S_{h}|}{|S_{h}|}\,,\quad\text{F1}=\frac{2\cdot\text{P}\cdot\text{R}}{\text{P}+\text{R}}\,.$ (1) In the experiments we configure entity summarizers to output at most $k$ triples and we set $k=|S_{h}|$, i.e., $k=5$ and $k=10$ are our two settings corresponding to the sizes of ground-truth summaries. We will trivially have P$=$R$=$F1 if $|S_{m}|=|S_{h}|$. However, some entity summarizers may output less than $k$ triples. For example, DIVERSUM [20] disallows an entity summary to contain triples having the same property. It is possible that an entity description contains less than $k$ distinct properties and hence DIVERSUM has to output less than $k$ triples. In this case, P$\neq$R and one should rely on F1. In the evaluation, for each entity in ESBM, we compare a machine-generated summary with each of the 6 ground-truth summaries by calculating F1, and take their aggregation value. Finally we report the mean F1 over all the entities. For aggregation function, we report the results of average, to show an overall match with all the different ground truths; on the website we also give the results of maximum, to show the best match with each individual ground truth. ### 5.2 Participating Entity Summarizers We not only evaluate existing entity summarizers but also compare them with two special entity summarizers we create: an oracle entity summarizer which is used to show the best possible performance on ESBM, and a new supervised learning based entity summarizer. Existing Entity Summarizers. We evaluate 9 out of the 12 general-purpose entity summarizers reviewed in Section 2. We re-implement RELIN [2], DIVERSUM [20], LinkSUM [23], FACES [7], FACES-E [8], and CD [28], while MPSUM [27], BAFREC [12], and KAFCA [11] are open source. We exclude SUMMARUM [24], ES-LDA [17], and ES-LDAext [16] because LinkSUM represents an extension of SUMMARUM, and MPSUM represents an extension of ES-LDA and ES-LDAext. We follow the original implementation and suggested configuration of existing entity summarizers as far as possible. However, for RELIN, we replace its Google-based relatedness measure with a string metric [19] because Google’s search API is no longer free. We also use this metric to replace the unavailable UMBC’s SimService used in FACES-E. For DIVERSUM, we ignore its witness count measure since it does not apply to ESBM. For LinkSUM, we obtain backlinks between entities in LinkedMDB via their corresponding entities in DBpedia. RELIN, CD, and LinkSUM compute a weighted combination of two scoring components. We tune these hyperparameters in the range of 0–1 in 0.01 increments. Since these summarizers are unsupervised, we use both the training set and the validation set described in Section 3.4 for tuning hyperparameters. Oracle Entity Summarizer. We implement an entity summarizer denoted by ORACLE to approximate the best possible performance on ESBM and form a reference point used for comparisons. ORACLE simply outputs $k$ triples that are selected by the most participants into ground-truth summaries. Supervised Learning Based Entity Summarizer. Existing general-purpose entity summarizers are unsupervised. We implement a supervised learning based entity summarizer with features that are used by existing entity summarizers. A triple with property $p$ and value $v$ describing entity $e$ is represented by the following features: * • $\mathtt{gf}_{\mathbb{T}}$: the number of triples in the dataset where $p$ appears [23, 12], * • $\mathtt{lf}$: the number of triples in the description of $e$ where $p$ appears [20, 23], * • $\mathtt{vf}_{\mathbb{T}}$: the number of triples in the dataset where $v$ appears [7, 8, 12], and * • $\mathtt{si}$: the self-information of the triple [2, 7, 8, 28]. We also add three binary features: * • $\mathtt{isC}$: whether $v$ is a class, * • $\mathtt{isE}$: whether $v$ is an entity, and * • $\mathtt{isL}$: whether $v$ is a literal. Based on the training and validation sets described in Section 3.4, we implement and tune 6 pointwise learning to rank models provided by Weka: SMOreg, LinearRegression, MultilayerPerceptron, AdditiveRegression, REPTree, and RandomForest. Each model outputs $k$ top-ranked triples as a summary. ### 5.3 Evaluation Results We first report the overall evaluation results to show which entity summarizer generally performs better. Then we break down the results into different entity types (i.e., classes) for detailed comparison. Finally we present and analyze the performance of our supervised learning based entity summarizer. Table 4: Average F1 over all the entities in a dataset. For the nine existing entity summarizers, significant improvements and losses over each other are indicated by $\blacktriangle$ and $\blacktriangledown$ ($p<0.05$), respectively. Insignificant differences are indicated by $\circ$. | DBpedia | LinkedMDB ---|---|--- | $k=5$ | $k=10$ | $k=5$ | $k=10$ RELIN | 0.242 ${}^{\text{-}\circ\circ\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.455 ${}^{\text{-}\blacktriangledown\circ\circ\blacktriangledown\circ\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.203 ${}^{\text{-}\circ\circ\blacktriangledown\circ\blacktriangle\blacktriangledown\circ\blacktriangledown}$ | 0.258 ${}^{\text{-}\blacktriangledown\circ\blacktriangledown\blacktriangledown\circ\blacktriangledown\blacktriangledown\blacktriangledown}$ DIVERSUM | 0.249 ${}^{\circ\text{-}\circ\circ\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.507 ${}^{\blacktriangle\text{-}\blacktriangle\circ\circ\circ\circ\circ\circ}$ | 0.207 ${}^{\circ\text{-}\circ\blacktriangledown\circ\blacktriangle\blacktriangledown\circ\blacktriangledown}$ | 0.358 ${}^{\blacktriangle\text{-}\blacktriangle\circ\circ\blacktriangle\blacktriangledown\circ\blacktriangledown}$ FACES | 0.270 ${}^{\circ\circ\text{-}\circ\circ\circ\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.428 ${}^{\circ\blacktriangledown\text{-}\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.169 ${}^{\circ\circ\text{-}\blacktriangledown\blacktriangledown\circ\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.263 ${}^{\circ\blacktriangledown\text{-}\blacktriangledown\blacktriangledown\circ\blacktriangledown\blacktriangledown\blacktriangledown}$ FACES-E | 0.280 ${}^{\blacktriangle\circ\circ\text{-}\circ\circ\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.488 ${}^{\circ\circ\blacktriangle\text{-}\circ\circ\circ\circ\circ}$ | 0.313 ${}^{\blacktriangle\blacktriangle\blacktriangle\text{-}\blacktriangle\blacktriangle\blacktriangledown\blacktriangle\circ}$ | 0.393 ${}^{\blacktriangle\circ\blacktriangle\text{-}\blacktriangle\blacktriangle\circ\circ\circ}$ CD | 0.283 ${}^{\blacktriangle\blacktriangle\circ\circ\text{-}\circ\blacktriangledown\circ\circ}$ | 0.513 ${}^{\blacktriangle\circ\blacktriangle\circ\text{-}\circ\circ\circ\circ}$ | 0.217 ${}^{\circ\circ\blacktriangle\blacktriangledown\text{-}\blacktriangle\blacktriangledown\circ\blacktriangledown}$ | 0.331 ${}^{\blacktriangle\circ\blacktriangle\blacktriangledown\text{-}\blacktriangle\blacktriangledown\blacktriangledown\blacktriangledown}$ LinkSUM | 0.287 ${}^{\blacktriangle\blacktriangle\circ\circ\circ\text{-}\blacktriangledown\circ\circ}$ | 0.486 ${}^{\circ\circ\blacktriangle\circ\circ\text{-}\circ\circ\circ}$ | 0.140 ${}^{\blacktriangledown\blacktriangledown\circ\blacktriangledown\blacktriangledown\text{-}\blacktriangledown\blacktriangledown\blacktriangledown}$ | 0.279 ${}^{\circ\blacktriangledown\circ\blacktriangledown\blacktriangledown\text{-}\blacktriangledown\blacktriangledown\blacktriangledown}$ BAFREC | 0.335 ${}^{\blacktriangle\blacktriangle\blacktriangle\blacktriangle\blacktriangle\blacktriangle\text{-}\circ\circ}$ | 0.503 ${}^{\blacktriangle\circ\blacktriangle\circ\circ\circ\text{-}\circ\circ}$ | 0.360 ${}^{\blacktriangle\blacktriangle\blacktriangle\blacktriangle\blacktriangle\blacktriangle\text{-}\blacktriangle\blacktriangle}$ | 0.402 ${}^{\blacktriangle\blacktriangle\blacktriangle\circ\blacktriangle\blacktriangle\text{-}\circ\circ}$ KAFCA | 0.314 ${}^{\blacktriangle\blacktriangle\blacktriangle\blacktriangle\circ\circ\circ\text{-}\circ}$ | 0.509 ${}^{\blacktriangle\circ\blacktriangle\circ\circ\circ\circ\text{-}\circ}$ | 0.244 ${}^{\circ\circ\blacktriangle\blacktriangledown\circ\blacktriangle\blacktriangledown\text{-}\circ}$ | 0.397 ${}^{\blacktriangle\circ\blacktriangle\circ\blacktriangle\blacktriangle\circ\text{-}\circ}$ MPSUM | 0.314 ${}^{\blacktriangle\blacktriangle\blacktriangle\blacktriangle\circ\circ\circ\circ\text{-}}$ | 0.512 ${}^{\blacktriangle\circ\blacktriangle\circ\circ\circ\circ\circ\text{-}}$ | 0.272 ${}^{\blacktriangle\blacktriangle\blacktriangle\circ\blacktriangle\blacktriangle\blacktriangledown\circ\text{-}}$ | 0.423 ${}^{\blacktriangle\blacktriangle\blacktriangle\circ\blacktriangle\blacktriangle\circ\circ\text{-}}$ ORACLE | 0.595 | 0.713 | 0.619 | 0.678 SMOreg | 0.279 | 0.543 | 0.403 | 0.472 LinearRegression | 0.319 | 0.556 | 0.401 | 0.471 MultilayerPerceptron | 0.340 | 0.560 | 0.390 | 0.477 AdditiveRegression | 0.345 | 0.558 | 0.415 | 0.510 REPTree | 0.392 | 0.570 | 0.455 | 0.538 RandomForest | 0.399 | 0.576 | 0.449 | 0.506 Overall Results of Existing Entity Summarizers. Table 4 presents the results of all the participating entity summarizers on two datasets under two size constraints. We compare nine existing summarizers using one-way ANOVA post-hoc LSD and we show whether the difference between each pair of them is statistical significant at the 0.05 level. Among existing summarizers, BAFREC achieves the highest F1 under $k=5$. It significantly outperforms six existing summarizers on DBpedia and outperforms all the eight ones on LinkedMDB. It is also among the best under $k=10$. MPSUM follows BAFREC under $k=5$ but performs slightly better under $k=10$. Other top-tier results belong to KAFCA on DBpedia and FACES-E on LinkedMDB. The F1 scores of ORACLE are in the range of 0.595–0.713. It is impossible for ORACLE or any other summarizer to reach $\text{F1}=1$, because for each entity in ESBM there are six ground-truth summaries which are often different and hence cannot simultaneously match a machine-generated summary. However, the gap between the results of ORACLE and the best results of existing summarizers is still as large as 0.20–0.26, suggesting that there is much room for improvement. Results on Different Entity Types. We break down the results of existing entity summarizers into 7 entity types (i.e., classes). When $k=5$ in Fig. 5, there is no single winner on every class, but BAFREC and MPSUM are among top three on 6 classes, showing relatively good generalizability over different entity types. Some entity summarizers have limited generalizability and they perform not well on certain classes. For example, RELIN and CD mainly rely on the self-information of a triple, while for Location entities their latitudes and longitudes are often unique in DBpedia but such triples with large self- information rarely appear in ground-truth summaries. Besides, most summarizers generate low-quality summaries for Agent, Film, and Person entities. This is not surprising since these entities are described in more triples and/or by more properties according to Fig. 2. Their summarization is inherently more difficult. When $k=10$ in Fig. 6, MPSUM is still among top three on 6 classes. KAFCA also shows relatively good generalizability—among top three on 5 classes. Figure 5: Average F1 over all the entities in each class under $k=5$. Figure 6: Average F1 over all the entities in each class under $k=10$. Results of Supervised Learning. As shown in Table 4, among the six supervised learning based methods, RandomForest and REPTree achieve the highest F1 on DBpedia and LinkedMDB, respectively. Four methods (MultilayerPerceptron, AdditiveRegression, REPTree, and RandomForest) outperform all the existing entity summarizers on both datasets under both size constraints, and two methods (SMOreg and LinearRegression) only fail to outperform in one setting. The results demonstrate the powerfulness of supervised learning for entity summarization. Further, recall that these methods only use standard models and rely on features that are used by existing entity summarizers. It would be reasonable to predict that better results can be achieved with specialized models and more advanced features. However, creating a large number of ground- truth summaries for training is expensive, and the generalizability of supervised methods for entity summarization still needs further exploration. Moreover, we are interested in how much the seven features contribute to the good performance of supervised learning. Table 5 shows the results of RandomForest after removing each individual feature. Considering statistical significance at the 0.05 level, two features $\mathtt{gf}_{\mathbb{T}}$ and $\mathtt{lf}$ show effectiveness on both datasets under both size constraints, and two features $\mathtt{vf}_{\mathbb{T}}$ and $\mathtt{si}$ are only effective on LinkedMDB. The usefulness of the three binary features $\mathtt{isC}$, $\mathtt{isE}$, and $\mathtt{isL}$ is not statistically significant. Table 5: F1 of RandomForest after removing each individual feature, its difference from using all features ($\Delta\%$), and the significance level for the difference ($p$). DBpedia | LinkedMDB ---|--- $k=5$ | $k=10$ | $k=5$ | $k=10$ | F1 | $\Delta\%$ | $p$ | | F1 | $\Delta\%$ | $p$ | | F1 | $\Delta\%$ | $p$ | | F1 | $\Delta\%$ | $p$ All | 0.399 | — | — | All | 0.576 | — | — | All | 0.449 | — | — | All | 0.506 | — | — -$\mathtt{gf}_{\mathbb{T}}$ | 0.346 | $-$5.360 | 0.000 | -$\mathtt{lf}$ | 0.546 | $-$0.030 | 0.000 | -$\mathtt{gf}_{\mathbb{T}}$ | 0.383 | $-$0.066 | 0.000 | -$\mathtt{lf}$ | 0.473 | $-$0.033 | 0.008 -$\mathtt{lf}$ | 0.366 | $-$3.307 | 0.000 | -$\mathtt{gf}_{\mathbb{T}}$ | 0.551 | $-$0.025 | 0.000 | -$\mathtt{lf}$ | 0.413 | $-$0.036 | 0.025 | -$\mathtt{vf}_{\mathbb{T}}$ | 0.477 | $-$0.029 | 0.010 -$\mathtt{isC}$ | 0.392 | $-$0.720 | 0.261 | -$\mathtt{vf}_{\mathbb{T}}$ | 0.569 | $-$0.007 | 0.198 | -$\mathtt{vf}_{\mathbb{T}}$ | 0.414 | $-$0.035 | 0.022 | -$\mathtt{gf}_{\mathbb{T}}$ | 0.479 | $-$0.027 | 0.007 -$\mathtt{isE}$ | 0.397 | $-$0.267 | 0.720 | -$\mathtt{isE}$ | 0.570 | $-$0.006 | 0.262 | -$\mathtt{si}$ | 0.442 | $-$0.007 | 0.574 | -$\mathtt{si}$ | 0.486 | $-$0.020 | 0.009 -$\mathtt{si}$ | 0.400 | $+$0.027 | 0.973 | -$\mathtt{isC}$ | 0.571 | $-$0.005 | 0.303 | -$\mathtt{isE}$ | 0.455 | $+$0.005 | 0.651 | -$\mathtt{isL}$ | 0.491 | $-$0.015 | 0.079 -$\mathtt{isL}$ | 0.401 | $+$0.160 | 0.816 | -$\mathtt{si}$ | 0.572 | $-$0.004 | 0.402 | -$\mathtt{isL}$ | 0.456 | $+$0.007 | 0.504 | -$\mathtt{isE}$ | 0.492 | $-$0.014 | 0.148 -$\mathtt{vf}_{\mathbb{T}}$ | 0.407 | $+$0.720 | 0.346 | -$\mathtt{isL}$ | 0.578 | $+$0.002 | 0.683 | -$\mathtt{isC}$ | 0.463 | $+$0.013 | 0.281 | -$\mathtt{isC}$ | 0.514 | $+$0.008 | 0.396 Conclusion. Among existing entity summarizers, BAFREC generally shows the best performance on ESBM while MPSUM seems more robust. However, none of them are comparable with our straightforward implementation of supervised learning, which in turn is still far away from the best possible performance represented by ORACLE. Therefore, entity summarization on ESBM is a non-trivial task. We invite researchers to experiment with new ideas on ESBM. ## 6 Discussion and Future work We identify the following limitations of our work to be addressed in future work. Evaluation Criteria. We compute F1 score in the evaluation, which is based on common triples but ignores semantic overlap between triples. A triple $t$ in a machine-generated summary $S$ may partially cover the information provided by some triple $t^{\prime}$ in the ground-truth summary. It may be reasonable to not completely penalize $S$ for missing $t^{\prime}$ but give some reward for the presence of $t$. However, it is difficult to quantify the extent of penalization for all possible cases, particularly when multiple triples semantically overlap with each other. In future work, we will explore more proper evaluation criteria. Representativeness of Ground Truth. The ground-truth summaries in ESBM are not supposed to represent the view of the entire user population. They are intrinsically biased towards their creators. Besides, these ground-truth summaries are created for general purposes. Accordingly, we use them to evaluate general-purpose entity summarizers. However, for a specific task, these summaries may not show optimality, and the participating systems may not represent the state of the art. Still, we believe it is valuable to evaluate general-purpose systems not only because of their wide range of applications but also because their original technical features have been reused by task- specific systems. In future work, we will extend ESBM to a larger scale, and will consider benchmarking task-specific entity summarization. Form of Ground Truth. ESBM provides ground-truth summaries, whereas some other benchmarks offer ground-truth scores of triples [22, 13, 1]. Scoring-based ground truth may more comprehensively evaluate an entity summarizer than our set-based ground truth because it not only considers the triples in a machine- generated summary but also assesses the rest of the triples. However, on the other hand, a set of top-scored triples may not equal an optimal summary because they may cover limited aspects of an entity and show redundancy. Therefore, both methods have their advantages and disadvantages. In future work, we will conduct scoring-based evaluation to compare with the current results. ## Acknowledgments This work was supported in part by the NSFC under Grant 61772264 and in part by the Qing Lan Program of Jiangsu Province. ## References * [1] Bobic, T., Waitelonis, J., Sack, H.: FRanCo - A ground truth corpus for fact ranking evaluation. In: SumPre 2015 & HSWI 2015 (2015) * [2] Cheng, G., Tran, T., Qu, Y.: RELIN: relatedness and informativeness-based centrality for entity summarization. In: ISWC 2011, Part I. pp. 114–129 (2011). https://doi.org/10.1007/978-3-642-25073-6_8 * [3] Cheng, G., Xu, D., Qu, Y.: C3D+P: A summarization method for interactive entity resolution. J. Web Sem. 35, 203–213 (2015). https://doi.org/10.1016/j.websem.2015.05.004 * [4] Cheng, G., Xu, D., Qu, Y.: Summarizing entity descriptions for effective and efficient human-centered entity linking. In: WWW 2015. pp. 184–194 (2015). https://doi.org/10.1145/2736277.2741094 * [5] Gottschalk, S., Demidova, E.: EventKG - the hub of event knowledge on the web \- and biographical timeline generation. Semantic Web 10(6), 1039–1070 (2019). https://doi.org/10.3233/SW-190355 * [6] Gunaratna, K.: Semantics-based Summarization of Entities in Knowledge Graphs. Ph.D. thesis, Wright State University (2017) * [7] Gunaratna, K., Thirunarayan, K., Sheth, A.P.: FACES: diversity-aware entity summarization using incremental hierarchical conceptual clustering. In: AAAI 2015. pp. 116–122 (2015) * [8] Gunaratna, K., Thirunarayan, K., Sheth, A.P., Cheng, G.: Gleaning types for literals in RDF triples with application to entity summarization. In: ESWC 2016. pp. 85–100 (2016). https://doi.org/10.1007/978-3-319-34129-3_6 * [9] Hasibi, F., Balog, K., Bratsberg, S.E.: Dynamic factual summaries for entity cards. In: SIGIR 2017. pp. 773–782 (2017). https://doi.org/10.1145/3077136.3080810 * [10] Hassanzadeh, O., Consens, M.P.: Linked movie data base. In: LDOW 2009 (2009) * [11] Kim, E.K., Choi, K.S.: Entity summarization based on formal concept analysis. In: EYRE 2018 (2018) * [12] Kroll, H., Nagel, D., Balke, W.T.: BAFREC: Balancing frequency and rarity for entity characterization in linked open data. In: EYRE 2018 (2018) * [13] Langer, P., Schulze, P., George, S., Kohnen, M., Metzke, T., Abedjan, Z., Kasneci, G.: Assigning global relevance scores to DBpedia facts. In: ICDE Workshops 2014. pp. 248–253 (2014). https://doi.org/10.1109/ICDEW.2014.6818334 * [14] Lehmann, J., Isele, R., Jakob, M., Jentzsch, A., Kontokostas, D., Mendes, P.N., Hellmann, S., Morsey, M., van Kleef, P., Auer, S., Bizer, C.: DBpedia - A large-scale, multilingual knowledge base extracted from Wikipedia. Semantic Web 6(2), 167–195 (2015). https://doi.org/10.3233/SW-140134 * [15] Liu, Q., Cheng, G., Gunaratna, K., Qu, Y.: Entity summarization: State of the art and future challenges. CoRR abs/1910.08252 (2019), http://arxiv.org/abs/1910.08252 * [16] Pouriyeh, S.A., Allahyari, M., Kochut, K., Cheng, G., Arabnia, H.R.: Combining word embedding and knowledge-based topic modeling for entity summarization. In: ICSC 2018. pp. 252–255 (2018). https://doi.org/10.1109/ICSC.2018.00044 * [17] Pouriyeh, S.A., Allahyari, M., Kochut, K., Cheng, G., Arabnia, H.R.: ES-LDA: entity summarization using knowledge-based topic modeling. In: IJCNLP 2017, Volume 1. pp. 316–325 (2017) * [18] Sim, S.E., Easterbrook, S.M., Holt, R.C.: Using benchmarking to advance research: A challenge to software engineering. In: ICSE 2003. pp. 74–83 (2003). https://doi.org/10.1109/ICSE.2003.1201189 * [19] Stoilos, G., Stamou, G.B., Kollias, S.D.: A string metric for ontology alignment. In: ISWC 2005. pp. 624–637 (2005). https://doi.org/10.1007/11574620_45 * [20] Sydow, M., Pikula, M., Schenkel, R.: The notion of diversity in graphical entity summarisation on semantic knowledge graphs. J. Intell. Inf. Syst. 41(2), 109–149 (2013). https://doi.org/10.1007/s10844-013-0239-6 * [21] Thalhammer, A.: Linked Data Entity Summarization. Ph.D. thesis, Karlsruher Institut für Technologie (2017) * [22] Thalhammer, A., Knuth, M., Sack, H.: Evaluating entity summarization using a game-based ground truth. In: ISWC 2012, Part II. pp. 350–361 (2012). https://doi.org/10.1007/978-3-642-35173-0_24 * [23] Thalhammer, A., Lasierra, N., Rettinger, A.: LinkSUM: Using link analysis to summarize entity data. In: ICWE 2016. pp. 244–261 (2016). https://doi.org/10.1007/978-3-319-38791-8_14 * [24] Thalhammer, A., Rettinger, A.: Browsing DBpedia entities with summaries. In: ESWC 2014 Satellite Events. pp. 511–515 (2014). https://doi.org/10.1007/978-3-319-11955-7_76 * [25] Thalhammer, A., Toma, I., Roa-Valverde, A.J., Fensel, D.: Leveraging usage data for linked data movie entity summarization. In: USEWOD 2012 (2012) * [26] Tonon, A., Catasta, M., Prokofyev, R., Demartini, G., Aberer, K., Cudré-Mauroux, P.: Contextualized ranking of entity types based on knowledge graphs. J. Web Sem. 37-38, 170–183 (2016). https://doi.org/10.1016/j.websem.2015.12.005 * [27] Wei, D., Gao, S., Liu, Y., Liu, Z., Huang, L.: MPSUM: Entity summarization with predicate-based matching. In: EYRE 2018 (2018) * [28] Xu, D., Zheng, L., Qu, Y.: CD at ENSEC 2016: Generating characteristic and diverse entity summaries. In: SumPre 2016 (2016)

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2003.03736

arxiv-papers

11institutetext: National Key Laboratory for Novel Software Technology, Nanjing University, China 11email<EMAIL_ADDRESS><EMAIL_ADDRESS> # DeepLENS: Deep Learning for Entity Summarization Qingxia Liu Gong Cheng Yuzhong Qu ###### Abstract Entity summarization has been a prominent task over knowledge graphs. While existing methods are mainly unsupervised, we present DeepLENS, a simple yet effective deep learning model where we exploit textual semantics for encoding triples and we score each candidate triple based on its interdependence on other triples. DeepLENS significantly outperformed existing methods on a public benchmark. ## 1 Introduction Entity summarization is the task of computing a compact summary for an entity by selecting an optimal size-constrained subset of entity-property-value triples from a knowledge graph such as an RDF graph [7]. It has found a wide variety of applications, for example, to generate a compact entity card from Google’s Knowledge Graph where an entity may be described in dozens or hundreds of triples. Generating entity summaries for general purposes has attracted much research attention, but existing methods are mainly unsupervised [2, 9, 3, 4, 13, 10, 6, 5, 11]. One research question that naturally arises is _whether deep learning can much better solve this task_. To the best of our knowledge, ESA [12] is the only supervised method in the literature for this task. ESA encodes triples using graph embedding (TransE), and employs BiLSTM with supervised attention mechanism. Although it outperformed unsupervised methods, the improvement reported in [12] was rather marginal, around $+7\%$ compared with unsupervised FACES-E [4] on the ESBM benchmark [8]. It inspired us to explore more effective deep learning models for the task of general-purpose entity summarization. In this short paper, we present DeepLENS,111https://github.com/nju- websoft/DeepLENS a novel Deep Learning based approach to ENtity Summarization. DeepLENS uses a simple yet effective model which addresses the following two limitations of ESA, and thus achieved significantly better results in the experiments. 1. 1. Different from ESA which encodes a triple using graph embedding, we use word embedding because we consider textual semantics more useful than graph structure for the entity summarization task. 2. 2. Whereas ESA encodes a set of triples as a sequence and its performance is sensitive to the chosen order, our aggregation-based representation satisfies permutation invariance and hence more suitable for entity summarization. In the remainder of the paper, Section 2 details DeepLENS, Section 3 presents experiment results, and Section 4 concludes the paper. ## 2 Approach #### 2.0.1 Problem Statement An RDF graph $T$ is a set of triples. The _description_ of entity $e$ in $T$, denoted by $\mathtt{Desc}(e)\subseteq T$, comprises triples where $e$ is the subject or object. Each triple $t\in\mathtt{Desc}(e)$ describes a property $\mathtt{prop}(t)$ which is the predicate of $t$, and gives a value $\mathtt{val}(t)$ which is the object or subject of $t$ other than $e$. For a size constraint $k$, a _summary_ of $e$ is a subset of triples $S\subseteq\mathtt{Desc}(e)$ with $|S|\leq k$. We aim to generate an optimal summary for general purposes. #### 2.0.2 Overview of DeepLENS Our approach DeepLENS generates an optimal summary by selecting $k$ most salient triples. As a supervised approach, it learns salience from labeled entity summaries. However, two issues remain unsolved. First, knowledge graph like RDF graph is a mixture of graph structure and textual content. The effectiveness of a learning-based approach to entity summarization relies on a _proper representation of entity descriptions of such mixed nature_. Second, the salience of a triple is not absolute but dependent on the context, i.e., the set of other triples in the entity description. It is essential to _represent their independence_. DeepLENS addresses these issues with the scoring model presented in Fig. 1. It has three modules which we will detail below: triple encoding, entity description encoding, and triple scoring. Finally, the model scores each candidate triple $t\in\mathtt{Desc}(e)$ in the context of $\mathtt{Desc}(e)$. Figure 1: Model of DeepLENS. #### 2.0.3 Triple Encoding For entity $e$, a triple $t\in\mathtt{Desc}(e)$ provides a property-value pair $\langle\mathtt{prop}(t),\mathtt{val}(t)\rangle$ of $e$. Previous research [12] leverages graph embedding to encode the structural features of $\mathtt{prop}(t)$ and $\mathtt{val}(t)$. By contrast, for the task of entity summarization we consider textual semantics more important than graph structure, and we _solely exploit textual semantics_ for encoding $t$. Specifically, for RDF resource $r$, we obtain its _textual form_ as follows. For an IRI or a blank node, we retrieve its rdfs:label if it is available, otherwise we have to use its local name; for a literal, we take its lexical form. We represent each word in the textual form by a pre-trained word embedding vector, and we average these vectors over all the words to represent $r$, denoted by $\text{Embedding}(r)$. For triple $t\in\mathtt{Desc}(e)$, we generate and concatenate such vector representations for $\mathtt{prop}(t)$ and $\mathtt{val}(t)$ to form $\boldsymbol{t}$, the _initial representation_ of $t$. Then $\boldsymbol{t}$ is fed into a multi-layer perceptron (MLP) to generate $\boldsymbol{h}$, the _final representation_ of $t$: $\boldsymbol{t}=\left[\text{Embedding}(\mathtt{prop}(t));~{}\text{Embedding}(\mathtt{val}(t))\right]\,,\quad\boldsymbol{h}=\text{MLP}_{\text{C}}(\boldsymbol{t})\,.\\\ $ (1) #### 2.0.4 Entity Description Encoding To score a candidate triple in the context of other triples in the entity description, previous research [12] captures the independence between triples in $\mathtt{Desc}(e)$ using BiLSTM to pass information. Triples are fed into BiLSTM as a sequence. However, $\mathtt{Desc}(e)$ is a set and the triples lack a natural order. The performance of this model is unfavourably sensitive to the order of input triples. Indeed, as we will show in the experiments, different orders could lead to considerably different performance. To generate a representation for $\mathtt{Desc}(e)$ that is _permutation invariant_ , we perform aggregation. Specifically, let $\boldsymbol{t_{1}},\ldots,\boldsymbol{t_{n}}$ be the initial representations of triples in $\mathtt{Desc}(e)$ computed by Eq. (1). We feed a MLP with each $\boldsymbol{t_{i}}$ for $1\leq i\leq n$ and generate their final representations $\boldsymbol{g_{1}},\ldots,\boldsymbol{g_{n}}$, which in turn are weighted using attention mechanism from $\boldsymbol{h}$ computed by Eq. (1), the final representation of the candidate triple $t$ to be scored. We calculate the sum of these weighted representations of triples to represent $\mathtt{Desc}(e)$, denoted by $\boldsymbol{d}$: $\boldsymbol{g_{i}}=\text{MLP}_{\text{D}}(\boldsymbol{t_{i}})\,,\quad a_{i}=\frac{\exp(\cos(\boldsymbol{h},\boldsymbol{g_{i}}))}{\sum_{j}\exp(\cos(\boldsymbol{h},\boldsymbol{g_{j}}))}\,,\quad\boldsymbol{d}=\sum_{i=1}^{n}{a_{i}\boldsymbol{g_{i}}}\,.\\\ $ (2) The result of summation is not sensitive to the order of triples in $\mathtt{Desc}(e)$. #### 2.0.5 Triple Scoring For each candidate triple $t\in\mathtt{Desc}(e)$ to be scored, we concatenate its final representation $\boldsymbol{h}$ and the representation $\boldsymbol{d}$ for $\mathtt{Desc}(e)$. We feed the result into a MLP to compute the context-based salience score of $t$: $\mathtt{score}(t|\mathtt{Desc}(e))=\text{MLP}_{\text{S}}(\left[\boldsymbol{h};~{}\boldsymbol{d}\right])\,.$ (3) Parameters of the entire model are jointly trained based on the mean squared error loss, supervised by labeled entity summaries. ## 3 Experiments ### 3.1 Datasets We used ESBM v1.2, the largest available benchmark for evaluating general- purpose entity summarization.222https://w3id.org/esbm For each of 125 entities in DBpedia and 50 entities in LinkedMDB, this benchmark provided 6 ground- truth summaries created by different human experts under $k=5$, and another 6 ground-truth summaries under $k=10$. We used the train-valid-test split specified in the benchmark to perform five-fold cross-validation. ### 3.2 Participating Methods We compared DeepLENS with 10 baseline methods. Unsupervised Methods. We compared with 9 unsupervised methods that had been tested on ESBM: RELIN [2], DIVERSUM [9], FACES [3], FACES-E [4], CD [13], LinkSUM [10], BAFREC [6], KAFCA [5], and MPSUM [11]. We directly presented their results reported on the ESBM website. Supervised Methods. We compared with ESA [12], the only supervised method in the literature to our knowledge. We reused its open-source implementation and configuration.333https://github.com/WeiDongjunGabriel/ESA We fed it with triples sorted in alphabetical order. For our approach DeepLENS, we used 300-dimensional fastText [1] word embedding vectors trained on Wikipedia to generate initial representations of triples. The numbers of hidden units in $\text{MLP}_{\text{C}}$, $\text{MLP}_{\text{D}}$, and $\text{MLP}_{\text{S}}$ were [64, 64], [64, 64], and [64, 64, 64], respectively. All hidden layers used ReLU as activation function. The final output layer of $\text{MLP}_{\text{S}}$ consisted of one linear unit. We trained the model using Adam optimizer with learning rate 0.01. For both ESA and DeepLENS, we performed early stopping on the validation set to choose the number of training epochs from 1–50. Oracle Method. ORACLE approximated the best possible performance on ESBM and formed a reference point used for comparisons. It outputted $k$ triples that most frequently appeared in ground-truth summaries. ### 3.3 Results Following ESBM, we compared machine-generated summaries with ground-truth summaries by calculating F1 score, and reported the mean F1 achieved by each method over all the test entities in a dataset. Table 1: Average F1 over all the test entities. Significant and insignificant differences ($p<0.01$) between DeepLENS and each baseline are indicated by $\blacktriangle$ and $\circ$, respectively. | DBpedia | LinkedMDB ---|---|--- | $k=5$ | $k=10$ | $k=5$ | $k=10$ RELIN [2] | 0.242 | 0.455 | 0.203 | 0.258 DIVERSUM [9] | 0.249 | 0.507 | 0.207 | 0.358 FACES [3] | 0.270 | 0.428 | 0.169 | 0.263 FACES-E [4] | 0.280 | 0.488 | 0.313 | 0.393 CD [13] | 0.283 | 0.513 | 0.217 | 0.331 LinkSUM [10] | 0.287 | 0.486 | 0.140 | 0.279 BAFREC [6] | 0.335 | 0.503 | 0.360 | 0.402 KAFCA [5] | 0.314 | 0.509 | 0.244 | 0.397 MPSUM [11] | 0.314 | 0.512 | 0.272 | 0.423 ESA [12] | 0.331 | 0.532 | 0.350 | 0.416 DeepLENS | 0.402 ▲▲▲▲▲▲▲▲▲▲ | 0.574 ▲▲▲▲▲▲▲▲▲▲ | 0.474 ▲▲▲▲▲▲▲▲▲▲ | 0.493 ▲▲▲▲▲▲▲▲▲▲ ORACLE | 0.595 | 0.713 | 0.619 | 0.678 Comparison with Baselines. As shown in Table 1, supervised methods were generally better than unsupervised methods. Our DeepLENS outperformed all the baselines including ESA. Moreover, two-tailed t-test showed that all the differences were statistically significant ($p<0.01$) in all the settings. DeepLENS achieved new state-of-the-art results on the ESBM benchmark. However, the notable gaps between DeepLENS and ORACLE suggested room for improvement and were to be closed by future research. Table 2: Average F1 over all the test entities achieved by different variants of ESA. | DBpedia | LinkedMDB ---|---|--- | $k=5$ | $k=10$ | $k=5$ | $k=10$ ESA | 0.331 | 0.532 | 0.350 | 0.416 ESA-text | 0.379 | 0.558 | 0.390 | 0.418 ESA-rnd | 0.116$\pm$0.008 | 0.222$\pm$0.007 | 0.113$\pm$0.015 | 0.219$\pm$0.011 Ablation Study. Compared with ESA, we attributed the better performance of DeepLENS to two improvements in our implementation: the exploitation of textual semantics, and the permutation invariant representation of triple set. They were demonstrated by the following ablation study of ESA. First, we compared two variants of ESA by encoding triples in different ways. For triple $t$, the original version of ESA encoded the structural features of $\mathtt{prop}(t)$ and $\mathtt{val}(t)$ using TransE. We implemented ESA- text, a variant that encoded both $\mathtt{prop}(t)$ and $\mathtt{val}(t)$ using fastText as in our approach. As shown in Table 2, ESA-text slightly outperformed ESA, showing the usefulness of textual semantics compared with graph structure used by ESA. Second, we compared two variants of ESA by feeding with triples in different orders. The default version of ESA was fed with triples sorted in alphabetical order for both training and testing. We implemented ESA-rnd, a variant that was fed with triples in alphabetical order for training but in random order for testing. We tested ESA-rnd 20 times and reported its mean F1 with standard deviation. In Table 2, the notable falls from ESA to ESA-rnd showed the unfavourable sensitivity of BiLSTM used by ESA to the order of input triples. ## 4 Conclusion We presented DeepLENS, a simple yet effective deep learning model for general- purpose entity summarization. It has achieved new state-of-the-art results on the ESBM benchmark, significantly outperforming existing methods. Thus, entity summarization becomes another research field where a combination of deep learning and knowledge graph is likely to shine. However, in DeepLENS we only exploit textual semantics. In future work, we will incorporate ontological semantics into our model. We will also revisit the usefulness of structural semantics. ## Acknowledgments This work was supported by the National Key R&D Program of China under Grant 2018YFB1005100 and by the Qing Lan Program of Jiangsu Province. ## References * [1] Bojanowski, P., Grave, E., Joulin, A., Mikolov, T.: Enriching word vectors with subword information. TACL 5, 135–146 (2017) * [2] Cheng, G., Tran, T., Qu, Y.: RELIN: relatedness and informativeness-based centrality for entity summarization. In: ISWC 2011, Part I. pp. 114–129 (2011) * [3] Gunaratna, K., Thirunarayan, K., Sheth, A.P.: FACES: diversity-aware entity summarization using incremental hierarchical conceptual clustering. In: AAAI 2015. pp. 116–122 (2015) * [4] Gunaratna, K., Thirunarayan, K., Sheth, A.P., Cheng, G.: Gleaning types for literals in RDF triples with application to entity summarization. In: ESWC 2016. pp. 85–100 (2016) * [5] Kim, E.K., Choi, K.S.: Entity summarization based on formal concept analysis. In: EYRE 2018 (2018) * [6] Kroll, H., Nagel, D., Balke, W.T.: BAFREC: Balancing frequency and rarity for entity characterization in linked open data. In: EYRE 2018 (2018) * [7] Liu, Q., Cheng, G., Gunaratna, K., Qu, Y.: Entity summarization: State of the art and future challenges. CoRR abs/1910.08252 (2019) * [8] Liu, Q., Cheng, G., Gunaratna, K., Qu, Y.: ESBM: An entity summarization benchmark. In: ESWC 2020 (2020) * [9] Sydow, M., Pikula, M., Schenkel, R.: The notion of diversity in graphical entity summarisation on semantic knowledge graphs. J. Intell. Inf. Syst. 41(2), 109–149 (2013) * [10] Thalhammer, A., Lasierra, N., Rettinger, A.: LinkSUM: Using link analysis to summarize entity data. In: ICWE 2016. pp. 244–261 (2016) * [11] Wei, D., Gao, S., Liu, Y., Liu, Z., Huang, L.: MPSUM: Entity summarization with predicate-based matching. In: EYRE 2018 (2018) * [12] Wei, D., Liu, Y., Zhu, F., Zang, L., Zhou, W., Han, J., Hu, S.: ESA: Entity summarization with attention. In: EYRE 2019. pp. 40–44 (2019) * [13] Xu, D., Zheng, L., Qu, Y.: CD at ENSEC 2016: Generating characteristic and diverse entity summaries. In: SumPre 2016 (2016)

2024-09-04T02:54:58.183852

2003.03751

arxiv-papers

# Classification of Doubly Distributive skew Hyperfields and Stringent hypergroups Nathan Bowler and Ting Su<EMAIL_ADDRESS>Department of Mathematics, Universität Hamburg, Germany<EMAIL_ADDRESS>Department of Mathematics, Universität Hamburg, Germany ###### Abstract. A hypergroup is stringent if $a\boxplus b$ is a singleton whenever $a\neq-b$. A hyperfield is stringent if the underlying additive hypergroup is. Every doubly distributive skew hyperfield is stringent, but not vice versa. We present a classification of stringent hypergroups, from which a classification of doubly distributive skew hyperfields follows. It follows from our classification that every such hyperfield is a quotient of a skew field. ###### Key words and phrases: hypergroup, hyperring, hyperfield, double distributivity ## 1\. Introduction The notion of hyperfield was first introduced by Krasner in [Kra57, Kra83]. It is an algebraic structure similar to a field except that its addition $\boxplus$ is multivalued. In [Vir10], Viro provided an excellent introduction to and motivation for hyperfields and introduced several good examples of hyperfields, including the tropical hyperfield $\mathbb{T}_{+}$, the tropical real hyperfield $\mathbb{TR}$ and the ultratriangle hyperfield $\mathbb{T}\triangle$. Viro has also illustrated the utility of $\mathbb{T}_{+}$ for the foundations of tropical geometry in several interesting papers (cf. [Vir10, Vir11]). In [BB16], Baker and Bowler presented an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, oriented matroids, and valuated matroids, and called the resulting objects matroids over hyperfields. A matroid over a field $F$ corresponds to a subspace of some $F^{n}$. A $\mathbb{K}$-matroid is just a matroid. An $\mathbb{S}$-matroid is an oriented matroid. And a $\mathbb{T}\triangle$-matroid is a valuated matroid, as defined in [DW92]. Baker and Bowler also provided two natural notions of matroids over a hyperfield $F$, weak $F$-matroids and strong $F$-matroids, and showed that the two notions coincide when $F$ has a property called double distributivity. A hyperfield $F$ is doubly distributive if $(a\boxplus b)(c\boxplus d)=ac\boxplus ad\boxplus bc\boxplus bd$ for any $a,b,c,d\in F$. Fields, $\mathbb{K}$, $\mathbb{S}$ and $\mathbb{T}\triangle$ are all doubly distributive. So too are the other two hyperfields mentioned above, $\mathbb{T}_{+}$ and $\mathbb{TR}$. It is these the results in tropical geometry and matroid theory which motivate our interest in doubly distributive hyperfields. More generally, we are also interested in doubly distributive hyperrings, which were also analysed by Baker and Bowler. In fact, rather than just hyperfields, they worked with a more general kind of algebraic object known as tracts (cf. [BB19]). The other important example of tracts other than hyperfields is given by partial fields, which have also been the subject of much fruitful study. Baker and Bowler defined a special class of tracts called partial hyperfields, objects based on hyperrings which generalize both hyperfields and partial fields in a natural way. The property of double distributivity also extends to hyperrings and thus to partial hyperfields. We will classify the doubly distributive skew hyperfields in Section 5. The classification itself will be described in Section 4, but has the following important consequence: ###### Definition 1.1. A valuation $\nu$ of a skew hyperfield $F$ is a map from $F$ to $G\cup\\{-\infty\\}$, where $(G,<)$ is a linearly ordered group, satisfying 1. (1) $\nu(x)=-\infty$ if and only if $x=0$. 2. (2) $\nu(xy)=\nu(x)\cdot\nu(y)$. 3. (3) $\nu(x)>\nu(y)$ implies $x\boxplus y=\\{x\\}$. ###### Theorem 1.2. For every doubly distributive skew hyperfield $F$, there is always a valuation $\nu$ of $F$ such that $\nu^{-1}(1_{G})$ is either the Krasner hyperfield, or the sign hyperfield, or a skew field. This compact description is from the paper [BP19]. In particular, since any nontrivial ordered group is infinite, it follows from our results that the only finite doubly distributive hyperfields are the Krasner hyperfield, the sign hyperfield and the finite fields. This classification has a number of applications. For example, we use it in Section 7 to show that any doubly distributive skew hyperfield is a quotient of a skew field. Bowler and Pendavingh used it in [BP19] to show that any doubly distributive skew hyperfield is perfect and to provide vector axioms for matroids over such skew hyperfields. Our classification uses a property of the underlying hypergroup which we call stringency. A hyperfield $F$ is stringent, if $a\boxplus b$ is a singleton whenever $a\neq-b$. ###### Proposition 1.3. Every doubly distributive skew hyperfield is stringent. ###### Proof. Let $F$ be a doubly distributive skew hyperfield. Let $a,b\in F^{\times}$ be such that $a\neq-b$. Let $x,y\in F^{\times}$ be such that $x,y\in a\boxplus b$. By double distributivity, we have $(a\boxplus b)(x^{-1}\boxplus-y^{-1})=(a\boxplus b)\cdot x^{-1}\boxplus(a\boxplus b)\cdot(-y^{-1})\supseteq x\cdot x^{-1}\boxplus y\cdot(-y^{-1})=1\boxplus-1\ni 0.$ As $a\neq-b$, then $x^{-1}=y^{-1}$, and so $x=y$. So $a\boxplus b$ is a singleton if $a\neq-b$. ∎ However, not every stringent skew hyperfield is doubly distributive. The following is a counterexample. ###### Example 1.4. Let $F:=\mathbb{Z}\cup\\{-\infty\\}$ be the stringent hyperfield with multiplication given by $a\odot b=a+b$ and multiplicative identity $0$. Hyperaddition is given by $a\boxplus b=\begin{cases}\\{\max(a,b)\\}&\text{ if $a\neq b$,}\\\ \\{c\,|\,c<a\\}&\text{ if $a=b$,}\end{cases}$ so that the additive identity is $-\infty$. Here we use the standard total order on $\mathbb{Z}$ and set $-\infty<x$ for all $x\in\mathbb{Z}$. $F$ is not doubly distributive because $\displaystyle(0\boxplus 0)\odot(0\boxplus 0)$ $\displaystyle=\\{z\,|\,z<0\\}\odot\\{z\,|\,z<0\\}=\\{z\,|\,z<-1\\},$ $\displaystyle 0\boxplus 0\boxplus 0\boxplus 0$ $\displaystyle=\\{z\,|\,z<0\\}\boxplus\\{z\,|\,z<0\\}=\\{z\,|\,z<0\\}.$ We use our classification of stringent skew hyperfields to derive a classification of stringent skew hyperrings in Section 6. However, this does not give a classification of doubly distributive skew hyperrings, since not every doubly distributive skew hyperring is stringent (see Example 6.2). In fact, we classify all stringent hypergroups, and our classification of doubly distributive skew hyperfields follows from this. ###### Definition 1.5. Let $(G,<)$ be a totally ordered set, let $(F_{g}\,|\,g\in G)$ be a family of hypergroups with a common identity element $0$ in each $F_{g}$ but otherwise disjoint, and let $\psi$ be the surjective function from $\bigcup_{g\in G}F_{g}^{\times}$ to $G$ sending $f$ in $F_{g}^{\times}$ to $g$. We denote the hyperaddition of $F_{g}$ by $\boxplus_{g}$. For any $g\in G$ we denote by $g\downarrow$ the set of $h\in G$ with $h<g$. Then the wedge sum $F=\bigvee_{g\in G}{F_{g}}$ is the hypergroup with ground set $\bigcup_{g\in G}F_{g}$ and hyperaddition given by $\displaystyle x\boxplus 0$ $\displaystyle=0\boxplus x=\\{x\\},$ $\displaystyle x\boxplus y$ $\displaystyle=\begin{cases}\\{x\\}&\text{if $\psi(x)>\psi(y)$,}\\\ \\{y\\}&\text{if $\psi(x)<\psi(y)$,}\\\ x\boxplus_{\psi(x)}y&\text{if $\psi(x)=\psi(y)$ and $0\not\in x\boxplus_{\psi(x)}y$,}\\\ (x\boxplus_{\psi(x)}y)\cup\psi^{-1}(\psi(x)\downarrow)&\text{if $\psi(x)=\psi(y)$ and $0\in x\boxplus_{\psi(x)}y$. }\end{cases}$ We can also define $\bigcup_{g\in G}F_{g}$ up to isomorphism if the $F_{g}^{\prime}s$ don’t have the same identity or aren’t otherwise disjoint, by replacing the $F_{g}^{\prime}s$ with suitably chosen isomorphic copies. We will show in Section 3 that this construction always yields a hypergroup, and we classify the stringent hypergroups as follows: ###### Theorem 1.6. Every stringent hypergroup is a wedge sum $\bigvee_{g\in G}{F_{g}}$ where each $F_{g}$ is either a copy of the Krasner hypergroup, or a copy of the sign hypergroup, or a group. This classification of hypergroups is used to derive the classification of doubly distributive skew hyperfields discussed above. ### 1.1. Structure of the paper After the classification of stringent hypergroups in Section 3, we show in Section 4 that every stringent skew hyperfield arises from a short exact sequence of groups, where the first group in the sequence is the multiplicative group of either the Krasner hyperfield or the sign hyperfield or a skew field, and the last group in the sequence is a totally ordered group. The underlying additive hypergroup is a wedge sum of isomorphic copies of hypergroups. Then we present the classification of doubly distributive skew hyperfields in Section 5 following from the classification of stringent skew hyperfields. We show the surprising result that every stringent skew hyperring is either a skew ring or a stringent skew hyperfield in Section 6. We use our classification to show that every stringent skew hyperfield is a quotient of a skew field by some normal subgroup in Section 7. In Appendix A we present a proof that a construction really gives a skew field and in Appendix B we talk about the semirings associated to doubly distributive hyperfields. ### Acknowledgements We thank Matthew Baker and Laura Anderson (second author’s PhD advisor) for introducing the two authors to each other. We thank Laura Anderson and Tom Zaslavsky, who gave us important comments on early versions of the work. Thanks also to Pascal Gollin for asking whether our classification might hold for all stringent hypergroups. ## 2\. Background ###### Notation 2.1. Throughout $G$ and $H$ denotes groups. For a hypergroup (or skew hyperring) $S$, $S^{\times}$ denotes $S-\\{0\\}$. For a function $f$ from a hypergroup (or skew hyperring) $A$ to a hypergroup (or skew hyperring) $B$, $\operatorname{supp}(f)$ denotes the set of support of $f$ (the elements of $A$ where the function value is not zero). ### 2.1. Hypergroups, hyperrings and hyperfields ###### Definition 2.2. A hyperoperation on a set $S$ is a map $\boxplus$ from $S\times S$ to the collection of non-empty subsets of $S$. If $A$, $B$ are non-empty subsets of $S$, we define $A\boxplus B:=\bigcup_{a\in A,b\in B}a\boxplus b$ and we say that $\boxplus$ is associative if $a\boxplus(b\boxplus c)=(a\boxplus b)\boxplus c$ for all $a,b,c\in S$. All hyperoperations in this paper will be associative. ###### Definition 2.3. [Vir10] A hypergroup is a tuple $(G,\boxplus,0)$ where $\boxplus$ is an associative hyperoperation on $G$ such that: 1. (1) $0\boxplus x=x\boxplus 0=\\{x\\}$ for all $x\in G$. 2. (2) For every $x\in G$ there is a unique element $x^{\prime}$ of $G$ such that $0\in x\boxplus x^{\prime}$ and there is a unique element $x^{\prime\prime}$ of $G$ such that $0\in x^{\prime\prime}\boxplus x$. Furthermore, $x^{\prime}=x^{\prime\prime}$. This element is denoted by $-x$ and called the hyperinverse of $x$. 3. (3) (Invertibility of sums) $x\in y\boxplus z$ if and only if $-x\in-z\boxplus-y$. A hypergroup is said to be commutative if 4. (4) $x\in y\boxplus z$ if and only if $x\in z\boxplus y$. ###### Theorem 2.4. [Vir10] In Definition 2.3, the axiom (3) can be replaced by (Reversibility property) $x\in y\boxplus z$ implies $y\in x\boxplus-z$ and $z\in-y\boxplus x$. The Reversibility property was introduced by Marshall in [Mar06]. ###### Definition 2.5. A skew hyperring is a tuple $(R,\odot,\boxplus,1,0)$ such that: 1. (1) $(R,\odot,1)$ is a monoid. 2. (2) $(R,\boxplus,0)$ is a commutative hypergroup. 3. (3) (Absorption rule) $x\odot 0=0\odot x=0$ for all $x\in R$. 4. (4) (Distributive Law) $a\odot(x\boxplus y)=(a\odot x)\boxplus(a\odot y)$ and $(x\boxplus y)\odot a=(x\odot a)\boxplus(y\odot a)$ for all $a,x,y\in R$. A hyperring is a skew hyperring with commutative multiplication. A skew hyperring $F$ is called a skew hyperfield if $0\neq 1$ and every non- zero element of $F$ has a multiplicative inverse. A hyperfield is then a skew hyperfield with commutative multiplication. ###### Definition 2.6. Let $F$ and $G$ be skew hyperrings. We may define a skew hyperring $F\times G$ with $(x_{1},y_{1})\boxplus(x_{2},y_{2})$ defined as $(x_{1}\boxplus_{F}x_{2})\times(y_{1}\boxplus_{G}y_{2})$ and multiplication defined pointwise. Its additive identity is $(0_{F},0_{G})$ and its multiplicative identity is $(1_{F},1_{G})$. We call $F\times G$ the product of $F$ and $G$. Let $x,y\in F$, we will sometimes write $xy$ instead of $x\odot y$ if there is no risk of confusion. ###### Example 2.7. In [Vir10], Viro provided a good introduction to hyperfields. Several of the following hyperfields were first introduced there. 1. (1) If $F$ is a field, then $F$ is a hyperfield with $a\odot b=a\cdot b$ and $a\boxplus b=\\{a+b\\}$, for any $a,b\in F$. 2. (2) The Krasner hyperfield $\mathbb{K}:=\\{0,1\\}$ has the usual multiplication rule and hyperaddition is defined by $0\boxplus x=\\{x\\}$ for $x\in\mathbb{K}$ and $1\boxplus 1=\\{0,1\\}$. 3. (3) The sign hyperfield $\mathbb{S}:=\\{0,1,-1\\}$ has the usual multiplication rule and hyperaddition is defined by $0\boxplus x=\\{x\\},x\boxplus x=\\{x\\}$ for $x\in\mathbb{S}$, and $1\boxplus-1=\\{0,1,-1\\}$. 4. (4) The triangle hyperfield $\triangle:=\mathbb{R}_{\geq 0}$ has the usual multiplication rule and hyperaddition is defined by $x\boxplus y=\\{z\,|\,|x-y|\leq z\leq x+y\\}$. 5. (5) The tropical hyperfield $\mathbb{T}_{+}:=\mathbb{R}\cup\\{-\infty\\}$ has multiplication defined by $x\odot y=x+y$ (with $-\infty$ as an absorbing element), for $x,y\in\mathbb{T}_{+}$. Hyperaddition is defined by $x\boxplus y=\begin{cases}\\{\max(x,y)\\}&\text{ if $x\neq y$,}\\\ \\{z\,|\,z\leq x\\}&\text{ if $x=y$.}\end{cases}$ Here we use the standard total order on $\mathbb{R}$ and set $-\infty<x$ for all $x\in\mathbb{R}$. The additive identity is $-\infty$ and the multiplicative identity is $0$. 6. (6) The tropical phase hyperfield $\Phi:=S^{1}\cup\\{0\\}$ has the usual multiplication rule and hyperaddition is defined by $0\boxplus x=\\{x\\}$, $x\boxplus-x=S^{1}\cup\\{0\\}$ and $x\boxplus y=\\{\frac{ax+by}{|ax+by|}\,|\,a,b\in\mathbb{R}_{\geq 0},a+b\neq 0\\}$ for $x,y\in S^{1}$ with $y\neq-x$.111This is called phase hyperfield in Viro’s paper, but more recent papers have often worked with the phase hyperfield (7) described next. The confusion on this point is exacerbated by the fact that Viro incorrectly claims that his phase hyperfield is the same as the quotient hyperfield of the complex numbers by the positive real numbers, but this construction actually gives the hyperfield (7). 7. (7) The phase hyperfield $\mathbb{P}:=S^{1}\cup\\{0\\}$ has the usual multiplication rule and hyperaddition is defined by $0\boxplus x=\\{x\\}$, $x\boxplus-x=\\{x,-x,0\\}$ and $x\boxplus y=\\{\frac{ax+by}{|ax+by|}\,|\,a,b\in\mathbb{R}_{>0}\\}$ for $x,y\in S^{1}$ with $y\neq-x$. 8. (8) The tropical real hyperfield $\mathbb{TR}:=\mathbb{R}$ has the usual multiplication rule and hyperaddition is defined by $x\boxplus y=\begin{cases}\\{x\\}&\text{ if $|x|>|y|$,}\\\ \\{y\\}&\text{ if $|x|<|y|$,}\\\ \\{x\\}&\text{ if $x=y$,}\\\ \\{z\,|\,|z|\leq|x|\\}&\text{ if $x=-y$.}\end{cases}$ 9. (9) The tropical complex hyperfield $\mathbb{TC}:=\mathbb{C}$ has the usual multiplication rule and hyperaddition is defined by $x\boxplus y=\begin{cases}\\{x\\}&\text{if $|x|>|y|$,}\\\ \\{y\\}&\text{if $|x|<|y|$,}\\\ \\{|x|\dfrac{ax+by}{|ax+by|}\,|\,a,b\in\mathbb{R}_{\geq 0},a+b\neq 0\\}&\text{if $|x|=|y|$ and $x\neq-y$,}\\\ \\{z\,|\,|z|\leq|x|\\}&\text{if $x=-y$.}\end{cases}$ 10. (10) The ultratriangle hyperfield $\mathbb{T}\triangle:=\mathbb{R}_{\geq 0}$ (denoted by $\mathbb{Y}_{\times}$ in [Vir10] and $\mathbb{T}$ in [BB19]) has the usual multiplication rule and hyperaddition is defined by $x\boxplus y=\begin{cases}\\{\max(x,y)\\}&\text{ if $x\neq y$,}\\\ \\{z\,|\,z\leq x\\}&\text{ if $x=y$.}\end{cases}$ ###### Definition 2.8. [Vir10, BB19] A skew hyperring $R$ is said to be doubly distributive if for any $a$, $b$, $c$ and $d$ in $R$, we have $(a\boxplus b)(c\boxplus d)=ac\boxplus ad\boxplus bc\boxplus bd.$ ###### Example 2.9. Fields, $\mathbb{K}$, $\mathbb{S}$, $\mathbb{T}_{+}$, $\mathbb{TR}$, $\mathbb{T}\triangle$ are all doubly distributive, but $\triangle$, $\mathbb{P}$, $\Phi$ and $\mathbb{TC}$ are not doubly distributive. ###### Definition 2.10. A hypergroup $G$ is said to be stringent if for any $a,b\in G$ the set $a\boxplus b$ is a singleton whenever $a\neq-b$. A skew hyperring is said to be stringent if the underlying hypergroup $F$ is stringent. ### 2.2. Homomorphism ###### Definition 2.11. [BB16, Pen18] A hypergroup homomorphism is a map $f:G\rightarrow H$ such that $f(0)=0$ and $f(x\boxplus y)\subseteq f(x)\boxplus f(y)$ for all $x,y\in G$. A skew hyperring homomorphism is a map $f:R\rightarrow S$ which is a homomorphism of additive commutative hypergroups as well as a homomorphism of multiplicative monoids (i.e., $f(1)=1$ and $f(x\odot y)=f(x)\odot f(y)$ for $x,y\in R$). A skew hyperfield homomorphism is a homomorphism of the underlying skew hyperrings. A hypergroup (resp. skew hyperring, skew hyperfield) isomorphism is a bijection $f:G\rightarrow H$ which is a hypergroup (resp. skew hyperring, skew hyperfield) homomorphism and whose inverse is also a hypergroup (resp. skew hyperring, skew hyperfield) homomorphism. ###### Example 2.12. The map $\exp:\mathbb{T}_{+}\rightarrow\mathbb{T}\triangle$ is a hyperfield isomorphism. ## 3\. Classification of stringent hypergroups Our aim in this section is to prove Theorem 1.6, the Classification Theorem for stringent hypergroups. We will work with the definition of wedge sums given as Definition 1.5. First we will show that $F:=\bigvee_{g\in G}F_{g}$ is indeed a hypergroup. ###### Lemma 3.1. $F$ is again a hypergroup. If every hypergroup in $(F_{g}\,|\,g\in G)$ is stringent, then so is $F$. If every hypergroup in $(F_{g}\,|\,g\in G)$ is commutative, then so is $F$. ###### Proof. For associativity, suppose we have $x_{1},x_{2},x_{3}\in F$. If any of them is 0, then associativity is clear, so suppose that each $x_{i}$ is in $H$. If one of the elements $\psi(x_{i})$ of $G$, say $\psi(x_{i_{0}})$, is bigger than the others, then $x_{1}\boxplus(x_{2}\boxplus x_{3})=\\{x_{i_{0}}\\}=(x_{1}\boxplus x_{2})\boxplus x_{3}$. If one of the $\psi(x_{i})$ is smaller than the others, then both $x_{1}\boxplus(x_{2}\boxplus x_{3})$ and $(x_{1}\boxplus x_{2})\boxplus x_{3}$ evaluate to the sum of the other two $x_{j}$. So we may suppose that all $\psi(x_{i})$ are equal, taking the common value $g$. If $0\not\in x_{1}\boxplus_{g}x_{2}\boxplus_{g}x_{3}$, then both $x_{1}\boxplus(x_{2}\boxplus x_{3})$ and $(x_{1}\boxplus x_{2})\boxplus x_{3}$ evaluate to $x_{1}\boxplus_{g}x_{2}\boxplus_{g}x_{3}$, whereas if $0\in x_{1}\boxplus_{g}x_{2}\boxplus_{g}x_{3}$, then both evaluate to $(x_{1}\boxplus_{g}x_{2}\boxplus_{g}x_{3})\cup\psi^{-1}(g\downarrow)$. The hyperinverse of 0 is 0 and the hyperinverse of any other $x$ is its hyperinverse in $F_{\psi(x)}$, and $0$ is the additive identity. For invertibility of sums, suppose we have $x,y,z\in F$. We would like to show that $x\in y\boxplus z$ if and only if $-x\in-z\boxplus-y$. It suffices to prove one direction, say if $x\in y\boxplus z$, then $-x\in-z\boxplus-y$. If $\psi(y)<\psi(z)$, then $x\in y\boxplus z=\\{z\\}$ and $\psi(-y)<\psi(-z)$. So $-z\boxplus-y=\\{-z\\}=\\{-x\\}$. Similarly, we have if $\psi(y)>\psi(z)$, then $-z\boxplus-y=\\{-x\\}$. If $\psi(x)=\psi(y)=\psi(z)$, then the statement holds by the reversibility of the hypergroup $(F_{\psi(y)},\boxplus_{\psi(y)},0)$. Otherwise we have $\psi(x)<\psi(y)=\psi(z)$, and so $y=-z$. Then $\psi(-x)<\psi(-y)=\psi(-z)$ and so $-x\in-z\boxplus-y$. Then, we would like to show that $F$ is stringent if every hypergroup $F_{g}$ in $(F_{g}\,|\,g\in G)$ is. By definition of $F$, we just need to show that for any $x,y\in F$ with $\psi(x)=\psi(y)$ and $0\not\in x\boxplus_{\psi(x)}y$, $x\boxplus y$ is a singleton. As $F_{\psi(x)}$ is stringent and $0\not\in x\boxplus_{\psi(x)}y$, then $x\boxplus_{\psi(x)}y$ is a singleton. So $x\boxplus y=x\boxplus_{\psi(x)}y$ is also a singleton. Finally, it is clear that $\boxplus$ is commutative if each $\boxplus_{g}$ is commutative. ∎ Now we begin the proof of the Classification Theorem. We first introduce a useful lemma. Note that this lemma automatically holds for stringent commutative hypergroups, so readers only interested in that case may skip the proof. ###### Lemma 3.2. Let $F$ be a stringent hypergroup. If $y\in x\boxplus y$, then $y\in y\boxplus x$. ###### Proof. We will divide the proof into four cases. _Case 1:_ If $x=y$, this is immediate. _Case 2:_ If $x=-y$, then by reversibility we get $y\in x\boxplus y\Rightarrow y\in-y\boxplus y\Rightarrow y\in y\boxplus y\Rightarrow y\in y\boxplus-y\Rightarrow y\in y\boxplus x.$ _Case 3:_ If $y=-y$, then by reversibility and case 2 we get $y\in x\boxplus y\Rightarrow x\in y\boxplus-y\Rightarrow x\in-y\boxplus y\Rightarrow y\in y\boxplus x.$ _Case 4:_ Now we suppose $x\notin\\{y,-y\\}$ and $y\neq-y$. Let $z\in F^{\times}$ be such that $y\boxplus x=\\{z\\}$ and let $t\in F^{\times}$ be such that $-y\boxplus-y=\\{t\\}$. Then by associativity we get $z\boxplus y\boxplus t=(y\boxplus x)\boxplus y\boxplus(-y\boxplus-y)=y\boxplus(x\boxplus y)\boxplus-y\boxplus-y=y\boxplus y\boxplus-y\boxplus-y\ni 0.$ So we get $0\in z\boxplus y\boxplus t$, thus $-z\in y\boxplus t$. As $t\in-y\boxplus-y$, we have $-y\in y\boxplus t$. So $-z,-y\in y\boxplus t$. Then by stringency we get either $z=y$ or $t=-y$. If $z=y$, then we are done. Now assume $t=-y$. Thus $-z\in y\boxplus t=y\boxplus-y$, and so $-y\in-y\boxplus z$. As $y\boxplus x=\\{z\\}$, we have $x\in-y\boxplus z$. So $-y,x\in-y\boxplus z$. Then by stringency we get either $x=-y$ or $z=y$. By case 2, the statement holds. ∎ Now we define a relation on $F^{\times}$ which roughly corresponds to the ordering of $G$. ###### Definition 3.3. We define a relation $<_{F}$ on $F^{\times}$ by $x<_{F}y$ if $x\boxplus y=y\boxplus x=\\{y\\}$ but $x\neq y$. ###### Lemma 3.4. $<_{F}$ is a strict partial order on $F^{\times}$. ###### Proof. Irreflexivity is built into the definition, so it remains to check transitivity. Suppose that $x<_{F}y<_{F}z$. Then $x\boxplus z=x\boxplus y\boxplus z=y\boxplus z=\\{z\\}$. Similarly, $z\boxplus x=\\{z\\}$. We cannot have $x=z$, since then $\\{y\\}=y\boxplus x=y\boxplus z=\\{z\\}$, so $y=z$, which is a contradiction. ∎ ###### Lemma 3.5. If $x<_{F}y$, then 1. (1) $\pm x<_{F}\pm y$. 2. (2) for any $z\in F^{\times}$ we have either $x<_{F}z$ or $z<_{F}y$. ###### Proof. 1. (1) It suffices to prove that $-x<_{F}y$ by invertibility of sums. As $x<_{F}y$, then $x\neq-y$ since $0\in-y\boxplus y$. As $x\boxplus y=\\{y\\}$, then $y\in-x\boxplus y$. By stringency, $-x\boxplus y=\\{y\\}$. Similarly, $y\boxplus-x=\\{y\\}$. So $-x<_{F}y$. 2. (2) By (1), we have $\pm x<_{F}\pm y$. Suppose that $z\not<_{F}y$. If $z\in\\{y,-y\\}$, then we have $x<_{F}z$. Otherwise, $y\not\in z\boxplus y$ and $y\notin y\boxplus z$ by Lemma 3.2. Then $0\not\in z\boxplus y\boxplus-y$ and $0\not\in-y\boxplus y\boxplus z$. So by stringency, we have $z\boxplus y\boxplus-y=\\{z\\}$ and $-y\boxplus y\boxplus z=\\{z\\}$. However, $x\in y\boxplus-y$ and $x\in-y\boxplus y$, since $x<_{F}y$. So $z\boxplus x=\\{z\\}$ and $x\boxplus z=\\{z\\}$. Now if $z\neq x$ this implies that $x<_{F}z$, but if $z=x$ then we have $z<_{F}y$. ∎ Now we define a relation $\sim_{F}$ on $F^{\times}$ by $x\sim_{F}y$ if and only if both $x\not<_{F}y$ and $y\not<_{F}x$. ###### Lemma 3.6. $\sim_{F}$ is an equivalence relation. ###### Proof. $\sim_{F}$ is clearly reflexive and symmetric. For transitivity, suppose that $x\sim_{F}y$ and $y\sim_{F}z$. If $x<_{F}z$ then either $x<_{F}y$, contradicting $x\sim_{F}y$, or else $y<_{F}z$, contradicting $y\sim_{F}z$, so this is impossible. Similarly we have $z\not<_{F}x$. So $x\sim_{F}z$. ∎ The following results are obvious and we will put them together. ###### Lemma 3.7. 1. (1) If $x\sim_{F}y<_{F}z$ or $x<_{F}y\sim_{F}z$, then $x<_{F}z$. 2. (2) The relation $<_{F}$ could lift to a relation (denoted by $<_{F}^{\prime}$) on the set $G$ of $\sim_{F}$-equivalence classes and $(G,<_{F}^{\prime})$ is a totally ordered set. 3. (3) For every $x\in F^{\times}$, $-x\sim_{F}x$. 4. (4) Let $x,y,z\in F^{\times}$ with $x\neq-y$, $y\neq-z$ and $z\neq-x$. If $0\in x\boxplus y\boxplus z$, then $x\sim_{F}y\sim_{F}z$. ###### Proof. For (1), the proof is trivial. (1) implies (2). (3) As $0\in x\boxplus-x$, we have $x\not<_{F}-x$ and $-x\not<_{F}x$. So $-x\sim_{F}x$. (4) If not, then without loss of generality we have $x<_{F}y$, and so $-z\in x\boxplus y=\\{y\\}$, giving $y=-z$, contradicting our assumptions. ∎ ###### Lemma 3.8. Let $(x_{i}\,|\,i\in I)$ be a finite family of elements of $F$, and $z\in F$ with $x_{i}<_{F}z$ for all $i\in I$. Then for any $y\in\boxplus_{i\in I}x_{i}$ we have $y<_{F}z$. ###### Proof. It suffices to prove this when $I$ has just two elements, say $x_{1}$ and $x_{2}$, since the general result then follows by induction. Suppose $x_{1},x_{2}<_{F}z$ and $y\in x_{1}\boxplus x_{2}$, then we have $y\boxplus z\subseteq x_{1}\boxplus x_{2}\boxplus z=x_{1}\boxplus z=\\{z\\}$ and $z\boxplus y\subseteq z\boxplus x_{1}\boxplus x_{2}=z\boxplus x_{2}=\\{z\\}.$ So $y\boxplus z=\\{z\\}$ and $z\boxplus y=\\{z\\}$. If $z\in x_{1}\boxplus x_{2}$ then $-x_{1}\in x_{2}\boxplus-z=\\{-z\\}$, contradicting $x_{1}<_{F}z$. So $z\not\in x_{1}\boxplus x_{2}$, and so $z\neq y$. So $y<_{F}z$. ∎ It follows from the above results that the sum $x\boxplus y$ is given by $\\{x\\}$ if $x>_{F}y$, by $\\{y\\}$ if $x<_{F}y$, by $\\{z\\}$ for some $z$ in the $\sim_{F}$-equivalence class of $x$ and $y$ if $x\sim_{F}y$ but $x\neq-y$, and by some subset of that class together with $\\{t\,|\,t<_{F}x\\}\cup\\{0\\}$ if $x=-y$. This looks very similar to the hyperaddition given in Definition 1.5. We now want to consider the structure of the equivalence classes. Let $g$ be an equivalence class in $G$ and let $F_{g}$ be the set $g\cup\\{0\\}$. We can define a multivalued binary operation $\boxplus_{g}$ on $F_{g}$ by $x\boxplus_{g}y=(x\boxplus y)\cap F_{g}$. ###### Lemma 3.9. For any element $g$ in $G$, $F_{g}$ is again a hypergroup, with hyperaddition given by $\boxplus_{g}$. ###### Proof. For every $x\in F_{g}$, we have $0\boxplus_{g}x=\\{x\\}\cap F_{g}=\\{x\\}$. Suppose $0\in x\boxplus_{g}y$, then $0\in x\boxplus y$, and so $y=-x$. Similarly, if $0\in y\boxplus_{g}x$, then $y=-x$. For invertibility of sums, let $x,y,z\in F_{g}$ with $x\in y\boxplus_{g}z$. Then we have $x\in y\boxplus z$. By invertibility of sums of $F$, $-x\in-z\boxplus-y$. So $-x\in-z\boxplus_{g}-y$. For associativity, suppose we have $x,y,z\in F_{g}$. We would like to show that $(x\boxplus_{g}y)\boxplus_{g}z=x\boxplus_{g}(y\boxplus_{g}z).$ Let $t\in F_{g}$. Let us first show that $t\in x\boxplus_{g}(y\boxplus_{g}z)$ if and only if $t\in x\boxplus(y\boxplus z)$. It is clear that $x\boxplus_{g}(y\boxplus_{g}z)\subseteq x\boxplus(y\boxplus z)$. So it suffices to prove the other direction. We suppose that $t\in x\boxplus(y\boxplus z)$. Then there exists $k\in F$ such that $k\in y\boxplus z$ and $t\in x\boxplus k$. If $k\in F_{g}$, then we are done. If not, we have $y=-z$ and $k<_{F}y$. So we also have $k<_{F}x$, and so $t=x\in x\boxplus_{g}0\subseteq x\boxplus_{g}(y\boxplus_{g}z)$. Similarly, we can also get $t\in(x\boxplus_{g}y)\boxplus_{g}z$ if and only if $t\in(x\boxplus y)\boxplus z$. By associativity of $F$, $(x\boxplus y)\boxplus z=x\boxplus(y\boxplus z)$. So $(x\boxplus_{g}y)\boxplus_{g}z=x\boxplus_{g}(y\boxplus_{g}z).$ ∎ ###### Lemma 3.10. For any element $g$ in $G$, $F_{g}$ is either isomorphic to $\mathbb{K}$ or isomorphic to $\mathbb{S}$ or is a group. ###### Proof. For any $y$ and any $x$ with $x\in y\boxplus-y$, we have $y\in x\boxplus y$ and so $x<_{F}y$ unless $x\in\\{-y,0,y\\}$. So for any $y\in F_{g}$ we have $y\boxplus_{g}-y\subseteq\\{-y,0,y\\}$. Now suppose that there is some $y\in F_{g}$ with $y\boxplus_{g}-y\neq\\{0\\}$. Then $y$ is nonzero and $y,-y\in-y\boxplus_{g}y$. Suppose for a contradiction that there is some $z\in F_{g}\setminus\\{-y,0,y\\}$, and let $t$ be the unique element of $-y\boxplus z$. Then by Lemma 3.5, $t\notin\\{y,-y\\}$, since $z\not<_{F}-y$. So $y\boxplus t=\\{z\\}$. Thus $y\in y\boxplus_{g}0\subseteq(-y\boxplus_{g}y)\boxplus_{g}(t\boxplus_{g}-t)=-y\boxplus_{g}(y\boxplus_{g}t)\boxplus_{g}-t=-y\boxplus_{g}z\boxplus_{g}-t=t\boxplus_{g}-t,$ and so $y\in\\{-t,0,t\\}$, which is the desired contradiction. So if there is any $y$ with $y\boxplus_{g}-y\neq\\{0\\}$, then $F_{g}=\\{-y,0,y\\}$. It is now not hard to check that in this case if $y=-y$ then $F_{g}\cong\mathbb{K}$, and if $y\neq-y$ then $F_{g}\cong\mathbb{S}$. On the other hand, if there is no such $y$ then the hyperaddition on $F_{g}$ is single-valued, and so $F_{g}$ is a group. ∎ We can finally prove the Classification Theorem. ###### Proof of Theorem 1.6. Let $H$ be $F^{\times}$, let $G$ be given as above and let $\psi$ be the map sending an element $h$ of $H$ to its equivalence class in $G$. For any $x$ and $y$ in $H$, if $\psi(x)>_{F}^{\prime}\psi(y)$ then $x>_{F}y$ and so $x\boxplus y=\\{x\\}$. Similarly if $\psi(x)<_{F}^{\prime}\psi(x)$ then $x\boxplus y=\\{y\\}$. If $\psi(x)=\psi(y)$ then $x\boxplus_{\psi(x)}y=(x\boxplus y)\cap F_{\psi(x)}$. So by the remarks following Lemma 3.8 we have that the hyperaddition of $F$ agrees with that of $\bigvee_{g\in G}F_{g}$ in this case as well. ∎ ## 4\. Classification of stringent skew hyperfields In this section, we will present the classification of stringent skew hyperfields. We will first introduce a construction of skew hyperfields arising from short exact sequences. ###### Definition 4.1. Let $F$ be a skew hyperfield and let $G$ be a totally ordered group. Suppose that we have a short exact sequence of groups $1\to F^{\times}\xrightarrow{\varphi}H\xrightarrow{\psi}G\to 1\,.$ Since $\varphi$ is injective, by replacing $H$ with an isomorphic copy if necessary we may (and shall) suppose that $\varphi$ is the identity. As usual, we define $x^{h}$ to be $h^{-1}\cdot x\cdot h$ for $x,h\in H$. We extend this operation by setting $0_{F}^{h}:=0_{F}$. We say that the short exact sequence has stable sums if for each $h\in H$ the operation $x\mapsto x^{h}$ is an automorphism of $F$ (as a skew hyperfield). Since this operation clearly preserves the multiplicative structure, this is equivalent to the condition that it is always an automorphism of the underlying additive hypergroup. Furthermore, any short exact sequence as above with $H$ abelian automatically has stable sums. Suppose now that we have a short exact sequence with stable sums as above. Then we may define a hyperfield with multiplicative group $H$ as follows. We begin by choosing some object not in $H$ to serve as the additive identity, and we denote this object by $0$. For each $g$ in $G$, let $A_{g}$ be $\psi^{-1}(g)\cup\\{0\\}$. For any $h$ in $\psi^{-1}(g)$ there is a bijection $\lambda_{h}$ from $F$ to $A_{g}$ sending $0_{F}$ to $0$ and $x$ to $h\cdot x$ for $x\in F^{\times}$, and so there is a unique hypergroup structure on $A_{g}$ making $\lambda_{h}$ an isomorphism of hypergroups. Furthermore, this structure is independent of the choice of $h$ since for $h_{1},h_{2}\in\psi^{-1}(g)$ the map $\lambda_{h_{1}}^{-1}\cdot\lambda_{h_{2}}$ is just left multiplication by $h_{1}^{-1}\cdot h_{2}$, which is an automorphism of the additive hypergroup of $F$. In this way we obtain a well defined hypergroup structure on $A_{g}$, whose hyperaddition we denote by $\boxplus_{g}$. Then the $G$-layering $F\rtimes_{H,\psi}G$ of $F$ along this short exact sequence has as ground set $H\cup\\{0\\}$. Multiplication is given by $x\cdot y=0$ if $x$ or $y$ is $0$ and by the multiplication of $H$ otherwise. $H\cup\\{0\\}$ is the underlying set of the hypergroup $\bigvee_{g\in G}A_{g}$, and we take the hyperaddition of $F\rtimes_{H,\psi}G$ to be given by that of this hypergroup. Explicitly; the hyperaddition is given by taking 0 to be the additive identity and setting $x\boxplus y=\begin{cases}\\{x\\}&\text{if }\psi(x)>\psi(y),\\\ \\{y\\}&\text{if }\psi(x)<\psi(y),\\\ x\boxplus_{\psi(x)}y&\text{if }\psi(x)=\psi(y)\text{ and }0\not\in x\boxplus_{\psi(x)}y,\\\ (x\boxplus_{\psi(x)}y)\cup\psi^{-1}(\psi(x)\downarrow)&\text{if }\psi(x)=\psi(y)\text{ and }0\in x\boxplus_{\psi(x)}y.\end{cases}$ ###### Lemma 4.2. $F\rtimes_{H,\psi}G$ is again a skew hyperfield. If $F$ is stringent, then so is $F\rtimes_{H,\psi}G$. ###### Proof. As shown in Lemma 3.1, $\bigvee_{g\in G}A_{g}$ is a commutative hypergroup. So it suffices to show that $\cdot$ distributes over $\boxplus$. For left distributivity, we must prove an equation of the form $x_{1}\cdot(x_{2}\boxplus x_{3})=x_{1}\cdot x_{2}\boxplus x_{1}\cdot x_{3}$. As usual, if any of the $x_{i}$ is 0, then this is trivial, so we suppose that each $x_{i}$ is in $H$. If $\psi(x_{2})>\psi(x_{3})$, then both sides are equal to $x_{1}\cdot x_{2}$. If $\psi(x_{2})<\psi(x_{3})$, then both sides are equal to $x_{1}\cdot x_{3}$. So we may assume that $\psi(x_{2})=\psi(x_{3})$ and we call their common value $g$. Then $x_{2}\boxplus_{g}x_{3}=\lambda_{x_{2}}(1\boxplus_{F}x_{2}^{-1}\cdot x_{3})$ and $(x_{1}\cdot x_{2})\boxplus_{\psi(x_{1})\cdot g}(x_{1}\cdot x_{3})=\lambda_{x_{1}\cdot x_{2}}(1\boxplus_{F}x_{2}^{-1}\cdot x_{3})$. So if $0\not\in x_{2}\boxplus_{g}x_{3}$, then also $0\not\in(x_{1}\cdot x_{2})\boxplus_{\psi(x_{1})\cdot g}(x_{1}\cdot x_{3})$, and so both sides of the equation are equal to $x_{1}\cdot(x_{2}\boxplus_{g}x_{3})$. If $0\in x_{2}\boxplus_{g}x_{3}$, then also $0\in(x_{1}\cdot x_{2})\boxplus_{\psi(x_{1})\cdot g}(x_{1}\cdot x_{3})$, and so both sides of the equation are equal to $x_{1}\cdot(x_{2}\boxplus_{g}x_{3})\cup x_{1}\cdot\psi^{-1}(g\downarrow)$. For the right distributivity, we need to consider bijections $\lambda_{h}^{\prime}\colon F\to A_{\psi(h)}$ similar to the $\lambda_{h}$. We take $\lambda_{h}^{\prime}(x)$ to be $x\cdot h$ for $x\in F^{\times}$ and to be $0$ for $x=0_{F}$. Then since $\lambda_{h}^{\prime}(x)=\lambda_{h}(x)^{h}$ for any $x$ and the short exact sequence has stable sums, the $\lambda_{h}^{\prime}$ are also hyperfield isomorphisms. So we may argue as above but with the $\lambda_{h}^{\prime}$ in place of the $\lambda_{h}$. Finally, we must show that $F\rtimes_{H,\psi}G$ is stringent if $F$ is. By definition of $F\rtimes_{H,\psi}G$, we just need to show that for $x,y\in F\rtimes_{H,\psi}G$ with $\psi(x)=\psi(y)$ and $0\not\in x\boxplus_{\psi(x)}y$, $x\boxplus y$ is a singleton. As $F$ is stringent and $0\not\in x\boxplus_{\psi(x)}y$, then $x\boxplus_{\psi(x)}y$ is a singleton. So $x\boxplus y=x\boxplus_{\psi(x)}y$ is also a singleton. ∎ Now let us see some interesting examples of hyperfields constructed in this way. ###### Example 4.3. If $F$ is the Krasner hyperfield, $G$ and $H$ are both the additive group of real numbers, and $\psi$ is the identity, then $F\rtimes_{H,\psi}G$ is the tropical hyperfield. ###### Example 4.4. $F:=\mathbb{Z}\cup\\{-\infty\\}$ in Example 1.4 can arise from the short exact sequence of groups $0\to GF(2)^{\times}\xrightarrow{\varphi}\mathbb{Z}\xrightarrow{\psi}\mathbb{Z}\to 0.$ ###### Example 4.5. In [AD19], Anderson and Davis drew a diagram encoding many popular and important hyperfields and homomorphisms, see as follows. $\mathbb{TR}$$\mathbb{TC}$$\mathbb{T}\triangle$$\mathbb{S}$$\mathbb{K}$$\Phi$$\mathbb{P}$$\mathbb{R}$$\mathbb{C}$$\triangle$$|\,\,|$$|\,\,|$$|\,\,|$$|\,\,|$phphphphphph The diagram with the solid arrows commutes. The four dashed arrows are inclusions giving sections (one-sided inverses). Here ph is the _phase map_ ph$(x)=x/|x|$ if $x=0$ and ph$(0)=0$. In each of the ten hyperfields, the underlying set is a subset of the complex numbers closed under multiplication. And in each hyperfield, multiplication, the additive identity, and the multiplicative identity coincides with that of the complex numbers. Our classification gives a good relationship between the hyperfields in each column and we can construct each hyperfield in the bottom row from the corresponding element of the row just above the bottom and the ordered group $\mathbb{R}_{>0}$. 1. (1) From the short exact sequence of groups $1\to\mathbb{S}^{\times}\rightarrow\mathbb{R}^{\times}\rightarrow\mathbb{R}_{>0}\to 1,$ we can get the tropical real hyperfield $\mathbb{TR}=\mathbb{S}\rtimes\mathbb{R}_{>0}$. 2. (2) From the short exact sequence of groups $1\to\Phi^{\times}\rightarrow\mathbb{C}^{\times}\rightarrow\mathbb{R}_{>0}\to 1,$ we can get the tropical complex hyperfield $\mathbb{TC}=\Phi\rtimes\mathbb{R}_{>0}$. 3. (3) From the short exact sequence of groups $1\to\mathbb{K}^{\times}\rightarrow\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}\to 1,$ we can get the ultratriangle hyperfield $\mathbb{T}\triangle=\mathbb{K}\rtimes\mathbb{R}_{>0}$. Since in each column the second element is obtained as a quotient of the first by a $\mathbb{R}_{>0}-$subgroup, this operation of putting back the factor of $\mathbb{R}_{>0}$ yields a hyperfield on the same ground set as the top element of the column. Our aim is to show that every stringent skew hyperfield is of the form $F\rtimes_{H,\psi}G$ with $F$ either the Krasner hyperfield or the sign hyperfield or a skew field. Let’s start with a stringent skew hyperring. Let $R$ be a stringent skew hyperring. By Theorem 1.6, we can classify $R$ to be the wedge sum $\bigvee_{g\in G}R_{g}$ with a surjective mapping $\psi$ from $R^{\times}$ to the set $G$ defined in last section and an ordering $<_{R}^{\prime}$ on $G$ by $\psi(x)<_{R}^{\prime}\psi(y)$ if and only if $x\boxplus y=\\{y\\}$ but $x\neq y$, where the hypergroup $R_{g}$ is either isomorphic to $\mathbb{K}$ or isomorphic to $\mathbb{S}$ or is a group. Thus by distributivity of $R$, $\psi(x)<_{R}^{\prime}\psi(y)$ if and only if $\psi(ax)<_{R}^{\prime}\psi(ay)$ if and only if $\psi(xa)<_{R}^{\prime}\psi(ya)$ for $a\in R^{\times}$. So the multiplication of $R$ lifts to a multiplication on $G$ respecting the ordering, with identity $\psi(1):=1_{G}$. By Lemma 3.7(2), we can easily get the following lemma. ###### Lemma 4.6. $(G,\cdot,<_{R}^{\prime})$ is a totally ordered monoid. If $R$ is a skew hyperfield, then $G$ is a totally ordered group. Now we want to consider the structure of $R_{g}$. ###### Lemma 4.7. $R_{1_{G}}$ is again a skew hyperring, with hyperaddition given by $\boxplus_{1_{G}}$ and multiplication by that of $R$. ###### Proof. By Lemma 3.9, it suffices to check the distributivity. To prove left distributivity we must show that any element $t\in R_{1_{G}}$ of $x\cdot(y\boxplus z)$ is also an element of the same expression evaluated in $R_{1_{G}}$. So let $w$ be an element of $y\boxplus z$ with $x\cdot w=t$. This second equation implies that the equivalence class of $w$ is $1_{G}$, as desired. The right distributivity is similar. ∎ ###### Lemma 4.8. If $R$ is a skew hyperfield, $R_{1_{G}}$ is either the Krasner hyperfield or the sign hyperfield or a skew field. ###### Proof. By Lemma 3.10 and Lemma 4.7, we can get $R_{1_{G}}$ is either the Krasner hyperfield or the sign hyperfield or a skew ring. Since $\sim_{R}$ respects the multiplication, the multiplicative inverse of anything equivalent to $1_{R}$ is again equivalent to $1_{R}$, so that $R_{1_{G}}$ is a skew field if it is a skew ring. ∎ ###### Lemma 4.9. For every $g\in G$, the hypergroup of $R_{g}$ is isomorphic to the hypergroup of $R_{1_{G}}$. ###### Proof. Let $a\in R_{g}^{\times}$. Define $f:R_{g}\rightarrow R_{1_{G}}$ by sending $0$ to $0$ and $x$ in $R_{g}^{\times}$ to $a^{-1}\cdot x$. Since $f$ has an inverse operation, namely left multiplication by $a$, this is a bijection. Now we would like to show $f(x\boxplus_{g}y)=f(x)\boxplus_{1_{G}}f(y)$. $\displaystyle f(x\boxplus_{g}y)$ $\displaystyle=a^{-1}\cdot(x\boxplus_{g}y)$ $\displaystyle=a^{-1}\cdot\big{(}(x\boxplus y)\cap R_{g}\big{)}$ $\displaystyle=\big{(}a^{-1}\cdot(x\boxplus y)\big{)}\cap(a^{-1}\cdot R_{g})$ $\displaystyle=\big{(}(a^{-1}\cdot x)\boxplus(a^{-1}\cdot y)\big{)}\cap R_{1_{G}}$ $\displaystyle=(a^{-1}\cdot x)\boxplus_{1_{G}}(a^{-1}\cdot y)$ $\displaystyle=f(x)\boxplus_{1_{G}}f(y)$ ∎ Now using the above results, we can classify the stringent skew hyperfields as follows. ###### Theorem 4.10. Any stringent skew hyperfield $R$ has the form $F\rtimes_{H,\psi}G$, where $F$ is either the Krasner hyperfield or the sign hyperfield or a skew field. ###### Proof. Let $F$ be $R_{1_{G}}$, let $H$ be $R^{\times}$ and let $G$ be given as above. Let $\varphi$ be the injection of $F^{\times}$ as a subgroup of $H$ and let $\psi$ be the map sending an element $h$ of $H$ to its equivalence class in $G$. Then $1\to F^{\times}\xrightarrow{\varphi}H\xrightarrow{\psi}G\to 1$ is a short exact sequence. For any $x$ and $y$ in $H$, if $\psi(x)>_{R}^{\prime}\psi(y)$, then $x>_{R}y$ and so $x\boxplus y=\\{x\\}$. Similarly if $\psi(x)<_{R}^{\prime}\psi(x)$, then $x\boxplus y=\\{y\\}$. If $\psi(x)=\psi(y)$, then $x\boxplus_{\psi(x)}y=x\cdot((1\boxplus x^{-1}\cdot y)\cap R_{1_{G}})=(x\boxplus y)\cap R_{\psi(x)}$. So by the remarks following Lemma 3.8 we have that the hyperaddition of $R$ agrees with that of $F\rtimes_{H,\psi}G$ in this case as well. ∎ Using results of Marshall’s paper [Mar06], we can show that the structure is even more constrained if the multiplication of $R$ is commutative (so that $R$ is a stringent hyperfield) and $R_{1_{G}}$ is the Krasner or the sign hyperfield. ###### Proposition 4.11. Let $R$ be a stringent skew hyperfield with $R_{1_{G}}=\mathbb{S}$ and let $a\in R^{\times}-\\{1,-1\\}$. Then $a^{2}\notin\\{1,-1\\}$. ###### Proof. As $a\notin\\{1,-1\\}$, then $a\not\sim_{R}1$. So $\psi(a)\neq 1$. Then $\psi(a^{2})=(\psi(a))^{2}\neq 1$, since $G$ is a totally ordered group. So $a^{2}\not\sim_{R}1$. That is $a^{2}\notin\\{1,-1\\}$. ∎ Following are some useful Lemmas in Marshall’s paper (cf. section 3,[Mar06]). ###### Definition 4.12. [Mar06] Let $R$ be a hyperfield. A subset $P$ of $R$ is called an ordering if $P\boxplus P\subseteq P,P\odot P\subseteq P,P\cup-P=R\text{ and }P\cap-P=\\{0\\}.$ ###### Definition 4.13. [Mar06] A hyperfield $R$ is said to be real if $-1\notin R^{2}\boxplus R^{2}$ where $R^{2}:=\\{a^{2}\,|\,a\in R\\}$. ###### Lemma 4.14. [Mar06, Lemma 3.3] Let $R$ be a hyperfield. $R$ has an ordering if and only if $R$ is real. ###### Lemma 4.15. [Mar06, Lemma 3.2, 3.3] Let $R$ be a hyperfield with $1\neq-1$. If $R$ has an ordering $P$, then $-1\notin P$. Based on above lemmas, we get the following. ###### Proposition 4.16. If $R$ is a stringent hyperfield with $R_{1_{G}}=\mathbb{S}$, then $R$ has an ordering. ###### Proof. By Lemma 4.14, we just need to show that $R$ is real. Suppose that $-1\in R^{2}\boxplus R^{2}$. Then there exist $a,b\in R$ such that $-1\in a^{2}\boxplus b^{2}$. By Proposition 4.11, $a^{2}\neq-1$ and $b^{2}\neq-1$. Thus $a\neq 0$ and $b\neq 0$. And by reversibility, $-b^{2}\in 1\boxplus a^{2}$. As $a^{2}\neq-1$, then $1\boxplus a^{2}\subseteq\\{1,a^{2}\\}$. Thus $-b^{2}=a^{2}$. Then $-1=a^{2}b^{-2}=(ab^{-1})^{2}$, a contradiction to Proposition 4.11. So $R$ is real, and therefore has an ordering. ∎ ###### Theorem 4.17. If $R$ is a stringent hyperfield with $R_{1_{G}}\in\\{\mathbb{K},\mathbb{S}\\}$, then $R$ arises from a short exact sequence $1\to R_{1_{G}}^{\times}\xrightarrow{\varphi}R_{1_{G}}^{\times}\times G\xrightarrow{\psi}G\to 1.$ ###### Proof. If $R_{1_{G}}=\mathbb{K}$, this is trivial. If $R_{1_{G}}=\mathbb{S}$, by Theorem 4.10, we may suppose $R=\mathbb{S}\rtimes_{H,\psi}G=H\cup\\{0\\}$ with a short exact sequence of groups $1\to\mathbb{S}^{\times}\xrightarrow{\varphi}H\xrightarrow{\psi}G\to 1.$ By Proposition 4.16, we know that $R$ has an ordering $P$. Let $O=P-\\{0\\}$. As $P\cup-P=R$ and $P\cap-P=\\{0\\}$, then $R=O\mbox{\hskip 1.49994pt$\cup$$\cdot$\hskip 3.99994pt}-O\cup\\{0\\}$. By Lemma 4.15, $-1\notin P$. Then $-1\notin O$, thus $1\in O$. And as $P\odot P\subseteq P$, then $O\odot O\subseteq O$. For any $a\in O$, $a^{-1}\in O$. Otherwise $-a^{-1}\in O$. Then $a\odot-a^{-1}=-1\in O$, which is a contradiction. So $O$ is a multiplicative group with $1\in O$ and $R=O\mbox{\hskip 1.49994pt$\cup$$\cdot$\hskip 3.99994pt}-O\cup\\{0\\}$, and $\psi\restriction O$ is an isomorphism from $O$ to $G$. Now we can identify $x\in H$ with $(1,\psi(x))$ if $x\in O$, and with $(-1,\psi(x))$ if $x\notin O$, giving a bijection from $H$ to $\mathbb{S}^{\times}\times G$. So $R\cong(\mathbb{S}^{\times}\times G)\cup\\{0\\}$. ∎ It is not clear whether this result extends to stringent skew hyperfields. ## 5\. Classification of Doubly Distributive Skew Hyperfields In this section, we will present the classification of doubly distributive skew hyperfields. ###### Proposition 5.1. The doubly distributive skew hyperfields are precisely those of the form $F\rtimes_{H,\psi}G$ of exactly one of the following types: 1. (1) $F$ is the Krasner hyperfield, 2. (2) $F$ is the sign hyperfield, 3. (3) $F$ is a skew field and $G$ satisfies $\\{ab\,|\,a,b<1_{G}\\}=\\{c\,|\,c<1_{G}\\}.$ The following example is a doubly distributive hyperfield of type (3) in Proposition 5.1. ###### Example 5.2. Let $F:=\mathbb{R}$ be the hyperfield with usual multiplication and hyperaddition given by $x\boxplus y=\begin{cases}x&\text{ if }|x|>|y|,\\\ y&\text{ if }|x|<|y|,\\\ \\{z\,|\,|z|<|x|\\}&\text{ if }|x|=|y|.\end{cases}$ This hyperfield is stringent and arises from the short exact sequence of groups $1\to GF(3)^{\times}\xrightarrow{\varphi}\mathbb{R}^{\times}\xrightarrow{\psi}\mathbb{R}_{>0}\to 1\,.$ Another natural example of doubly distributive skew hyperfields of type (3) in Proposition 5.1 can be found in [Pen18], which is built around a (noncommutative) such hyperfield (the one called $L^{\sigma}$). Before showing the proof of Proposition 5.1, We’ll first introduce a useful lemma. ###### Lemma 5.3. Let $R$ be a stringent skew hyperfield. $R$ is doubly distributive if and only if $(1\boxplus-1)(1\boxplus-1)=1\boxplus-1\boxplus 1\boxplus-1.$ ###### Proof. By Definition 2.8, $R$ is doubly distributive if and only if $(a\boxplus b)(c\boxplus d)=ac\boxplus ad\boxplus bc\boxplus bd$, for any $a,b,c,d\in R$. As $R$ is stringent, we have $u\boxplus v$ is a singleton if $u\neq-v$. So if either $a\neq-b$ or $c\neq-d$, then the equation above is just about distributivity. It already holds. If both $a=-b$ and $c=-d$, then $(a\boxplus b)(c\boxplus d)=(a\boxplus-a)(c\boxplus-c)=a(1\boxplus-1)(1\boxplus-1)c$ and $ac\boxplus ad\boxplus bc\boxplus bd=ac\boxplus-ac\boxplus-ac\boxplus ac=a(1\boxplus-1\boxplus 1\boxplus-1)c.$ So $R$ is doubly distributive if and only if $(1\boxplus-1)(1\boxplus-1)=1\boxplus-1\boxplus 1\boxplus-1.$ ∎ Now we will present the proof of Proposition 5.1. ###### Proof of Proposition 5.1. By Proposition 1.3 and Theorem 4.10, we know that a doubly distributive skew hyperfield $R$ also has the form $F\rtimes_{H,\psi}G$, where $F$ is either the Krasner hyperfield or the sign hyperfield or a skew field, with a short exact sequence of groups $1\to F^{\times}\xrightarrow{\varphi}H\xrightarrow{\psi}G\to 1\,.$ So it suffices to show that the hyperfields of type (1) and (2) are doubly distributive and the hyperfields with $F$ a skew field are doubly distributive if and only if they are of type (3). _Case 1:_ When $F=\mathbb{K}=\\{1,0\\}$, hyperaddition is defined by $\displaystyle x\boxplus 0$ $\displaystyle=\\{x\\},$ $\displaystyle x\boxplus y$ $\displaystyle=\begin{cases}\\{x\\}&\text{ if $\psi(x)>\psi(y)$,}\\\ \\{y\\}&\text{ if $\psi(x)<\psi(y)$,}\\\ \\{z\,|\,\psi(z)\leq\psi(x)\\}\cup\\{0\\}&\text{ if $\psi(x)=\psi(y)$, that is $x=y$.}\end{cases}$ By Lemma 5.3, $R$ is doubly distributive if and only if $(1\boxplus 1)(1\boxplus 1)=1\boxplus 1\boxplus 1\boxplus 1.$ $(1\boxplus 1)(1\boxplus 1)=(\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\})\cdot(\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\})=\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\},$ and $1\boxplus 1\boxplus 1\boxplus 1=(\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\})\boxplus(\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\})=\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\}.$ So $R$ is doubly distributive when $F=\mathbb{K}$. _Case 2:_ When $F=\mathbb{S}=\\{1,-1,0\\}$, hyperaddition is defined by $\displaystyle x\boxplus 0$ $\displaystyle=\\{x\\},$ $\displaystyle x\boxplus y$ $\displaystyle=\begin{cases}\\{x\\}&\text{ if $\psi(x)>\psi(y)$,}\\\ \\{y\\}&\text{ if $\psi(x)<\psi(y)$,}\\\ \\{x\\}&\text{ if $x=y$,}\\\ \\{z\,|\,\psi(z)\leq\psi(x)\\}\cup\\{0\\}&\text{ if $x=-y$.}\end{cases}$ By Lemma 5.3, $R$ is doubly distributive if and only if $(1\boxplus-1)(1\boxplus-1)=1\boxplus-1\boxplus 1\boxplus-1.$ $(1\boxplus-1)(1\boxplus-1)=(\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\})\cdot(\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\})=\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\},$ and $1\boxplus-1\boxplus 1\boxplus-1=(\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\})\boxplus(\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\})=\\{z\,|\,\psi(z)\leq 1\\}\cup\\{0\\}.$ So $R$ is doubly distributive when $F=\mathbb{S}$. _Case 3:_ When $F$ is a skew field, hyperaddition is defined by $\displaystyle x\boxplus 0$ $\displaystyle=\\{x\\},$ $\displaystyle x\boxplus y$ $\displaystyle=\begin{cases}\\{x\\}&\text{ if $\psi(x)>\psi(y)$,}\\\ \\{y\\}&\text{ if $\psi(x)<\psi(y)$,}\\\ x\boxplus_{\psi(x)}y&\text{ if $\psi(x)=\psi(y)$ and $0\notin x\boxplus_{\psi(x)}y$,}\\\ \\{z\,|\,\psi(z)<\psi(x)\\}\cup\\{0\\}&\text{ if $\psi(x)=\psi(y)$ and $0\in x\boxplus_{\psi(x)}y$.}\\\ \end{cases}$ By Lemma 5.3, $R$ is doubly distributive if and only if $(1\boxplus-1)(1\boxplus-1)=1\boxplus-1\boxplus 1\boxplus-1.$ $(1\boxplus-1)(1\boxplus-1)=(\\{z\,|\,\psi(z)<1\\}\cup\\{0\\})\cdot(\\{z\,|\,\psi(z)<1\\}\cup\\{0\\})=\\{xy\,|\,\psi(x),\psi(y)<1\\}\cup\\{0\\},$ and $1\boxplus-1\boxplus 1\boxplus-1=(\\{z\,|\,\psi(z)<1\\}\cup\\{0\\})\boxplus(\\{z\,|\,\psi(z)<1\\}\cup\\{0\\})=\\{z\,|\,\psi(z)<1\\}\cup\\{0\\}.$ So $R$ is doubly distributive if and only if $\\{xy\,|\,\psi(x),\psi(y)<1\\}\cup\\{0\\}=\\{z\,|\,\psi(z)<1\\}\cup\\{0\\}.$ We claim that $\\{xy\,|\,\psi(x),\psi(y)<1\\}\cup\\{0\\}=\psi^{-1}(\psi(1)\downarrow)\cup\\{0\\}=\\{z\,|\,\psi(z)<1\\}\cup\\{0\\},$ if and only if $\\{ab\,|\,a,b<1_{G}\\}=\\{c\,|\,c<1_{G}\\}.$ $(\Rightarrow):$ If $\\{xy\,|\,\psi(x),\psi(y)<1\\}\cup\\{0\\}=\\{z\,|\,\psi(z)<1\\}\cup\\{0\\}$, the direction $\subseteq$ is clear. We just need to consider the other direction. Let $c\in G$ be such that $c<1_{G}$ and let $z\in\psi^{-1}(c)$. Then there exist $x,y\in H$ such that $z=xy$ and $\psi(x),\psi(y)<1$ by our assumption. So $c=\psi(z)=\psi(xy)=\psi(x)\psi(y)$. We have $c\in\\{ab\,|\,a,b<1_{G}\\}$. $(\Leftarrow):$ If $\\{ab\,|\,a,b<1_{G}\\}=\\{c\,|\,c<1_{G}\\}$, the direction $\subseteq$ is also clear. We just need to consider the other direction. Let $z\in H$ be such that $\psi(z)<1$ and let $c=\psi(z)$. Then there exist $a,b\in G$ such that $c=ab$ and $a,b<1_{G}$ by our assumption. Let $x\in H$ be such that $\psi(x)=a<1_{G}$ and let $y=x^{-1}z$. We have $\psi(y)=\psi(x^{-1}z)=a^{-1}c=b<1_{G}$ and $z=xy$. So $z\in\\{xy\,|\,\psi(x),\psi(y)<1\\}$. So $R$ is doubly distributive if and only if $\\{ab\,|\,a,b<1_{G}\\}=\\{c\,|\,c<1_{G}\\}.$ ∎ ## 6\. Reduction of stringent skew hyperrings to hyperfields In this section, we will show that stringent skew hyperrings are very restricted. ###### Theorem 6.1. Every stringent skew hyperring is either a skew ring or a stringent skew hyperfield. ###### Proof. If $G$ is trivial, then $R=R_{1_{G}}$. So $R$ is either $\mathbb{K}$, or $\mathbb{S}$, or a skew ring. If $G$ is nontrivial, we would like to show that every element $x$ in $R^{\times}$ is a unit. Now let $s$ and $t$ in $R^{\times}$ be such that $x\cdot s>_{R}1$ and $t\cdot x>_{R}1$. Then by the remarks after Lemma 3.8, we have $1\in x\cdot s\boxplus-x\cdot s=x\cdot(s\boxplus-s),$ $1\in t\cdot x\boxplus-t\cdot x=(t\boxplus-t)\cdot x.$ So there exists $y\in s\boxplus-s$ and $z\in t\boxplus-t$ such that $1=x\cdot y=z\cdot x$. Thus $y=(z\cdot x)\cdot y=z\cdot(x\cdot y)=z$. So $x$ has a multiplicative inverse $y$ in $R$. Then $x$ is a unit of $R$. So every stringent skew hyperring is either a skew ring or a stringent skew hyperfield. ∎ We cannot classify doubly distributive hyperring using our classification because not every doubly distributive hyperring is stringent. The following is a counterexample. ###### Example 6.2. The hyperring $\mathbb{K}\times\mathbb{K}$ that is the square of the Krasner hyperfield is doubly distributive but not stringent. ## 7\. Every stringent skew hyperfield is a quotient of a skew field In this section, we would like to show that every stringent skew hyperfield is a quotient of a skew field by some normal subgroup. In particular, every stringent hyperfield is a quotient of a field by some special kind of subgroups, called ‘hüllenbildend’. This kind of subgroups was studied by Diller and Grenzdörffer in [DG73] when they tried to unify the treatment of various notions of convexity in projective spaces over a field $K$ by introducing for any subgroup $U\leq K^{\times}$ the notion of $U$-convexity. They showed that this notion is reasonably well behaved if and only if $U$ is as follows. ###### Definition 7.1. [DG73] Let $K$ be a field and let $U\leq K^{\times}$. $U$ is called $U$-‘hüllenbildend’ (hull producing) if $U$ satisfies $x,y\in K,x+y-xy\in U\rightarrow x\in U\text{ or }y\in U.$ (1) In [Dre77], Dress presented a simple complete classification of such ‘hüllenbildend’ subgroups $U$. We will combine our classification of stringent (skew) hyperfields into three types with Dress’s classification of such subgroups. ###### Theorem 7.2. [Dre77, Theorem 1] Let $U\leq K^{\times}$ satisfy (1) and let $S_{U}=\\{x\in K\,|\,x\notin U\text{ and }x+U\subseteq U\\}$. Then $S_{U}$ is the maximal ideal of a valuation ring $R=R_{U}(=\\{x\in K\,|\,x\cdot S_{U}\subseteq S_{U}\\})$ in $K$, $U$ is contained in $R$, $\overline{U}=\\{\overline{x}\in\overline{K}_{U}=R_{U}/S_{U}\,|\,x\in U\\}$ is either a domain of positivity in $\overline{K}_{U}$ (if $-1\notin U$, $2\in U$) or $\overline{U}=\\{\overline{1}\\}$ or $\overline{U}=\overline{K}_{U}^{\times}$ and, in any case, $U=\\{x\in R_{U}\,|\,\overline{x}\in\overline{U}\\}$. We will first explain how to choose the suitable subgroup $U$ in the case of stringent hyperfield and then give the proof in full generality for stringent skew hyperfields. From our classification in Theorem 4.10, we know that every stringent hyperfield $F$ has the form $M\rtimes_{H,\psi}G$, where $M$ is either $\mathbb{K}$, or $\mathbb{S}$, or a field. To show that $F$ is a quotient by some subgroup $U$, we will choose $U$ with $\overline{U}=\\{\overline{1}\\}$ if $M$ is $\mathbb{K}$, choose $U$ with $\overline{U}$ a domain of positivity if $M$ is $\mathbb{S}$, and choose $U$ with $\overline{U}=\overline{K}_{U}^{\times}$ if $M$ is a field. Now we begin the proof that every stringent skew hyperfield is a quotient of a skew field $K$ by some normal subgroup $U$. First, we recall the definition of quotient hyperfield. The quotient hyperfield $K/U=\\{[g]=gU\,|\,g\in K\\}$ was introduced by Krasner in [Kra83] with multiplication given by $[g]\cdot[h]=[gh]$, for $[g],[h]\in K/U$. Hyperaddition is given by $[g]\boxplus[0]=[g]$ and $[g]\boxplus[h]=\\{[f]\subseteq K/U\,|\,f\in gU+hU\\}$, for $[g],[h]\in(K/U)^{\times}$. As the subgroup $U$ we are choosing would be normal, so this quotient works in the skew case. We may suppose that a stringent skew hyperfield $F=M\rtimes_{H,\psi}G$ arises from a short exact sequence of groups $1\to M^{\times}\xrightarrow{\varphi}H\xrightarrow{\psi}G\to 1\,,$ where $G$ is a totally ordered group equipped with a total order $\leq$ and $M$ is either $\mathbb{K}$, or $\mathbb{S}$, or a skew field. We define an order $\leq^{\prime}$ on $G$ such that $x\leq^{\prime}y$ if and only if $y\leq x$. So $\leq^{\prime}$ is also a total order on $G$. Similarly as in the non- skew case, we will also choose $U$ with $\overline{U}=\\{\overline{1}\\}$ if $M$ is $\mathbb{K}$, choose $U$ with $\overline{U}$ a domain of positivity if $M$ is $\mathbb{S}$, and choose $U$ with $\overline{U}=\overline{K}_{U}^{\times}$ if $M$ is a skew field. Our difficulty now is to choose a suitable skew field $K$ for the quotient hyperfield corresponding to each $U$. We will introduce two different constructions of skew fields, as follows. ###### Example 7.3. [FS01] (Construction 1) Let $k$ be an arbitrary field. Define $K=k((G))$ to be the ring of formal power series whose powers come from $G$, that is, the elements of $K$ are functions from $G$ to $k$ such that the support of each function is a well-ordered subset of $(G,\leq^{\prime})$. Addition is pointwise, and multiplication is the Cauchy product or convolution, that is the natural operation when viewing the functions as power series $\sum_{{a\in G}}p(a)x^{a}$ It is well known (and easy to check) that $K$ is a skew field. We will construct a skew field $K=k((G))$ as in Example 7.3 by choosing $k$ to be an arbitrary field when $M$ is $\mathbb{K}$ and choosing $k$ to be the field $\mathbb{R}$ of real numbers (or any other ordered field) when $M$ is $\mathbb{S}$. The second construction is for a stringent skew hyperfield $F=M\rtimes_{H,\psi}G$ when $M$ is a skew field. ###### Example 7.4. (Construction 2) We define $K=M[[G]]$ to be the set of formal sums of elements of $H$ all from different layers such that the support is well-ordered, that is, an element of $K$ is a function $p$ from $G$ to $H$ such that for any $g$ in $G$, $p(g)\in\psi^{-1}(g)\cup\\{0\\}=A_{g}$ and the support of each function is a well-ordered subset of $(G,\leq^{\prime})$. As $M$ is a skew field and each $\lambda_{h}$ with $h\in H$ is an isomorphism of hypergroups, then $(A_{g},\boxplus_{g},0)$ is always an abelian group. We claim that $K$ is a skew field, viewing functions as power series $\sum_{{a\in G}}p(a)x^{a}\,,$ with addition $+_{K}$ given by $\sum_{{a\in G}}p(a)x^{a}+_{K}\sum_{{a\in G}}q(a)x^{a}=\sum_{{a\in G}}(p(a)\boxplus_{a}q(a))x^{a},$ and the additive identity is $\sum_{{a\in G}}0x^{a}$. Multiplication $\cdot_{K}$ is given by $\Big{(}\sum_{{a\in G}}p(a)x^{a}\Big{)}\cdot_{K}\big{(}\sum_{{a\in G}}q(a)x^{a}\Big{)}=\sum_{{s\in G}}\Big{(}\underset{g\cdot_{G}h=s}{\underset{h\in\operatorname{supp}(q),}{\underset{g\in\operatorname{supp}(p),}{\boxplus_{s}}}}p(g)\cdot_{H}q(h)\Big{)}x^{s},$ and the multiplicative identity is $1x^{1_{G}}$. Since the proof that this really gives a skew field is a long calculation and is very similar to that for $k((G))$, we do not give it here but in Appendix A. Now we divide the proof into three cases and show that $F\cong K/U$ in each case. For simplicity, we denote $\min(\operatorname{supp}(p))$ by $m_{p}$ for $p\in K^{\times}$. 1. _Case_ 1: If $M$ is $\mathbb{K}$, then let $U=\\{p\in K^{\times}\,|\,m_{p}=1_{G}\\}$. It’s easy to check that $U$ is normal. Then the quotient hyperfield $K/U=\\{[q]=qU\,|\,q\in K\\}$ has $\displaystyle[q]$ $\displaystyle=\\{p\in K^{\times}\,|\,m_{p}=m_{q}\\},$ $\displaystyle[0]$ $\displaystyle=\\{0_{K}\\}.$ So we can identify $[q]$ in $(K/U)^{\times}$ with $m_{q}$ in $G$ and identify $[0]$ in $K/U$ with $0$. So we have $K/U\cong(G\cup\\{0\\},\boxplus,\cdot)$ with multiplication given by $\displaystyle 0\cdot g$ $\displaystyle=0,$ $\displaystyle g\cdot h$ $\displaystyle=g\cdot_{G}h,$ where $g,h\in G$. And hyperaddition is given by $\displaystyle g\boxplus 0$ $\displaystyle=\\{g\\},$ $\displaystyle g\boxplus h$ $\displaystyle=\begin{cases}\\{g\\}&\text{ if $g<^{\prime}h$, that is $g>h$, }\\\ \\{h\\}&\text{ if $g>^{\prime}h$, that is $g<h$, }\\\ \\{f\in G\,|\,f\geq^{\prime}g\\}\cup\\{0\\}=\\{f\in G\,|\,f\leq g\\}\cup\\{0\\}&\text{ if $g=h$,}\end{cases}$ where $g,h\in G$. Now it is clear to see that $K/U\cong(G\cup\\{0\\},\boxplus,\cdot)\cong\mathbb{K}\rtimes_{H,\psi}G=F$. 2. _Case_ 2: If $M$ is $\mathbb{S}$, $k=\mathbb{R}$ (or any other ordered field) and $K=k((G))$, then let $U=\\{p\in K^{\times}\,|\,m_{p}=1_{G}\text{ and }p(1_{G})>0\\}$. It’s easy to check that $U$ is normal. Then the quotient hyperfield $K/U=\\{[q]=qU\,|\,q\in K\\}$ has $\displaystyle[q]$ $\displaystyle=\\{p\in K^{\times}\,|\,m_{p}=m_{q}\text{ and }p(m_{p})>0\\}\text{ if }q(m_{q})>0,$ $\displaystyle[q]$ $\displaystyle=\\{p\in K^{\times}\,|\,m_{p}=m_{q}\text{ and }p(m_{p})<0\\}\text{ if }q(m_{q})<0,$ $\displaystyle[0]$ $\displaystyle=\\{0_{K}\\}.$ We can identify $[q]$ in $(K/U)^{\times}$ with $(1,m_{q})$ if $q(m_{q})>0$, identify $[q]$ in $(K/U)^{\times}$ with $(-1,m_{q})$ if $q(m_{q})<0$, and identify $[0]$ with $0$. So we have $K/U\cong((\mathbb{S}^{\times}\times G)\cup\\{0\\},\boxplus,\cdot)$ with multiplication given by $\displaystyle(r,g)\cdot 0$ $\displaystyle=0,$ $\displaystyle(r_{1},g_{1})\cdot(r_{2},g_{2})$ $\displaystyle=(r_{1}\cdot_{\mathbb{S}}r_{2},g_{1}\cdot_{G}g_{2}),$ where $r,r_{1},r_{2}\in\mathbb{S}^{\times}$ and $g,g_{1},g_{2}\in G$. And hyperaddition is given by $\displaystyle(r,g)\boxplus 0$ $\displaystyle=\\{(r,g)\\},$ $\displaystyle(r_{1},g_{1})\boxplus(r_{2},g_{2})$ $\displaystyle=\begin{cases}\\{(r_{1},g_{1})\\}&\text{if $g_{1}<^{\prime}g_{2}$, that is $g_{1}>g_{2}$, }\\\ \\{(r_{2},g_{2})\\}&\text{if $g_{1}>^{\prime}g_{2}$, that is $g_{1}<g_{2}$, }\\\ \\{(r_{1},g_{1})\\}&\text{if $g_{1}=g_{2}$ and $r_{1}=r_{2}$ }\\\ \\{(r,f)\,|\,f\geq^{\prime}g\\}\cup\\{0\\}=\\{(r,f)\,|\,f\leq g\\}\cup\\{0\\}&\text{if $g_{1}=g_{2}$ and $r_{1}=-r_{2}$,}\end{cases}$ where $r,r_{1},r_{2}\in\mathbb{S}^{\times}$ and $g,g_{1},g_{2}\in G$. So by Theorem 4.17, $K/U\cong((\mathbb{S}^{\times}\times G)\cup\\{0\\},\boxplus,\cdot)\cong\mathbb{S}\rtimes_{H,\psi}G=F$. 3. _Case_ 3: If $M$ is a skew field and $K=M[[G]]$, then let $U=\\{p\in K^{\times}\,|\,m_{p}=1_{G}\text{ and }p(1_{G})=1\\}$. It’s easy to check that $U$ is normal. Then the quotient hyperfield $K/U=\\{[q]=qU\,|\,q\in K\\}$ has $\displaystyle[q]$ $\displaystyle=\\{p\in K^{\times}\,|\,m_{p}=m_{q}\text{ and }p(m_{p})=q(m_{q})\\},$ $\displaystyle[0]$ $\displaystyle=0_{K}.$ We can identify $[q]$ in $(K/U)^{\times}$ with $q(m_{q})$ in $H$ (clearly $\psi(q(m_{q}))=m_{q}$) and identify $[0]$ with $0_{F}$. So we have $K/U\cong F$ with multiplication given by $\displaystyle[q]\cdot 0$ $\displaystyle=0,$ $\displaystyle[q]\cdot[h]$ $\displaystyle=\\{p\in K^{\times}\,|\,m_{p}=m_{q}\cdot_{G}m_{h}\text{ and }p(m_{p})=q(m_{q})\cdot_{H}h(m_{h})\\}=[p]$ Hyperaddition is given by $[q]\boxplus 0=[q],$ $\displaystyle[q]\boxplus[h]$ $\displaystyle=$ $\displaystyle\begin{cases}\\{[q]\\}&\text{if $m_{q}<^{\prime}m_{h}$, that is $m_{q}>m_{h}$, }\\\ \\{[h]\\}&\text{if $m_{q}>^{\prime}m_{h}$, that is $m_{q}<m_{h}$, }\\\ \\{[p]=\\{p\in K^{\times}\,|\,m_{p}=m_{q}\text{ and }p(m_{p})=q(m_{q})\boxplus_{m_{q}}h(m_{h})\\}\\}&\text{if $m_{q}=m_{h}$ and $0\notin q(m_{q})\boxplus_{m_{q}}h(m_{h})$,}\\\ \\{[p]\in(K/U)^{\times}\,|\,m_{p}>^{\prime}m_{q}\\}\cup\\{0\\}=\\{[p]\in(K/U)^{\times}\,|\,m_{p}<m_{q}\\}\cup\\{0\\}&\text{if $m_{q}=m_{h}$ and $0\in q(m_{q})\boxplus_{m_{q}}h(m_{h}).$}\end{cases}$ where $[q],[h]\in(K/U)^{\times}$. So $K/U\cong M\rtimes_{H,\psi}G=F.$ ###### Theorem 7.5. Every stringent skew hyperfield is a quotient of a skew field. ###### Corollary 7.6. Every doubly distributive skew hyperfield is a quotient of a skew field. It follows from the construction that the same statements with all instances of the word ‘skew’ removed also hold. ## Appendix A The construction 2 in Example 7.4 gives a skew field ###### Lemma A.1. Let $F=M\rtimes_{H,\psi}G$ be a stringent skew hyperfield arising from a short exact sequence of groups $1\to M^{\times}\xrightarrow{\varphi}H\xrightarrow{\psi}G\to 1\,,$ where $G$ is a totally ordered group and $M$ is a skew field. Define $K=k[[G]]$ as we did in section 7. Then $K$ is a skew field. ###### Proof. The commutativity and associativity of $(K,+_{K},\sum_{{a\in G}}0x^{a})$ follow from those of $(H\cup\\{0\\},\boxplus,0)$. So we only need to show the associativity of $(K,\cdot_{K},1x^{1_{G}})$, the existence of a multiplicative inverse for every element and the distributivity. An important principle which we will need again and again as we go along is a kind of distributivity of the composition of $H$ over the various additions $\boxplus_{g}$. To express it cleanly, we begin by extending $\cdot_{H}$ to $H\cup\\{0\\}$ by setting $x\cdot 0=0\cdot x=0$ for all $x\in H\cup\\{0\\}$. Suppose that we have elements $x$ and $y_{1},y_{2}\ldots y_{n}$ of $H$ with $\psi(y_{i})=u\in G$ for all $i$, so that $\boxplus_{i=1}^{n}y_{i}$ is defined. Let $v\in G$ be such that $v=\psi(x)$. Then $z\mapsto x\cdot_{H}z$ is a bijection from $A_{u}$ to $A_{v\cdot u}$ whose composition with $\lambda_{y_{1}}$ is $\lambda_{x\cdot_{H}y_{1}}$, so it must also be an isomorphism of hypergroups. Thus $x\cdot_{H}\big{(}\underset{1\leq i\leq n}{\boxplus_{u}}y_{i}\big{)}=\underset{1\leq i\leq n}{\boxplus_{v\cdot u}}x\cdot_{H}y_{i}\,.$ A similar argument using the $\lambda^{\prime}_{h}$ defined in the proof of Lemma 4.2 shows $\big{(}\underset{1\leq i\leq n}{\boxplus_{u}}y_{i}\big{)}\cdot_{H}x=\underset{1\leq i\leq n}{\boxplus_{u\cdot v}}y_{i}\cdot_{H}x\,.$ To show the associativity of $(K,\cdot_{K},1x^{1_{G}})$, we let $p,q,w\in K$. Then for $s\in G$, $\displaystyle\big{(}(p\cdot_{K}q)\cdot_{K}w\big{)}(s)$ $\displaystyle=\underset{g\cdot_{G}c=s}{\underset{c\in\operatorname{supp}(w)}{\boxplus_{s}}}\Big{(}\big{(}\underset{a\cdot_{G}b=g}{\underset{b\in\operatorname{supp}(q)}{\underset{a\in\operatorname{supp}(p)}{\boxplus_{g}}}}p(a)\cdot_{H}q(b)\big{)}\cdot_{H}w(c)\Big{)}$ $\displaystyle=\underset{g\cdot_{G}c=s}{\underset{c\in\operatorname{supp}(w)}{\boxplus_{s}}}\Big{(}\underset{a\cdot_{G}b=g}{\underset{b\in\operatorname{supp}(q)}{\underset{a\in\operatorname{supp}(p)}{\boxplus_{s}}}}p(a)\cdot_{H}q(b)\cdot_{H}w(c)\Big{)}$ $\displaystyle=\underset{a\cdot_{G}b\cdot_{G}c=s}{\underset{c\in\operatorname{supp}(w)}{{\underset{b\in\operatorname{supp}(q)}{\underset{a\in\operatorname{supp}(p)}{\boxplus_{s}}}}}}p(a)\cdot_{H}q(b)\cdot_{H}w(c),$ $\displaystyle=\underset{a\cdot_{G}h=s}{\underset{a\in\operatorname{supp}(p)}{\boxplus_{s}}}\Big{(}\underset{b\cdot_{G}c=h}{\underset{c\in\operatorname{supp}(w)}{\underset{b\in\operatorname{supp}(q)}{\boxplus_{s}}}}p(a)\cdot_{H}q(b)\cdot_{H}w(c)\Big{)}$ $\displaystyle=\underset{a\cdot_{G}h=s}{\underset{a\in\operatorname{supp}(p)}{\boxplus_{s}}}\Big{(}p(a)\cdot_{H}\big{(}\underset{b\cdot_{G}c=h}{\underset{c\in\operatorname{supp}(w)}{\underset{b\in\operatorname{supp}(q)}{\boxplus_{h}}}}q(b)\cdot_{H}w(c)\big{)}\Big{)}$ $\displaystyle=\big{(}p\cdot_{K}(q\cdot_{K}w)\big{)}(s)\,.$ So $(p\cdot_{K}q)\cdot_{K}w=p\cdot_{K}(q\cdot_{K}w)$. Next we will show that each element of $K$ has a multiplicative inverse. We do this first for those $p\in K=k[[G]]$ such that $m_{p}=1_{G}$ and $p(m_{p})=1$. Let $S$ be the set of finite sums of elements of $\operatorname{supp}(p)$. $S$ is well founded. Define $q\in K=k[[G]]$ such that $q(1_{G}):=1$, $q(s):=0$ for $s\notin S$ and, for $s\in S$, define $q(s)$ recursively by $q(s):=-\Big{(}\underset{g\cdot_{G}h=s}{\underset{h\in S-\\{s\\}}{\underset{g\in\operatorname{supp}(p)-\\{1_{G}\\}}{\boxplus_{s}}}}p(g)\cdot_{H}q(h)\Big{)}.$ So $\displaystyle p\cdot_{K}q(1_{G})$ $\displaystyle=1,$ $\displaystyle p\cdot_{K}q(s)$ $\displaystyle=0$ if $s\notin S$, $\displaystyle p\cdot_{K}q(s)$ $\displaystyle=\underset{g\cdot_{G}h=s}{\underset{h\in\operatorname{supp}(q)}{\underset{g\in\operatorname{supp}(p)}{\boxplus_{s}}}}p(g)\cdot_{H}q(h)$ $\displaystyle=\Big{(}\underset{g\cdot_{G}h=s}{\underset{h\in\operatorname{supp}(q)-\\{s\\}}{\underset{g\in\operatorname{supp}(p)-\\{1_{G}\\}}{\boxplus_{s}}}}p(g)\cdot_{H}q(h)\Big{)}\boxplus_{s}p(1)\cdot_{H}q(s)$ $\displaystyle=\Big{(}\underset{g\cdot_{G}h=s}{\underset{h\in\operatorname{supp}(q)-\\{s\\}}{\underset{g\in\operatorname{supp}(p)-\\{1_{G}\\}}{\boxplus_{s}}}}p(g)\cdot_{H}q(h)\Big{)}\boxplus_{s}q(s)$ $\displaystyle=0$ if $s\in S-\\{1_{G}\\}$. So $p\cdot_{K}q$ is the identity. Therefore, $q$ is the multiplicative inverse of $p$. Next we consider elements of $K$ with only a single summand, that is, those of the form $ax^{g}$. It is clear that each such element also has a multiplicative inverse, namely $a^{-1}x^{g^{-1}}$. Now every element of $K$ can be expressed as a product $p_{1}\cdot p_{2}$, with $m_{p_{1}}=1_{G}$ and $p_{1}(m_{p_{1}})=1$ and such that $p_{2}$ has only a single summand. As seen above, each of $p_{1}$ and $p_{2}$ has a multiplicative inverse, and hence $p_{1}\cdot p_{2}$ also has one, namely $p_{2}^{-1}\cdot p_{1}^{-1}$. For distributivity, we first would like to show that $p\cdot_{K}(q+_{K}w)=p\cdot_{K}q+_{K}p\cdot_{K}w$. For $s\in G$, $\displaystyle(p\cdot_{K}(q+_{K}w))(s)$ $\displaystyle=\underset{g\cdot_{G}h=s}{\boxplus_{s}}p(g)\cdot_{H}(q(h)\boxplus_{h}w(h))$ $\displaystyle=\underset{g\cdot_{G}h=s}{\boxplus_{s}}\big{(}p(g)\cdot_{H}q(h)\boxplus_{s}p(g)\cdot_{H}w(h)\big{)}$ $\displaystyle=\big{(}\underset{g\cdot_{G}h=s}{\boxplus_{s}}p(g)\cdot_{H}q(h)\big{)}\boxplus_{s}\big{(}\underset{g\cdot_{G}h=s}{\boxplus_{s}}p(g)\cdot_{H}w(h)\big{)},$ $\displaystyle=(p\cdot_{K}q+_{K}p\cdot_{K}w)(s)\,.$ So $p\cdot_{K}(q+_{K}w)=p\cdot_{K}q+_{K}p\cdot_{K}w$. A similar calculation shows that $(p+_{K}q)\cdot_{K}w=p\cdot_{K}w+_{K}q\cdot_{K}w$. So $K=k[[G]]$ is a skew field. ∎ ## Appendix B The semirings associated to doubly distributive hyperfields In [GJL17], Lemma 6.2(2) provides a way to build a semiring out of a doubly distributive hyperfield.222In [Row16, Theorem 2.5], Rowen extended the theory of constructing the semiring to every hyperfield. In this section, we would like to talk about these semirings. For any doubly distributive hyperfield $H$ we can define binary operations $\oplus$ and $\odot$ on $\mathcal{P}H$ by setting $A\oplus B:=\bigcup_{a\in A,b\in B}a\boxplus b$ (this is just the extension of $\boxplus$ to subsets of $H$ from Definition 2.2) and $A\odot B:=\\{ab\colon a\in A,b\in B\\}$. Let $\langle H\rangle$ be the substructure of $(\mathcal{P}H,\oplus,\odot)$ generated from the singletons of elements of $H$. So $\langle H\rangle$ is a semiring. We will refer $\langle H\rangle$ as the associated semiring to $H$. Using our classification, we can easily determine all such associated semirings. Surprisingly, some of the basic examples have already been intensively studied and play an important role in the foundations of tropical geometry. In each case, we find that $\langle H\rangle$ contains only few elements in addition to the singletons of elements of $H$. We have seen that any doubly distributive hyperfield has the form $F\rtimes_{H,\psi}G$, where $F$ is the Krasner hyperfield, the sign hyperfield or a field. We divide into cases according the value of $F$. ### B.1. Supertropical semirings If $F$ is the Krasner hyperfield then $\psi\colon H^{\times}\to G$ is an isomorphism, and we can take it to be the identity. Then the elements of $\langle H\rangle$ are the singletons of elements of $H$ and the sets $g^{\nu}:=\\{h\in G\colon h\leq g\\}\cup\\{0\\}$. To simplify the definition of the addition we define an operation $\nu$ on $\langle H\rangle\setminus\\{\\{0\\}\\}$ by $\nu(\\{g\\})=\nu(g^{\nu})=g^{\nu}$ and we transfer the total order of $G$ to the $g^{\nu}$ in the obvious way. Then addition is given by $x\oplus\\{0\\}=x$ for any $x$ and otherwise by $x\oplus y=\begin{cases}x&\text{if $\nu(x)>\nu(y)$,}\\\ y&\text{if $\nu(x)<\nu(y)$,}\\\ \nu(x)&\text{if $\nu(x)=\nu(y)$.}\\\ \end{cases}$ Multiplication is given by $x\odot\\{0\\}=\\{0\\}$, by $\\{g\\}\odot\\{h\\}=\\{g\cdot h\\}$, by $\\{g\\}\odot h^{\nu}=(g\cdot h)^{\nu}$ and by $g^{\nu}\odot h^{\nu}=(g\cdot h)^{\nu}$. In the case that $G$ is the ordered group of real numbers, this is simply the supertropical semiring introduced by Izhakian in [Izh09]. This associated semiring has also been studied by Rowen in [Row16]. It would be reasonable to call such semirings in general supertropical semirings. ### B.2. Symmetrised $(\max,+)$-semirings If $F$ is the sign hyperfield then by Theorem 4.17 without loss of generality it arises from a short exact sequence $1\to\mathbb{S}^{\times}\to\mathbb{S}^{\times}\times G\to G\to 1\,.$ The elements of $\langle H\rangle$ then have the form $0:=\\{0_{H}\\}$, $\oplus g:=\\{(1,g)\\}$, $\ominus g:=\\{(-1,g)\\}$, or $g^{\circ}:=\\{(i,h)\colon i\in\mathbb{S}^{\times},h\leq g\\}\cup\\{0_{H}\\}$. There is an obvious projection map $\pi$ from $\langle H\rangle\setminus\\{0\\}$ to $G$. Then addition is given by $x\oplus 0=x$ for any $x$, by $x\oplus y=x$ if $\pi(x)>\pi(y)$, by $x\oplus g^{\circ}=g^{\circ}$ if $\pi(x)=g$, by $(\oplus g)\oplus(\oplus g)=\oplus g$, by $(\ominus g)\oplus(\ominus g)=\ominus g$ and by $(\oplus g)\oplus(\ominus g)=g^{\circ}$. Multiplication is given by $x\odot 0=0$ for any $x$, by $x\odot g^{\circ}=(\pi(x)\cdot g)^{\circ}$, by $(\oplus g)\odot(\oplus h)=\oplus(g\cdot h)$, by $(\ominus g)\odot(\ominus h)=\oplus(g\cdot h)$ and by $(\oplus g)\odot(\ominus h)=\ominus(g\cdot h)$. In the case that $G$ is the ordered group of real numbers, this is simply the symmetrised $(\max,+)$-semiring introduced by Akian et al in [ACG+91]. So it would be reasonable to call such semirings in general symmetrised $(max,+)$-semirings. ### B.3. Linearised $(\max,+)$-semirings If $F$ is a field, then the elements of $\langle H\rangle$ are the singletons of elements of $H$ (which are in canonical bijection with $H$) and the sets $\psi^{-1}(g\downarrow)\cup\\{0\\}$ (which are in canonical bijection with $G$). So $\langle H\rangle$ is isomorphic to the semiring on $H\cup G$ with $x\oplus y$ for $x,y\in H$ given by the unique element of $x\boxplus y$ if this set is a singleton and by $\psi(x)$ otherwise, with $x\oplus g$ for $x\in H$ and $g\in G$ given by $x$ if $\psi(x)\geq g$ and by $g$ otherwise, and with $g\oplus h$ for $g,h\in G$ given by $\max(g,h)$. For multiplication, $x\odot y=x\cdot y$ for $x,y\in H$ and $x\odot g=\psi(x)\cdot g$ for $x\in H$ and $y\in G$ and finally $g\odot h=g\cdot h$ for $g,h\in G$. By analogy to the previous construction, we could refer to such semirings as linearised $(\max,+)$-semirings. So far as we know, such semirings have not yet been seriously investigated. ## References * [ACG+91] Marianne Akian, G. Cohen, S. Gaubert, Ramine Nikoukhah, and J.P. Quadrat. Linear systems in (max, +) algebra. In Proceedings of the 29th IEEE Conference on Decision and Control, pages 151 – 156 vol.1, 01 1991. * [AD19] Laura Anderson and James F. Davis. Hyperfield Grassmannians. Adv. Math., 341:336–366, 2019. * [BB16] Matthew Baker and Nathan Bowler. Matroids over hyperfields. arXiv:1601.01204, 2016. * [BB19] Matthew Baker and Nathan Bowler. Matroids over partial hyperstructures. Adv. Math., 343:821–863, 2019. * [BP19] Nathan Bowler and Rudi Pendavingh. Perfect matroids over hyperfields. arXiv:1908.03420, 2019. * [DG73] Justus Diller and Jochen Grenzdörffer. $G$-Hüllen metrischer Teilräume. Math. Ann., 200:151–164, 1973. * [Dre77] A. Dress. On orderings and valuations of fields. Geometriae Dedicata, 6(3):259–266, 1977. * [DW92] Andreas W. M. Dress and Walter Wenzel. Valuated matroids. Adv. Math., 93(2):214–250, 1992. * [FS01] László Fuchs and Luigi Salce. Modules over non-Noetherian domains, volume 84 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001. * [GJL17] Jeffrey Giansiracusa, Jaiung Jun, and Oliver Lorscheid. On the relation between hyperrings and fuzzy rings. Beitr. Algebra Geom., 58(4):735–764, 2017. * [Izh09] Zur Izhakian. Tropical arithmetic and matrix algebra. Communications in Algebra, 37(4):1445–1468, 2009. * [Kra57] Marc Krasner. Approximation des corps valués complets de caractéristique $p\not=0$ par ceux de caractéristique $0$. In Colloque d’algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques, pages 129–206. Établissements Ceuterick, Louvain; Librairie Gauthier-Villars, Paris, 1957. * [Kra83] Marc Krasner. A class of hyperrings and hyperfields. Internat. J. Math. Math. Sci., 6(2):307–311, 1983. * [Mar06] M. Marshall. Real reduced multirings and multifields. J. Pure Appl. Algebra, 205(2):452–468, 2006. * [Pen18] Rudi Pendavingh. Field extensions, derivations, and matroids over skew hyperfields. arXiv:1802.02447, 2018. * [Row16] Louis Halle Rowen. Algebras with a negation map. arXiv:1602.00353, 2016. * [Vir10] Oleg Viro. Hyperfields for tropical geometry i. hyperfields and dequantization. arXiv:1006.3034v2, 2010. * [Vir11] O. Ya. Viro. On basic concepts of tropical geometry. Tr. Mat. Inst. Steklova, 273(Sovremennye Problemy Matematiki):271–303, 2011.

2024-09-04T02:54:58.201400

2003.03837

arxiv-papers

# Hardware Architecture Proposal for TEDA algorithm to Data Streaming Anomaly Detection Lucileide M. D. da Silva<EMAIL_ADDRESS>Maria G. F. Coutinho <EMAIL_ADDRESS>Carlos E. B. Santos<EMAIL_ADDRESS>Mailson R. Santos<EMAIL_ADDRESS>Luiz Affonso Guedes<EMAIL_ADDRESS>M. Dolores Ruiz<EMAIL_ADDRESS>Marcelo A. C. Fernandes111Present address: John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA<EMAIL_ADDRESS>Laboratory of Machine Learning and Intelligent Instrumentation, Federal University of Rio Grande do Norte, Natal 59078-970, Brazil. Federal Institute of Education, Science and Technology of Rio Grande do Norte, Paraiso, Santa Cruz, RN, 59200-000, Brazil. Department of Statistics and Operations Research, University of Granada, Spain. Department of Computer Engineering and Automation, Federal University of Rio Grande do Norte, Natal, RN, 59078-970, Brazil. ###### Abstract The amount of data in real-time, such as time series and streaming data, available today continues to grow. Being able to analyze this data the moment it arrives can bring an immense added value. However, it also requires a lot of computational effort and new acceleration techniques. As a possible solution to this problem, this paper proposes a hardware architecture for Typicality and Eccentricity Data Analytic (TEDA) algorithm implemented on Field Programmable Gate Arrays (FPGA) for use in data streaming anomaly detection. TEDA is based on a new approach to outlier detection in the data stream context. In order to validate the proposals, results of the occupation and throughput of the proposed hardware are presented. Besides, the bit accurate simulation results are also presented. The project aims to Xilinx Virtex-6 xc6vlx240t-1ff1156 as the target FPGA. ###### keywords: FPGA , TEDA , data streaming , reconfigurable computing ††journal: arXiv.org ## 1 Introduction Outlier detection or anomaly detection consists in detect rare events in a data set. It is a central problem in many application areas such as time series forecasting, data mining and industrial process monitoring. Due the increasing number of sensors in the most diverse areas and applications, there is a huge raise in the availability of data from time series. Thus, outlier detection for temporal data has become a central problem [1], especially when data are captured and processed continuously in online way. In this case, the data are considered as data streams [2]. Some important aspects need to be considered when choosing an anomaly detection method, such as the computational effort to handle large streaming data. Since the received information need to be stored and analyzed without compromising memory and run-time. Many of the solutions presented in the literature require prior knowledge of the process and system, such as mathematical models, data distribution, and predefined parameters [3]. Anomaly detection is traditionally done from statistical analysis, using probability and making a series of initial assumptions that in most cases are not in practice applied. A disadvantage of the traditional statistical method is comparing a single point with the average of all points rather than comparing with sample or data pairs. This way, the information is no longer punctual and local. Moreover, probability theory was developed from examples where processes and variables are purely random. However, real processes are not purely random and shows dependency between samples. Thus, real problems are addressed from offline processes, where the entire data set needs to be known. Being a potential problem of the traditional method. Another problem with traditional approaches is that they often use an offline dataset. Thus, all samples must be previously available from the beginning of the algorithm execution [4], making it impossible to use in real-time and data stream applications. This type of data presents new technical challenges and opportunities in new fields of work. Detecting real-time anomalies can provide valuable information in critical scenarios, but it is a high computational demand problem that still lacks reliable solutions capable of providing high processing capabilities. Typicality and Eccentricity Data Analytic (TEDA) is based on new approach to outlier detection in data stream context [5] and it can applied with an algorithm to detect autonomous behavior in industrial process operation, for example. TEDA analyzes the density of each sample of data read, calculated according to the distance from the sample to the other samples previously read. It is an online algorithm that learns autonomously without the need for prior knowledge about the process or parameters. Therefore, the computational effort required is smaller, allowing the use in real time applications [3]. TEDA can be used as an alternative statistical framework for analyzing most data, except for purely random processes. It is based on new metrics, all based on similarity/proximity of data in the data space, not in density or entropy, as in traditional methods. The metrics used with TEDA are typicality, defined in [5] as the extent to which objects are ?good examples? of a concept, and eccentricity, defined as how distinct the object is from the rest of the group. A high eccentricity data has a low typicality and is usually an outlier [3]. Eccentricity can be very useful for anomaly detection, image processing, fault detection, particle physics, etc. Allows analysis for data samples (which can also be done in real time for data stream) [6]. It is also relevant in clustering processes, since elements of a cluster are naturally opposed to the atypical [5]. Another area where anomaly detection has been increasingly used is in industry 4.0 projects. One of the challenges of the Industry 4.0 is the detection of production failures and defects [7]. New technologies aim to add value and increase process productivity, but face difficulties in performing complex and massive-scale computing due to the large amount of data generated [8]. The huge accumulation of real time data to flow in a network, for example, can quickly overload traditional computing systems due to the large amount of data that originates from the sensors and the requirement for intensive processing and high performance. The development of specialized hardware presents itself as a possible solution to overcome the bottlenecks, making it possible to create solutions for mass data processing and, at the same time, consider ultra-low-latency, low-power, high-throughput, security and ultra-reliable conditions, important requirements for increasing productivity and quality in industry 4.0 processes. Thinking about the challenges presented, this work proposes a specialized hardware architecture of TEDA for anomaly detection. The development of the hardware technique allows systems to be made even faster than their software counterparts, extending the possibilities of use for situations where time constraints are even more severe. In addition allowing its use in applications with large data processing. The works [9, 10, 11, 12, 13] were developed in hardware, specifically on FPGA, for the acceleration of complex algorithms. The development of machine learning algorithms in hardware has grown significantly. This is justified from performance data with respect to system sampling times compared to software equivalents. One of the motivations for this work is the possibility of accelerating the TEDA algorithm and handling large data streams, such as streaming and real-time. In this work, all validation and synthesis results was made using a FPGA Virtex 6 xc6vlx240t1ff1156. The FPGA choice was because it has high performance. Modern FPGAs can deliver performance and density comparable to Application Specific Integrated Circuits (ASICs), without the disadvantages of high development time and enabling reprogramming, as FPGAs have a flexible architecture. The rest of this paper is organized as follows: This first section presented a introduction about the work explaining the motivation behind it and major contributions. Section 2 discusses some related works and the state of the art. In Section 3 will be presented a theoretical foundation regarding the TEDA technique. Section 4 presents the implementation description details for the architecture proposed. Section 5 will present the validation and synthesis results of the proposed hardware, as well as comparisons with software implementations. Finally, Section 6 will present the conclusions regarding the obtained results. ## 2 Related work Real-time anomaly detection in data stream has potential applications in many areas. Such as: preventive maintenance, fault detection, fraud detection, signals monitoring, among others. Concepts that can be used in many different ranges of industry, such as information technology, finance, medicine, security, energy, e-commerce, agriculture, social media, among others. In the literature there are some uses of the TEDA technique for anomaly detection and even for classification. The article presented in [6] shows a proposal for a new TEDA-based anomaly detection algorithm. The proposed method, called by the author $\sigma$ gap, combines the accumulated proximity information for all samples with the comparison of specific point pairs suspected of being anomalies. Using local spatial distribution information about the vicinity of the suspect point. In the journal, TEDA is compared to an approach using traditional statistical methods, emphasizing that the set of initial assumptions is different. TEDA has been shown to be a generalization of traditional statistics compared to a known analysis, n $\sigma$, which is a widely used principle for threshold anomaly detection. The same result was obtained for both approaches, although TEDA does not need the initial assumptions. In addition, for various types of proximity measurements (such as Euclidean, Cosine, Mahalanobis), it has been shown that due to the recursion feature, TEDA is computationally more efficient and suitable for online and real-time applications. In the work [14] a study is presented about the use of TEDA for fault detection in industrial processes. The work is pioneering the use of this approach for real industry data. For the experiments, TEDA was applied online to the dataset provided by the DAMADICS (Development and Application of Methods for Actuator Diagnosis in Industrial Control Systems) database, one of the most widely used benchmarks in fault detection and diagnosis applications. The experiments showed good results both in accuracy and execution time, which shows the suitability of this approach for real-time industrial applications. Finally, it was found that the TEDA algorithm is capable of dealing with the limitations and particularities of the industrial environment. The paper of [15] is intended to enable the use of TEDAClass, which consists of the TEDA algorithm for classification, in big data processing. The main feature of the proposed algorithm, called TEDAClassBDp, is the processing of block data, where each block uses the TEDAClass so that all blocks operate in parallel. As with TEDAClass, the proposed algorithm does not require information from previous data, and its operation occurs recursively, online and in real-time. The results indicated a reduction in time and computational complexity, without significantly compromising its accuracy, which indicates the strong possibility of using the proposal in problems where it is necessary to process large volumes of data quickly. The work presented in [16] proposes a new non-frequency and density-based data analysis tool. Classified by the author as a further development of TEDA and an effective alternative to the probability distribution function (pdf). Typicality Distribution Function (TDF) can provide valuable information for extreme process analysis, fault detection and identification, where the number of extreme event or fault observations is often disproportionately small. The proposed offers a closed non-parametric analytical (quadratic) description, extracted from the actual realizations of the data exactly in contrast to the usual practice in which these distributions are being assumed or approximated. In addition, for various types of proximity and similarity measures (such as Euclidean, Mahalonobis, and cosine distances), it can be recursively calculated, thus computationally efficient and suitable for online and real- time algorithms. As a fundamental theoretical innovation, TDF and TEDA application areas can range from anomaly detection, grouping, classification, prediction, control, filter regression (similar to Kalman). Practical applications may be even broader, so it is difficult to list them all. The paper presented in [3] proposes the application of TEDA for fault detection in industrial processes. The effectiveness of the proposal has been demonstrated with two real industrial plants, using data streaming, and compared with traditional failure detection methods. This paper presents a practical application of the TEDA algorithm for two fault detection problems of real industrial plants. The first application uses a well-known database, DAMADICS, a database that provides actual data on the water evaporation process of an operating plant of a Polish sugar manufacturing plant. The second application was made from data analysis of a pilot plant of the authors’ university laboratory. A plant equipped with real industrial instruments used for process control. The work of [4] presents a new proposal for unsupervised fuzzy classifier, capable of aggregating the main characteristics of evolving classifiers, as well as making fuzzy classifications of real time data streams completely online. The proposed algorithm uses TEDA concepts, replacing traditional clusters with data clouds, granular structures without shape or predefined boundaries. For data classification, the proposed approach uses the concepts of soft-labeling rather than mutually exclusive classes. Experiments performed using data obtained from different operational failures of a real industrial plant, showed very significant results regarding unsupervised as well as semi- supervised learning, requiring minimal human interaction. The manuscript presented in [2] brings a new algorithm for detecting anomalies based on an online memory sequence algorithm called Hierarchical Temporal Memory (HTM). The performance of the proposed algorithm was evaluated and compared with a set of real time anomaly detection algorithms. Comparative analysis was performed as a way to evaluate anomaly detection algorithms for data streaming. All analyzes were performed from the Numenta Anomaly Benchmark (NAB) [17], which is a benchmark of actual streaming data. The paper published by [18] brings a study for anomaly detection in TCP / IP networks. The purpose of the paper is to detect computer network anomalies in the process of virtual machine (VM) live migration from local to cloud, by comparing this approach between TEDA, clustering K-Means, and static analysis. They used the tuple - Source IP, Destination IP, Source Port, and Destination Port \- to create a signature process and validate errors, including those of traffic flow hidden in the legitimate network. Testing was done using the SECCRIT (SEcure Cloud Computing for CRitical Infrastructure IT - http://www.seccrit.eu) project dataset, which allows anomalies or environmental attacks to be analyzed with Live Migration and other background traffic conditions. The results demonstrate that the proposed method makes it possible to automatically and successfully detect anomalies in attacks, network port scan (NPS) and network scan (NS). A major difficulty is distinguishing a high-volume attack from a denial of service (DoS) attack, for example. Accuracy and false negative rate calculations were made for comparison with K-Means and the proposed solution, with TEDA having better rates in almost all measurements performed. As the amount of data that needs to be processed grows exponentially and autonomous systems become increasingly important and necessary. Implementation of machine learning and streaming algorithms have been studying in literature. The work presented in [19] describes how to use run-time reconfiguration on FPGAs to improve the efficiency of streaming data transmission in shared communication channel with real-time applications. The reconfigurable architecture proposed consists of two subsystems: the reconfiguration subsystem, which running the modules, and the scheduling subsystem, that controls which modules are loaded to the reconfiguration subsystem. Besides, many works in the literature have been studied fault and anomaly detection in hardware. In work [20], an implementation of target and anomaly detection algorithms for real-time hyper-spectral imaging was proposed on FPGA. The algorithms were implemented in streaming fashion, similar to this work. The results, obtained from a Kintex-7 FPGA using fixed point structure, were very satisfactory and demonstrated that the implementation can be used in different detection circumstances. The work [21] presented a study of the impact of Neural Network architectures compared to statistical methods in the implementation of an Electrocardiogram (ECG) anomaly detection algorithm on FPGA. The fixed point implementation contributes to reduce the amount of needed resources. However, the design was made with High Level Sinthesys (HLS), witch could not optimize the FPGA resources consumption. In relation to the TEDA algorithm, no studies in the literature aimed at exploring its hardware implementation on FPGA were identified to date this paper had been write, which this work proposes to accomplish in a pioneering manner. ## 3 TEDA TEDA was introduced by [22] as a statistical framework, influenced by recursive density estimation algorithms. However, unlike algorithms that uses data density as a measure of similarity, TEDA uses concepts of typicity and eccentricity to infer whether a given sample is normal or abnormal to the dataset. The methodology used in TEDA does not require the use of a previous data information, and can be applied to problems involving fault detection, clustering, classification, among others [22]. TEDA is a data structure-based anomaly detection algorithm that aims to generalize and avoid the need for well-known, but very restrictive, initial conditions inherent in traditional statistics and probability theory [23]. The approach presented in the TEDA has some advantages over traditional statistical anomaly detection methods. Its recursive feature allows it to handle large volumes of data, such as data streams, with low computational cost and online, enabling faster processing. TEDA main features include [6]: * 1. It is entirely based on data and its distribution in data spaces; * 2. No previous assumptions are made; * 3. Limits and parameters does not need to be pre-specified; * 4. No sample independence required; * 5. An infinite number of observations are not required. The typicality of TEDA is the similarity of a given data sample to the rest of the dataset samples to which it belongs. Eccentricity, on the other hand, is the opposite of typicality, which indicates how much a sample is dissociated from the other samples in its set. Thus, an outlier can be defined as a sample with high eccentricity and low typicality, considering a threshold established for comparison. It is important to note that for eccentricity and typicality calculations no parameter or threshold is required. To calculate the eccentricity of each sample, TEDA uses the sum of the geometric distances between the analyzed sample $\bm{x}_{k}$ and the other samples in the set. Thus, the higher this value, the greater the eccentricity of the sample, and consequently, the lower its typicality. [6] proposed recursively calculating eccentricity. Thus, the eccentricity, $\xi$ can be expressed as $\xi_{k}(x)=\frac{1}{k}+\frac{(\bm{\mu}_{k}^{x}-\bm{x}_{k})^{T}(\bm{\mu}^{x}_{k}-\bm{x}_{k})}{k[\sigma^{2}]^{x}_{k}},[\sigma^{2}]^{x}_{k}>0$ (1) where $k$ is discreization instant; $\bm{x}_{k}$ is a input set of N elements in the k-th iteration, $\bm{x}_{k}=[x_{k}^{1}\ x_{k}^{2}\ ...\ x_{k}^{N}]$; $\bm{\mu}_{x}^{k}$ is also a N elements vector, equal to the average of $\bm{x}_{k}$ at the $k$-th iteration and $[\sigma^{2}]^{x}_{k}$ is the variance of $\bm{x}_{k}$ at the $k$-th iteration. The calculation of $\bm{\mu}_{x}^{k}$ and $[\sigma^{2}]^{x}_{k}$ is also recursively done, using the following equation $\bm{\mu}^{x}_{k}=\frac{(k-1)}{k}\bm{\mu}^{x}_{k-1}+\frac{1}{k}\bm{x}_{k},\ k\geq 1,\ \bm{\mu}^{x}_{0}=0$ (2) and $[\sigma^{2}]^{x}_{k}=\frac{(k-1)}{k}[\sigma^{2}]^{x}_{k-1}+\frac{1}{k}\left\|\bm{x}_{k}-\bm{\mu}_{k}\right\|^{2},\ k\geq 1,\ [\sigma^{2}]^{x}_{0}=0.$ (3) The typicality of a given sample $\bm{x}_{k}$, at the $k$-th iteration, can be expressed as a complement to eccentricity [6], as follows $\tau_{k}(x)=1-\xi_{k}(x).$ (4) In addition, [6] also defined that normalized eccentricity can be calculated as $\zeta_{k}(x)=\frac{\xi_{k}(x)}{2},\sum^{k}_{i=1}\xi_{k}(x)=1,\ k\geq 2.$ (5) In order to separate normal state data from abnormal state data, it is necessary to define a comparison threshold. For anomaly detection, the use of the $m\sigma$ [24] threshold is widespread. However, this principle must first assume the distribution of the analyzed data, such as the Gaussian distribution [6]. Chebyshev inequality can be used for any data distribution, assuming that the probability that the data samples are more than $m\sigma$ from the average is less than or equal to $1/m^{2}$, where $\sigma$ is the standard deviation of the data [25]. The condition that produces the same results as Chebyshev’s inequality, discarding any assumptions about data and its independence, can be expressed as [6] $\zeta_{k}>\frac{m^{2}+1}{2k},\ m>0$ (6) where $m$ corresponds to the comparison threshold. For a better understanding of the hardware implemented technique in this work, the Algorithm 1 details the operation of TEDA, based on the equations presented above. Input: $\mathbf{x}_{k}$: $k$-th sample; $m$: threshold Output: outlier: sample classification as abnormal or normal 1 begin 2 while _receive $\mathbf{x}$_ do 3 if _k=1_ then 4 $\bm{\mu}^{x}_{k}\leftarrow\mathbf{x}_{k}$; 5 $[\sigma^{2}]^{x}_{k}\leftarrow 0$; 6 7 else 8 update $\bm{\mu}^{x}_{k}$ using equation 2; 9 update $[\sigma^{2}]_{k}^{x}$ using equation 3; 10 update $\xi_{k}(x)$ using equation 1; 11 update $\zeta_{k}(x)$ using equation 5; 12 if _$\zeta_{k}(x) >\frac{m^{2}+1}{2k}$_ then 13 $outlier\leftarrow true$; 14 15 else 16 $outlier\leftarrow false$; 17 18 19 $k\leftarrow k+1$; 20 21 22 Algorithm 1 TEDA As presented in the Algorithm 1, only input data samples, $\bm{x}_{k}$, and a comparison threshold, $m$, are used as input to the algorithm. The output for each entry, $\bm{x}_{k}$, is the indication of the sample’s classification as abnormal (outlier = true) or normal (outlier = false). ## 4 Implementation description In this work, a TEDA FPGA proposal was implemented using Register Transfer Level (RTL) such as works presented in [9, 10, 11, 12, 13]. In the following section characteristics of the proposal will be presented, as well as details regarding processing time. A design overview can be seen in Figure 1. ### 4.1 Architecture proposal overview As illustrated in the Figure 1, the proposed implementation of TEDA has four different block structures: The MEAN module, which implements the average described in Equation 2; The VARIANCE module, responsible for calculate the variance as presented at the equation 3; The ECCENTRICITY module, which calculates the eccentricity, as presented in the equation 1; and the OUTLIER module, a block used to normalize the eccentricity as in equation 5 and compare with the threshold, as showed in equation 6. The architecture was developed in an attempt to pipeline the operations presented in Algorithm 1 in order to decrease the TEDA processing time. So, the output of the ECCENTRICITY and OUTLIER modules are one clock cycle delayed in relation to VARIANCE module and two in relation to MEAN module. As well as VARIANCE module is one clock cycle delayed in relation to MEAN module. Each of the modules are detailed later in the following sections. The implementation has the Algorithm 1 as reference. The system receives the FPGA clock and the $k$-th sample vector $\mathbf{x}_{k}$ as inputs. The $k$-th iteration number is updated from the increment of a counter and the $m$ threshold is used as a constant, stored at OUTLIER module. As in the algorithm line 1, the MEAN module do each single element average of $\mathbf{x}_{k}$ vector. It is possible to observe that there are $N$ MEAN blocks, where $N$ is the vector size. This block will be detailed in section 4.2. After this step, moving to the next line (1), the calculation of variance is done in VARIANCE Module, this block is detailed in the section 4.3. ECCENTRICITY block has as inputs the signals that left the block VARIANCE and $k$, as referred at line 1 and detailed in subsection 4.4. OUTLIER block is detailed in subsection 4.5. It receives the ECCENTRICITY, $\xi_{k}(x)$, and calculate the normalized eccentricity to compare with the threshold as presented in lines 1, 1 and 1. Figure 1: General architecture overview. ### 4.2 Module I - MEAN Each n-th MEAN module computes the average of each one of n-th elements vector $\bm{x}_{k}$ acquired at run time. The implementations is based on Equation 2 and it is detailed in Figure 2. In addition to receiving the n-th element of vector $\bm{x}_{k}$ as an input, the MEAN block uses a counter to define the number of sample interaction, k. The implementation uses a comparator block identified at the Figure 2 as MCOMPn witch is used to verify if the system is in the first iteration as Line 1 of Algorithm 1. The MMUXn is a multiplexer that acts as a conditional evaluation, using as selecting value the output of MCOMPn comparator. The register MREGn is storing the n-th $\bm{\mu}^{x}_{k}$ element ($\mu^{n}_{k}$). The $\mu^{n}_{k}$ value stored in MREGn is multiplied with $\frac{k-1}{k}$ in MMULT1n and added in MSUMn with the output of MMULT2n that has as input $x^{n}_{k}$ and the inverse value of $k$. Each n-th element of vector $\bm{x}_{k}$, $x^{n}_{k}$, requires a MEAN block. Figure 2: MEAN module. ### 4.3 Module II - VARIANCE The VARIANCE module is illustrated in Figure 3. It computes the variance of $\bm{x}_{k}$ vector samples by receiving the $\bm{x}_{k}$ vector itself and its average, $\bm{\mu}^{x}_{k}$, calculated in the previous MEAN blocks. The VARIANCE module, as the MEAN module, uses a comparator identified at the Figure 3 as VCOMP1 also to verify if the system is in the first iteration (Line 1 of Algorithm 1). The VMUX1 is a multiplexer that also implements a conditional evaluation to release the value $0$ in the register output VREG1 in the first iteration. The register VREG1 stores the variance value, $[\sigma^{2}]^{x}_{k}$, from the second iteration. The other registers in the block, VREG2 register and the N VREG$n$ registers, are used to delay by one clock cycle the iteration number $k$ and the elements of $\bm{x}_{k}$ respectively. Figure 3: VARIANCE module. As demonstrated in Equation 3, the variance calculation is done recursively. It is necessary to calculate $\left\|\bm{x}_{k}-\bm{\mu}_{k}\right\|^{2}$ and to do that, N subtractors (VSUB$n$) and N multipliers (VMULT1_$n$) are used, as well a adder (VSUM1) with N inputs. Each element of vector $\bm{\mu}^{x}_{k}$ is subtracted from its respective element in vector $\bm{x}_{k}$ and the result of this operation is multiplied by itself (squared) and then added to the other results. The $\left\|\bm{x}_{k}-\bm{\mu}_{k}\right\|^{2}$ value is the multiplied (at VMULT2) by $1/k$. It is then added at VSUM2 adder with the variance calculated in the previous iteration, $[\sigma^{2}]^{x}_{k}$, multiplied (VMULT3) by $(k-1)/k$. From the second iteration on, this value passes through the VMUX1 multiplexer to the VREG1 register, delivering the calculation of the variance value at the VARIANCE block output. The values of $\left\|\bm{x}_{k}-\bm{\mu}_{k}\right\|^{2}$ and $1/k$ are also delivered at the output of the VARIANCE block to avoid redundant operations as they will be used in the next block, the ECCENTRICITY block. ### 4.4 Module III - ECCENTRICITY The ECCENTRICITY module is a simpler block than those previously presented. This is because it uses operations already performed in the VARIANCE block to calculate eccentricity. The geometric distance $\left\|\bm{x}_{k}-\bm{\mu}_{k}\right\|^{2}$ (equivalent to $(\bm{\mu}_{k}^{x}-\bm{x}_{k})^{T}(\bm{\mu}^{x}_{k}-\bm{x}_{k})$) is stored in register EREG3 and $1/k$ is stored in EREG4 register. As the ECCENTRICITY module is the architecture design of Equation 1 (Algorithm 1 line 1), the variance value $[\sigma^{2}]^{x}_{k}$ is multiplied by $k$ (EMULT1) and used to divise (EDIV1) the geometric distance $(\bm{\mu}_{k}^{x}-\bm{x}_{k})^{T}(\bm{\mu}^{x}_{k}-\bm{x}_{k})$. This operation output is added to $1/k$ in the ESUM1 adder, calculating the eccentricity of the samples ($\xi_{k}(x)$) and delivering to the ECCENTRICITY block output. Figure 4: ECCENTRICITY module. ### 4.5 Module IV - OUTLIER Finally, in the OUTLIER block, the samples are classified into abnormal (outlier = true) or normal (outlier = false). The design module can be seen in Figure 5. To classify the samples, the OUTLIER block normalizes eccentricity by dividing (ODIV1) by a constant, as shown in Equation 5, and compares (OCOMP1 this normalized eccentricity with a threshold as shown in the Lines 1, 1, 1 and 1 of the Algorithm 1. The register OREG1 and OREG2 are used to synchronize the iteration number $k$, since as the modules act in pipeline, the operations carried out in the OUTLIER block (as well as in ECCENTRICITY module) are delayed by two clock cycles in relation to the system input. Figure 5: OUTLIER module. ### 4.6 Processing time The proposed architecture has an initial delay, $d$, that can be expressed as $d=3\times t_{c}$ (7) where $t_{c}$ is the system critical path time. The execution time of the circuit implemented for TEDA algorithm is determined by the system critical path time, $t_{c}$. So, after the initial delay, the execution time of the proposed TEDA, $t_{TEDA}$, can be expressed as $t_{TEDA}=t_{c}$ (8) thus, in every $t_{TEDA}$ it is possible to obtain the output of a sample inserted, that is, the sample classification as abnormal or normal. The throughput of the implementation, $th_{TEDA}$, in samples per second (SPS) can be expressed as $th_{TEDA}=\frac{1}{t_{TEDA}}.$ (9) ## 5 Results In this section will be presented the hardware validation and synthesis results for the architecture proposed in this work. All cases were validated and synthesized on floating point. Validation results were used to verify the hardware functionality, while synthesis results allow the system to be analyzed for important parameters for the design of hardware architectures such as hardware occupancy and processing time, considering factors such as throughput and speedup. ### 5.1 Validation results To validate the hardware architecture of the TEDA algorithm, we used the DAMADICS (Development and Application of Methods of the Actuator Diagnosis in Industrial Control Systems) benchmark dataset [26]. The benchmark provides a real data set of the water evaporation process in a Polish sugar factory. It is a plant with three actuators; a control valve, which controls the flow of water in the pipes; a pneumatic motor, which controls variable valve openings and a positioner. This dataset has faults at different times of the day on specific days. There are four different fault types, as shown in Table 1. Table 1: Fault types [26]. Fault | Description ---|--- f16 | Positioner supply pressure drop f17 | Unexpected pressure change across the valve f18 | Fully or partly opened bypass valves f19 | Flow rate sensor fault Artificial failures were introduced on specific days to plant operation data. The dataset has a set of $19$ faults in these $3$ actuators. As a way to validate the architecture, actuator $1$ failures were simulated. Table 2 shows a detailed description of some introduced faults for actuator $1$. Table 2: List of artificial failures introduced to actuator 1 [26]. Item | Fault | Sample | Date | Description ---|---|---|---|--- 1 | f18 | 58800-59800 | Oct 30, 2001 | Partly opened bypass valve 2 | f16 | 57275-57550 | Nov 9, 2001 | Positioner supply pressure drop 3 | f18 | 58830-58930 | Nov 9, 2001 | Partly opened bypass valve 4 | f18 | 58520-58625 | Nov 9, 2001 | Partly opened bypass valve 5 | f18 | 54600-54700 | Nov 17, 2001 | Partly opened bypass valve 6 | f16 | 56670-56770 | Nov 17, 2001 | Positioner supply pressure drop 7 | f17 | 37780-38400 | Nov 20, 2001 | Unexpected pressure drop across the valve Figure 6 shows the results obtained for the item 1 signal of Table 2. Figure 6(a) illustrates the behavior of two simulated input variables in hardware architecture ($\bm{x}_{k}=[x_{k}^{1}\ x_{k}^{2}]$). It is possible to observe that a failure happens between the moments $k$=58900 and $k$=59800. In Figure 6(b) it is possible to observe that there is a sudden change in the behavior of the eccentricity (black curve), surpassing the value of the comparison threshold with $m=3$ (red curve). (a) Fault item 1 - input vector $\bm{x}_{k}$. (b) Fault item 1 - normalized eccentricity $\zeta_{k}(x)$ with $5/k$ $(m=3)$ threshold. Figure 6: Detection of outliers in the dataset: Behavior of fault item 1. In Figure 7 it is possible to observe the results obtained for the item 7 signal, from Table 2. As within Figure 6, Figures 7(a) illustrates the behavior of two elements of input $\bm{x}_{k}=[x^{1}_{k}\ x^{2}_{k}]$ in hardware architecture and in Figure 7(b) it is possible to observe that there is a change of eccentricity (black curve), surpassing the value of the comparison threshold (red curve) also to $m=3$. The failure happens between moments $k=37700$ and $k=38400$. (a) Fault item 7 - input vector $\bm{x}_{k}$. (b) Fault item 7 - normalized eccentricity $\zeta_{k}(x)$ with $5/k$ $(m=3)$ threshold. Figure 7: Detection of outliers in the dataset: Behavior of fault item 7. Validation results in hardware architecture were compared with results obtained in a python software implementation of the algorithm TEDA. The hardware architecture was designed with floating point number format. ### 5.2 Synthesis results After performing to validate the implemented circuit, the hardware synthesis was performed to obtain the FPGA resource occupation report, as well as the critical time information used to calculate the proposed implementation processing time. The floating point synthesis results were obtained for a Xilinx Virtex 6 xc6vlx240t-1ff1156 FPGA. #### 5.2.1 Hardware occupation Table 3 presents data related to the hardware occupation of the circuit implemented in the target FPGA. The first column shows the number of multipliers used, the second column displays the number of registers, and the third column shows the number of logical cells used as LUT ($n_{LUT}$) throughout the circuit. Table 3: Hardware occupation. Multipliers | | Registers --- $n_{LUT}$ $27$ ($3$%) | $414$ ($<1$%) | $11.567$ ($7$%) Analyzing the data presented in Table 3 it can be seen that even using a floating point resolution, which demands a greater amount of hardware resources than a fixed point implementation, only a small portion of the resources were occupied from the target FPGA, with a total of only about $3\%$ from multipliers, less than $1\%$ from registers, and about $7\%$ from logical cells used as LUT. With this, we found that the proposed circuit could also be applied in low cost FPGAs, where the amount of available hardware resources is even smaller. In addition, multiple TEDA modules could be applied in parallel for anomaly detection in the same dataset, in order to further reduce processing time. #### 5.2.2 Processing time Table 4 presents information about the processing time (from Line 1 to Line 1 in Algorithm 1) of the architecture implemented for the TEDA technique. The first column indicates the circuit critical time, $t_{c}$, the second column shows the initial delay, expressed by Equation 7, the third column the TEDA run-time, expressed by Equation 8, and the last column the implementation throughput in samples per second (SPS), expressed by Equation 9, which consists of the amount of samples processed and classified (as normal or outlier) by TEDA every second. Table 4: Processing time. Critical time | | Delay --- TEDA time | Throughput $138\,\text{ns}$ | $414\,\text{ns}$ | $138\,\text{ns}$ | $7.2$ MSPS The data presented in Table 4 are quite expressive. The circuit critical time, which also corresponds in the TEDA run-time, was only $t_{c}=138\,\text{ns}$. Thus, after the $414\,\text{ns}$ delay, it is possible to get output for a processed sample sorted every $138\,\text{ns}$, which guarantees a throughput of $7.2$ million sorted samples per second. These results indicate the feasibility of using the proposal presented in this work to manipulate large data flows in real time. ### 5.3 Platforms comparison To date, no previous literature has been found to explore TEDA hardware implementations. Thus, this paper presents, for the first time, a proposal to implement the TEDA technique on FPGA. To verify the advantages of the hardware application proposed here over implementations on other software platforms, some comparisons of the FPGA processing time with the processing time of other software implementations were made. Table 5 presents the results of the comparisons made. The first column indicates the hardware used, the second presents the processing time required to obtain the classification of each sample, and the third column, the speedup achieved by the proposal presented in this paper. Table 5: Software implementations comparison. Platform | Time | Speedup ---|---|--- This work proposal on FPGA | $138\,\text{ns}$ | $-$ Python (Colab without GPU) | $435\,\text{ms}$ | $3\text{,}000\text{,}000\times$ Python (Colab with Tesla K80 GPU) | $39.2\,\text{ms}$ | $280\text{,}000\times$ Python (Local execution with 940 MX GPU) | $23.1\,\text{ms}$ | $167\text{,}000\times$ The data presented in Table 5 reaffirm the importance of this work. The hardware implementation on FPGA proposed here has been able to achieve speedups of up to $3$ million times compared to a Pyhton TEDA implementation using the Colab tool (without GPU processing). For the same Pyhton implementation using the Tesla K80 GPU processing Colab tool, a speedup of $280$ thousand times was obtained. In addition, when compared to a Python implementation on Intel(R) Core(TM) i7-7500U<EMAIL_ADDRESS>with 16 GB of RAM and GeForce 940 MX GPU, the hardware implementation on FPGA still had a $167$ thousand times advantage. Results that prove the advantages of using the proposal presented in this work to accelerate the TEDA technique, through the implementation on FPGA. ## 6 Conclusion This work presented a proposal for hardware implementation of the TEDA data streaming anomaly detection technique. The hardware was implemented in RTL using floating point format. Synthesis results were obtained for a Xilinx Virtex 6 xc6vlx240t-1ff1156 FPGA. The proposed implementation used a small portion of the target FPGA resources, besides allowing the results to be obtained in a short processing time. 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URL http://diag.mchtr.pw.edu.pl/damadics/

2024-09-04T02:54:58.219827

2003.03867

arxiv-papers

 # Strategic Abilities of Asynchronous Agents: Semantic Side Effects and How to Tame Them Wojciech Jamroga1,2 Wojciech Penczek1 Teofil Sidoruk1,3 1Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland 2Interdisciplinary Centre on Security, Reliability and Trust, SnT, University of Luxembourg 3Faculty of Mathematics and Information Science, Warsaw University of Technology {jamroga, penczek<EMAIL_ADDRESS> ###### Abstract Recently, we have proposed a framework for verification of agents’ abilities in asynchronous multi-agent systems (MAS), together with an algorithm for automated reduction of models (?). The semantics was built on the modeling tradition of distributed systems. As we show here, this can sometimes lead to counterintuitive interpretation of formulas when reasoning about the outcome of strategies. First, the semantics disregards finite paths, and yields unnatural evaluation of strategies with deadlocks. Secondly, the semantic representations do not allow to capture the asymmetry between proactive agents and the recipients of their choices. We propose how to avoid the problems by a suitable extension of the representations and change of the execution semantics for asynchronous MAS. We also prove that the model reduction scheme still works in the modified framework. ## 1 Introduction Modal logics of strategic ability. _Alternating-time temporal logic_ $\mathbf{ATL_{\mathrm{}}^{*}}$ (?; ?; ?) is probably the most popular logic to describe interaction of agents in multi-agent systems. Formulas of $\mathbf{ATL_{\mathrm{}}^{*}}$ allow to express statements about what agents (or groups of agents) can achieve. For example, $\langle\\!\langle{taxi}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\neg\mathsf{{fatality}}$ says that the autonomous cab can drive in such a way that nobody is ever killed, and $\langle\\!\langle{taxi,passg}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{destination}}$ expresses that the cab and the passenger have a joint strategy to arrive at the destination, no matter what any other agents do. Such statements allow to express important functionality and safety requirements in a simple and intuitive way. Moreover, the provide input to algorithms and tools for verification of strategic abilities, that have been in constant development for over 20 years (?; ?; ?; ?; ?; ?; ?; ?; ?; ?; ?; ?; ?; ?; ?; ?). Still, there are two caveats. First, all the realistic scenarios of agent interaction, that one may want to specify and verify, involve imperfect information. That is, the agents in the system do not always know exactly the global state of the system, and thus they have to make their decisions based on their local view of the situation. Unfortunately, verification of agents with imperfect information is hard to very hard – more precisely, $\mathbf{{\Delta_{{2}}^{\mathbf{{P}}}}}$-complete to undecidable, depending on the syntactic and semantic variant of the logic (?; ?; ?). Also, the imperfect information semantics of $\mathbf{ATL_{\mathrm{}}^{*}}$ does not admit alternation-free fixpoint characterizations (?; ?; ?), which makes incremental synthesis of strategies impossible, or at least difficult to achieve (?; ?; ?; ?; ?). Secondly, the semantics of strategic logics is traditionally based on synchronous concurrent game models. In other words, one implicitly assumes the existence of a global clock that triggers subsequent global events in the system; at each tick of the clock, all the agents choose their actions, and the system proceeds accordingly with a global transition. However, many real- life systems are inherently asynchronous, and do not operate on a global clock that perfectly synchronizes the atomic steps of all the components. Moreover, many systems that are synchronous at the implementation level can be more conveniently modeled as asynchronous on a more abstract level. In many scenarios, both aspects combine. For example, when modeling an anti-poaching operation (?), one may take into account the truly asynchronous nature of events happening in different national parks, but also the best level of granularity for modeling the events happening within a single nature reserve. Asynchronous semantics and partial-order reduction. We have recently proposed how to adapt the semantics of $\mathbf{ATL_{\mathrm{}}^{*}}$ to asynchronous MAS (?). We also showed that the technique of _partial order reduction (POR)_ (?; ?; ?; ?; ?; ?; ?) can be adapted to verification of strategic abilities in asynchronous MAS. In fact, the (almost 30 years old) POR for linear time logic $\mathbf{LTL}$ can be taken off the shelf and applied to a significant part of $\mathbf{ATL^{*}_{\mathrm{ir}}}$, the variant of $\mathbf{ATL_{\mathrm{}}^{*}}$ based on strategies with imperfect information and imperfect recall. This is very important, as the practical verification of asynchronous systems is often impossible due to the state- and transition- space explosion resulting from interleaving of local transitions. POR allows for a significant, sometimes even exponential, reduction of the models. Semantic side effects. While the result is appealing, there is a sting in its tail: the $\mathbf{ATL_{\mathrm{}}^{*}}$ semantics in (?) leads to counterintuitive interpretation of strategic properties. First, it disregards finite paths, and evaluates some intuitively losing strategies as winning (and vice versa). Secondly, it provides a flawed interpretation of the _concurrency fairness_ assumption. Thirdly, the representations and their execution semantics do not allow to capture the asymmetry between the agents that control which synchronization branch will be taken, and those influenced by their choices. We tentatively indicated some of the problems in the extended abstract (?). In this paper, we demonstrate them carefully, and propose how they can be avoided. Contribution. Our contribution is threefold. First, we discuss in detail the semantic side effects of adding strategic reasoning on top of classical models of concurrent systems (?). We identify the reasons, and demonstrate the problematic phenomena on simple examples. Secondly, we show how to avoid these pitfalls by extending the class of representations and slightly changing the execution semantics of strategies. Specifically, we add “silent” $\epsilon$-transitions in the models and on outcome paths of strategies, and allow for nondeterministic choices in the agents’ repertoires. We also identify a family of fairness-style conditions, suitable for the interaction of proactive and reactive agents. No less importantly, we prove that partial order reduction is still correct in the modified framework. Motivation. The variant of $\mathbf{ATL_{\mathrm{}}^{*}}$ for asynchronous systems in (?) was proposed mainly as a framework for formal verification. This was backed by the results showing that it submits to partial order reduction. However, a verification framework is only useful if it allows to specify requirements in an intuitive way, so that the property we _think_ we are verifying is indeed _the one being verified_. In this paper, we show that this was not the case. We also propose how to overcome the problems without spoiling the efficient reduction scheme. The solutions are not merely technical. In fact, they lead to a better understanding of how strategic activity influences the overall behavior of the system, and how it should be integrated with the traditional models of asynchronous interaction. ## 2 Models of Multi-agent Systems We first recall the models of asynchronous interaction in MAS, proposed in (?) and inspired by (?; ?; ?). ### 2.1 Asynchronous Multi-agent Systems In logical approaches to MAS, one usually assumes synchronous actions of all the agents (?; ?). However, many agent systems are inherently asynchronous, or it is useful to model them without assuming precise timing relationships between the actions of different agents. As an example, consider a team of logistic robots running in a factory (?). Often no global clock is available to all the robots, and even if there is one, the precise relative timing for robots operating in different places is usually irrelevant. Such a system can be conveniently represented with a set of automata that execute asynchronously by interleaving local transitions, and synchronize their moves whenever a shared event occurs. The idea is to represent the behavior of each agent by a finite automaton where the nodes and transitions correspond, respectively, to the agent’s local states and the events in which it can take part. Then, the global behavior of the system is obtained by the interleaving of local transitions, assuming that, in order for a shared event to occur, all the corresponding agents must execute it in their automata. This motivates the following definition. ###### Definition 2.1 (Asynchronous MAS). An _asynchronous multi-agent system (AMAS)_ S consists of $n$ agents ${\mathbb{A}\mathrm{gt}}=\\{{1,\dots,n}\\}$,111 We do not consider the environment component, which may be added with no technical difficulty. each associated with a tuple $A_{i}=(L_{i},\iota_{i},\mathit{Evt}_{i},R_{i},T_{i}{,\mathcal{PV}_{i},V_{i}})$ including a set of _possible local states_ $L_{i}=\\{l_{i}^{1},l_{i}^{2},\dots,l_{i}^{n_{i}}\\}$, an _initial state_ $\iota_{i}\in L_{i}$, and a set of _events_ $\mathit{Evt}_{i}=\\{\alpha_{i}^{1},\alpha_{i}^{2},\ldots,\alpha_{i}^{m_{i}}\\}$. An agent’s _repertoire of choices_ 222 In interpreted systems, this function is usually referred to as a _protocol_. Here, we opt for a different name to avoid possible confusion, e.g., with security protocols. $R_{i}:L_{i}\to 2^{\mathit{Evt}_{i}}\setminus\\{\emptyset\\}$ selects the events available at each local state. $T_{i}:L_{i}\times\mathit{Evt}_{i}\rightharpoonup L_{i}$ is a (partial) _local transition function_ such that $T_{i}(l_{i},\alpha)$ is defined iff $\alpha\in R_{i}(l_{i})$. That is, $T_{i}(l,\alpha)$ indicates the result of executing event $\alpha$ in local state $l$ from the perspective of agent $i$. Let $\mathit{Evt}=\bigcup_{i\in{\mathbb{A}\mathrm{gt}}}\mathit{Evt}_{i}$ be the set of all events, and $Loc=\bigcup_{i\in{\mathbb{A}\mathrm{gt}}}L_{i}$ be the set of all local states in the system. For each event $\alpha\in\mathit{Evt}$, $Agent(\alpha)=\\{{i\in{\mathbb{A}\mathrm{gt}}\mid\alpha\in\mathit{Evt}_{i}}\\}$ is the set of agents which have $\alpha$ in their repertoires; events shared by multiple agents are jointly executed by all of them. We assume that each agent $i$ in the AMAS is endowed with a disjoint set of its _local propositions $\mathcal{PV}_{i}$_, and their valuation $V_{i}:L_{i}\rightarrow 2^{\mathcal{PV}_{i}}$. The overall set of propositions $\mathcal{PV}=\bigcup_{i\in{\mathbb{A}\mathrm{gt}}}\mathcal{PV}_{i}$ collects all the local propositions. As our working example, we use the following scenario. ###### Example 2.2 (Conference in times of epidemic). Consider the AMAS in Figure 1, consisting of the Steering Committee Chair ($sc$), the General Chair ($gc$), and the Organizing Committee Chair ($oc$). Faced with the Covid-19 epidemics, $sc$ can decide to give up the conference, or send a signal to $gc$ to proceed and open the meeting. Then, $gc$ and $oc$ jointly decide whether the conference will be run on site or online. In the former case, the epidemiologic risk is obviously much higher, indicated by the atomic proposition $\mathsf{{epid}}$. The set of events, the agents’ repertoires of choices, and the valuation of atomic propositions can be easily read from the graph. For easier reading, all the private events are shown in grey. Note that event $proceed$ is shared by agents $sc$ and $gc$, and can only be executed jointly. Similarly, $onsite$ and $online$ are shared by $gc$ and $oc$. All the other events are private, and do not require synchronization. gc | oc | sc ---|---|--- | $0$$1$$\mathsf{{open}}$$2$$3$$\boldsymbol{proceed}$$\boldsymbol{onsite}$${online}$$\boldsymbol{rest}$$\boldsymbol{rest}$ --- | $0$$1$$\mathsf{{epid}}\quad{}$$2$$3$$\mathsf{{closed}}\quad{}$$onsite$$\boldsymbol{online}$$\boldsymbol{handle}$$\boldsymbol{handle}$$\boldsymbol{idle}$ --- | $0$$1$$2$${proceed}$$giveup$$proceed$$giveup$ --- Figure 1: Simple asynchronous MAS: agents $gc$, $oc$, and $sc$. A joint strategy of agents $\\{{gc,oc}\\}$ is highlighted. ### 2.2 Interleaved Interpreted Systems To understand the interaction between asynchronous agents, we use the standard execution semantics from concurrency models, i.e., interleaving with synchronization on shared events. To this end, we compose the network of local automata (i.e., AMAS) to a single automaton based on the notions of _global states_ and _global transitions_ , see below. ###### Definition 2.3 (Model). Let $S$ be an AMAS with $n$ agents. Its _model_ $IIS(S)$ extends $S$ with: (i) the set of global states $St\subseteq L_{1}\times\ldots\times L_{n}$, including the _initial state_ $\iota=(\iota_{1},\dots,\iota_{n})$ and all the states reachable from $\iota$ by $T$ (see below); (ii) the _global transition function_ $T:St\times\mathit{Evt}\rightharpoonup St$, defined by $T(g_{1},\alpha)=g_{2}$ iff $T_{i}(g_{1}^{i},\alpha)=g^{i}_{2}$ for all $i\in Agent(\alpha)$ and $g_{1}^{i}=g^{i}_{2}$ for all $i\in{\mathbb{A}\mathrm{gt}}\setminus Agent(\alpha)$; (iii) the _global valuation_ of propositions $V:St\rightarrow 2^{\mathcal{PV}}$, defined as $V(l_{1},\dots,l_{n})=\bigcup_{i\in{\mathbb{A}\mathrm{gt}}}V_{i}(l_{i})$. Models, sometimes called _interleaved interpreted systems_ (IIS), are used to provide an execution semantics to AMAS, and consequently provide us with semantic structures to reason about AMAS. Intuitively, the global states in $IIS(S)$ can be seen as the possible configurations of local states of all the agents. Moreover, the transitions are labeled by events that are simultaneously selected (in the current configuration) by all the agents that have the event in their repertoire. Clearly, private events (i.e., events such that $Agent(\alpha)$ is a singleton) require no synchronization. ###### Example 2.4 (Conference). The model for the asynchronous MAS of Example 2.2 is shown in Figure 1. We say that event $\alpha\in\mathit{Evt}$ is _enabled_ at $g\in St$ if $T(g,\alpha)=g^{\prime}$ for some $g^{\prime}\in St$. The set of events enabled at $g$ is denoted by $enabled(g)$. The global transition function is assumed to be serial, i.e., at each $g\in St$ there exists at least one enabled event. Discussion. This modeling approach is standard in theory of concurrent systems, where it dates back to the early 1980s and the idea of APA Nets (asynchronous, parallel automata nets) (?). Note that APA Nets and their models were _not_ proposed with causal interpretation in mind. In particular, they were _not_ meant to capture the interaction of purposeful agents that freely choose their strategies, but rather a set of reactive components converging to a joint behavior. Despite superficial differences, the same applies to process-algebraic approaches to concurrency, such as CSP (?), CCS (?), ACP (?), and $\pi$-calculus (?). Definition 2.1 extends that with the repertoire functions from synchronous models of MAS (?; ?). Agent $i$’s repertoire lists the events available to $i$, and is supposed to define the space of $i$’s strategies. As we show further, this is not enough in case of asynchronous MAS. ## 3 Reasoning About Abilities: ATL* _Alternating-time temporal logic_ $\mathbf{ATL_{\mathrm{}}^{*}}$ (?; ?; ?) generalizes the branching-time temporal logic $\mathbf{CTL^{*}}$ (?) by replacing the path quantifiers $\mathsf{E},\mathsf{A}$ with _strategic modalities_ $\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\gamma$, expressing that agents $A$ can enforce the temporal property $\gamma$. While the semantics of $\mathbf{ATL_{\mathrm{}}^{*}}$ is typically defined for models of synchronous systems, a variant for asynchronous MAS was proposed recently (?). We summarize the main points in this section. ### 3.1 Syntax Let $\mathcal{PV}$ be a set of propositional variables and ${\mathbb{A}\mathrm{gt}}$ the set of all agents. The language of $\mathbf{ATL_{\mathrm{}}^{*}}$ is defined as below. $\varphi::=\mathsf{{p}}\mid\neg\varphi\mid\varphi\wedge\varphi\mid\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\gamma$, $\gamma::=\varphi\mid\neg\gamma\mid\gamma\land\gamma\mid\mathrm{X}\,\gamma\mid\gamma\,\mathrm{U}\,\gamma$, where $\mathsf{p}\in\mathcal{PV}$, $A\subseteq{\mathbb{A}\mathrm{gt}}$, $\mathrm{X}\,$ stands for “next”, and $\,\mathrm{U}\,$ for “strong until” ($\gamma_{1}\,\mathrm{U}\,\gamma_{2}$ denotes that $\gamma_{1}$ holds until $\gamma_{2}$ becomes true). The other Boolean operators and constants are defined as usual. “Release” can be defined as $\gamma_{1}\,\mathrm{R}\,\gamma_{2}\equiv\neg((\neg\gamma_{1})\,\mathrm{U}\,(\neg\gamma_{2}))$. “Eventually” and “always” can be defined as $\mathrm{F}\,\gamma\equiv\mathit{true}\,\mathrm{U}\,\gamma$ and $\mathrm{G}\,\gamma\equiv\mathit{false}\,\mathrm{R}\,\gamma$. Moreover, the $\mathbf{CTL^{*}}$ operator “for all paths” can be defined as $\mathsf{A}\gamma\equiv\langle\\!\langle{\emptyset}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\gamma$. ###### Example 3.1 (Conference). Formula $\langle\\!\langle{sc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{open}}$ expresses that the Steering Chair can enforce that the conference is eventually opened. Moreover, formula $\langle\\!\langle{gc,oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\neg\mathsf{{epid}}$ says that the General Chair and the Organizing Chair have a joint strategy to avoid high epidemiological risk. ### 3.2 Strategies and Outcomes We adopt Schobbens’ taxonomy and notation for strategy types (?): $\mathrm{ir}$, $\mathrm{Ir}$, $\mathrm{iR}$, and $\mathrm{IR}$, where _I_ (resp. _i_) denotes perfect (resp. imperfect) _information_ , and _R_ (resp. _r_) denotes perfect (resp. imperfect) _recall_. In particular, an _imperfect information/imperfect recall strategy ($\mathrm{ir}$-strategy) for $i$_ is a function $\sigma_{i}\colon L_{i}\to\mathit{Evt}_{i}$ s.t. $\sigma_{i}(l)\in R_{i}(l)$ for each $l\in L_{i}$. We denote the set of such strategies by $\Sigma_{i}^{\mathrm{ir}}$. A _collective strategy_ $\sigma_{A}$ for a coalition $A=(1,\dots,m)\subseteq{\mathbb{A}\mathrm{gt}}$ is a tuple of strategies, one per agent $i\in A$. The set of $A$’s collective $\mathrm{ir}$ strategies is denoted by $\Sigma_{A}^{\mathrm{ir}}$. We will sometimes use $\sigma_{A}(g)=(\sigma_{a_{1}}(g),\dots,\sigma_{a_{m}}(g))$ to denote the tuple of $A$’s selections at state $g$. ###### Example 3.2 (Conference). A collective strategy for the General Chair and the OC Chair in the conference scenario is shown in Figure 1. An infinite sequence of global states and events $\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}g_{2}\dots$ is called an (interleaved) _path_ if $g_{i}\stackrel{{\scriptstyle\alpha_{i}}}{{\longrightarrow}}g_{i+1}$ for every $i\geq 0$. $\mathit{Evt}(\pi)=\alpha_{0}\alpha_{1}\alpha_{2}\ldots$ is the sequence of events in $\pi$, and $\pi[i]=g_{i}$ is the $i$-th global state of $\pi$. $\Pi_{M}(g)$ denotes the set of all paths in model $M$ starting at $g$. Intuitively, the outcome of $\sigma_{A}$ in $g$ is the set of all the paths that can occur when the agents in $A$ follow $\sigma_{A}$ and the agents in ${\mathbb{A}\mathrm{gt}}\setminus A$ freely choose events from their repertoires. To define it formally, we first refine the concept of an enabled event, taking into account the choices of $A$ in strategy $\sigma_{A}$. ###### Definition 3.3 (Enabled events). Let $A=(1,\dots,m)$, $g\in St$, and let $\overrightarrow{\alpha}_{A}=(\alpha_{1},\dots,\alpha_{m})$ be a tuple of events such that every $\alpha_{i}\in R_{i}(g^{i})$. That is, every $\alpha_{i}$ can be selected by its respective agent $i$ at state $g$. We say that event $\beta\in\mathit{Evt}$ is _enabled by $\overrightarrow{\alpha}_{A}$ at $g\in St$_ iff * • for every $i\in Agent(\beta)\cap A$, we have $\beta=\alpha_{i}$, and * • for every $i\in Agent(\beta)\setminus A$, it holds that $\beta\in R_{i}(g^{i})$. Thus, $\beta$ is enabled by $\overrightarrow{\alpha}_{A}$ if all the agents that “own” $\beta$ can choose $\beta$ for execution, even when $\overrightarrow{\alpha}_{A}$ has been selected by the coalition $A$. We denote the set of such events by $enabled(g,\overrightarrow{\alpha}_{A})$. Clearly, $enabled(g,\overrightarrow{\alpha}_{A})\subseteq enabled(g)$. ###### Example 3.4 (Conference). Consider state $g=000$ and the choices of agents $A=\\{{gc,oc}\\}$ shown in Figure 1, i.e., $\overrightarrow{\alpha}_{A}=(proceed,online)$. The only events enabled by $\overrightarrow{\alpha}_{A}$ are $proceed$ and $giveup$. Event $onsite$ is not enabled because $A$ chose different events for execution; $online$ is not enabled because it requires synchronization which is impossible at $000$. ###### Definition 3.5 (Outcome paths). The _outcome_ of strategy $\sigma_{A}\in\Sigma_{A}^{\mathrm{ir}}$ in state $g\in St$ is the set $\mathit{out}_{M}(g,\sigma_{A})\subseteq\Pi_{M}(g)$ such that $\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}g_{2}\dots\in\mathit{out}_{M}(g,\sigma_{A})$ iff $g_{0}=g$, and $\forall i\geq 0\quad\alpha_{i}\in enabled(\pi[i],\sigma_{A}(\pi[i]))$. One often wants to look only at paths that do not consistently ignore agents whose choice is always enabled. Formally, a path $\pi$ satisfies _concurrency- fairness_ (CF) if there is no event $\alpha$ enabled in all states of $\pi$ from $\pi[n]$ on and such that for every $\alpha_{i}$ actually executed in $\pi[i]$, $i=n,n+1,\dots$, we have $Agent(\alpha)\cap Agent(\alpha_{i})=\emptyset$. We denote the set of all such paths starting at $g$ by $\Pi_{M}^{\textbf{CF}}(g)$. ###### Definition 3.6 (CF-outcome). The _CF -outcome_ of $\sigma_{A}\in\Sigma_{A}^{\mathrm{ir}}$ is defined as $\mathit{out}^{\textbf{CF}}_{M}(g,\sigma_{A})=\mathit{out}_{M}(g,\sigma_{A})\cap\Pi_{M}^{\textbf{CF}}(g)$. ### 3.3 Strategic Ability for Asynchronous Systems The semantics of $\mathbf{ATL_{\mathrm{ir}}^{*}}$ in AMAS is defined by the following clause for strategic modalities (?): $M,g\models_{{}_{\mathrm{ir}}}\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\gamma$ iff there is a strategy $\sigma_{A}\in\Sigma_{A}^{\mathrm{ir}}$ s.t. $\mathit{out}_{M}(g,\sigma_{A})\neq\emptyset$ and, for each path $\pi\in\mathit{out}_{M}(g,\sigma_{A})$, we have $M,\pi\models_{{}_{\mathrm{ir}}}\gamma$. The clauses for Boolean and temporal operators are standard. Moreover, the _concurrency-fair semantics_ $\models_{{}_{\mathrm{ir}}}^{\textbf{CF}}$ of $\mathbf{ATL_{\mathrm{}}}$ and $\mathbf{ATL_{\mathrm{}}^{*}}$ is obtained by replacing $\mathit{out}_{M}(g,\sigma_{A})$ with $\mathit{out}_{M}^{\textbf{CF}}(g,\sigma_{A})$ in the above clause. ###### Example 3.7 (Conference). Clearly, formula $\langle\\!\langle{gc,oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\neg\mathsf{{epid}}$ holds in $(M_{\mathit{conf}},000)$, in both $\models_{{}_{\mathrm{ir}}}$ and $\models_{{}_{\mathrm{ir}}}^{\textbf{CF}}$ semantics. To see that, fix $\sigma_{gc}(0)=proceed$ and $\sigma_{gc}(1)=\sigma_{oc}(0)=online$ in the collective strategy of $\\{{gc,oc}\\}$. Note also that $M_{\mathit{conf}},000\models_{{}_{\mathrm{ir}}}\neg\langle\\!\langle{gc,oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{closed}}$ because, after executing $proceed$ and $online$ (or $onsite$), event $rest$ may be selected forever. On the other hand, such paths are not concurrency- fair, and thus $M_{\mathit{conf}},000\models_{{}_{\mathrm{ir}}}^{\textbf{CF}}\langle\\!\langle{gc,oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{closed}}$. Discussion. Strategic play assumes proactive attitude: the agents in $\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}$ are free to choose _any_ available strategy $\sigma_{A}$. This is conceptually consistent with the notion of agency (?). At the same time, it is somewhat at odds with the standard semantics of concurrent processes, where the components cannot stubbornly refuse to synchronize if that is the only way to proceed with a transition. This seems a minor problem, but it is worrying that a strategy can have the empty set of outcomes, and equally worrying that such strategies are treated differently from the other ones. Indeed, as we will show in the subsequent sections, the semantics proposed in (?) leads to a counterintuitive interpretation of strategic formulas. ## 4 Semantic Problems and How to Avoid Them $000$$101$$\mathsf{{open}}$$002$$211$$\mathsf{{epid}}$$321$$231$$\mathsf{{closed}}$$331$$\mathsf{{closed}}$$proceed$$giveup$$onsite$$online$$giveup$$rest$$handle$$rest$$handle$$rest$$idle$$rest$$idle$ --- Figure 2: Model $M_{\mathit{conf}}$ for the conference scenario. We highlight the transitions enabled by the strategy in Figure 1, and the resulting reachable states. Starting with this section, we describe some problematic phenomena that follow from the straightforward combination of strategic ability with models of concurrent systems, proposed in (?). We also show how to extend the representations and modify their execution semantics to avoid the counterintuitive interpretation of formulas. ### 4.1 Deadlock Strategies and Finite Paths An automata network is typically required to produce no deadlock states, i.e., every global state in its composition must have at least one outgoing transition. Then, all the maximal paths are infinite, and it is natural to refer to only infinite paths in the semantics of temporal operators. In case of AMAS, the situation is more delicate. Even if the AMAS as a whole produces no deadlocks, some strategies might, which makes the interpretation of strategic modalities cumbersome. We illustrate this on the following example. ###### Example 4.1 (Conference). Recall the 3-agent AMAS of Figure 1, together with its model $M_{\mathit{conf}}$ (Figure 2). Clearly, $M_{\mathit{conf}}$ has no deadlock states. Let us now look at the collective strategies of coalition $\\{{gc,oc}\\}$, with agent $sc$ serving as the opponent. It is easy to see that the coalition has no way to prevent the opening of the conference, i.e., it cannot prevent the system from reaching state $101$. However, the strategy depicted in Figure 1 produces only one _infinite_ path: $(000\,giveup\,002\,giveup\,\dots)$. Since the $\mathbf{ATL_{\mathrm{}}^{*}}$ semantics in Section 3 disregards finite paths, we get that $M_{\mathit{conf}},000\models\langle\\!\langle{gc,oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\neg\mathsf{{open}}$, which is counterintuitive. Things can get even trickier. In particular, the outcome of a strategy can be empty – in fact, it may even happen that a coalition has only strategies with empty outcomes. | $0$$1$$2$$3$$\mathsf{{voted_{a}}}$$4$$\mathsf{{voted_{b}}}$$vote_{a}$$vote_{b}$$send$$send$$idle_{v}$$idle_{v}$ --- | $0$$1$$vote_{a}$$vote_{b}$$send$$idle_{ebm}$ --- Figure 3: Casting a ballot: voter $v$ (left) and EBM $ebm$ (right) ###### Example 4.2 (Voting). Consider the AMAS in Figure 3 that depicts a simple voting scenario. A voter $v$ can fill in an electronic ballot with a vote for candidate $\mathsf{{a}}$ or $\mathsf{{b}}$, and then push the $send$ button. The Electronic Ballot Machine $ebm$ duly registers the choices of the voter. Note that all the _joint_ strategies of $\\{{v,ebm}\\}$ produce only finite sequences of transitions. This is because $ebm$ must choose a single event at location $0$ in a memoryless strategy, and thus $v$ and $ebm$ are bound to “miscoordinate” either at the first or at the second step. Since finite paths are not included in the outcome sets, and the semantics in Section 3.3 rules out strategies with empty outcomes, we get that $IIS(S_{vote}),00\models\neg\langle\\!\langle{v,ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\top$, which is quite strange. Notice that removing the non-emptiness requirement from the semantic clause in Section 3.3 does not help. In that case, any joint strategy of $\\{{v,ebm}\\}$ could be used to demonstrate that $\langle\\!\langle{v,ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\bot$. ### 4.2 Solution: Adding Silent Transitions To deal with the problem, we augment the model of the system with special “silent” transitions, labeled by $\epsilon$, that are fired whenever no “real” transition can occur. In our case, the $\epsilon$-transitions account for the possibility that some agents miscoordinate and thus block the system. Moreover, we redefine the outcome set of a strategy so that an $\epsilon$-transition is taken whenever such miscoordination occurs. ###### Definition 4.3 (Undeadlocked IIS). Let $S$ be an AMAS, and assume that no agent in S has $\epsilon$ in its alphabet of events. The _undeadlocked model of S_ , denoted $M^{\text{$\epsilon$}}=IIS^{\text{$\epsilon$}}(S)$, extends the model $M=IIS(S)$ as follows: * • $\mathit{Evt}_{M^{\text{$\epsilon$}}}=\mathit{Evt}_{M}\cup\\{{\epsilon}\\}$, where $Agent(\epsilon)=\emptyset$; * • For each $g\in St$, we add the transition $g\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}g$ iff there is a selection of agents’ choices $\overrightarrow{\alpha}_{A}=(\alpha_{1},\dots,\alpha_{k})$, $\alpha_{i}\in R_{i}(g)$, such that $enabled_{M}(g,\overrightarrow{\alpha}_{A})=\emptyset$. In that case, we also fix $enabled_{M^{\text{$\epsilon$}}}(g,\overrightarrow{\alpha}_{A})=\\{{\epsilon}\\}$. In other words, “silent” loops are added in the states where a combination of the agents’ actions can block the system. Paths are defined as in Section 2.2. The following is trivial. ###### Proposition 4.4. For any AMAS $S$, any state $g\in IIS^{\text{$\epsilon$}}(S)$, and any strategy $\sigma_{A}$, we have that $enabled_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A}(state))\neq\emptyset$. ###### Example 4.5 (Conference). The undeadlocked model $M_{\mathit{conf}}^{\text{$\epsilon$}}$ of the conference scenario (Example 2.2) extends the model in Figure 2 with one $\epsilon$-loop at state $101$. The loop models the situation when the agents choose $(onsite,online,proceed)$ or $(online,onsite,proceed)$. We leave it for the reader to check that, at the other states, all the combinations of choices enable at least one transition. For the strategy in Example 4.1, notice that its outcome in $M_{\mathit{conf}}^{\text{$\epsilon$}}$ contains _two_ infinite paths: not only $(000\,giveup\,002\,giveup\,002\dots)$, but also $(000\,proceed\,101\,\epsilon\,101\dots)$. Since the latter path invalidates the temporal formula $\mathrm{G}\,\neg\mathsf{{open}}$, we get that $M_{\mathit{conf}},000\not\models\langle\\!\langle{gc,oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\neg\mathsf{{open}}$, as expected. $00$$10$$20$$31$$\mathsf{{voted_{a}}}$$41$$\mathsf{{voted_{b}}}$$\epsilon$$vote_{a}$$vote_{b}$$\epsilon$$send$$\epsilon$$send$$\genfrac{}{}{0.0pt}{}{idle_{v}}{idle_{ebm}}$$\genfrac{}{}{0.0pt}{}{idle_{v}}{idle_{ebm}}$ Figure 4: Undeadlocked IIS for the voting scenario ###### Example 4.6 (Voting). The undeadlocked model for the voting scenario is presented in Figure 4. Note that formula $\neg\langle\\!\langle{v,ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\top$ does not hold anymore, because the joint strategies of $\\{{v,ebm}\\}$ have nonempty outcomes in $IIS^{\text{$\epsilon$}}(S_{vote})$. On the other hand, the formula $\langle\\!\langle{v}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{voted_{a}}}$ (and even $\langle\\!\langle{v,ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{voted_{a}}}$) does not hold, which is contrary to the intuition behind the modeling. We will come back to this issue in Section 7. Discussion. Adding “silent” transitions to account for the control flow when no observable event occurs is pretty standard. The crucial issue is _where_ to add them. Here, we add the $\epsilon$-transitions whenever a subset of agents might choose to miscoordinate (and stick to their choices). Again, this is in line with the notion of agency and strategic play in MAS (?; ?). In the next section, we will discuss a concept of “agent fairness” where the addition of $\epsilon$-transitions is constrained by the assumption that only a given subset of agents is fully proactive. The examples used in this section expose an important feature of agent systems. The execution semantics of concurrent processes is often defined by a state-transition graph (or, alternatively, by the tree of paths generated by the graph, i.e., the tree unfolding of the graph). For systems that involve proactive agents, this is not enough. Rather, the execution semantics should map from the possible coalitions and their available strategies to the outcome sets of those strategies. In this sense, the possible behaviors of an agent system should be understood via the _set of possible execution trees_ , rather than a single tree. This is consistent with the theoretical model of MAS in (?), based on path effectivity functions. An alternative way out of the problem is to include finite maximal paths in the outcomes of strategies. However, the interpretation of strategic modalities over finite paths is rather nonstandard (?) and may pose new problems in the asynchronous setting. Moreover, our approach allows to reuse the existing techniques and tools, which are typically built for infinite path semantics, including the verification and partial order reduction functionalities of tools like SPIN (?) and STV (?). In general, this is a design dilemma between changing the logical semantics of the formulas vs. updating the execution semantics of the representations. Here, we choose the latter approach. ## 5 Playing Against Reactive Opponents The solution proposed in Section 4.2 is based on the assumption that an agent is free to choose any event in its repertoire – even one that prevents the system from executing anything. The downside is that, for most systems, only safety goals can be achieved (i.e., properties specified by $\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\varphi$). For reachability, there is often a combination of the opponents’ choices that blocks the execution early on, and prevents the coalition from reaching their goal. In this section, we define a fairness-style condition that constrains the choices of more “reactive” opponents. We also show a construction to verify the abilities of the coalition over the resulting paths in a technically simpler way. ### 5.1 Opponent-Reactiveness Given a strategy $\sigma_{A}$, the agents in $A$ are by definition assumed to be proactive. Below, we propose an execution semantics for $\sigma_{A}$ which assumes that $A$ cannot be stalled forever by miscoordination on the part of the opponents. ###### Definition 5.1 (Opponent-reactiveness). A path $\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}g_{2}\dots$ in $IIS^{\text{$\epsilon$}}(S)$ is _opponent-reactive for strategy $\sigma_{A}$_ iff we have that $\alpha_{n}=\epsilon$ implies $enabled(g_{n},\sigma_{A}(g_{n}))=\\{{\epsilon}\\}$. In other words, whenever the agents outside $A$ have a way to proceed, they must proceed. The _reactive outcome_ (or _React-outcome_) of $\sigma_{A}$ in $g$, denoted $\mathit{out}^{\textup{React}}(g,\sigma_{A})$, is the restriction of $\mathit{out}(g,\sigma_{A})$ to its opponent-reactive paths. ###### Example 5.2 (Conference). Consider the undeadlocked model $M_{\mathit{conf}}^{\text{$\epsilon$}}$ of Example 4.5. Path $(000\,proceed\,101\,\epsilon\,101\dots)$ is opponent- reactive for the strategy of agents $\\{{gc,oc}\\}$ shown in Figure 1. On the other hand, consider coalition $\\{{gc,sc}\\}$, and the following strategy of theirs: $\sigma_{gc}(0)=proceed,\sigma_{gc}(1)=onsite,\sigma_{sc}(0)=proceed$. The same path is _not_ opponent-reactive for the strategy because the only opponent ($oc$) has a response at state $101$ that enables a “real” transition ($onsite$). ###### Proposition 5.3. In $\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$, the only possible occurrence of $\epsilon$ is as an infinite sequence of $\epsilon$-transitions following a finite prefix of “real” transitions. ###### Proof. Take any $\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}g_{2}\dots\in\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$ such that $\epsilon$ occurs on $\pi$, and let $i$ be the first position on $\pi$ st. $\alpha_{i}=\epsilon$. By Definition 5.1, we get that $enabled(g_{i},\sigma_{A}(g_{i}))=\\{{\epsilon}\\}$. Moreover, $state_{i+1}=g_{i}$, so also $enabled(g_{i+1},\sigma_{A}(g_{i+1}))=\\{{\epsilon}\\}$. Thus, $\alpha_{i+1}=\epsilon$. It follows by simple induction that $\alpha_{j}=\epsilon$ for every $j\geq i$. ∎ The _opponent-reactive semantics_ $\models_{{}_{\mathrm{ir}}}^{\textup{React}}$ of $\mathbf{ATL_{\mathrm{}}^{*}}$ is obtained by replacing $\mathit{out}_{M}(g,\sigma_{A})$ with $\mathit{out}_{M}^{\textup{React}}(g,\sigma_{A})$ in the semantic clause presented in Section 3.3. ### 5.2 Encoding Strategic Deadlock-Freeness Under Opponent-Reactiveness in AMAS If we adopt the assumption of opponent-reactiveness for coalition $A$, there is an alternative, technically simpler way to obtain the same semantics of strategic ability as in Section 4.2. The idea is to introduce the “silent” transitions already at the level of the AMAS. ###### Definition 5.4 (Undeadlocked AMAS). The _undeadlocked variant of $S$_ is constructed from $S$ by adding an auxiliary agent $A_{\epsilon}$ with $L_{\epsilon}=\\{{q_{0}^{\epsilon}}\\}$, $\iota_{\epsilon}=q_{0}^{\epsilon}$, $\mathit{Evt}_{\epsilon}=\\{{\epsilon}\\}$, $R_{\epsilon}(q_{0}^{\epsilon})=\\{{\epsilon}\\}$, $T_{i}(q_{0}^{\epsilon},\epsilon)=q_{0}^{\epsilon}$, and $\mathcal{PV}_{\epsilon}=\emptyset$. In other words, we add a module with a single local state and a “silent” loop labeled by $\epsilon$, as in Figure 5. We will denote the undeadlocked variant of $S$ by $S^{\text{$\epsilon$}}$. Note that $S^{\text{$\epsilon$}}$ can be seen as a special case of AMAS. Thus, the outcome sets and reactive outcomes of strategies in $IIS(S^{\text{$\epsilon$}})$ are defined exactly as before. $q_{0}^{\epsilon}$$\epsilon$ Figure 5: The auxiliary agent added in $S^{\text{$\epsilon$}}$ ###### Example 5.5 (Voting). The undeadlocked AMAS $\mathrm{ASV}_{1,2}^{\epsilon}$ is obtained by augmenting $\mathrm{ASV}_{1,2}$ with the auxiliary agent in Figure 5. Obviously, the extra agent adds $\epsilon$-loops to the model of $S$, i.e., to $IIS(S)$. We show now that, under the assumption of opponent-reactiveness, the view of $A$’s strategic ability in the undeadlocked AMAS $S^{\text{$\epsilon$}}$ corresponds precisely to $A$’s abilities in the undeadlocked model of the original AMAS $S$, i.e., $IIS^{\text{$\epsilon$}}(S)$. This allows to deal with deadlocks and finite paths without redefining the execution semantics for AMAS, set in Definition 2.3, and thus use the existing tools such as SPIN (?) in a straightforward way. ###### Proposition 5.6. Let $A\subseteq{\mathbb{A}\mathrm{gt}}$. In $\mathit{out}^{\textup{React}}_{IIS(S^{\text{$\epsilon$}})}(g,\sigma_{A})$, the only possible occurrence of $\epsilon$ is as an infinite suffix of $\epsilon$-transitions. ###### Proof. Analogous to Proposition 5.3. ∎ ###### Theorem 5.7. For every strategy $\sigma_{A}$ in $S$, we have that $\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})=\mathit{out}^{\textup{React}}_{IIS(S^{\text{$\epsilon$}})}(g,\sigma_{A}).$ ###### Proof. $\boldsymbol{\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})\subseteq\mathit{out}^{\textup{React}}_{IIS(S^{\text{$\epsilon$}})}(g,\sigma_{A})}$: Consider any $\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}g_{2}\dots\in\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$. If there are no $\epsilon$-transitions on $\pi$, we have that $\pi\in\mathit{out}^{\textup{React}}_{IIS(S)}(g,\sigma_{A})\subseteq\mathit{out}^{\textup{React}}_{IIS(S^{\text{$\epsilon$}})}(g,\sigma_{A})$, QED. Suppose that $\pi$ includes $\epsilon$-transitions, with $\alpha_{i}$ being the first one. Then, we have that $\alpha_{j}\neq\epsilon$ and $\alpha_{j}\in enabled_{IIS^{\text{$\epsilon$}}(S)}(g_{j},\sigma_{A}(g_{j}))$ for every $j<i$, hence also $\alpha_{j}\in enabled_{IIS(S)}(g_{j},\sigma_{A}(g_{j}))\subseteq enabled_{IIS(S^{\text{$\epsilon$}})}(g_{j},\sigma_{A}(g_{j}))$. (*) By Proposition 5.3, $g_{j}=g_{i}$ and $\alpha_{j}=\epsilon$ for every $j\geq i$. By Definition 5.1, $enabled_{IIS^{\text{$\epsilon$}}(S)}(g_{j},\sigma_{A}(g_{j}))=\\{{\epsilon}\\}$. Hence, $enabled_{IIS(S)}(g_{j},\sigma_{A}(g_{j}))=\emptyset$ and $enabled_{IIS(S^{\text{$\epsilon$}})}(g_{j},\sigma_{A}(g_{j}))=\\{{\epsilon}\\}$. (**) Thus, by (*) and (**), $\pi\in\mathit{out}^{\textup{React}}_{IIS(S^{\text{$\epsilon$}})}(g,\sigma_{A})$, QED. $\boldsymbol{\mathit{out}^{\textup{React}}_{IIS(S^{\text{$\epsilon$}})}(g,\sigma_{A})\subseteq\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})}$: Analogous, with Proposition 5.6 used instead of Proposition 5.3. ∎ Discussion. Opponent-reactiveness is to strategic properties what fairness conditions are to temporal properties of asynchronous systems. If an important property cannot be satisfied in all possible executions, it may at least hold under some reasonable assumptions about which events can be selected by whom in response to what. Clearly, the condition can be considered intuitive by some and problematic by others. The main point is, unlike in the previous semantics, now it is made explicit, and can be adopted or rejected depending on the intuition. Note that the semantic extensions proposed in this paper (silent transitions and nondeterministic choices for strategies) make sense both with and without opponent-reactiveness. Note that, under the reactiveness assumption, we have that $M_{\mathit{conf}}^{\text{$\epsilon$}},000\models_{{}_{\mathrm{ir}}}^{\textup{React}}\langle\\!\langle{gc,sc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{epid}}$ and $M_{\mathit{conf}}^{\text{$\epsilon$}},000\models_{{}_{\mathrm{ir}}}^{\textup{React}}\langle\\!\langle{oc}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{G}\,\neg\mathsf{{epid}}$. This seems to contradict the commonly accepted requirement of _regularity_ in games (?). However, the contradiction is only superficial, as the two formulas are evaluated _under different execution assumptions_ : for the former, we assume agent $oc$ to be reactive, whereas the latter assumes $gc$ and $sc$ to react to the strategy of $oc$. ## 6 Concurrency-Fairness Revisited In Def. 3.6, we recalled the notion of concurrency-fair outcome of (?). The idea was to remove from $out(g,\sigma_{A})$ the paths that consistently ignore agents whose events are enabled _at the level of the whole model_. Unfortunately, the definition has unwelcome side effects, too. ### 6.1 Problems with Concurrency-Fairness We first show that, contrary to intuition, Definition 3.6 automatically disregards _deadlock paths_ , i.e., paths with finitely many “real” transitions. ###### Proposition 6.1. Consider an AMAS $S$ and a path $\pi$ in $IIS^{\text{$\epsilon$}}(S)$ such that, from some point $i$ on, $\pi$ includes only $\epsilon$-transitions. Then, for every strategy $\sigma_{A}$ in $S$, we have that $\pi\notin\mathit{out}^{\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$. ###### Proof. Take $\pi$ as above, i.e., $\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}\dots g_{i}\epsilon g_{i}\epsilon g_{i}\dots$. Since the transition function in $IIS^{\text{$\epsilon$}}(S)$ is serial, there must be some event $\beta\neq\epsilon$ enabled in $g_{i}$. In consequence, $\beta$ is always enabled from $i$ on, but none of its “owners” in $Agent(\beta)$ executes an event on $\pi$ after $i$. Hence, $\pi$ does not satisfy CF, and does not belong to $\mathit{out}^{\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$ for any strategy $\sigma_{A}$. ∎ Thus, the CF condition eliminates all the deadlock paths from the outcome of a strategy (for instance, the path $(000\,proceed\,101\,\epsilon\,101\dots)$ in Example 4.5). In consequence, reasoning about concurrency-fair paths suffers from the problems that we identified in Section 4.1, even for undeadlocked models. Moreover, combining the temporal and strategic fairness (i.e., CF and React) collapses the undeadlocked execution semantics altogether, see below. ###### Proposition 6.2. Reasoning about reactive _and_ fair outcomes in an undeadlocked model reduces to reasoning about the fair executions in the original model without $\epsilon$-transitions. Formally, let $\mathit{out}^{\textup{React},\textbf{CF}}_{M}(g,\sigma_{A})=\mathit{out}^{\textup{React}}_{M}(g,\sigma_{A})\cap\mathit{out}^{\textbf{CF}}_{M}(g,\sigma_{A})$. For any AMAS $S$ and any strategy $\sigma_{A}$ in $S$, we have: $\mathit{out}^{\textup{React},\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})=\mathit{out}^{\textbf{CF}}_{IIS(S)}(g,\sigma_{A})$. ###### Proof. Clearly, we have $\mathit{out}^{\textbf{CF}}_{IIS(S)}(g,\sigma_{A})\subseteq\mathit{out}^{\textup{React},\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$, since $\mathit{out}^{\textup{React},\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$ can only add to $\mathit{out}^{\textbf{CF}}_{IIS(S)}(g,\sigma_{A})$ new paths that include $\epsilon$-transitions. For the other direction, take any $\pi\in\mathit{out}^{\textup{React},\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$, and suppose that it contains an $\epsilon$-transition. By Proposition 5.3, it must have an infinite suffix consisting only of $\epsilon$-transitions. Then, by Proposition 6.1, $\pi\notin\mathit{out}^{\textbf{CF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$, which leads to a contradiction. Thus, $\pi$ contains only transitions from $IIS(S)$, and hence $\pi\in\mathit{out}^{\textbf{CF}}_{IIS(S)}(g,\sigma_{A})$, QED. ∎ ### 6.2 Strategic Concurrency-Fairness So, how should fair paths be properly defined for strategic reasoning? The answer is simple: in relation to the outcome of the strategy being executed. ###### Definition 6.3 (Strategic CF). $\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}g_{2}\dots$ is a _concurrency-fair path for strategy $\sigma_{A}$ and state $g$_ iff $g_{0}=g$, and there is no event $\alpha$ s.t., for some $n$ and all $i\geq n$, we have $\alpha\in enabled(\pi[i],\sigma_{A}(\pi[i]))$ and $Agent(\alpha)\cap Agent(\alpha_{i})=\emptyset$. That is, agents with an event always enabled _by $\sigma_{A}$_ cannot be ignored forever. The _SCF -outcome_ of $\sigma_{A}\in\Sigma_{A}^{\mathrm{ir}}$ is defined as $\mathit{out}^{\textbf{SCF}}_{M}(g,\sigma_{A})=\\{\pi\in\mathit{out}_{M}(g,\sigma_{A})\mid\pi\text{ is concurrency-fair for }\sigma_{A},g\\}$. The following formal results show that SCF does not suffer from the problems demonstrated in Section 6.1. ###### Proposition 6.4. There is an AMAS $S$, a strategy $\sigma_{A}$ in $S$, and a deadlock path $\pi$ in $IIS^{\text{$\epsilon$}}(S)$ such that $\pi$ is concurrency-fair for $\sigma_{A}$. ###### Proof. To demonstrate the property, it suffices to take the AMAS and the strategy of $\\{{gc,oc}\\}$ depicted in Figure 1, and the path $\pi=(000\,proceed\,101\,\epsilon\,101\dots)$. ∎ ###### Theorem 6.5. Opponent-reactiveness and strategic concurrency-fairness are incomparable. Formally, there exists an AMAS $S$, a state $g$ in $IIS^{\text{$\epsilon$}}(S)$, and a strategy $\sigma_{A}$ such that $\mathit{out}^{\textbf{SCF}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})\not\subseteq\mathit{out}^{\textup{React}}_{IIS^{\text{$\epsilon$}}(S)}(g,\sigma_{A})$, and vice versa. ###### Proof. Consider the undeadlocked model $M_{\mathit{conf}}^{\text{$\epsilon$}}$ in Example 4.5, and the strategy discussed in Example 5.2: $\sigma_{gc}(0)=proceed$, $\sigma_{gc}(1)=onsite$, $\sigma_{sc}(0)=proceed$. Let $\pi_{1}=(000\,proceed\,101\,\epsilon\,101\,onsite\,211\,rest\,211\linebreak handle\,211\,rest\,211\dots)$. We have $\pi_{1}\in\mathit{out}^{\textbf{SCF}}_{M_{\mathit{conf}}^{\text{$\epsilon$}}}(g,\sigma_{A})$, but $\pi_{1}\notin\mathit{out}^{\textup{React}}_{M_{\mathit{conf}}^{\text{$\epsilon$}}}(g,\sigma_{A})$. On the other hand, for path $\pi_{2}=(000\,proceed\,101\,onsite\,211\,rest\,211\,rest\,\dots)$, we have that $\pi_{2}\notin\mathit{out}^{\textbf{SCF}}_{M_{\mathit{conf}}^{\text{$\epsilon$}}}(g,\sigma_{A})$, but $\pi_{2}\in\mathit{out}^{\textup{React}}_{M_{\mathit{conf}}^{\text{$\epsilon$}}}(g,\sigma_{A})$. ∎ Discussion. Theorem 6.5 suggests that reactiveness and fairness conditions arise from orthogonal concerns. The two concepts refer to different factors that influence which sequences of events can occur. Opponent-reactiveness constrains the choices that (a subset of) the agents can select. Concurrency- fairness and its strategic variant restrict the way in which the “scheduler” (Nature, Chance, God…) can choose from the events selected by the agents. ## 7 Strategies in Asymmetric Interaction Now, we point out that AMAS are too restricted to model the strategic aspects of asymmetric synchronization in a natural way (e.g., a sender sending a message to a receiver). ### 7.1 Simple Choices are Not Enough We demonstrate the problem on an example. ###### Example 7.1 (Voting). As already pointed out, we have $IIS^{\text{$\epsilon$}}(S_{vote}),00\not\models\langle\\!\langle{v,ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{voted_{a}}}$ in the model of Example 4.2. This is because receiving a vote for $\mathsf{{a}}$, a vote for $\mathsf{{b}}$, and the signal to send the vote, belong to _different choices_ in the repertoire of the EBM, and the agent can only select one of them in a memoryless strategy. Moreover, formula $\langle\\!\langle{ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{voted_{a}}}$ holds under the condition of opponent-reactiveness, i.e., the EBM can force a reactive voter to vote for a selected candidate. Clearly, it was not the intention behind the AMAS: the EBM is supposed to _listen_ to the choice of the voter. No matter whose strategies are considered, and who reacts to whose actions, the EBM should have no influence on what the voter votes for. The problem arises because the repertoire functions in AMAS are based on the assumption that the agent can choose any single event in $R_{i}(l_{i})$. This does not allow for natural specification of situations when the exact transition is determined by another agent. For the AMAS in Example 4.2, the decision to vote for candidate $\mathsf{{a}}$ or $\mathsf{{b}}$ (or to press $send$) should belong solely to the voter. Thus, setting the EBM repertoire as $R_{ebm}(0)=\\{{vote_{a},vote_{b},send}\\}$ does not produce a good model of strategic play in the scenario. ### 7.2 AMAS with Explicit Control As a remedy, we extend the representations so that one can indicate which agent(s) control the choice between events. ###### Definition 7.2 (AMAS with explicit control). Everything is exactly as in Definition 2.1, except for the repertoires of choices, which are now functions $R_{i}:L_{i}\to 2^{2^{\mathit{Evt}_{i}}\setminus\\{\emptyset\\}}\setminus\\{\emptyset\\}$. That is, $R_{i}(l)$ lists nonempty subsets of events $X_{1},X_{2},\dots\subseteq\mathit{Evt}_{i}$, each capturing an available choice of $i$ at the local state $l$. If the agent chooses $X_{j}=\\{{\alpha_{1},\alpha_{2},\dots}\\}$, then only an event in that set can be executed within the agent’s module; however, the agent has no firmer control over which one will be fired. Accordingly, we assume that $T_{i}(l,\alpha)$ is defined iff $\alpha\in\bigcup R_{i}(l)$.333 For a set of sets $X$, we use $\bigcup X$ to denote its “flattening” $\bigcup_{x\in X}x$. Notice that the AMAS of Definition 2.1 can be seen as a special case where $R_{i}(l)$ is always a list of singletons. The definitions of IIS and undeadlocked IIS stay the same, as agents’ repertoires of choices are not actually used to generate the state-transition structure for the model of $S$. Moreover, undeadlocked AMAS with explicit control can be obtained analogously to Definition 5.4 by adding the auxiliary “epsilon”-agent with $R_{\epsilon}(q_{0}^{\epsilon})=\\{{\\{{\epsilon}\\}}\\}$ in its sole local state. Strategies still assign choices to local states; hence, the type of agent $i$’s strategies is now $\sigma_{i}\colon L_{i}\to 2^{\mathit{Evt}_{i}}\setminus\\{\emptyset\\}$ s.t. $\sigma_{i}(l)\in R_{i}(l)$. The definition of the outcome set is updated accordingly, see below. ###### Definition 7.3 (Outcome sets for AMAS with explicit control). First, we lift the set of events enabled by $\overrightarrow{\alpha}_{A}=(\alpha_{1},\dots,\alpha_{m})$ at $g$ to match the new type of repertoires and strategies. Formally, $\beta\in enabled(g,\overrightarrow{\alpha}_{A})$ iff: (1) for every $i\in Agent(\beta)\cap A$, we have $\beta\in\alpha_{i}$, and (2) for every $i\in Agent(\beta)\setminus A$, it holds that $\beta\in\bigcup R_{i}(g^{i})$. The outcome, React-outcome, and SCF-outcome of $\sigma_{A}$ in $M,g$ are given as in Definitions 3.5, 5.1, and 6.3. ###### Example 7.4 (Voting). We improve our voting model by assuming repertoires of choices for the voter and the EBM as follows: $R_{ebm}(0)=\\{{\\{{vote_{a},vote_{b},send}\\}}\\}$, $R_{v}(0)=\\{{\\{{vote_{a}}\\},\\{{vote_{b}}\\}}\\}$, $R_{v}(1)=R_{v}(2)=\\{{\\{{send}\\}}\\}$, etc. That is, the voter’s choices are as before, but the EBM only listens to what the voter selects. Clearly, $\langle\\!\langle{v,ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{voted_{a}}}$ holds in the new AMAS. Moreover, $\langle\\!\langle{ebm}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\mathrm{F}\,\mathsf{{voted_{a}}}$ does not hold anymore, even assuming opponent-reactiveness. It is easy to see that Propositions 4.4, 5.3, 5.6, and 6.4, as well as Theorems 5.7 and 6.5 still hold in AMAS with explicit control. Discussion. When reasoning about strategic play of asynchronous agents, two kinds of asymmetry come into the picture. On the one hand, the processes (agents) being modeled often synchronize in an asymmetric way. For example, the sender chooses which message to send to the receiver. On the other hand, the agents $A$ in formula $\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\varphi$ choose the strategy and thus push the other agents to respond accordingly. The variant of AMAS introduced in (?) does not allow to capture the former kind of asymmetry. In consequence, the choice between the available synchronization branches belongs solely to the agents indicated by the formula. Unfortunately, there is no natural way to model the converse situation, i.e., when the agents in $\langle\\!\langle{A}\rangle\\!\rangle_{{}_{\\!\mathit{}}}$ are forced by the choices of their opponents. With the new variant of AMAS, we extend the representations so that the modeler can explicitly specify the degree of autonomy of each participating agent. Without that, the degree of autonomy is implicit and comes from the formula being evaluated. Related modeling approaches. Various forms of asymmetric synchronization are present in most process algebras. For example, $\pi$-calculus distinguishes between the action $\overline{c}\langle a\rangle$ of sending the value $a$ on channel $c$, and action $c(x)$ of listening on channel $c$ and storing whatever comes in variable $x$. CSP goes further, and allows for a similar degree of flexibility to ours through suitable combinations of deterministic choice, nondeterministic choice, and interface parallel operators. Other synchronization primitives are also possible, see e.g. (?) for an overview. Instead of allowing for multiple synchronization primitives, we come up with a single general primitive that can be instantiated to cover different kinds of interaction. We note in passing the similarity of our new repertoire functions in Definition 7.2 to state effectivity functions (?; ?) and especially alternating transition systems (?). ## 8 Partial Order Reduction Still Works _Partial order reduction (POR)_ has been defined for temporal and temporal- epistemic logics without “next” (?; ?; ?; ?), and recently extended to strategic specifications (?). The idea is to take a network of automata (AMAS in our case), and use depth-first search through the space of global states to generate a reduced model that satisfies exactly the same formulas as the full model. Essentially, POR removes paths that change only the interleaving order of an “irrelevant” event with another event. Importantly, the method generates the reduced model directly from the representation, without generating the full model at all. ### 8.1 Correctness of POR in the New Semantics POR is a powerful technique to contain state-space explosion and facilitate verification, cf. e.g. the experimental results in (?). In this paper, we extend the class of models, and modify their execution semantics. We need to show that the reduction algorithm in (?), defined for the flawed semantics of ability, is still correct after the modifications. Our main technical result in this respect is Theorem A.11, presented below. The detailed definitions, algorithms and proofs are technical (and rather tedious) adaptations of those in (?). We omit them here for lack of space, and refer the inquisitive reader to Appendix A. Theorem A.11. Let $M=\mathit{IIS}(S^{\text{$\epsilon$}})$, $M^{\text{$\epsilon$}}=IIS^{\text{$\epsilon$}}(S)$ and let $A\subseteq{\mathbb{A}\mathrm{gt}}$ be a subset of agents. Moreover, let ${M{{}^{\prime}}}\subseteq M$ and $M^{\text{$\epsilon$}}{{}^{\prime}}\subseteq M^{\text{$\epsilon$}}$ be the reduced models generated by DFS with the choice of enabled events $E(g^{\prime})$ given by conditions C1, C2, C3 and the independence relation $I_{A,\mathit{PV}}$. For each $\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$ formula $\varphi$ over $\mathit{PV}$, that refers only to coalitions $\hat{A}\subseteq A$, we have: 1. 1. $M,\iota\models_{{}_{\mathrm{ir}}}^{\textup{React}}\varphi$ iff ${M{{}^{\prime}}},\iota^{\prime}\models_{{}_{\mathrm{ir}}}^{\textup{React}}\varphi$, and 2. 2. $M^{\text{$\epsilon$}},\iota\models_{{}_{\mathrm{ir}}}\varphi$ iff $M^{\text{$\epsilon$}}{{}^{\prime}},\iota^{\prime}\models_{{}_{\mathrm{ir}}}\varphi$. Thus, the reduced models can be used to model-check the $\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$ properties of the full models. Proof idea. We aim at showing that the full model $M$ and the reduced one $M^{\prime}$ satisfy the same formulas of $\mathbf{ATL_{\mathrm{\mathrm{ir}}}^{*}}$ referring only to coalitions $\hat{A}\subseteq A$ and containing no nested strategic operators. Thanks to the restriction on the formulas, the proof can be reduced to showing that ${M{{}^{\prime}}}$ satisfies the condition $\textbf{AE}_{A}$, which states that for each strategy and for each path of the outcome of this strategy in $M$ there is an equivalent path in the outcome of the same strategy in $M^{\prime}$. In order to show that $\textbf{AE}_{A}$ holds, we use the conditions on the selection of events $E(g^{\prime})$ to be enabled at state $g^{\prime}$ in $M^{\prime}$. The conditions include the requirement that $\epsilon$ is always selected, together with the three conditions ${\bf C1,C2,C3}$ adapted from (?; ?; ?). Intuitively, ${\bf C1}$ states that, along each path $\pi$ in $M$ which starts at $g^{\prime}$, each event that is dependent on an event in $E(g^{\prime})$ cannot be executed in $M$ unless an event in $E(g^{\prime})$ is executed first in $M$. ${\bf C2}$ says that $E(g^{\prime})$ either contains all the events, or only events that do not change the values of relevant propositions. ${\bf C3}$ guarantees that for every cycle in $M^{\prime}$ containing no $\epsilon$-transitions, there is at least one node $g^{\prime}$ in the cycle for which all the enabled events of $g^{\prime}$ are selected. First, we show that $M$ and $M^{\prime}$ are stuttering-equivalent, i.e., they have the same sets of paths modulo stuttering (that is, finite repetition of states on a path). The crucial observation here is that the reduction of $M$ under the conditions C1, C2, C3 is equivalent to the reduction of $M$ without the $\epsilon$-loops under the conditions C1, C2, C3 of (?), and then adding the $\epsilon$-loops to all the states of the reduced model. Therefore, for the paths without $\epsilon$-loops the stuttering equivalence can be shown similarly to (?, Theorem 12) while for the paths with $\epsilon$-loops we need more involved arguments in the proof. It turns out that in addition to the fact that $M$ and $M^{\prime}$ are stuttering equivalent, we can show that stuttering equivalent paths of $M$ and $M^{\prime}$ have the same maximal sequence of visible events. From that, we can prove that $\textbf{AE}_{A}$ holds. ## 9 Conclusions In this paper, we reconsider the asynchronous semantics of strategic ability for multi-agent systems, proposed in (?). We have already hinted at certain problems with the semantics in the extended abstract (?). Here, we demonstrate in detail how the straightforward combination of strategic reasoning and models of distributed systems leads to counterintuitive interpretation of formulas. We identify three main sources of problems. First, the execution semantics does not handle reasoning about deadlock-inducing strategies well. Secondly, fairness conditions need to be redefined for strategic play. Thirdly, the class of representations lacks constructions to resolve the tension between the asymmetry imposed by strategic operators on the one hand, and the asymmetry of interaction, e.g., between communicating parties. We deal with the problems as follows. First, we change the execution semantics of strategies in asynchronous MAS by adding “silent” $\epsilon$-transitions in states where no “real” event can be executed. We also propose and study the condition of _opponent-reactiveness_ that assumes the agents outside the coalition to not obstruct the execution of the strategy forever. Note that, while the assumption may produce similar interpretation of formulas as in (?), it is now explicit – as opposed to (?), where it was “hardwired” in the semantics. The designer or verifier is free to adopt it or reject it, depending on their view of how the agents in the system behave and choose their actions. Secondly, we propose a new notion of _strategic concurrency-fairness_ that selects the fair executions of a strategy. Thirdly, we allow for nondeterministic choices in agents’ repertoires. This way, we allow to explicitly specify that one agent has more control over the outcome of an event than the other participants of the event. The main technical result consists in proving that partial order reduction for strategic abilities (?) is still correct after the semantic modifications. Thus, the new, more intuitive semantics admits efficient verification. Beyond $\mathbf{ATL_{\mathrm{ir}}}$. In this study, we have concentrated on the logic $\mathbf{ATL^{*}_{\mathrm{ir}}}$, i.e., the variant of $\mathbf{ATL_{\mathrm{}}^{*}}$ based on memoryless imperfect information strategies. Clearly, the concerns raised here are not entirely (and not even not primarily) logical. $\mathbf{ATL^{*}_{\mathrm{ir}}}$ can be seen as a convenient way to specify the players and the winning conditions in a certain class of games (roughly speaking, $1.5$-player games with imperfect information, positional strategies, and $\mathbf{LTL}$ objectives). The semantic problems, and our solutions, apply to all such games interpreted over arenas given by asynchronous MAS. Moreover, most of the claims presented here are not specific to $\mathrm{ir}$-strategies. In fact, we conjecture that our examples of semantic side effects carry over to the other types of strategies (except for the existence of coalitions whose all strategies have empty outcomes, which can happen for neither perfect information nor perfect recall). Similarly, our technical results should carry over to the other strategy types (except for the correctness of POR, which does not hold for agents with perfect information). We leave the formal analysis of those cases for future work. Other issues. An interesting question concerns the relationship between asynchronous and synchronous models. We conjecture that AMAS with explicit control can be simulated by concurrent game structures and alternating transition systems. Similarly, it should be possible to simulate CGS and ATS by AMAS with explicit control, at the expense of using a huge space of fully synchronized actions. 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Electronic Notes in Theoretical Computer Science 85(2):82–93. ## Appendix A Partial Order Reduction: Details All the results in this appendix are formulated and proved for the semantics of $\mathbf{ATL_{\mathrm{ir}}}$ over undeadlocked AMAS with explicit control. Also, we restrict the formulas to $\mathbf{ATL_{\mathrm{}}^{*}}$ without nested strategic modalities and the next step operator $\mathrm{X}\,$ (“simple $\mathbf{ATL_{\mathrm{}}^{*}}$”, or $\mathbf{sATL_{\mathrm{}}^{*}}$). As noted in (?), $\mathbf{sATL_{\mathrm{}}^{*}}$ is sufficient for most practical specifications and much more expressive than $\mathbf{LTL}$. Yet, as we prove below, it enjoys the same efficiency of partial order reduction. We begin by introducing the relevant notions of equivalence. Then, we propose conditions on reduced models that preserve the stuttering equivalence with and without the assumption of _opponent-reactiveness_ (React). We point out algorithms that generate such models, and prove their correctness. It should be stressed that the reduction scheme proposed here is general, in the sense that it preserves equivalent representatives of both fair and unfair paths in the model. In particular, we do _not_ propose a variant of POR, optimized for strategic concurrency-fair paths, analogous to reductions of (?) for CF. A variant of POR for $\mathbf{sATL_{\mathrm{\mathrm{ir}}}}$ under the SCF assumption is planned for future work. ### A.1 Properties of Submodels Given an undeadlocked AMAS $S^{\text{$\epsilon$}}$, partial order reduction attempts to generate only a subset of states and transitions that is sufficient for verification of $S^{\text{$\epsilon$}}$, i.e., a relevant _submodel_ of $\mathit{IIS}(S^{\text{$\epsilon$}})$. ###### Definition A.1 (Submodel). Let models $M,{M{{}^{\prime}}}$ extend the same AMAS $S^{\text{$\epsilon$}}$, so that $St^{\prime}\subseteq St$, $\iota\in St^{\prime}$, $T$ is an extension of $T^{\prime}$, and $V^{\prime}=V|_{St^{\prime}}$. Then, we write ${M{{}^{\prime}}}\subseteq M$ and call ${M{{}^{\prime}}}$ a _submodel_ of $M$. Note that, for each $g\in St^{\prime}$, we have $\Pi_{{M{{}^{\prime}}}}(g)\subseteq\Pi_{M}(g)$. ###### Lemma A.2. Let ${M{{}^{\prime}}}\subseteq M$, $A\in{\mathbb{A}\mathrm{gt}}$, $\sigma_{A}\in\Sigma_{A}^{\mathrm{ir}}$. Then, we have $\mathit{out}^{\textup{React}}_{{M{{}^{\prime}}}}(\iota,\sigma_{A})=\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{A})\cap\Pi_{{M{{}^{\prime}}}}(\iota)$. _Proof._ Note that each joint $\mathrm{ir}$-strategy in $M$ is also a well defined $\mathrm{ir}$-joint strategy in ${M{{}^{\prime}}}$ as it is defined on the local states of each agent of an AMAS which is extended by both $M$ and ${M{{}^{\prime}}}$. The lemma follows directly from the definition of React- outcome (Def. 5.1 and 7.3), plus the fact that $\Pi_{{M{{}^{\prime}}}}(\iota)\subseteq\Pi_{M}(\iota)$. $\blacksquare$ ###### Lemma A.3. Let $M$ be a model, $\pi,\pi^{\prime}\in\Pi_{M}(\iota)$, and for some $i\in{\mathbb{A}\mathrm{gt}}:$ $\mathit{Evt}(\pi)\mid_{\mathit{Evt}_{i}}=\mathit{Evt}(\pi^{\prime})\mid_{\mathit{Evt}_{i}}$. Then, for each $\mathrm{ir}$-strategy $\sigma_{i}$, we have $\pi\in\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{i})$ iff $\pi^{\prime}\in\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{i})$. _Proof._ Let $\mathit{Evt}(\pi)\mid_{\mathit{Evt}_{i}}=b_{0}b_{1}\ldots$ be the sequence of the events of agent $i$ in $\pi$. For each $b_{j}$ let $\pi[b_{j}]$ denote the global state from which $b_{j}$ is executed in $\pi$. By induction we can show that for each $j\geq 0$, we have $\pi[b_{j}]^{i}=\pi^{\prime}[b_{j}]^{i}$. For $j=0$ it is easy to see that $\pi[b_{0}]^{i}=\pi[b_{0}]^{i}=\iota^{i}$. Assume that the thesis holds for $j=k$. The induction step follows from the fact the local evolution $T_{i}$ is a function, so if $\pi[b_{k}]^{i}=\pi^{\prime}[b_{k}]^{i}=l$ for some $l\in L_{i}$, then $\pi[b_{k+1}]^{i}=\pi^{\prime}[b_{k+1}]^{i}=T_{i}(l,b_{k})$. Thus, by Def. 5.1 and 7.3, for each $\mathrm{ir}$-strategy $\sigma_{i}$ we have $\pi\in\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{i})$ iff $\pi^{\prime}\in\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{i})$, which concludes the proof. $\blacksquare$ Lemma A.3 can be easily generalized to joint strategies $\sigma_{A}\in\Sigma_{A}^{\mathrm{ir}}$. ### A.2 Stuttering Equivalence Let $M$ be a model, ${M{{}^{\prime}}}\subseteq M$, and $\mathit{PV}\subseteq\mathcal{PV}$ a subset of propositions. Stuttering equivalence says that two paths can be divided into corresponding finite segments, each satisfying exactly the same propositions. Stuttering path equivalence444 The property is usually called _stuttering trace equivalence_ (?). We use a slightly different name to avoid confusion with Mazurkiewicz traces, also used in this paper. requires two models to always have corresponding, stuttering-equivalent paths. ###### Definition A.4 (Stuttering equivalence). Two paths $\pi\in\Pi_{M}(\iota)$ and $\pi^{\prime}\in\Pi_{{M{{}^{\prime}}}}(\iota)$ are stuttering equivalent, denoted $\pi\equiv_{s}\pi^{\prime}$, if there exists a partition $B_{0}=(\pi[0],\dots,\pi[i_{1}-1]),\ B_{1}=(\pi[i_{1}],\dots,\pi[i_{2}-1]),\ \ldots$ of the states of $\pi$, and an analogous partition $B^{\prime}_{0},B^{\prime}_{1},\ldots$ of the states of $\pi^{\prime}$, s.t. for each $j\geq 0:$ $B_{j}$ and $B^{\prime}_{j}$ are nonempty and finite, and $V(g)\cap\mathit{PV}=V^{\prime}(g^{\prime})\cap\mathit{PV}$ for every $g\in B_{j}$ and $g^{\prime}\in B^{\prime}_{j}$. Models $M$ and ${M{{}^{\prime}}}$ are stuttering path equivalent, denoted $M\equiv_{s}{M{{}^{\prime}}}$ if for each path $\pi\in\Pi_{M}(\iota)$, there is a path $\pi^{\prime}\in\Pi_{{M{{}^{\prime}}}}(\iota)$ such that $\pi\equiv_{s}\pi^{\prime}$.555Typically, the definition also contains the symmetric condition which in our case always holds for $M$ and its submodel ${M{{}^{\prime}}}$, as $\Pi_{{M{{}^{\prime}}}}(\iota)\subseteq\Pi_{M}(\iota)$. ###### Theorem A.5 ((?)). If $M\equiv_{s}{M{{}^{\prime}}}$, then we have $M,\iota\models\varphi$ iff ${M{{}^{\prime}}},\iota^{\prime}\models\varphi$, for any $\mathbf{LTL_{-X}}$ formula $\varphi$ over $\mathit{PV}$. ### A.3 Independence of Events Intuitively, an event is invisible iff it does not change the valuations of the propositions.666 This concept of invisibility is technical, and is not connected to the view of any agent in the sense of (?). Additionally, we can designate a subset of agents $A$ whose events are visible by definition. Furthermore, two events are independent iff they are not events of the same agent and at least one of them is invisible. ###### Definition A.6 (Invisible events). Consider a model $M$, a subset of agents $A\subseteq{\mathbb{A}\mathrm{gt}}$, and a subset of propositions $\mathit{PV}\subseteq\mathcal{PV}$. An event $\alpha\in\mathit{Evt}$ is invisible wrt. $A$ and $\mathit{PV}$ if $Agent(\alpha)\cap A=\emptyset$ and for each two global states $g,g^{\prime}\in St$ we have that $g\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}g^{\prime}$ implies $V(g)\cap\mathit{PV}=V(g^{\prime})\cap\mathit{PV}$. The set of all invisible events for $A,\mathit{PV}$ is denoted by $Invis_{A,\mathit{PV}}$, and its closure – of visible events – by $Vis_{A,\mathit{PV}}=\mathit{Evt}\setminus Invis_{A,\mathit{PV}}$. ###### Definition A.7 (Independent events). The notion of _independence_ $I_{A,\mathit{PV}}\subseteq\mathit{Evt}\times\mathit{Evt}$ is defined as: $I_{A,\mathit{PV}}=\\{(\alpha,\alpha^{\prime})\in\mathit{Evt}\times\mathit{Evt}\mid Agent(\alpha)\cap Agent(\alpha^{\prime})=\emptyset\\}\ \setminus\ (Vis_{A,\mathit{PV}}\times Vis_{A,\mathit{PV}})$. Events $\alpha,\alpha^{\prime}\in\mathit{Evt}$ are called dependent if $(\alpha,\alpha^{\prime})\not\in I_{A,\mathit{PV}}$. If it is clear from the context, we omit the subscript $\mathit{PV}$. ### A.4 Preserving Stuttering Equivalence Rather than generating the full model $M=\mathit{IIS}(S^{\text{$\epsilon$}})$, one can generate a reduced model ${M{{}^{\prime}}}$ satisfying the following property: $\textbf{AE}_{A}:\>\forall\sigma_{A}\\!\in\\!\Sigma_{A}^{\mathrm{ir}}\quad\forall\pi\\!\in\\!\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{A})$ $\qquad\exists\pi^{\prime}\\!\in\\!\mathit{out}^{\textup{React}}_{{M{{}^{\prime}}}}(\iota,\sigma_{A})\quad\pi\\!\equiv_{s}\\!\pi^{\prime}$. We define a class of algorithms that generate reduced models satisfying $\textbf{AE}_{A}$ (Section A.4), and then prove that these models preserve $\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$ (Section A.4). Algorithms for partial order reduction. POR is used to reduce the size of models while preserving satisfaction for a class of formulas. The standard DFS (?) or DDFS (?) is modified in such a way that from each visited state $g$ an event $\alpha$ to compute the successor state $g_{1}$ such that $g\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}g_{1}$, is selected from $E(g)\cup\\{{\epsilon}\\}$ such that $E(g)\subseteq enabled(g)\setminus\\{{\epsilon}\\}$. That is, the algorithm always selects $\epsilon$, plus a subset of the enabled events at $g$. Let $A\subseteq{\mathbb{A}\mathrm{gt}}$. The conditions on the heuristic selection of $E(g)$ given below are inspired by (?; ?; ?). C1 Along each path $\pi$ in $M$ that starts at $g$, each event that is dependent on an event in $E(g)$ cannot be executed in $\pi$ without an event in $E(g)$ being executed first in $\pi$. Formally, $\forall\pi\in\Pi_{M}(g)$ such that $\pi=g_{0}\alpha_{0}g_{1}\alpha_{1}\ldots$ with $g_{0}=g$, and $\forall b\in\mathit{Evt}$ such that $(b,c)\notin I_{A}$ for some $c\in E(g)$, if $\alpha_{i}=b$ for some $i\geq 0$, then $\alpha_{j}\in E(g)$ for some $j<i$. C2 If $E(g)\neq enabled(g)\setminus\\{{\epsilon}\\}$, then $E(g)\subseteq Invis_{A}$. C3 For every cycle in ${M{{}^{\prime}}}$ containing no $\epsilon$-transitions, there is at least one node $g$ in the cycle for which $E(g)=enabled(g)\setminus\\{{\epsilon}\\}$, i.e., for which all the successors of $g$ are expanded. ###### Theorem A.8. Let $A\subseteq{\mathbb{A}\mathrm{gt}}$, $M=\mathit{IIS}(S^{\text{$\epsilon$}})$, and ${M{{}^{\prime}}}\subseteq M$ be the reduced model generated by DFS with the choice of $E(g^{\prime})$ for $g^{\prime}\in St^{\prime}$ given by conditions C1, C2, C3 and the independence relation $I_{A}$. Then, ${M{{}^{\prime}}}$ satisfies $\textbf{AE}_{A}$. _Proof._ Let ${M{{}^{\prime}}}\subseteq M=\mathit{IIS}(S^{\text{$\epsilon$}})$ be the reduced model generated as specified. Notice that the reduction of $M$ under the conditions C1, C2, C3 above is equivalent to the reduction of $M$ without the $\epsilon$-loops under the conditions C1, C2, C3 of (?), and then adding the $\epsilon$-loops to all the states of the reduced model. Although the setting is slightly different, it can be shown similarly to (?, Theorem 12) that the conditions C1, C2, C3 guarantee that the models: (i) $M$ without $\epsilon$-loops and (ii) ${M{{}^{\prime}}}$ without $\epsilon$-loops are stuttering path equivalent. More precisely, for each path $\pi=g_{0}a_{0}g_{1}a_{1}\cdots$ with $g_{0}=\iota$ (without $\epsilon$-transitions) in $M$ there is a stuttering equivalent path $\pi^{\prime}=g^{\prime}_{0}a^{\prime}_{0}g^{\prime}_{1}a^{\prime}_{1}\cdots$ with $g^{\prime}_{0}=\iota$ (without $\epsilon$-transitions) in $M^{\prime}$ such that $\mathit{Evt}(\pi)|_{Vis_{A}}=\mathit{Evt}(\pi^{\prime})|_{Vis_{A}}$, i.e., $\pi$ and $\pi^{\prime}$ have the same maximal sequence of visible events for $A$. (*) We will now prove that this implies $M\equiv_{s}{M{{}^{\prime}}}$. Removing the $\epsilon$-loops from $M$ eliminates two kinds of paths: (a) paths with infinitely many “proper” events, and (b) paths ending with an infinite sequence of $\epsilon$-transitions. Consider a path $\pi$ of type (a) from $M$. Notice that the path $\pi_{1}$, obtained by removing the $\epsilon$-transitions from $\pi$, is stuttering-equivalent to $\pi$. Moreover, by (*), there exists a path $\pi_{2}$ in ${M{{}^{\prime}}}$ without $\epsilon$-transitions, which is stuttering-equivalent to $\pi_{1}$. By transitivity of the stuttering equivalence, we have that $\pi_{2}$ is stuttering equivalent to $\pi$. Since $\pi_{2}$ must also be a path in ${M{{}^{\prime}}}$, this concludes this part of the proof. Consider a path $\pi$ of type (b) from $M$, i.e., $\pi$ ends with an infinite sequence of $\epsilon$-transitions. Let $\pi_{1}$ be the sequence obtained from $\pi$ after removing $\epsilon$-transitions, and $\pi_{2}$ be any infinite path without $\epsilon$-transitions such that $\pi_{1}$ is its prefix. Then, it follows from (*) that there is a stuttering equivalent path $\pi_{2}^{\prime}=g^{\prime}_{0}a^{\prime}_{0}g^{\prime}_{1}a^{\prime}_{1}\cdots$ with $g^{\prime}_{0}=\iota$ in $M^{\prime}$ such that $\mathit{Evt}(\pi_{2})|_{Vis_{A}}=\mathit{Evt}(\pi_{2}^{\prime})|_{Vis_{A}}$. Consider the minimal finite prefix $\pi_{1}^{\prime}$ of $\pi_{2}^{\prime}$ such that $\mathit{Evt}(\pi_{1}^{\prime})|_{Vis_{A}}=\mathit{Evt}(\pi_{1})|_{Vis_{A}}$. Clearly, $\pi_{1}^{\prime}$ is a sequence in $M^{\prime}$ and can be extended with an infinite number of $\epsilon$-transitions to the path $\pi^{\prime}$ in $M^{\prime}$. It is easy to see that $\pi$ and $\pi^{\prime}$ are stuttering equivalent. So far, we have shown that our reduction under the conditions C1, C2, C3 guarantees that the models $M$ and ${M{{}^{\prime}}}$ are stuttering path equivalent, and more precisely that for each path $\pi=g_{0}a_{0}g_{1}a_{1}\cdots$ with $g_{0}=\iota$ in $M$ there is a stuttering equivalent path $\pi^{\prime}=g^{\prime}_{0}a^{\prime}_{0}g^{\prime}_{1}a^{\prime}_{1}\cdots$ with $g^{\prime}_{0}=\iota$ in $M^{\prime}$ such that $\mathit{Evt}(\pi)|_{Vis_{A}}=\mathit{Evt}(\pi^{\prime})|_{Vis_{A}}$, i.e., $\pi$ and $\pi^{\prime}$ have the same maximal sequence of visible events for $A$. To show that ${M{{}^{\prime}}}$ satisfies $\textbf{AE}_{A}$, consider an $\mathrm{ir}$-joint strategy $\sigma_{A}$ and $\pi\in\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{A})$. As demonstrated above, there is $\pi^{\prime}\in\Pi_{{M{{}^{\prime}}}}(\iota)$ such that $\pi\equiv_{s}\pi^{\prime}$ and $\mathit{Evt}(\pi)|_{Vis_{A}}=\mathit{Evt}(\pi^{\prime})|_{Vis_{A}}$. Since $\mathit{Evt}_{i}\subseteq Vis_{A}$ for each $i\in A$, the same sequence of events of each $\mathit{Evt}_{i}$ is executed in $\pi$ and $\pi^{\prime}$. Thus, by the generalization of Lemma A.3 to $\mathrm{ir}$-joint strategies we get $\pi^{\prime}\in\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{A})$. So, by Lemma A.2 we have $\pi^{\prime}\in\mathit{out}^{\textup{React}}_{M{{}^{\prime}}}(\iota,\sigma_{A})$. $\blacksquare$ Algorithms generating reduced models, in which the choice of $E(g)$ is given by similar conditions, can be found for instance in (?; ?; ?; ?; ?; ?). POR for proactive opponents. The same reduction still works without the assumption of opponent-reactiveness (React). ###### Theorem A.9. Let $M^{\text{$\epsilon$}}=IIS^{\text{$\epsilon$}}(S)$ be an undeadlocked IIS. Then, its reduced model $M^{\text{$\epsilon$}}{{}^{\prime}}$, generated as in Theorem A.8, satisfies $\textbf{AE}_{A}$. _Proof (Sketch)._ In this setting, there is no auxiliary agent in the AMAS, and $\epsilon$-transitions are added directly to the IIS in accordance with Definition 4.3. Hence, not every global state of $M^{\text{$\epsilon$}}$ necessarily has an $\epsilon$ loop, but only those where a miscoordinating combination of events exists. However, this does not impact the reduction itself. First, note that Lemma A.2 still holds, directly from the definition of outcome (Definition 3.5). Furthermore, because in the undeadlocked IIS $M^{\text{$\epsilon$}}$ the $\epsilon$-transitions do not belong to any agent, Lemma A.3, where sequences of some agent $i$’s events are considered, also holds. Note that the React condition only restricts the outcome sets, and not the model itself: both $M=IIS(S^{\epsilon})$ and $M^{\text{$\epsilon$}}$ contain the same two types (a) and (b) of paths with $\epsilon$-transitions as discussed in Theorem A.8. Hence, following its reasoning, it can first be shown that models $M^{\text{$\epsilon$}}$ and $M^{\text{$\epsilon$}}{{}^{\prime}}$ without their $\epsilon$-transitions are stuttering path equivalent, and that it remains the case also when both types of paths including $\epsilon$ loops are included. Note that the remark about ${M{{}^{\prime}}}$ being equivalent to reducing $M$ without $\epsilon$ loops and adding them to each global state obviously does not apply to $M^{\text{$\epsilon$}}$ (not every global state of $M^{\text{$\epsilon$}}$ has them in the first place). However, this observation has no bearing on the proof. As before, $\epsilon$ is explicitly stated to be selected for the subset $E(g)$, ensuring preservation of representative paths with $\epsilon$ in $M^{\text{$\epsilon$}}{{}^{\prime}}$. $\blacksquare$ Correctness of reductions satisfying $\textbf{AE}_{A}$. We show that the reduced models satisfying $\textbf{AE}_{A}$ preserve $\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$. ###### Theorem A.10. Let $A\subseteq{\mathbb{A}\mathrm{gt}}$, and let models ${M{{}^{\prime}}}\subseteq M$, $M^{\text{$\epsilon$}}{{}^{\prime}}\subseteq M^{\text{$\epsilon$}}$ satisfy $\textbf{AE}_{A}$. For each $\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$ formula $\varphi$ over $\mathit{PV}$, that refers only to coalitions $\hat{A}\subseteq A$, we have that: 1. 1. $M,\iota\models_{{}_{\mathrm{ir}}}^{\textup{React}}\varphi$ iff ${M{{}^{\prime}}},\iota^{\prime}\models_{{}_{\mathrm{ir}}}^{\textup{React}}\varphi$, and 2. 2. $M^{\text{$\epsilon$}},\iota\models_{{}_{\mathrm{ir}}}\varphi$ iff $M^{\text{$\epsilon$}}{{}^{\prime}},\iota^{\prime}\models_{{}_{\mathrm{ir}}}\varphi$. _Proof._ Proof by induction on the structure of $\varphi$. We show the case $\varphi=\langle\\!\langle{\hat{A}}\rangle\\!\rangle_{{}_{\\!\mathit{}}}\gamma$. The cases for $\neg,\land$ are straightforward. Notice that $\mathit{out}^{\textup{React}}_{{M{{}^{\prime}}}}(\iota,\sigma_{\hat{A}})\subseteq\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{\hat{A}})$, which together with the condition $\textbf{AE}_{A}$ implies that the sets $\mathit{out}^{\textup{React}}_{M}(\iota,\sigma_{\hat{A}})$ and $\mathit{out}^{\textup{React}}_{{M{{}^{\prime}}}}(\iota,\sigma_{\hat{A}})$ are stuttering path equivalent. Analogously, this is the case for $\mathit{out}_{{M{{}^{\prime}}}}(\iota,\sigma_{\hat{A}})\subseteq\mathit{out}_{M}(\iota,\sigma_{\hat{A}})$, i.e. without the React assumption. Hence, (1) and (2) follow from Theorem A.5. $\blacksquare$ Together with Theorems A.8 and A.9, we obtain the following. ###### Theorem A.11. Let $M=\mathit{IIS}(S^{\text{$\epsilon$}})$, $M^{\text{$\epsilon$}}=IIS^{\text{$\epsilon$}}(S)$ and let ${M{{}^{\prime}}}\subseteq M$ and $M^{\text{$\epsilon$}}{{}^{\prime}}\subseteq M^{\text{$\epsilon$}}$ be the reduced models generated by DFS with the choice of $E(g^{\prime})$ for $g^{\prime}\in St^{\prime}$ given by conditions C1, C2, C3 and the independence relation $I_{A,\mathit{PV}}$. For each $\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$ formula $\varphi$ over $\mathit{PV}$, that refers only to coalitions $\hat{A}\subseteq A$, we have: 1. 1. $M,\iota\models_{{}_{\mathrm{ir}}}^{\textup{React}}\varphi$ iff ${M{{}^{\prime}}},\iota^{\prime}\models_{{}_{\mathrm{ir}}}^{\textup{React}}\varphi$, and 2. 2. $M^{\text{$\epsilon$}},\iota\models_{{}_{\mathrm{ir}}}\varphi$ iff $M^{\text{$\epsilon$}}{{}^{\prime}},\iota^{\prime}\models_{{}_{\mathrm{ir}}}\varphi$. This concludes the proof that the adaptation of POR for $\mathbf{LTL_{-X}}$ to $\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$, originally presented in (?), remains sound in the updated semantics proposed in Sections 4 and 7. That is, the structural condition $\textbf{AE}_{A}$ is sufficient to obtain correct reductions for $\mathbf{sATL_{\mathrm{\mathrm{ir}}}^{*}}$ with and without the new opponent-reactiveness assumption (Theorem A.11). Thanks to that, one can potentially reuse or adapt the existing POR algorithms and tools for $\mathbf{LTL_{-X}}$, and the actual reductions are likely to be substantial.

2024-09-04T02:54:58.237523

2003.03891

arxiv-papers

# Adaptive Extremum Seeking Using Recursive Least Squares Nursefa Zengin, Baris Fidan School of Engineering, University of Waterloo, ON, Canada<EMAIL_ADDRESS> ###### Abstract Extremum seeking (ES) optimization approach has been very popular due to its non-model based analysis and implementation. This approach has been mostly used with gradient based search algorithms. Since least squares (LS) algorithms are typically observed to be superior, in terms of convergence speed and robustness to measurement noises, over gradient algorithms, it is expected that LS based ES schemes will also provide faster convergence and robustness to sensor noises. In this paper, with this motivation, a recursive least squares (RLS) estimation based ES scheme is designed and analysed for application to scalar parameter and vector parameter static map and dynamic systems. Asymptotic convergence to the extremum is established for all the cases. Simulation studies are provided to validate the performance of proposed scheme. ## I Introduction Extremum seeking (ES) is a popular technique for adaptive optimization of the performance of dynamic systems by tuning certain system parameters based on measurements. The main advantage of this technique is that limited or no knowledge of the plant model is required. ES is suitable for optimization of the performance of systems with complex dynamics, unavailable suitable measurements to validate the model, and time-varying disturbances that are difficult to model accurately ([1]). The most common ES algorithm used in the literature is the classical band-pass filtering based one, in which the gradient of the output with respect to the input will determine the direction of adjusting the input variables. This method was successfully applied to different application areas including biochemical reactors [[2, 3]], ABS control in automotive brakes ([4, 1, 5, 6, 7]), mobile robots ([8, 9, 10]), mobile sensor networks ([11, 12, 13]). Among other types of ES algorithms, perturbation based ES relies on added perturbation signals to estimate the gradient of the output by correlating the perturbations. To overcome the implementation drawbacks of introducing perturbation signals, some methods that are free of perturbation signals have been developed by [14, 15, 16]. Convergence rate of conventional ES algorithms is a limiting factor in many applications. Recursive Least Squares (RLS) based estimation has significant potential in relaxing this limitation and improving robustness to measurement noises. [17, 15, 18] used certain LS based techniques in their ES algorithms to obtain better convergence results. [17] estimated the gradient of the output with respect to the input using a LS based adaptive law for a class of nonlinear dynamic systems together with a sinusoidal perturbation signal. [15] used past data of a performance map to estimate the gradient of this performance map by a first order LS fit. The proposed method used no dither signal, but utilized a time window of history data of the performance map. [18] provided general results and a framework for the design of ES schemes applied to systems with parametric uncertainties and used LS algorithm to estimate unknown parameters of the known system. In absence of the parameter knowledge, a series of control/optimization schemes have been proposed in the literature utilizing certain ES tools such as switching methods ([19]), signal perturbation for persistence excitation, and band pass filtering ([19],[20],[21],[18]. [22] and [23] used a discrete time ES scheme to estimate the gradient as a time-varying parameter using LS like update laws. They removed the need for averaging system in order to achieve the convergence of ES. The designs are simulated for static unknown maps, systems with unknown discrete-time dynamics and sampled-data systems. In this paper, a continuous time RLS parameter estimation based ES scheme is designed and analysed for scalar parameter and vector parameter static map and dynamic systems. Asymptotic convergence to the extremum is established for each case. Numerical simulation examples are provided to validate the performance of proposed scheme comparing the results with gradient parameter estimation based one. A specific simulation example, antilock braking systems (ABS), in [1] is studied to compare the performance of RLS estimation based ES with classical gradient based ES. Contents of this paper are as follows. Section II is dedicated to the problem statement. In Section III, existing classical perturbation based ES is reviewed. Proposed RLS estimation based adaptive ES is developed for scalar parameter systems in Section IV, and for vector parameter systems in Section V. Comparative simulation examples are presented in Section VI. Finally, conclusions of the paper are given in Section VII. ## II Problem Statement The ES problem of interest is defined for static map systems and dynamic systems separately in the following subsections. ### II-A Static Maps Consider a concave static map system $\displaystyle y=h_{s}(u)=\bar{h}_{s}(\theta^{*},u),\quad\theta^{*}=\begin{bmatrix}\theta^{*}_{1}&\cdots&\theta^{*}_{N}\end{bmatrix}^{T},$ (1) where $\theta^{*}\in\mathbb{R}^{N}$ is a fixed unknown parameter vector, $u\in\mathbb{R}^{m}$ is the input and $y\in\mathbb{R}$ is the output of the system. Assume that the control input signal $u$ is generated by a smooth control law $u=\alpha(\theta)$ (2) parametrized by a control parameter vector $\theta\in\mathbb{R}^{N}$. ###### Assumption 1 The static map $\bar{h}_{s}(\theta^{*},u)$ is smoothly differentiable. ###### Assumption 2 $h_{s}(u)=\bar{h}_{s}(\theta^{*},u)$ has a single extremum (maximum) $y^{*}$ at $u=\alpha(\theta^{*}).$ The control objective is to maximize the steady-state value of $y$ but without requiring the knowledge of $\theta^{*}$ or the system function $h_{s}$. ### II-B Dynamic Systems Consider a general multi-input-single-output (MISO) nonlinear system $\dot{x}=f(x,u)=\bar{f}(\theta^{*},x,u),$ (3) $y=h_{d}(x)=\bar{h}_{d}(\theta^{*},\theta)=h(\theta),$ (4) $\theta=\pi(x)$ (5) where $x\in\mathbb{R}^{n}$ is the state, $u\in\mathbb{R}^{m}$ is the input, $y\in\mathbb{R}$ is the output, all measurable, and $f:\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ and $h_{d}=h\circ\pi$ are smooth functions. Assume that the control input signal $u$ is in the form (2), the control parameter $\theta\in\mathbb{R}^{N}$ is dependant on $x$ through a map $\pi(.):\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}$. The closed loop system can be written as follows: $\dot{x}=f(x,\alpha(\theta))=f(x,\alpha(\pi(x)).$ (6) The equilibria of (6) can be parameterized by $\theta$. The following assumptions about the closed loop system (3) are made, similarly to [21]. ###### Assumption 3 There exists a smooth function $l:\mathbb{R}^{N}\rightarrow\mathbb{R}^{m}$ such that $f(x,\alpha(x,\theta))=0\quad\text{if and only if}\quad x=l(\theta),$ (7) for any $(x,\theta)\in\mathbb{R}^{m}\times\mathbb{R}^{N}.$ For each $\theta\in\mathbb{R}^{N}$, the equilibrium $x_{e}=l(\theta)$ of the system (6) is locally exponentially stable with decay and overshoot constants uniformly dependent on $\theta$. ###### Assumption 4 There exists $\theta^{*}\in\mathbb{R}^{N}$ such that for all admissible $x$ values, $h_{d}(x)$ has its unique maximum at $x=x^{*}=l(\theta^{*}),$ $y^{\prime}(x^{*})={\frac{\partial h}{\partial x}}\Bigr{|}_{\begin{subarray}{c}x=x^{*}\end{subarray}}=0,$ (8) and the $m\times m$ Hessian matrix $y^{\prime\prime}(x^{*})={\frac{\partial^{2}h}{\partial x^{2}}}\Bigr{|}_{\begin{subarray}{c}x=x^{*}\end{subarray}}$ is negative definite. The control objective is to maximize the steady-state value of $y$ but without requiring the knowledge of $\theta^{*}$ or the system functions $h_{d},f$. This objective could be perfectly performed if $\theta^{*}$ was known and substituted in (2). The control parameter vector estimation can be done in different ways, leading to different ES schemes, even for the fixed control structure (2). The assumption that $h$ has a maximum is without loss of generality, considering a maximum seeking task. Minimum seeking case would be treated identically, replacing $y$ with $-y$ in the subsequent feedback design. In the next section, existing classical perturbation based ES approach will be reviewed to give an idea about our proposed design and to later use in simulation comparisons. ## III Classical Perturbation Based Extremum Seeking for Dynamic Systems In the classical ES approach shown in Fig.1, a high pass filter, a multiplier, and a lowpass filter are used to find the extremum. A general single input nonlinear system is considered in the design of [21]. A multi input ES approach is examined in [24]. Figure 1: Classic perturbation based ES scheme for multi input dynamic systems given by [24]. In the approach in [1, 21], the control law (2) feeding the plant (3) is tuned via the time-varying parameter $\theta=\begin{bmatrix}\theta_{1},\theta_{2},\cdots,\theta_{N}\end{bmatrix}^{T}$ that is produced by $\displaystyle\theta(t)=\hat{\theta}(t)+S(t),$ (9) where $\displaystyle S(t)$ $\displaystyle=\begin{bmatrix}a_{1}sin(\omega_{1}t)&a_{2}sin(\omega_{2}t)&\cdots&a_{N}sin(\omega_{N}t)\end{bmatrix}^{T},$ (10) and $\hat{\theta}(t)$ is generated by $\displaystyle\dot{\hat{\theta}}(t)$ $\displaystyle=k\hat{G}(t),$ (11) $\displaystyle\dot{\hat{G}}(t)$ $\displaystyle=\omega_{l}M(t)(y(t)-\eta(t))-\omega_{l}\hat{G}(t),$ $\displaystyle\dot{\eta}(t)$ $\displaystyle=\omega_{h}\left(y(t)-\eta(t)\right).$ Perturbation signal is selected as $M(t)=\begin{bmatrix}\frac{2}{a_{1}}sin(\omega_{1}t)&\frac{2}{a_{2}}sin(\omega_{2}t)&...&\frac{2}{a_{N}}sin(\omega_{N}t)\end{bmatrix}^{T}$. In the next two sections, we develop RLS estimation based ES scheme with forgetting factor instead of the approach of Section 3. Our proposed RLS estimation based adaptive ES scheme will be separately developed for two cases: for scalar parameter $(N=1)$ systems and for vector parameter $(N>1)$ systems, in Sections 4 and 5, respectively. ## IV RLS based ES Design for Scalar Parameter Systems ### IV-A Static Maps Consider the static map (1) and the control law (2) for scalar case, $N=1$, under Assumptions 1 and 2 about the closed-loop system. The proposed scheme is depicted in Fig. 2. RLS estimation based ES block shown in Fig. 2 consists of two parts: an RLS based adaptive parameter identifier estimating the gradient $h_{\theta}=\frac{\partial y}{\partial\theta}$ and a control law to be fed by this estimate. Figure 2: RLS based ES scheme for scalar parameter static maps. Consider the static map equation (1). In this equation, the time derivative of the output $y$ is given by $\dot{y}=h_{\theta}\dot{\theta}.$ (12) Design of the RLS based estimator to generate $\hat{h}_{\theta}$ considers the relation (12) that is in the linear parametric model form. $z=h_{\theta}\phi.$ (13) where $z=\dot{y},\quad\phi=\dot{\theta}.$ (14) If $\dot{y}$ is not available for measurement, then the regressor signals can be generated as $\displaystyle z=\frac{s}{s+\omega_{l}}[y],\quad\phi=\frac{1}{s+\omega_{l}}[\dot{\theta}],$ (15) i.e., $\displaystyle\dot{z}=-\omega_{l}z+\dot{y},\quad\dot{\phi}=-\phi\omega_{l}+\dot{\theta},$ (16) where $\omega_{l}>0$ is a constant design parameter. The control law generating $\theta$ is proposed to be $\dot{\theta}=k\hat{h}_{\theta},\quad k>0.$ (17) Assuming that the time variation of $h_{\theta}$ is sufficiently slow, we design an RLS estimator for the parametric model (13) as follows: $\dot{\hat{h}}_{\theta}=p\epsilon\phi,$ (18) $\dot{p}=\beta p-p^{2}\phi^{2},$ (19) $\epsilon=z-\hat{h}_{\theta}\phi,$ (20) where $\beta>0$ is forgetting factor and $p$ is the covariance term. The overall ES scheme producing $\theta(t)$ can be summarized by (17), (18), (19), and (20). ### IV-B Dynamic Systems The RLS estimation based ES control scheme (17)-(20) applies to the dynamic system (3)-(5) for $N=1$ with the control law (2) under Assumptions 3 and 4. The proposed ES scheme is depicted in Fig. 3. Figure 3: RLS based ES scheme for scalar parameter dynamic systems. ### IV-C Stability Analysis In this section, stability proof of the proposed schemes in Sections IV-A and IV-B will be presented. We know that $\theta^{*}$ is the equilibrium point and the estimated gradient will be $h_{\theta}=0$ at the equilibrium point $\theta=\theta^{*}$. We can write our stability result as follows: ###### Theorem IV.1 Consider the RLS estimation based ES scheme given in Figs. 2, 3 and defined in (17) - (20) with $z$ and $\phi$ as given in (14) or (15), and Assumptions 1 \- 4. For any initial condition $\hat{\theta}(0)\in\mathbb{R}^{N}$ and adaptation gain $k$, $\theta(t)$ asymptotically converges to small neighborhood of extremum parameter $\theta^{*}$. ###### Proof: We consider the Lyapunov function as $V(\theta(t))=\frac{1}{2}\left(\theta(t)-\theta^{*}\right)^{2}=\frac{1}{2}\tilde{\theta}^{2}.$ (21) We write the time derivative of $V$ along the solutions of (17) as $\dot{V}=\dot{\theta}\left(\theta(t)-\theta^{*}\right)=\dot{\theta}\tilde{\theta}.$ (22) Substituting (17) into (22), we obtain $\dot{V}=k\hat{h}_{\theta}\tilde{\theta}.$ (23) For the maximum case, $k>0$. Negative definiteness of (23) depends on the initial condition $\theta_{0}$ that determines the signs of $\hat{h}_{\theta}$ and $\tilde{\theta}$. If $\theta(0)<\theta^{*}$, then $\hat{h}_{\theta}>0$ and $\tilde{\theta}<0$. On the other hand, if $\theta(0)>\theta^{*}$, then $\hat{h}_{\theta}<0$ and $\tilde{\theta}>0$. Hence, for both cases $\dot{V}<0$. We also need to examine the forgetting factor $\beta$ and the persistent excitation (PE) of $\phi$. If $\phi$ is PE, then (17) guarantees that $p\in\mathcal{L}_{\infty}$ and $\theta(t)\to\theta^{*}$ as $t\to\infty$. When $\beta>0$, the convergence of $\theta(t)\to\theta^{*}$ is exponential ([25]). ∎ ## V RLS based ES Design for Vector Parameter Systems In this section, the proposed RLS estimation based ES scheme is extended to the systems with vector parameters $(N>1)$. Similar to the classical gradient based analysis, small sinusoidal perturbation signals with different frequencies ($\omega_{1},\cdots,\omega_{N}$) are added to the control signals to provide sufficiently rich excitation. ### V-A Static Maps Figure 4: RLS based ES scheme for vector parameter static maps. Consider the block diagram in Fig. 4 for the static map in (1). The time derivative of (1) is given by $\dot{y}=h_{\theta}^{T}\dot{\theta},$ (24) which, similarly to (13), can be written in the linear parametric form $z=h_{\theta}^{T}\phi,$ (25) where $z$ and $\phi$ are again defined by either (14) or (15). The control law (17) is used for updating $\theta$ in the vector case as well. The design of the RLS estimator to produce $\hat{h}_{\theta}$ is based on the parametric model (25) and is given as follows ([25]): $\dot{\hat{h}}_{\theta}=P\epsilon\phi,$ (26) $\dot{P}=\beta P-P\phi\phi^{T}P,$ (27) $\epsilon=z-\hat{h}_{\theta}^{T}\phi,$ (28) where $\beta$ is the forgetting factor and $P$ is the covariance matrix of the RLS algorithm. The control law generating $\theta$ is proposed to be $\dot{\hat{\theta}}=k\hat{h}_{\theta},\quad k>0.$ (29) $\theta(t)=\hat{\theta}(t)+S(t),$ (30) where $S(t)$ is defined as in (10). Different from scalar parameter systems, we use perturbation signals, $S(t)$. The need to use of dither signals in vector parameter systems is that dither signals with different frequencies can be implemented on each input signal to achieve overall PE. ### V-B Dynamic Systems Figure 5: RLS based ES scheme for vector parameter dynamic systems. The RLS estimation based ES scheme (26) - (29) applies to the dynamic system (3)-(5) with control law (2) under Assumptions 3 and 4 for vector parameter systems. Block diagram of the proposed ES scheme is given in Fig.5. ### V-C Stability Analysis The intuition in (30) is to satisfy persistence of excitation for $N$-dimensional $\phi$ by introducing at least one distinct dither frequency for each input, following the standard perturbation based ES control approaches mentioned in Section III. Similar to the analysis in Section IV-C, consider the Lyapunov function as $V(\tilde{\theta}(t))=\frac{1}{2}\tilde{\theta}^{T}\tilde{\theta}.$ (31) We write the time derivative of $V$ along the solutions of (29) as $\dot{V}=\tilde{\theta}^{T}\dot{\tilde{\theta}}=\tilde{\theta}^{T}\dot{\theta}.$ (32) Substituting (30) into (32), we obtain $\dot{V}=\tilde{\theta}^{T}(k\hat{h}_{\theta}+\dot{S}).$ (33) The relationship between $\tilde{\theta}$ and $\hat{h}_{\theta}$ in Section 4.3 applies to vector parameter case. The stability again depends on $k$, initial condition $\theta(0)$, forgetting factor $\beta$, and PE of $\phi$, that is guaranteed by addition of dither signals in (30). Hence, $P\in\mathcal{L}_{\infty}$ and $\theta(t)\to\theta^{*}$ as $t\to\infty$. ## VI Simulations In this section, we present simulation results to show the validity of the proposed schemes. We will present two examples for scalar parameter and vector parameter cases with their comparison results with classical ES method in Section III. ### VI-A Scalar Parameter Simulation Example Consider the following model $\displaystyle y$ $\displaystyle=10m(u),$ (34) $\displaystyle m(u)$ $\displaystyle=k_{1}\left(1-e^{-k_{2}u}\right)-k_{3}u$ $\displaystyle u$ $\displaystyle=\theta,$ where $\theta^{*}=0.3$. $\theta_{0}=0.01$ is chosen as initial value for both schemes. $k_{1}=1.05,k_{2}=23,k_{3}=0.52$ are given. For RLS estimation based ES scheme, the following parameters are used: $k_{ls}=0.01$, $p_{0}=10^{3}$, and $\beta=0.98$ are given. For classical ES scheme, the following parameters are given: $k=0.08$, $\omega_{h}=0.6$, $\omega_{l}=0.8$, $S(t)=0.01\sin 3t$, and $M(t)=sin3t$. We apply the Gaussian measurement noise as ($\sigma=0.05$) for both gradient and RLS algorithms. We apply RLS estimation based ES scheme in Fig.3. The results for this example is given in Fig.6. It is obvious that proposed scheme can reach a neighborhood of the extremum point $\theta^{*}=0.3$ at $y^{*}=8.85$ less than 2 second while classical ES finds the extremum point very late and cannot maintain that extremum point under measurement noise. Figure 6: Single parameter RLS estimation based ES results. ### VI-B Vector Parameter Simulation Example Consider the following model $\displaystyle y$ $\displaystyle=y_{1}+y_{2},$ (35) $\displaystyle y_{1}$ $\displaystyle=am(u_{1}),\leavevmode\nobreak\ m(u_{1})=(2m^{*}_{1}u^{*}_{1}u_{1})/(u^{*2}_{1}+u^{2}_{1}),$ $\displaystyle y_{2}$ $\displaystyle=am(u_{2}),\leavevmode\nobreak\ m(u_{2})=(2m^{*}_{2}u^{*}_{2}u_{2})/(u^{*2}_{2}+u^{2}_{2}),$ $\displaystyle u$ $\displaystyle=[u_{1},\leavevmode\nobreak\ u_{2}]=[\theta_{1},\leavevmode\nobreak\ \theta_{2}].$ where $[\theta^{*}_{1},\leavevmode\nobreak\ \theta^{*}_{2}]=[0.2,\leavevmode\nobreak\ 0.3]$. For both schemes, initial values are given as $u_{0}=[0.1,\leavevmode\nobreak\ 0.1].$ We aim to reach $y^{*}_{1}(\theta^{*}_{1})=5$ and $y^{*}_{2}(\theta^{*}_{2})=9$. For RLS estimation based ES scheme, the following parameters are used: $k=[0.01,\leavevmode\nobreak\ 0.01]$, $P_{0}=10^{4}$, $\beta=0.98$, and $S(t)=[0.01\sin 7t,\leavevmode\nobreak\ 0.01\sin 10t]$ are given. For classical ES scheme, the following parameters are given: $k=[0.02,\leavevmode\nobreak\ 0.01]$, $\omega_{h}=[0.6,\leavevmode\nobreak\ 0.6]$, $\omega_{l}=[0.8,\leavevmode\nobreak\ 0.8]$, $S(t)=[0.01\sin t,\leavevmode\nobreak\ 0.01\sin 2t]$, and $M(t)=[4.5\sin 5t,\leavevmode\nobreak\ 11\sin 5t]$. We apply the Gaussian measurement noise as ($\sigma=0.05$) for both gradient and RLS algorithms. Simulation results are given in Fig.7 for both RLS estimation based and classical ES schemes. It is clear that the results taken with RLS can converge the extremum point and find the maximized output $y^{*}$ while classical ES scheme has difficulty to reach the extremum point. One reason for this difficulty is that in classical ES scheme has many tuning parameters that must be tuned accordingly. For vector case, we also emphasize the need to apply perturbation terms to the scheme in order to observe multiple input channels separately. When there is no perturbation signal applied, the inputs cannot be distinguished and converge to an average value that caused to reach a value near the maximum. Similar to scalar case, RLS estimation based ES scheme outweighs classical ES scheme in terms of reaching extremum under measurement noises. ### VI-C ABS Simulation Example In this section, we also tested our ES scheme in ABS using MATLAB/Simulink. Then, we compared its performance with gradient based ES scheme developed by [1]. The wheel characteristics are given by the following set of equations Figure 7: Vector parameter RLS estimation based ES results. $\displaystyle m\dot{\nu}$ $\displaystyle=N\mu(\lambda),$ (36) $\displaystyle I\dot{\omega}$ $\displaystyle=-B\omega-NR\mu(\lambda)+\tau,$ where $v,\omega,m,N,R,I$ are linear velocity, angular velocity, the mass, the weight, radius, and the moment of inertia of the wheel, respectively. $B\omega$ is the bearing friction torque, $\tau$ is braking torque, $\mu(\lambda)$ is the friction force coefficient. $\lambda$ is the wheel slip which is defined as $\lambda(v,\omega)=\frac{R\omega-\nu}{\nu}.$ (37) Controller design procedure are identical to the design in [1]. The parameters that are identical in both schemes are given as follows: $m=400kg$, $R=0.3m$, $I=1.7kgm^{2}$, $B=0.01kg/s$. Perturbation signal amplitude and frequency is selected as $a=0.01$, $\omega=3$, high pass, low pass and regulation gain are selected as $\omega_{h}=0.6$, $\omega_{l}=0.8$, $k=6$ in gradient based scheme equations (9), (10), and (11). $k=-0.01$ is used in ABS case and $\beta=0.95$ is selected for RLS based scheme. The simulation for both gradient and RLS schemes is performed under the Gaussian noise ($\sigma=0.1$) in longitudinal acceleration measurement, $\dot{v}$. Initial conditions are selected the same in both schemes for a fair comparison. We use the approximation model (38) in simulations to see the effect of the proposed schemes. $\mu(\lambda)=2\mu_{max}\frac{\lambda^{*}\lambda}{{{\lambda}^{*2}}+\lambda^{2}},$ (38) where (38) has a maximum at $\lambda=\lambda^{*}$ with $\mu(\lambda^{*})=\mu_{m}$. For simulation, we choose wet road since it is one of the safety critical conditions. Simulation results of ABS for gradient/RLS based scheme comparison are given in Fig.8. Results show that vehicle stopping time of RLS parameter estimation based ES in an emergency situation is less than that of gradient one. Slip ratio estimation is almost 2 sec quicker with RLS parameter estimation, can be seen in Fig. 8(a). RLS based ES scheme gives better results under measurement noise and can reach the maximum deceleration in less time. (a) Friction force coefficient and estimated slip results for ABS. (b) Braking torque, velocity and deceleration results for ABS. Figure 8: Wet road comparison results for ABS. ## VII Conclusion This paper focuses on designing an RLS parameter estimation based ES scheme for scalar parameter and vector parameter static map and dynamic systems. Their stability conditions are stated for each case. The proposed ES scheme does not need perturbation signals for scalar parameter systems; however, the proposed ES scheme needs perturbation signals with different frequencies for vector parameter systems. 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2024-09-04T02:54:58.256546

2003.03956

arxiv-papers

# HEPLike: an open source framework for experimental likelihood evaluation Jihyun Bhom Marcin Chrzaszcz Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Krak´ow, Poland ###### Abstract We present a computer framework to store and evaluate likelihoods coming from High Energy Physics experiments. Due to its flexibility it can be interfaced with existing fitting codes and allows to uniform the interpretation of the experimental results among users. The code is provided with large open database, which contains the experimental measurements. The code is of use for users who perform phenomenological studies, global fits or experimental averages. ###### keywords: experimental high energy physics , likelihoods ††journal: Computer Physics Communications PROGRAM SUMMARY/NEW VERSION PROGRAM SUMMARY Program Title: HEPLike Licensing provisions(please choose one): GPLv3 Programming language: C++ Supplementary material: Journal reference of previous version: FIRST VERSION OF PROGRAM Nature of problem(approx. 50-250 words): Provide a uniform way of store, share and evaluate experimental likelihoods in a proper statistical manner. The code can be easily interfaced with existing global fitting codes. In addition a large database with the measurements is published. The program targets users who perform in their scientific work: phenomenological studies, global fits or measurements averages. The HEPLike has been created for FlavBit project[1], which was used to perform several analysis[2,3] and here we present an updated version, which can be used in standalone mode. Solution method(approx. 50-250 words): C++ code that evaluates the statistical properties of the measurements without user intervention. The large open database is provided as well. The measurements are stored in YAML files allowing for easy readability and extensions. ## References * [1] arXiv: 1705.07933 * [2] arXiv: 1705.07935 * [3] arXiv: 1705.07917 ## 1 Introduction In the High Energy Physics (HEP) the experimental measurements are performed by several collaborations, which measure various different observables. The experimental results are presented in various ways; some being as simple as a measurement with an Gaussian error, some more complicated as multiple correlated measurements with asymmetric errors or in some places even a full likelihood function is being published. To make things more complicated in some cases multiple representations of the same measurement are being published. All of this makes it hard to directly use and compare various different results. It also leaves a room for misinterpreting the results by theorists, which use these inputs to their studies. It happens that the asymmetric errors are being symmetrized, instead of using the full likelihood only central value with approximated asymmetric error is being used. The High Energy Physics Likelihoods (HEPLike) is a computer program that allows to store and share the likelihoods of various measured quantities. The published code can be useful for users performing phenomenological studies using experimental results, global fitting collaborations or experimental averages. Thanks to its structure it is easy to be interface with existing codes. It simplifies the work of people as instead of looking up the appropriate measurement and coding up their own likelihood they can download the database of measurements and choose the one they need. Furthermore, it shifts the burden of constructing the proper likelihood functions back to the experimentalists, which performed the measurement at the first place and are clearly the most appropriate people to handle this task. The computer code described in this paper is written in C++, making it useful for majority of fitting programs available on the market [1, 2, 3, 4, 5, 6]. The library can be used in both the $\chi^{2}$ and likelihood fits. Moreover, it contains a statistical module with useful functions that can be used in the statistical analysis. Besides the computer code a database with the likelihoods is being published. The measurements are stored in the YAML files making them easy to read by both the machine and human. This database can be easily extended by adding new YAML files if new measurement becomes available. With the software we provide useful utilities, which allows to perform searches inside the database, create BiBtex containing publications, which have been in the fit, etc. The paper is organized as follows: in Sec. 2 construction of the likelihood functions is presented. Sec. 3 explains the detailed code implementations and data storage, while Sec. 4 describes how to install and use HEPLike software. ## 2 Likelihood constructions In this section we will present how likelihoods in HEPLike are stored and constructed. Each measurement is stored in separate YAML file. There are several ways collaborations published their results depending on the measurements itself: * 1. Upper limits, * 2. Single measurement with symmetric uncertainty, * 3. Single measurement with asymmetric uncertainty, * 4. Multiple measurements with symmetric uncertainty, * 5. Multiple measurements with asymmetric uncertainty, * 6. One dimensional likelihood function, * 7. n-dimensional likelihood function. In addition, there is growing interest from the community that the experimental collaborations instead of only the results of the analysis publish also the dataset that has been used to obtain the result. For this future occasion we have also implement a way that this data can be directly used in the fits. Each of these cases has a different module of HEPLike that is designed to evaluate the likelihood functions. In this section we will present the statistical treatment of the above cases and the modules that are responsible for their evaluation. Each of the YAML files is required to have the following information (here for example we use the $R_{\mathup{{{K}}^{\scriptstyle{\ast}}}}$ measurement [7]): ⬇ 1BibCite: Aaij:2017vbb 2BibEntry: ’@article{Aaij:2017vbb, 3 author = ”Aaij, R. and others”, 4 title = ”{Test of lepton universality 5 with $B^{0} \rightarrow 6 K^{*0}\ell^{+}\ell^{-}$ decays}”, 7 collaboration = ”LHCb”, 8 journal = ”JHEP”, 9 volume = ”08”, 10 year = ”2017”, 11 pages = ”055”, 12 doi = ”10.1007/JHEP08(2017)055”, 13 eprint = ”1705.05802”, 14 archivePrefix = ”arXiv”, 15 primaryClass = ”hep-ex”, 16 reportNumber = ”LHCB-PAPER-2017-013, 17 CERN-EP-2017-100”, 18 SLACcitation = ”%%CITATION = ARXIV:1705.05802;%%” 19 } 20 ’ 21DOI: 10.1007/JHEP08(2017)055 22Process: R_{Kstar^{*}} 23FileName: RKstar.yaml 24Name: RKstar 25Source: HEPDATA 26SubmissionYear: 2017 27PublicationYear: 2018 28Arxiv: 1705.05802 29Collaborations: LHCb 30Kinematics: q2>1.1 && q2<6\. 31HLAuthor: Gal Anonim 32HLEmail<EMAIL_ADDRESS> 33HLType: HL_ProfLikelihood The above informations are used to store the information relevant for the bookkeeping. For instance the entries BibCite and BibEntry correspond to the information that are used to generate a BiBtex citation file with the measurements that have been used in the studies. The DOI corresponds to the digital object identifier of the publication. The Decay defines the process that has been studied. It can also be replaced by the Process entry. The Name is a unique name of this measurement type. If the measurement gets updated with more data or by other collaboration the Name entry in the new YAML file should be the same as in the old one. Source entry corresponds to the source of the measurement. This can be either a HEPData or the collaboration itself. The SubmissionYear (PublicationYear) refers to the year of appearance (publication) of the result. The Arxiv codes the Arxiv number, while the Collaborations stores the information which experimental collaboration has performed the measurement. Finally, the Kinematics stores additional information about the kinematic region that has been measured. The HLAuthor and HLEmail encode the information about the YAML file author and his email in case user needs further information about the encoded measurement. Last but not least the entry HLType contains the information about which HEPLike object should be used to read the file. Reading of this content in the YAML is implemented in the HL_Data class. All other classes that construct the likelihood functions inherit from this class its capabilities. Please note that if the information is missing in the YAML file the program will omit reading this entry. The only exception is the FileName, which is mandatory. If a user wants to be notified by the program that some informations are missing the HL_debug_yaml variable has to be set to true (default value is false). ### 2.1 Upper limits In case where the measurement did not observe a significant access of signal candidates the collaborations usually report an upper limit on the measured quantity. Commonly $90\%$ or $95\%$ upper limits are quoted. Experiments use various statistical approaches to compute this limits. It can be the $\rm CL_{s}$ method [8], Feldman–Cousins [9] or some variation of Bayesian methods [10]. Publication of only an upper limits does not provide enough information to use the result in global fits. However, nowadays experiments besides the aforementioned upper limits publish a full p-value scans. Examples of such scans are shown in Fig. 1. The plots are usually available in digital format, which allows the information to be extracted and used in computer program. Figure 1: Example of p-value scans for the $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{s}}}}}^{\scriptstyle{0}}}\to\mathup{{{\tau}}^{\scriptstyle{-}}}\mathup{{{\tau}}^{\scriptstyle{+}}}$ [11] (left) and $\mathup{{{D}}}\to\mathup{{{e}}}\mathup{{{\mu}}}$ [8] (right). Please note that the $\rm CL_{s}$ value can be interpreted as p-value as explained in [12]. The black line corresponds to the observed $\rm CL_{s}$/p-value. In HEPLike a class HL_Limit is responsible for handling this type of measurements. It reads the YAML file that contains the standard information about the measurement (see Sec. 2 for details). The additional information of the observed $\rm CL_{s}$/p-value is stored in the YAML file in the following way111Please note that the besides this information the previous information from Sec. 2 should be included.: ⬇ 1Cls: 2- [0.0, 1.0] 3- [1.0e-10, 0.977091694706] 4- [2.0e-10, 0.954375824297] 5- [3.0e-10, 0.93200355343] 6- [4.0e-10, 0.910630700546] 7- [5.0e-10, 0.889382721809] The Cls can be replaced in the YAML file by p-value as they correspond to the same information. The first number in each array is the value of tested hypothesis (for example branching fraction), while the second is the corresponding $\rm CL_{s}$/p-value. These values are then interpreted using a $\chi^{2}$ distribution with one degree of freedom: $pdf(x)=\frac{1}{2^{1/2}\Gamma(1/2)}x^{1/2-1}e^{-x/2},$ (1) which had the cumulative distribution function defined as: $cdf(x)=\frac{1}{\Gamma(1/2)}\gamma(1/2,x/2).$ (2) In the above equations the $\Gamma(x)$ and $\gamma(k,x)$ correspond to Gamma and incomplete gamma functions. By revering the $cdf(x)$ one can obtain the $\chi^{2}$ value: $\chi^{2}=cdf^{-1}(1-p),$ (3) where p corresponds to the p-value of a given x hypothesis. This $\chi^{2}$ can be then translated to the log-likelihood via Wilks theorem [13]: $-\log(\mathcal{L})=\frac{1}{2}\chi^{2},$ (4) where the $\mathcal{L}$ is the likelihood. The user can choose if he wants to obtain the $\chi^{2}$, likelihood or a log-likelihood value of a given hypothesis. ### 2.2 Single measurement with symmetric uncertainties The simplest case of a published experimental result is a single value with a symmetric uncertainty. This is for example a typical result of an PDG of HFLAV average [14, 15]. The measurement is coded in the YAML file as: ⬇ 1Observables: 2- [ ”Br_A2BCZ”, 0.1, 0.05, 0.01 ] The first argument in the array ”Br_A2BCZ” corresponds to the observable name. Then the first number corresponds to the measured central value. The 2nd and the 3rd number are the statistical and systematic uncertainties. In case where only one uncertainty is available the 3rd number should be omitted and it will be automatically set to 0 in the software. We have decided to keep the plural Observables to be more uniform in case where more observables are measured. The module responsible for reading this YAML file is called HL_Gaussian, it calculates the $\chi^{2}$ for an $x$ hypothesis: $\chi^{2}=\frac{(x_{obs}-x)^{2}}{\sigma_{stat}^{2}+\sigma_{syst}^{2}},$ (5) where the $x_{obs}$ correspond to the measured central value in the YAML file and the $\sigma_{stat}$ and $\sigma_{syst}$ are the statistical and systematic uncertainties respectively. This can be the again translated to the likelihood and log-likelihood value using Eq. 4. ### 2.3 Single measurement with asymmetric uncertainties A simple extension of the Gaussian uncertainty is when an asymmetric uncertainty is reported. This type of measurements although less frequent appear in the literature. The publication in this case reports the central value and two uncertainties: $\sigma_{+}$ and $\sigma_{-}$, which correspond to the right (for values larger than the measured central value) and left (for values smaller than the measured central value) uncertainty. In HEPLike we have created a HL_BifurGaussian class, which reads the following entry in the YAML file: ⬇ 1Observables: 2- [ ”Br_A2BCZ”, 0.1, 0.05, -0.06, 0.01, -0.02 ] The first argument is again the name of the observable and the second one is its central value. The third and fourth arguments correspond to the statistical $\sigma_{+}$ and $\sigma_{-}$ uncertainties, while the fifth and sixth to the systematical $\sigma_{+}$ and $\sigma_{-}$ uncertainties. It is important to keep the minus sign before the left side uncertainties. The code will indicate the error in case of missing sign. In some cases the systematical uncertainty is reported to be symmetric. In such case the last number can be omitted in the YAML entry. In the literature there exist number of ways to interpret asymmetric uncertainties [16]. We have chosen the most commonly used one which is the so- called bifurcated Gaussian: $\displaystyle\chi^{2}=\begin{cases}\frac{(x_{obs}-x)^{2}}{\sigma_{+}^{2}},&\text{if }x\geq x_{obs}\\\ \frac{(x_{obs}-x)^{2}}{\sigma_{-}^{2}},&\text{if }x<x_{obs},\\\ \end{cases}$ (6) where the $\sigma_{\pm}^{2}$ is the sum of squared statistical and systematic uncertainties for right/left case. Once $\chi^{2}$ is calculated it can be translated to the log-likelihood using Eq. 4. ### 2.4 Multiple measurements with symmetric uncertainties Nowadays the most common are simultaneous measurements of various quantities, which are correlated between each other. For instance cross section measurements in different kinematic bins, or measurements of angular coefficients in heavy mesons decays. In HEPLike the class responsible for handling these cases is called HL_nDimGaussian. It reads the following information from the YAML file: ⬇ 1Observables: 2- [ ”BR1”, 0.1, 0.02] 3- [ ”BR2”, 0.2, 0.01, 0.01] 4- [ ”BR3”, 0.4, 0.04] 5Correlation: 6- [ ”BR1”, ”BR2”, ”BR3”] 7- [ 1. , 0.2 , 0 ] 8- [ 0.2, 1., 0. ] 9- [ 0 , 0., 1. ] The information in the “Observables” entry is exactly the same as in the HL_Gaussian class. Please note that similarly to the previous class the systematic uncertainty is not mandatory and in case if it is not provided the code will treat it as 0. The next entry in the YAML file is the “Correlation”, which encodes the correlation matrix. The first ”row” is the names of the variables it is important to keep the same order of variables as in the “Observables” entry. The HL_nDimGaussian evaluates the $\chi^{2}$ in the following way: $\displaystyle\chi^{2}=V^{T}{\rm Cov}^{-1}V,$ (7) where V is a column matrix, which is the difference between the measured and the tested i-th observable value. The $\rm Cov$ is a square matrix, constructed from the correlation matrix (${\rm Corr}$): ${\rm Cov}_{i,j}={\rm Corr}_{i,j}\sigma_{i}\sigma_{j}$. Often a user does not want to use the full set of measured quantities but just their subset. In this case a function Restrict(vector<string>) can be used. By passing in a form of vector the list of observables to be used, the program will create new smaller covariance matrix, which will be used to evaluate the $\chi^{2}$. In a similar manner the $\chi^{2}$ can be translated to the likelihood and log-likelihood value by Eq. 4. ### 2.5 Multiple measurements with asymmetric uncertainties More complicated case is when multiple correlated measurements are reported with asymmetric uncertainties. The case is similar to the one discussed in Sec. 2.3 and same statistic comments apply in this case. The YAML file encoding such a measurement will contain the following entries: ⬇ 1Observables: 2- [ ”BR1”, 0.1, +0.02, -0.01, 0.02] 3- [ ”BR2”, 0.2, +0.01, -0.05, +0.03, -0.02] 4- [ ”BR3”, 0.3, +0.04, -0.03, 0.05] 5Correlation: 6- [ ”BR1”, ”BR2”, ”BR3”] 7- [ 1. , 0.1 , 0.2 ] 8- [ 0.1, 1., 0.1 ] 9- [ 0.2 , 0.1, 1. ] The meaning of the “Observables” entry is the same as in the previous class (cf. Sec. 2.3) and the “Correlation” encodes the same information as in the HL_nDimGaussian class (cf.. Sec. 2.4). The rules about the minus sign and symmetric systematic uncertainty are the same as in case of the HL_BifurGaussian (cf. Sec. 2.3). The difference arises when one evaluates the $\chi^{2}$, namely the $\rm cov$ matrix is constructed depending if $\sigma_{+}$ and $\sigma_{-}$ uncertainty is relevant: $\displaystyle{\rm Cov}_{i,j}=\begin{cases}{\rm Corr}_{i,j}~{}\sigma^{i}_{+}\sigma^{j}_{+},&\text{if }x^{i}\geq x^{i}_{obs}\text{ and }x^{j}\geq x^{j}_{obs}\\\ {\rm Corr}_{i,j}~{}\sigma^{i}_{+}\sigma^{j}_{-},&\text{if }x^{i}\geq x^{i}_{obs}\text{ and }x^{j}<x^{j}_{obs}\\\ {\rm Corr}_{i,j}~{}\sigma^{i}_{-}\sigma^{j}_{+},&\text{if }x^{i}<x^{i}_{obs}\text{ and }x^{j}\geq x^{j}_{obs}\\\ {\rm Corr}_{i,j}~{}\sigma^{i}_{-}\sigma^{j}_{-},&\text{if }x^{i}<x^{i}_{obs}\text{ and }x^{j}<x^{j}_{obs}\\\ \end{cases}$ (8) The obtained $\rm Cov$ matrix is then used to calculate the $\chi^{2}$ using Eq. 7. The rest follows the same procedure as described in Sec. 2.4. ### 2.6 One dimensional likelihood function The best way a result can be published is by providing the (log-)likelihood function. This type of results are more and more common in the literature. The most easy is the one-dimensional likelihood scans as can be presented in form of a figure, which examples are shown in Fig. 2. Figure 2: Examples of published one-dimensional likelihoods in the Lepton Universality Violation of the $\mathup{{{B}}}\to\mathup{{{K}}^{\scriptstyle{\ast}}}\ell\ell$ [7] (left) and $\mathup{{{B}}}\to\mathup{{{K}}}\ell\ell$ [17] (right). The biggest advantage of publishing the results in this form is its completeness. The (log-)likelihood curve contains all the information about all the non-Gaussian effects and incorporates the systematic uncertainties. The technical problem is how to publish such information. Usually plots are published in the pdf or png formats which makes them hard to be used. Since experiments are mostly using ROOT [18] framework the plots are saved also in the C format, which contains the points in the form of arrays. This of course makes the points accessible however it is not easy to automate retrieving this data from the C file. The best solution is provided by the HEPData portal [19]. It allows to download the data in a user preferred format. In HEPLike we have chosen to use the ROOT format by default, in which the data points are saved in the form of a TGraph object, which is also the way experimentalists like to store this information. In the YAML file we specify the path of the ROOT in the following way: ⬇ 1ROOTData: data/HEPData-ins1599846-v1-Table_1.root 2TGraphPath: ”Table 1/Graph1D_y1” The ROOTData encodes the location of the ROOT file, while the TGraphPath encodes the location of the TGraph object in that ROOT file. In HEPLike the class HL_ProfLikelihood is responsible for reading and encoding this likelihood. The value of the log-likelihood can be ten translated again into the $\chi^{2}$ with Eq. 4. ### 2.7 n-dimensional likelihood function The natural extension of one dimensional likelihood is an n-dim likelihood, where $n\geq 2$. Currently experimental collaborations publish only 2-dim likelihood functions (cf. Fig. 3). Figure 3: Examples of published two-dimensional likelihoods. The $\mathcal{B}(\mathup{{{B}}{}_{\scriptstyle{\mathup{{{s}}}}}^{\scriptstyle{0}}}\to\mu\mu)$ vs $\mathcal{B}(\mathup{{{B}}{}_{\scriptstyle{\mathup{{{d}}}}}^{\scriptstyle{0}}}\to\mu\mu)$ likelihood [20] (left) and $\sigma(\mathup{{{t}}}\mathup{{\overline{{t}}}}\mathup{{{Z}}})$ vs $\sigma(\mathup{{{t}}}\mathup{{\overline{{t}}}}\mathup{{{W}}})$ likelihood [21] (right). The natural way of encoding such information is a histogram: TH2D or TH3D and we have chosen this way to store this information. The corresponding entry in the YAML file looks as following: ⬇ 1ROOTData: data/LHCb/RD/Bs2mumu_5fb/histB2mumu.root 2TH2Path: ”h_2DScan” Similar to the one dimensional likelihood (Sec. 2.6) the ROOTData encodes the location of the ROOT file, while the TH2Path(TH3Path) encodes the location of the TH2D(TH3D) object. In the long run the community will have to address the question how to publish higher dimensional likelihoods and this module (HL_nDimLikelihood) will have to be extended for such use cases. ### 2.8 Fits to experimental data It is possible that in the future experimental collaborations besides the results will made the datasets public. The procedure and the form in which the data should be published is not decided and there is an ongoing debate if the published data should correspond to the raw detector data, the final selected points used in the analysis or something between? Clearly publishing a raw data is problematic, as people outside the collaboration do not have the necessary knowledge about the calibration and efficiency correction procedures or data taking conditions. The most useful way to publish the dataset is to allow the experimentalists to perform all the selection, all the necessary efficiency corrections and publish the final dataset that has been used for analysis. This would allow the theory community to use the dataset directly in their fits without knowing the technicalities about the experimental data analysis. For this case in HEPLike we have implemented such a class HL_ExpPoints. The data are stored in the TTree structure located in the ROOT file. The YAML file encodes this information in form: ⬇ 1ROOTData: data/toy/data.root 2TTreePath: t 3Observables: 4- [ x ] 5- [ y ] 6- [ z ] 7Weight: w where the ROOTData points to the ROOT file and the TTreePath stores the information of the TTree location inside the ROOT file. It is assumed that the experiments will provide all the corrections in form of event-by-event weights. The name of the weight inside the TTree is encoded in the Weight entry. In general the data points are elements of $\mathcal{R}^{n}$ vector space, which coordinates are stored in the Observables entry. The only thing that user needs to provide to the HL_ExpPoints object is a pointer to the function to be fitted. The function should have a form: double (*fun)(vector<double> par , vector<double> point), where the par vector encodes the parameters that want to be fitted and the point corresponds to a data point. The HL_ExpPoints will then evaluate the likelihood: $\mathcal{L}(\omega)=f(\textbf{x}|\omega)^{w(\textbf{x})}$ (9) for the whole dataset. In the above the x correspondents to the n-dimensional point, $\omega$ denotes the parameters that want to be fitted par, and $f$ denotes the fitting function (fun). The HEPLike does not provide a minimalizer or a scanner tool as it is not purpose of this type of software. It has to be interfaced with proper scanner tool for example [1]. Again the user can decide if he/she prefers to perform a $\chi^{2}$ or log-likelihood fit. The biggest advantage of such format is the compatibility with the experimental analysis. Experimentalist can in principle publish as well the function that they have used to fit this data and therefore a theorists reproduce the experimental result and start where the experimentalists finished. ## 3 Code implementation In this section we will discuss the implementation of the code used to create likelihoods discussed in Sec. 2. The code is build in several classes: * 1. HL_Data: base class from which other classes inherit their base functionality. * 2. HL_Limit: class that handles the upper limit measurements. * 3. HL_Gaussian: class that handles measurements with Gaussian uncertainty. * 4. HL_BifurGaussian: class that handles measurements with asymmetric uncertainty. * 5. HL_nDimGaussian: class that handles measurements with n-dimensional Gaussian uncertainties. * 6. HL_nDimBifurGaussian: class that handles measurements with n-dimensional asymmetric uncertainties. * 7. HL_ProfLikelihood: class that handles measurements with one-dimensional likelihood function. * 8. HL_nDimLikelihood: class that handles measurements with 2(3)-dimensional likelihood function. * 9. HL_ExpPoints: class that allows to perform the fits to experimental datasets. In Tab. LABEL:tab:functions we present the functionality of these classes. In addition we present the hierarchy of the structure of the class inheritance in Fig. 4. Table 1: Functions available in the HEPLike software. Function | Description ---|--- HL_Data() | Constructor of the HL_Data class. HL_Data(string) | Constructor of the HL_Data class. The argument that is taken by constructor is the path for the YAML file encoding the measurement. HL_Limit() | Constructor of the HL_Limit class. HL_Limit(string) | Constructor of the HL_Limit class. The argument that is taken by constructor is the path for the YAML file encoding the measurement. HL_Gaussian() | Constructor of the HL_Gaussian class. HL_Gaussian(string) | Constructor of the HL_Gaussian class. The argument that is taken by constructor is the path for the YAML file encoding the measurement. HL_BifurGaussian() | Constructor of the HL_BifurGaussian class. HL_BifurGaussian(string) | Constructor of the HL_Gaussian class. The argument that is taken by constructor is the path for the YAML file encoding the measurement. HL_nDimGaussian() | Constructor of the HL_nDimGaussian class. HL_nDimGaussian(string ) | Constructor of the HL_nDimGaussian class. The argument that is taken by constructor is the path for the YAML file encoding the measurement. HL_nDimBifurGaussian() | Constructor of the HL_nDimBifurGaussian class. HL_nDimBifurGaussian(string) | Constructor of the HL_nDimBifurGaussian class. The argument that is taken by constructor is the path for the YAML file encoding the measurement. HL_ProfLikelihood() | Constructor of the HL_ProfLikelihood class. HL_ProfLikelihood(string) | Constructor of the HL_ProfLikelihood class. The argument that is taken by constructor is the path for the YAML file encoding the measurement. HL_nDimLikelihood() | Constructor of the HL_nDimLikelihood class. HL_ProfLikelihood(string) | Constructor of the HL_nDimLikelihood class. The argument that is taken by constructor is the path for the YAML file encoding the measurement. HL_ExpPoints() | Constructor of the HL_ExpPoints class. HL_ExpPoints(string) | Constructor of the HL_ExpPoints class. The argument that is taken by constructor is the path for the YAML file encoding the measurement. read_standard() | Function that reads the general information about the measurement from the YAML file. set_debug_yaml(bool) | Function that enables debugging the YAML file. By default the debugging is switched off and can be switched on by passing a true bool argument to this function. Debugging will print a message that for a given information in the YAML file is missing. Read() | Function reading the YAML file. The function GetChi2(double) | Function that returns the $\chi^{2}$ value for a given point (passed to the function as double). Function is available for all classes besides HL_Data. GetLogLikelihood(double) | Function that returns the log-likelihood value for a given point (passed to the function as double). Function is available for all classes besides HL_Data. GetLikelihood(double) | Function that returns the likelihood value for a given point (passed to the function as double). Function is available for all classes besides HL_Data. GetCLs(double) | Function that returns $\rm CL_{s}$ or p-value for a given point (passed to the function as double). The function is a member of the HL_Limit class. Restrict(vector<string>) | Function that restricts number of observables from the YAML file. Function is a member of the HL_nDimGaussian, HL_nDimBifurGaussian and HL_nDimLikelihood classes. InitData() | Function of HL_ExpPoints class that reads to the memory the data from the TTree object. Profile() | Function of HL_nDimLikelihood class that creates the profile log-likelihood projections. SetFun() | Function of HL_ExpPoints class, that sets the pointer to the function to be fitted. Figure 4: Diagram of class inheritance of the HEPLike package. ## 4 Installation and usage In this chapter we will present the requirements and installation for the HEPLike package. The software is distributed via the github site: https://github.com/mchrzasz/HEPLike. In order to compile HEPLike the following packages (and the minimal version) needs to be installed: * 1. git * 2. cmake, 2.8 * 3. yaml-cpp, 1.58.0 * 4. gsl, 2.1 * 5. Boost, 1.58.0 * 6. ROOT, 6.08 The compilation is done in the following way: ⬇ 1cd <instalation dir> 2git clone https://github.com/mchrzasz/HEPLike.git 3cd HEPLike 4mkdir build 5cd build 6cmake .. 7make In the above the make can be replaced with make -jN, where N is the number of threads that user wants to be used for compilation. Please note that in case of non standard installation of some packages one might have to provide cmake with a proper path to the library. After successful compilation a libHEPLike.a and libHEPLike.solibraries will be created in the build directory. The HEPLike is provided with seven examples: * 1. Br_example.cc: example program showing the usage of the HL_Gaussian class. * 2. BrBifurGaussian_example.cc: example program showing the usage of the HL_BifurGaussian class. * 3. Data_Fit_example.cc: example program showing the usage of the HL_ExpPoints class. * 4. Limit_example.cc: example program showing the usage of the HL_Limit class. * 5. Ndim_BifurGaussian_example.cc: example program showing the usage of the HL_nDimBifurGaussian class. * 6. Ndim_Gaussian.cc: example program showing the usage of the HL_nDimGaussian class. * 7. Ndim_Likelihood_example.cc: example program showing the usage of the HL_nDimLikelihood class. * 8. ProfLikelihood_example.cc: example program showing the usage of the HL_ProfLikelihood class. To compile them a proper variable has to be set during the cmake stage: ⬇ 1 cd build 2 cmake -DEXECUTABLE=TRUE .. 3 make After the compilation in the build directory will contain executables from the following examples. The HEPLike package comes also with test procedures for each of the classes. To perform the tests user has to perform the command: ⬇ ctest or an equivalent: ⬇ make test If the HEPLike was successfully installed the output will look as following: ⬇ Test project /storage/github/HEPLike/build Start 1: HL_Test_YAML 1/7 Test #1: HL_Test_YAML ………………… Passed 0.01 sec Start 2: HL_Limit 2/7 Test #2: HL_Limit ……………………. Passed 0.27 sec Start 3: HL_Br_example 3/7 Test #3: HL_Br_example ……………….. Passed 0.02 sec Start 4: HL_BrBifurGaussian_example 4/7 Test #4: HL_BrBifurGaussian_example ……. Passed 0.01 sec Start 5: HL_Ndim_Gaussian 5/7 Test #5: HL_Ndim_Gaussian …………….. Passed 0.01 sec Start 6: HL_ProfLikelihood_example 6/7 Test #6: HL_ProfLikelihood_example …….. Passed 0.25 sec Start 7: HL_Ndim_BifurGaussian_example 7/7 Test #7: HL_Ndim_BifurGaussian_example …. Passed 0.01 sec 100% tests passed, 0 tests failed out of 7 Total Test time (real) = 0.57 sec ### 4.1 Available measurement The YAML files that contain the stored measurements are located in a second independent repository. The reason for this separation is that the YAML files are expected to be updated more frequently then the code itself. It is expected that users and experiments will contribute to this repository. By implementing such model it is ensured that the repository will contain the most up to date measurements. The repository can be found at: https://github.com/mchrzasz/HEPLikeData. The repository should be downloaded or cloned: ⬇ 1cd <some new dir> 2git clone https://github.com/mchrzasz/HEPLikeData.git Since the repository contains only YAML files there is no need for any compilation. The repository contains a directory data, where all the YAML files are kept. It should be linked by a symbolic link to the HEPLike package. Inside the data the measurements are grouped by experiments (ex. LHCb, ATLAS, CMS, etc.). Inside the experiment directory the measurements are grouped according to type of measurement in the collaborations, for example: RD, Semileptonic, Charmless, Exotica, etc. The names of the YAML files should be named accordingly to publication report number. For example: CERN- EP-2018-331.yaml. If a single publication produced more independent measurements, user might code them in the independent files and give further information at the end of the file, for example:CERN-PH- EP-2015-314_q2_01_0.98.yaml. Currently we are publishing the measurements that have been used by us in other projects [22, 23, 24]. The list of YAML files with the context is presented in Tab. LABEL:tab:yaml. Table 2: Functions available in the HEPLike software. | ---|--- File | Description CERN-EP-2017-100.yaml | YAML file encoding the measurement of branching fraction of the $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{d}}}}}^{\scriptstyle{0}}}\to\mu\mu$ and $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{s}}}}}^{\scriptstyle{0}}}\to\mu\mu$ decays [20]. PH-EP-2015-314_q2_0.1_0.98.yaml PH-EP-2015-314_q2_11.0_12.5.yaml PH-EP-2015-314_q2_1.1_2.5.yaml PH-EP-2015-314_q2_15.0_19.yaml PH-EP-2015-314_q2_2.5_4.0.yaml PH-EP-2015-314_q2_4.0_6.0.yaml PH-EP-2015-314_q2_6.0_8.0.yaml | YAML files encoding the measurements of the angular coefficients of $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{d}}}}}^{\scriptstyle{0}}}\to\mathup{{{K}}^{\scriptstyle{\ast}}}\mu\mu$ decay in different $q^{2}$ regions [25]. CERN-EP-2016-141_q2_0.1_0.98.yaml CERN-EP-2016-141_q2_11.0_12.5.yaml CERN-EP-2016-141_q2_1.1_2.5.yaml CERN-EP-2016-141_q2_15.0_19.yaml CERN-EP-2016-141_q2_2.5_4.0.yaml CERN-EP-2016-141_q2_4.0_6.0.yaml CERN-EP-2016-141_q2_6.0_8.0.yaml | YAML files encoding the measurements of the branching fraction of the $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{d}}}}}^{\scriptstyle{0}}}\to\mathup{{{K}}^{\scriptstyle{\ast}}}\mu\mu$ decay in different $q^{2}$ regions [26]. CERN-EP-2016-215_q2_0.1_0.98.yaml CERN-EP-2016-215_q2_1.1_2.5.yaml CERN-EP-2016-215_q2_2.5_4.yaml CERN-EP-2016-215_q2_4_6.yaml CERN-EP-2016-215_q2_6_8.yaml | YAML files encoding the measurements of the branching fraction of the $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{d}}}}}^{\scriptstyle{0}}}\to\mathup{{{K}}}\mathup{{{\pi}}}\mu\mu$ decay in different $q^{2}$ regions [27]. CERN-PH-EP-2015-145_0.1_2.yaml CERN-PH-EP-2015-145_11_12.5.yaml CERN-PH-EP-2015-145_15_19.yaml CERN-PH-EP-2015-145_1_6.yaml CERN-PH-EP-2015-145_2_5.yaml CERN-PH-EP-2015-145_5_8.yaml | YAML files encoding the measurements of the branching fraction of the $\mathup{{{B}}{}_{\scriptstyle{\mathup{{{s}}}}}^{\scriptstyle{0}}}\to\mathup{{{\phi}}}\mu\mu$ decay in different $q^{2}$ regions [27]. CERN-EP-2019-043.yaml | YAML file encoding the measurement of the $R_{K}$ [28]. CERN-EP-2017-100_q2_0.045_1.1.yaml CERN-EP-2017-100_q2_1.1_6.yaml | YAML file encoding the measurement of the $R_{\mathup{{{K}}^{\scriptstyle{\ast}}}}$ [7]. b2sgamma.yaml | YAML file encoding the HFLAV average of the $\mathup{{{b}}}\to\mathup{{{s}}}\mathup{{{\gamma}}}$ [15]. RD_RDstar.yaml | YAML file encoding the HFLAV average of the $R(\mathup{{{D}}})$ and $R(\mathup{{{D}}^{\scriptstyle{\ast}}})$ [15]. HFLAV_2016_157.yaml HFLAV_2016_160.yaml HFLAV_2016_161.yaml HFLAV_2016_162.yaml HFLAV_2016_164.yaml HFLAV_2016_165.yaml HFLAV_2016_166.yaml HFLAV_2016_167.yaml HFLAV_2016_168.yaml HFLAV_2016_169.yaml HFLAV_2016_170.yaml HFLAV_2016_171.yaml HFLAV_2016_176.yaml HFLAV_2016_177.yaml HFLAV_2016_178.yaml HFLAV_2016_179.yaml HFLAV_2016_180.yaml HFLAV_2016_181.yaml HFLAV_2016_182.yaml HFLAV_2016_183.yaml HFLAV_2016_211.yaml HFLAV_2016_212.yaml | YAML files encoding the upper limits of $\tau$ Lepton Flavour Violation decays [27]. As already mentioned the measurements are constantly growing and there is expected that the community will contribute to develop this repository. When a new YAML file is wrote before merging it with the repository it should be checked if it contains all the necessary information. It can be checked with the Test_YAML.cc program. It can be used in the following way: ⬇ 1cd HEPLike 2./build/Test_YAML <PATH_TO_YAML> If an entry is missing the user will be notified by a printout. The HEPLikeData repository contains also a template YAML (data/template.yaml) file which can be used to create new measurements YAML files. As already mentioned we provide useful utilities for the encoded measurements. The first is the ability to create BiBtex file for the measurements that have been used. The user should store the BiBtex items or YAML file names: ⬇ 1Aaij:2017vbb 2b2mumu.yaml To prepare the BiBtex file user should run the make_citations.py script located in the utils directory: ⬇ 1cd utils 2python make_citations.py list.txt after this command a new file references.bib, will be created, which will contain the full BiBtex entries. This can be directly used in preparing the publication. Another useful feature of HEPLike is the ability to search the measurement database for relevant measurements. The script allowing for that utility is also located in the utils. Currently the database can be searched for using the year of publication, Arxiv number, author of the YAML file or the unique name of the measurements. The syntax for running a search is the following: ⬇ 1python lookup.py –Arxiv 1705.05802 2Found files: 3../data/examples/RKstar_lowq2.yaml To see all available search options in the following script user can run it with help option: python lookup.py -h. ## 5 Summary We have presented a computer program HEPLike that enables to construct and evaluate experimental likelihoods. The software is designed to handle the interpretation of wide range of published results. It also allows to perform direct fits to data once it is provided by the experimental collaborations. The program can be easily interfaced with other computer programs and is aimed to help users, who perform fits to experimental results in their scientific work. 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2024-09-04T02:54:58.268761

2003.03977

arxiv-papers

# Wide-minima Density Hypothesis and the Explore-Exploit Learning Rate Schedule Nikhil Iyer Microsoft Research India<EMAIL_ADDRESS>V Thejas 11footnotemark: 1 Atlassian India<EMAIL_ADDRESS>Nipun Kwatra Microsoft Research India<EMAIL_ADDRESS>Ramachandran Ramjee Microsoft Research India<EMAIL_ADDRESS>Muthian Sivathanu Microsoft Research India<EMAIL_ADDRESS>Work done during an internship at Microsoft Research India ###### Abstract Several papers argue that wide minima generalize better than narrow minima. In this paper, through detailed experiments that not only corroborate the generalization properties of wide minima, we also provide empirical evidence for a new hypothesis that the density of wide minima is likely lower than the density of narrow minima. Further, motivated by this hypothesis, we design a novel explore-exploit learning rate schedule. On a variety of image and natural language datasets, compared to their original hand-tuned learning rate baselines, we show that our explore-exploit schedule can result in either up to 0.84% higher absolute accuracy using the original training budget or up to 57% reduced training time while achieving the original reported accuracy. For example, we achieve state-of-the-art (SOTA) accuracy for IWSLT’14 (DE-EN) dataset by just modifying the learning rate schedule of a high performing model. Keywords: deep learning, generalization, learning rate schedule, optimization ## 1 Introduction One of the fascinating properties of deep neural networks (DNNs) is their ability to generalize well, i.e., deliver high accuracy on the unseen test dataset. It is well-known that the learning rate (learning rate) schedules play an important role in the generalization performance (Keskar et al., 2016; Wu et al., 2018; Goyal et al., 2017). In this paper, we study the question, what are the key properties of a learning rate schedule that help DNNs generalize well during training? We start with a series of experiments training Resnet18 on Cifar-10 over 200 epochs. We vary the number of epochs trained at a high learning rate of $0.1$, called the explore epochs, from 0 to 100 and divide up the remaining epochs equally for training with learning rates of $0.01$ and $0.001$. Note that the training loss typically stagnates around 50 epochs with $0.1$ learning rate. Despite that, we find that as the number of explore epochs increase to 100, the average test accuracy also increases. We also find that the minima found in higher test accuracy runs are wider than the minima from lower test accuracy runs, corroborating past work on wide-minima and generalization (Keskar et al., 2016; Hochreiter and Schmidhuber, 1997; Jastrzebski et al., 2017; Wang et al., 2018). Moreover, what was particularly surprising was that, even when using fewer explore epochs, a few runs out of many trials still resulted in high test accuracies! Thus, we not only find that an initial exploration phase with a high learning rate is essential to the good generalization of DNNs, but that this exploration phase needs to be run for sufficient time, even if the training loss stagnates much earlier. Further, we find that, even when the exploration phase is not given sufficient time, a few runs still see high test accuracy values. To explain these observations, we hypothesize that, in the DNN loss landscape, the density of narrow minima is significantly higher than that of wide minima. Intuitively, a large learning rate can escape narrow minima easily (as the optimizer can jump out of them with large steps). However, once it reaches a wide minima, it is likely to get stuck in it (if the ”width” of the wide minima is large compared to the step size). With fewer explore epochs, a large learning rate might still get lucky occasionally in finding a wide minima but invariably finds only a narrower minima due to their higher density. As the explore duration increases, the probability of eventually landing in a wide minima also increases. Thus, a minimum duration of explore is necessary to land in a wide minimum with high probability. An observation on the rarity of wide minima has been hinted at by prior work (Wu et al., 2018; Baldassi et al., 2020) based on theoretical analysis of simple neural networks (see Section 2). In this paper, we add significant empirical evidence to these theoretical observations. We believe that all these results together constitute sufficient evidence for this observation to now be classified as a hypothesis, that we term the wide-minima density hypothesis. The hypothesis helps explain not only our experiments but also the generalization out-performance of prior heuristic-based learning rate decay schemes such as cosine decay (Loshchilov and Hutter, 2016). Cosine decay implicitly maintains a higher learning rate during the first half of training compared to schemes like linear decay. Based on the hypothesis, the higher learning rate allows cosine decay to find wider minima with higher probability, resulting in cosine decay’s better generalization compared to linear decay. Apart from helping explain empirical observations, the hypothesis also enables a principled learning rate schedule design that explicitly accounts for the requisite explore duration. Motivated by the hypothesis, we design a novel Explore-Exploit learning rate schedule, where the initial explore phase optimizes at a high learning rate in order to arrive in the vicinity of a wide minimum. This is followed by an exploit phase which descends to the bottom of this wide minimum. We give explore phase enough time so that the probability of landing in a wide minima is high. For the exploit phase, we experimented with multiple schemes, and found a simple, parameter-less, linear decay to zero to be effective. Thus, our proposed learning rate schedule optimizes at a constant high learning rate for a given duration, followed by a linear decay to zero. We call this learning rate schedule the Knee schedule. We extensively evaluate the Knee schedule across a wide range of models and datasets, ranging from NLP (BERT pre-training, Transformer on WMT’14(EN-DE) and IWSLT’14 (DE-EN)) to CNNs (ImageNet on ResNet-50, Cifar-10 on ResNet18), and spanning multiple optimizers: SGD Momentum, Adam, RAdam, and LAMB. In all cases, Knee schedule improves the test accuracy of state-of-the-art hand-tuned learning rate schedules, when trained using the original training budget. The explore duration is a hyper-parameter in Knee schedule but even if we set the explore duration to a fixed 50% fraction of total training budget, we find that it still outperforms prior schemes. We also experimented with reducing the training budget, and found that Knee schedule can achieve the same accuracy as the baseline under significantly reduced training budgets. For the BERTLARGE pretraining, WMT’14(EN-DE) and ImageNet experiments, we are able to train in 33%, 57% and 44% less training budget, respectively, for the same test accuracy. This corresponds to significant savings in GPU compute, e.g. savings of over 1000 V100 GPU-hours for BERTLARGE pretraining. The main contributions of our work 111Our work is available at: https://github.com/nikhil-iyer-97/wide-minima-density-hypothesis are: 1. nosep A hypothesis of lower density of wide minima in the DNN loss landscape, backed by extensive experiments, that explains why a high learning rate needs to be maintained for sufficient duration to achieve good generalization. 2. nosep The hypothesis explains the good performance of heuristic-based schemes such as cosine decay, and promotes a principled design of learning rate decay schemes. 3. nosep Motivated by the hypothesis, we design an Explore-Exploit learning rate schedule called Knee schedule that outperforms prior heuristic-based learning rate schedules, including achieving state-of-the-art results on the IWSLT’14 (DE-EN) dataset. ## 2 Related Work Generalization. There has been a lot of work on understanding the generalization characteristics of DNNs. Kawaguchi (2016) found that DNNs have many local minima, but all local minima were also the global minima. It has been observed by several authors that wide minima generalize better than narrow minima (Arora et al., 2018; Hochreiter and Schmidhuber, 1997; Keskar et al., 2016; Jastrzebski et al., 2017; Wang et al., 2018) but there have been other works questioning this hypothesis as well (Dinh et al., 2017; Golatkar et al., 2019; Guiroy et al., 2019; Jastrzebski et al., 2019; Yoshida and Miyato, 2017). Keskar et al. (2016) found that small batch SGD generalizes better and lands in wider minima than large batch SGD. However, recent work has been able to generalize quite well even with very large batch sizes (Goyal et al., 2017; McCandlish et al., 2018; Shallue et al., 2018), by scaling the learning rate linearly as a function of the batch size. Jastrzebski et al. (2019) analyze how batch size and learning rate influence the curvature of not only the SGD endpoint but also the whole trajectory. They found that small batch or large step SGD have similar characteristics, and yield smaller and earlier peak of spectral norm as well as smaller largest eigenvalue. Chaudhari et al. (2019); Baldassi et al. (2019) propose methods to drive the optimizer to wide minima. Wang et al. (2018) analytically show that generalization of a model is related to the Hessian and propose a new metric for the generalization capability of a model that is unaffected by model reparameterization of Dinh et al. (2017). Yoshida and Miyato (2017) argue that regularizing the spectral norm of the weights of the neural network help them generalize better. On the other hand, Arora et al. (2018) derive generalization bounds by showing that networks with low stable rank (high spectral norm) generalize better. Guiroy et al. (2019) looks at generalization in gradient-based meta-learning and they show experimentally that generalization and wide minima are not always correlated. Finally, Golatkar et al. (2019) show that regularization results in higher test accuracy specifically when it is applied during initial phase of training, similar to the importance of Knee schedule’s explore phase during initial phase of training. In a similar vein, Li et al. (2019) explain the regularization benefits of the initial higher learning rate by showing that higher learning rate helps networks learn easier-to-fit general patterns. Neural network loss landscapes. The landscape of loss in neural networks have been extensively studied (Draxler et al., 2018; Freeman and Bruna, 2016; Garipov et al., 2018; Sagun et al., 2017). These papers point out that the loss landscape contains both wide and narrow minima, and there may even exist a path from one minima to another without barriers. However, there are multiple paths between these minima and some paths indeed face barriers (e.g., see Figure 1 in Draxler et al. (2018)). Since we don’t know which path SGD and other optimizers might follow, even if wide and narrow minima are part of a single basin, SGD and other optimizers might still require higher learning rates to navigate from narrow to wide minima. Lower density of wide minima. Wu et al. (2018) compares the sharpness of minima obtained by full-batch gradient descent (GD) with different learning rates for small neural networks on FashionMNIST and Cifar10 datasets. They find that GD with a given learning rate finds the theoretically sharpest feasible minima for that learning rate. Thus, in the presence of several flatter minimas, GD with lower learning rates does not find them, leading to the conjecture that density of sharper minima is perhaps larger than density of wider minima. Baldassi et al. (2020) show analytically for simple, two- layer non-convex networks that wide minima exists and are rare, compared to narrow minima, local minima and saddle points. In this paper, we add significant evidence to these theoretical observations based on empirical results obtained on large-scale, state-of-the-art neural networks through carefully designed experiments. ## 3 Wide-Minima Density Hypothesis Many popular learning rate schedules, such as the step decay schedules for image datasets, start the training with high learning rate, and then reduce the learning rate periodically. For example, consider the case of Cifar-10 on Resnet-18, trained using a typical step learning rate schedule of $0.1,0.01,$ and $0.001$ for 100, 50, 50 epochs each. In many such schedules, even though training loss stagnates after several epochs of high learning rate, one still needs to continue training at high learning rate in order to get good generalization. For example, Figure 2 shows the training loss for Cifar-10 on Resnet-18, trained with a fixed learning rate of 0.1 (orange curve), compared to a model trained via a step schedule with learning rate reduced at epoch 50 (blue curve). As can be seen from the figure, the training loss stagnates after $\approx$ 50 epochs for the orange curve, and locally it makes sense to reduce the learning rate to decrease the loss. However, as shown in Table 2, generalization is directly correlated with duration of training at high learning rate, with the highest test accuracy achieved when the high learning rate is used for 100 epochs, well past the point where training loss stagnates. Note that the final training loss remains similar for all runs. To understand the above phenomena, we perform another experiment. We train Cifar-10 on Resnet-18 for 200 epochs, using a high learning rate of $0.1$ for only 30 epochs and then use learning rate of $0.01$ and $0.001$ for 85 epochs each. We repeat this training 50 times with different random weight initializations. On an average, as expected, this training yields a low test accuracy of $94.81$. However, in 1 of the 50 runs, we find that the test accuracy reaches $95.24$, even higher than the average accuracy of $95.1$ obtained while training at high learning rate for 100 epochs! Figure 1: Training loss for Cifar-10 on Resnet-18. Orange plot uses a fixed learning rate of 0.1, while in blue plot, the learning rate is reduced from 0.1 to 0.01 at epoch 50. Figure 2: Cifar-10 on Resnet-18 trained for 200 epochs with Momentum. A learning rate of 0.1 is used for the explore epochs. Half the remaining epochs are trained at 0.01 and the other half at 0.001. Reported results are average over 4 runs. Epochs at | Test Accuracy | Train Loss ---|---|--- 0.1 LR | Avg. (Std. Dev) | Avg. (Std. Dev.) 0 | 94.34 (0.13) | 0.0017 (8e-5) 30 | 94.81 (0.15) | 0.0017 (8e-5) 40 | 94.91 (0.14) | 0.0018 (9e-5) 60 | 95.01 (0.14) | 0.0018 (1e-4) 80 | 95.05 (0.15) | 0.0019 (1e-4) 100 | 95.10 (0.14) | 0.0021 (1e-4) ### 3.1 Hypothesis To explain the above observations, i.e., using a high learning rate for short duration results in low average test accuracy with rare occurrences of high test accuracy, while using the same high learning rate for long duration achieves high average test accuracy and frequent occurrences of high test accuracy, we introduce a new hypothesis. We hypothesize that, in the DNN loss landscape, the density of narrow minima is significantly higher than that of wide minima. Intuitively, a large learning rate can escape narrow minima “valleys” easily (as the optimizer can jump out of them with large steps). However, once it reaches a wide minima “valley”, it is likely to get stuck in it (if the “width” of the wide valley is large compared to the step size). This intuition is backed by theoretical results from Xie et al. (2020) that show that the time to escape a minimum using SGD is exponential in the inverse of learning rate as well as inverse of the sharpness (measured by eigenvalue of the Hessian at the minima). Thus, large learning rates escape narrow minima exponentially faster than wide minima. If wide and narrow minima were uniformly distributed, SGD with a large LR would be able to quickly escape the narrow minima, land on a wide minima and get stuck there. Yet, we see that we need to maintain large LR for significant duration for landing in a wide minima with high probability. On the other hand, if our hypothesis is true, i.e., wide minima are much fewer than narrow minima, the probability of landing in a wide minima after escaping a narrow minima is low, and the optimizer needs to take a lot of steps to have a high probability of eventually landing in a wide minimum. Thus, the hypothesis is a better explanation for the observation in Table 2, where the average accuracy continues to improve as we increase the number of high learning rate training steps. The hypothesis also explains why very few (just 1) of the 50 runs trained at $0.1$ learning rate for just 30-epochs also manages to attain high accuracy—these runs just got lucky in a probabilistic sense and landed in a wide minimum even with a shorter duration of explore. | | | ---|---|---|--- (a) Explore 0 | (b) Explore 30 | (c) Explore 60 | (d) Explore 100 Figure 3: Histogram of minima sharpness (Keskar et al., 2016) for 50 random trials of Cifar-10 on Resnet-18. Each figure shows histograms for runs with different number of explore epochs. The distribution moves toward lower sharpness and tightens as the number of explore epochs increase. | | | ---|---|---|--- (a) Explore 0 | (b) Explore 30 | (c) Explore 60 | (d) Explore 100 Figure 4: Histogram of test accuracy for 50 random trials of Cifar-10 on Resnet-18. Each figure shows histograms for runs with different number of explore epochs. The distribution moves toward higher test accuracy and sharpens as the number of explore epochs increase. To validate this hypothesis further, we run experiments similar to the one in Table 2. Specifically, we train Cifar-10 on Resnet-18 model for 200 epochs using a standard step schedule with learning rate of $0.1,0.01,0.001$. We vary the number of epochs trained using the high learning rate of 0.1, called the explore epochs, from 0 to 100 epochs, and divide up the rest of the training equally between 0.01 and 0.001. For each experimental setting, we conduct 50 random trials and plot the distributions of final test accuracy and the minima sharpness as defined by the metric in Keskar et al. (2016) (see section 3.2). If our hypothesis is true, then the more you explore, the higher the probability of landing (and getting stuck) in a wide minima region, which should cause the distribution to tighten and move towards wider minima (lower sharpness), as the number of explore steps increase. This is exactly what is observed in Figure 4. Also since wide minima correlate with higher test accuracy, we should see the test accuracy distribution move towards higher accuracy and sharpen, as the number of explore steps increase. This is confirmed as well in Figure 4. Longer training with low learning rate is not sufficient. Finally, to verify whether explore at high learning rate is essential, we train Cifar-10 for 10,000 epochs at a fixed lower learning rate of 0.001. The training loss converged but the final test accuracy was only 93.9, compared to an accuracy of over 95% in 200 epochs in Table 2. Thus, even training $50\times$ longer at low learning rate is not sufficient to achieve good generalization. Again, this observation ties in well with the theoretical results from Xie et al. (2020) where the authors show that the time to escape a minimum using SGD is exponential in the inverse of learning rate. Thus, this result adds further evidence to our density hypothesis, since even training $50\times$ longer at a low learning rate is not sufficient to land in a wide minima. Multi-scale. Given the importance of explore at high learning rate, a natural question that may arise is whether explore is necessary at smaller learning rate as well. To answer this, we train the same network for a total of 200 epochs with an initial high learning rate of $0.1$ for 100 epochs, but now we vary the number of epochs trained with the learning rate of $0.01$ (we call this finer-scale explore), and train with learning rate of $0.001$ for the remaining epochs. As can be seen from Table 1, although the final training loss remains similar, we find that finer-scale explore also plays a role similar to the initial explore in determining the final test accuracy. This indicates that our hypothesis about density of wide/narrow regions indeed holds at multiple scales. Table 1: Cifar-10 on Resnet-18 trained for 200 epochs. A learning rate of 0.1 is used for the first 100 epochs. We then vary the number of epochs trained with learning rate of $0.01$ (called finer-scale explore), and train the remaining epochs with a learning rate of $0.001$. We report averages values over 3 runs. Explore Epochs (Finer-scale) | Test Accuracy | Training Loss | Sharpness ---|---|---|--- 10 | 94.78 | 0.0031 | 5.48 20 | 94.91 | 0.0026 | 4.47 30 | 95.00 | 0.0023 | 4.02 40 | 95.02 | 0.0021 | 3.91 50 | 95.10 | 0.0021 | 3.54 ### 3.2 Minima Sharpness Our hypothesis predicts that higher explore helps the optimizer land in a wider minimum, which in turn helps generalization. We demonstrated this empirically in Figure 4, where we plotted the distribution of the minima sharpness, as measured by the sharpness metric introduced by (Keskar et al., 2016). In this section, we describe Keskar’s sharpness metric in detail. We also introduce a simple projected gradient ascent scheme to compute this metric efficiently, which scales well to large networks. Finally, we also evaluate our hypothesis with a different metric for minima sharpness, the Fisher Score, which is based on the Fisher information matrix. #### 3.2.1 Keskar’s Sharpness Metric Keskar’s sharpness metric is based on measuring the maximum jump in the network’s output function $F$ in a small neighborhood around the minimum. After a few simplifications, Keskar’s metric for sharpness around a point $x$ can be written as: $S_{x,F}(\epsilon):=\frac{(max_{y\in C_{\epsilon}(x)}F(x+y))-F(x)}{1+F(x)}\times 100,$ (1) where $C_{\epsilon}(x)$ is an $\epsilon$ neighborhood around $x$. Keskar et al. (2016) mentions that under certain conditions and for small values of $\epsilon$, $S_{x,F}$ is proportional to the largest eigenvalue of the Hessian. Please see Keskar et al. (2016) for more details. For our measurements we choose an $\epsilon$ of $1e^{-4}$. For solving the maximization problem in Equation 1, Keskar et al. (2016) uses a second-order L-BFGS-B (Byrd et al., 2003) optimization scheme. However, in our experiments we found the method to be very slow. To combat this, Keskar et al. (2016) limited their runs to 10 iterations but we found that results were suboptimal using few iterations. Instead, we employed a projected gradient ascent scheme to solve Equation 1. In each optimization step, we took a small step with a learning rate of 0.001 in the gradient direction and projected the updated point to lie inside $C_{\epsilon}(x)$. Because of the first order nature, this method is much faster. We found that even 1000 iterations were fast to compute and the results were much better than the second order method in all cases we evaluated. Using Keskar’s sharpness metric, we had shown in Figure 4 that the distribution of minima sharpness moves towards lower values as the number of explore epochs increase. In Table 2, we also report the average sharpness of the minima for varying explores. As predicted by our hypothesis, average sharpness decreases as number of explore epochs increase. Table 2: Keskar’s sharpness metric for Cifar-10 on Resnet-18 trained for 200 epochs with Momentum. A learning rate of 0.1 is used for the explore epochs. Half the remaining epochs are trained at 0.01 and the other half at 0.001. We report the average sharpness over 50 different trials. Explore Epochs | Sharpness ---|--- 0 | 10.56 30 | 5.43 60 | 3.86 100 | 3.54 #### 3.2.2 Fisher Score The maximum Eigen value of the Fisher Information Matrix (FIM) estimates the highest curvature at a point, and is used as another metric to measure minima sharpness (Sokol and Park, 2018). We used an unbiased estimate of the true Fisher matrix (see Kunstner et al. (2019)) using 10 unbiased samples per training data. Table 3 shows the average Fisher scores for the Cifar-10 experiments at varying explores. Again, the sharpness measured by the Fisher score decreases as the number of explore epochs increase. Table 3: Fisher Score for Cifar-10 on Resnet-18 trained for 200 epochs with Momentum. A learning rate of 0.1 is used for the explore epochs. Half the remaining epochs are trained at 0.01 and the other half at 0.001. We report the average Fisher score over 10 different trials. Explore Epochs | FIM score ---|--- 0 | 0.051 30 | 0.046 60 | 0.043 100 | 0.042 ## 4 Explore-Exploit Learning Rate Schedule Given that we need to explore at multiple scales for good generalization, how do we go about designing a good learning rate schedule? The search space of the varying learning rate steps and their respective explore duration is enormous. Fortunately, since the explore at the initial scale is searching over the entire loss surface while explore at finer-scales is confined to exploring only the wide-minima region identified by the initial explore, the former is more crucial. In our experiments as well, we found that the initial portion of training is much more sensitive to exploration and needs a substantial number of explore steps, while after this initial phase, several decay schemes worked equally well. This is similar to the observations in (Golatkar et al., 2019) where the authors found that regularization such as weight-decay and data augmentation mattered significantly only during the initial phase of training. The above observations motivate our Explore-Exploit learning rate schedule, where the explore phase first optimizes at a high learning rate for some minimum time in order to land in the vicinity of a wide minima. We should give the explore phase enough time (a hyper-parameter), so that the probability of landing in a wide minima is high. After the explore phase, we know with a high probability, that the optimizer is in the vicinity of a wide region. We now start the exploit phase to descend to the bottom of this wide region while progressively decreasing the learning rate. Any smoothly decaying learning rate schedule can be thought of as doing micro explore-exploit at progressively reduced scales. A steady descent would allow more explore duration at all scales, while a fast descent would explore less at higher learning rates. We experimented with multiple schedules for the exploit phase, and found a simple linear decay to zero, that does not require any hyper- parameter, to be effective in all the models/datasets we tried. We call our proposed learning rate schedule which starts at a constant high learning rate for some minimum time, followed by a linear decay to zero, the Knee schedule. Note that any learning rate decay scheme incorporates an implicit explore during the initial part, where the learning rate stays high enough. To evaluate the benefit of an explicit explore phase, we compare Knee schedule against several decay schemes such as linear and cosine. Interestingly, the results depend on the length of training. For long budget experiments, simple decay schemes perform comparable to Knee schedule in some experiments, since the implicit explore duration is also large, helping these schemes achieve good generalization. However for short budget experiments, these schemes perform significantly worse than Knee schedule, since the implicit explore duration is much shorter. See Table 4 , 5 and 6 for the comparison. Warmup. Some optimizers such as Adam use an initial warmup phase to slowly increase the learning rate. However, as shown in Liu et al. (2019), learning rate warmup is needed mainly to reduce variance during initial training stages and can be eliminated with an optimizer such as RAdam. Learning rate warmup is also used for large-batch training (Goyal et al., 2017). Here, warmup is necessary since the learning rate is scaled to a very large value to compensate for the large batch size. This warmup is complementary and can be incorporated into Knee schedule. For example, we do this for BERTLARGE pretraining experiment where a large 16k batch size was used. ## 5 Evaluation In this section we present extensive empirical evaluation of Knee schedule on multiple models and datasets across various optimizers, and compare Knee schedule against the original hand-tuned learning rate baselines. We first provide an overview of our main results followed by detailed experimental results. We then run further experiments to validate our wide-minima density hypothesis, as well as run sensitivity analysis of seed learning rate on the Knee schedule. Note that, for completeness, we present a detailed comparison of Knee schedule with many other learning rate schedules in literature such as linear decay, cosine decay (Loshchilov and Hutter, 2016), one-cycle (Smith, 2018) in Appendix A. ### 5.1 Experiments We evaluate Knee schedule on multiple models and datasets spanning both vision and NLP problems. The training of these models spanned various optimizers including SGD Momentum, Adam (Kingma and Ba, 2014a), RAdam (Liu et al., 2019) and LAMB (You et al., 2019). For all experiments, we used an out of the box policy, where we only change the learning rate schedule, without modifying anything else. We evaluate on multiple image datasets – Imagenet on Resnet-50, Cifar-10 on Resnet-18; as well as various NLP datasets – pretraining BERTLARGE on Wikipidea+BooksCorpus and fine-tuning it on SQuADv1.1; and WMT’14 (EN-DE), IWSLT’14 (DE-EN) on Transformers. ### 5.2 Results Overview In all our experiments, we find that Knee schedule shows an improvement in test accuracy over the original hand-tuned learning rate baseline as well as various other learning rate schedules in the literature. Further, we also find that Knee schedule can achieve the same accuracy as the baseline with a much reduced training budget. Table 4: We report the top-1 accuracy for ImageNet and Cifar-10, BLEU score for IWSLT’14 and WMT’14 and F1 score for BERT on SQuAD. All values are averaged over multiple runs for each experiment. Experiment details are mentioned in the individual sections of the experiments. | | | Knee | | | | ---|---|---|---|---|---|---|--- Experiment | Training | Knee | Schedule | Baseline | One-Cycle | Cosine | Linear | Budget | Schedule | (Fixed 50% | | Decay | Decay | | (epochs) | | explore) | | | | ImageNet | 90 | 76.71 | 76.58 | 75.87 | 75.39 | 76.41 | 76.54 Cifar-10 | 200 | 95.26 | 95.26 | 95.10 | 94.09 | 95.23 | 95.18 IWSLT | 50 | 35.53 | 35.23 | 34.97 | 34.77 | 35.21 | 34.97 WMT’14 | 70 | 27.53 | 27.41 | 27.29 | 27.19 | 27.35 | 27.29 BERTLARGE | 31250 (iters) | 91.51 | 91.51 | 91.34 | - | - | 91.34 Table 5: Shorter budget training: Test accuracy on all learning rate schedules tried in this paper, but trained with a shortened budget. We report same metrics as Table 4. Knee schedule achieves the same accuracy as baseline schedules using much lower budget, saving precious GPU-hours. | Shortened Training | Knee | | Cosine | Linear | Saving ---|---|---|---|---|---|--- Experiment | Budget | Schedule | One-Cycle | Decay | Decay | ( V100 GPU | (epochs) | | | | | hours) ImageNet | 50 | 75.92 | 75.36 | 75.71 | 75.82 | 27 Cifar-10 | 150 | 95.14 | 93.84 | 95.06 | 95.02 | 0.25 IWSLT | 35 | 35.08 | 34.43 | 34.46 | 34.16 | 0.75 WMT’14 | 30 | 27.28 | 26.80 | 26.95 | 26.77 | 80 BERTLARGE | 20854 (iters) | 91.29 | - | - | 90.64 | 1002 Table 6: Epochs required by different LR schedules to reach the target accuracy. The target accuracy is chosen based on Knee schedule’s results with a reduced budget. Experiment | Target BLEU Score | Knee schedule | Cosine Decay | Linear Decay ---|---|---|---|--- IWSLT | 35.08 | 35 | 45 | 60 WMT’14 | 27.28 | 30 | 60 | 70 Table 4 shows the test accuracies of the various experiments, when trained with the original budget; while Table 5 shows the results when trained with a reduced budget. As shown, for the original budget runs, Knee schedule improves on the test accuracies in all experiments. Note that in Knee schedule, the explore duration is a hyperparameter. To avoid tuning this hyperparameter, we experimented with a fixed 50% explore duration for the full budget runs. Even the fixed 50% explore Knee schedule outperforms all the other baselines. Also noteworthy is that Knee schedule is able to achieve the same test accuracies as the baseline’s full budget runs with a much lower training budget, saving precious GPU cycles (Table 5). While the difference in accuracy values between the various schedules might appear deceptively small in absolute terms, achieving these gains require a large amount of compute. For example, the number of epochs needed by each scheme to reach the target BLEU score for IWSLT’14 DE-EN and WMT’14 EN-DE with the Transformer network is shown in Table 6. One can see that Knee schedule is significantly more efficient as compared to say Cosine Decay, which takes 100% more training time to achieve the same accuracy for WMT‘14 EN-DE. Thus, the accuracy and/or compute gains achieved by Knee schedule is significant. A summary of our main experimental results is as follows: 1. 1. Imagenet on Resnet-50: We show an absolute gain of 0.8% in top-1 accuracy against the competitive step schedule baseline for this model. Also, Knee schedule can achieve the same accuracy as baseline in $\sim$45% less training epochs. 2. 2. BERTLARGE pre-training on Wikipedia+BooksCorpus dataset: Compared to the baseline of You et al. (2019), we improve the F1 score on SQuAD v1.1 fine- tuning task by 0.2% (91.51 compared to 91.34). Also, we were able to achieve similar accuracy as baseline in 33% less training steps (a saving of $\sim$1002 V100 GPU-hours!). 3. 3. WMT’14 and IWSLT machine translation on Transformers: Compared to competitive baselines, we were able to improve the BLEU scores by 0.24 and 0.56 points for the two tasks. Moreover, Knee schedule was able to achieve the same accuracy as baselines in 57% and 30% less training times. 4. 4. State of the Art (SOTA) Results: We also attain state of the art results on the IWSLT’14(DE-EN) machine translation dataset by simply replacing the learning rate schedule of the current SOTA model (Shen et al., 2020) with Knee. We were able to improve the BLEU score by 0.18, reaching a new SOTA score of 37.78. Moreover, Knee can achieve the current SOTA baseline value in 30% less training time. ### 5.3 Detailed Results We now describe each of our main experimental results in detail. #### 5.3.1 ImageNet Image Classification on Resnet-50 We train ImageNet dataset (Russakovsky et al., 2015) on Resnet-50 network (He et al., 2016) which has 25 million parameters, with a batch size of 256 and a seed learning rate of 0.1. Random cropping and random horizontal flipping augmentations were applied to the training dataset. We use SGD optimizer with momentum of 0.9 and weight decay of $1e^{-4}$. For baseline runs, we used the standard hand-tuned step learning rate schedule of 0.1, 0.01 and 0.001 for 30 epochs each. For Knee schedule we used a seed learning rate of 0.1 (same as baseline). We trained with the original budget of 90 epochs as well as with a reduced budget of 50 epochs. We used 30 explore epochs for the two experiments. 222We used the opensource implementation at: https://github.com/cybertronai/imagenet18_old Table 7 shows the training loss and test accuracies for our experiments. Knee schedule comfortably beats the test accuracy of baseline in the full budget run (with absolute gains of 0.8% and 0.4% in top-1 and top-5 accuracy, respectively), while meeting the baseline accuracy even with a much shorter budget. The fact that the baseline schedule takes almost $80\%$ more training time than Knee schedule for the same test accuracy, shows the effectiveness of our Explore-Exploit scheme. See Figure 6 in Appendix B for training curves. Table 7: ImageNet on Resnet-50 results. We report mean (stddev) over 3 runs. LR Schedule | Test Top 1 Acc. | Test Top 5 Acc. | Training Loss | Training Epochs ---|---|---|---|--- Baseline | 75.87 (0.035) | 92.90 (0.015) | 0.74 (1e-3) | 90 Knee | 76.71 (0.097) | 93.32 (0.031) | 0.79 (1e-3) | 90 Knee (short budget) | 75.92 (0.11) | 92.90 (0.085) | 0.90 (3e-3) | 50 #### 5.3.2 Cifar-10 Image Classification on Resnet-18 We train Cifar-10 dataset (Krizhevsky et al., 2009) on Resnet-18 network (He et al., 2016) which has around 11 million parameters. SGD optimizer is used with momentum of 0.9 and weight decay of $5e^{-4}$. Random cropping and random horizontal flipping augmentations were applied to the training dataset. 333We used the open-source implementation at: https://github.com/kuangliu/pytorch- cifar. For baseline, we used the hand-tuned step learning rate schedule of 0.1, 0.01 and 0.001 for 100, 50, 50 epochs, respectively. With Knee schedule, we train the network with the original budget of 200 epochs, as well as a reduced budget of 150 epochs. We used 100 explore epochs for both runs, and a seed learning rate of 0.1 (same as baseline). Table 8 shows the training loss and test accuracies for the experiments. Knee schedule beats the test accuracy of baseline in the full budget run, while meeting the baseline test accuracy in $25\%$ less budget. Refer to figure 7 in Appendix B for detailed comparisons of training loss, test accuracy, and learning rate. Table 8: Training loss and Test accuracy for Cifar-10 on Resnet-18. We report mean (stddev) over 7 runs. LR Schedule | Test Accuracy | Training Loss | Training Epochs ---|---|---|--- Baseline | 95.10 (0.14) | 0.002 (1e-4) | 200 epochs Knee | 95.26 (0.11) | 0.002 (1e-4) | 200 epochs Knee (short budget) | 95.14 (0.18) | 0.004 (3e-4) | 150 epochs #### 5.3.3 BERTLARGE Pre-training We pretrain on BERTLARGE on Wikipedia+BooksCorpus dataset with LAMB optimizer (You et al. (2019)). BERTLARGE has around 330 million parameters and the pre- training is divided into two phases with different sequence lengths. The first phase consists of 90% steps with sequence length of 128 and the second phase consists of the remaining 10% steps with sequence length of 512 (Devlin et al. (2018)). We used a batch size of 16384 in both phases of training 444We used the open-source implementation at: https://github.com/NVIDIA/DeepLearningExamples/tree/master/PyTorch/LanguageModeling/BERT. We use the same training budget of 31250 steps mentioned in (You et al. (2019)). We also train the model on a shortened training budget of 2/3rd the original steps (20854 steps). Since large batch training requires learning rate warmup (see Goyal et al. (2017)), we incorporate it into the Knee schedule by first doing a warmup of 10% as suggested in (You et al., 2019) followed by the explore-exploit phases. We used an explore of 50% of the total steps available for both phases of BERT training. For baseline, we use the warmup (10%) + linear decay (90%) schedule (You et al., 2019; Devlin et al., 2018). The pre-trained models are evaluated on the SQuAD v1.1 (Rajpurkar et al., 2016) dataset by fine-tuning on the dataset for 2 epochs. See Table 9 for the results. For the full budget run, Knee schedule improves the baseline by 0.2%, while for the reduced budget we achieved similar fine-tuning accuracy as baseline. The baseline schedule achieves a much lower accuracy with shorter budget training, showing the efficacy of Knee schedule. BERT pre-training is extremely compute expensive and takes around 47 hours on 64 V100 GPUs (3008 V100 GPU-hrs) on cloud VMs. The reduced budget amounts to a saving of approximately 1002 V100 GPU-hours! Table 9: BERTLARGE results. We report the pre-training train loss, and the test F1 accuracy on SQuAD v1.1 after fine-tuning. See figure 9 in Appendix B for training curves. LR Schedule | F1 score on SQuAD v1.1 | Training loss | Total Training Steps ---|---|---|--- Knee | 91.51 | 1.248 | 31250 Baseline (You et al., 2019) | 91.34 | - | 31250 Baseline (short budget) | 90.64 | 1.336 | 20854 Knee (short budget) | 91.29 | 1.275 | 20854 #### 5.3.4 Machine Translation on Transformer Network with WMT’14 and IWSLT In the second NLP task, we train the Transformer (base model) (Vaswani et al., 2017) on the IWSLT’14 (De-En) (Cettolo et al., 2014) and WMT’14 (En-De) (Bojar et al., 2014) datasets with the RAdam (Liu et al., 2019) optimizer. ##### WMT’14 (EN-DE): We use the default implementation provided by the fairseq package (Ott et al., 2019) 555https://github.com/pytorch/fairseq. We train WMT’14 (EN-DE) dataset on the TransformerBASE (Vaswani et al., 2017) model which has around 86 million parameters and use the RAdam (Liu et al., 2019) optimizer with $\beta_{1}$ of 0.9 and $\beta_{2}$ of 0.999. Label smoothed cross entropy was used as the objective function with an uncertainty of 0.1. A dropout of 0.1, clipping norm of 25 and weight decay of $1e^{-4}$ is used. Each training batch contains approximately 30000 tokens. The baseline schedule uses a linear decay for 70 epochs (Liu et al., 2019). With Knee schedule, we trained with the original budget of 70 epochs, as well as a reduced budget of 30 epochs. We used 50 and 25 explore epochs for the two runs, respectively and a seed learning rate of $3e^{-4}$ for both Knee schedule and baseline. In all cases we use the model checkpoint with least loss on the validation set for computing BLEU scores on the test set. Table 10 shows the training loss and test accuracy averaged over 3 runs. Knee schedule improves the test BLEU score of baseline in the full budget run by 0.24 points. In the shorter budget run, Knee schedule matches the test accuracy of the baseline while taking $57\%$ less training time (a saving of 80 V100 GPU- hours!). See Figure 10 in Appendix B for training curves. ##### IWSLT’14 (DE-EN): For IWSLT’14 (DE-EN) we use the same configuration as WMT’14 (EN-DE), except for a dropout of 0.3 following Fairseq’s out-of-box implementation. Each training batch contains approximately 4000 tokens. For Knee schedule, we choose explore as 30 epochs for short budget runs and 40 epochs for full budget runs. The baseline schedule uses a linear decay for 50 epochs (Liu et al., 2019). With Knee schedule, we trained with the original budget of 50 epochs, as well as a reduced budget of 35 epochs. We used 40 and 30 explore epochs for the two runs, respectively and a seed learning rate of $3e^{-4}$ for both Knee schedule and baseline. In all cases we use the model checkpoint with least loss on the validation set for computing BLEU scores on the test set. Knee schedule improves the baseline test BLEU score by 0.56 points in the full budget run. In the shorter budget run, Knee schedule matches the test accuracy of the baseline schedule while taking $30\%$ less training time. See Figure 11 in Appendix B for training curves. Table 10: Results for WMT’14 (EN-DE) on Transformer networks. The test BLEU scores are computed on the checkpoint with the best validation perplexity. We report mean (stdev) over 3 runs. LR Schedule | Test BLEU | Train | Validation | Training ---|---|---|---|--- Score | Perplexity | Perplexity | Epochs Baseline | 27.29 (0.06) | 3.87 (0.017) | 4.89 (0.02) | 70 Knee | 27.53 (0.12) | 3.89 (0.017) | 4.87 (0.006) | 70 Knee (short budget) | 27.28 (0.17) | 4.31 (0.02) | 4.92 (0.007) | 30 Table 11: Training, validation perplexity and test BLEU scores for IWSLT on Transformer networks. The test BLEU scores are computed on the checkpoint with the best validation perplexity. We report the mean and standard deviation over 3 runs. LR Schedule | Test BLEU | Train | Validation | Training ---|---|---|---|--- Score | Perplexity | Perplexity | Epochs Baseline | 34.97 (0.035) | 3.36 (0.001) | 4.91 (0.035) | 50 Knee | 35.53 (0.06) | 3.00 (0.044) | 4.86 (0.02) | 50 Knee (short budget) | 35.08 (0.12) | 3.58 (0.049) | 4.90 (0.063) | 35 #### 5.3.5 SQuAD-v1.1 fine-tuning on BERTBASE We also evaluate Knee schedule on the task of fine-tuning BERTBASE model Devlin et al. (2018) on SQuAD v1.1 Rajpurkar et al. (2016) with the Adam Kingma and Ba (2014b) optimizer 666We used the implementation at: https://github.com/huggingface/transformers. BERT fine-tuning is prone to overfitting because of the huge model size compared to the small fine-tuning dataset, and is typically run for only a few epochs. For baseline we use the linear decay schedule mentioned in Devlin et al. (2018). We use a seed learning rate of $3e^{-5}$ and train for 2 epochs. For Knee schedule, we train the network with 1 explore epoch with the same seed learning rate of $3e^{-5}$. Table 12 shows our results over 3 runs. We achieve a mean EM score of 81.4, compared to baseline’s 80.9, a 0.5% absolute improvement. We don’t do a short budget run for this example, as the full budget is just 2 epochs. Please refer to Figure 14 in Appendix B for the training loss, test accuracy and learning rate curves. Table 12: SQuAD fine-tuning on BERTBASE. We report the average training loss, and average test EM, F1 scores over 3 runs. LR Schedule | EM | F1 | Train Loss | Training Epochs ---|---|---|---|--- Baseline | 80.89 (0.15) | 88.38 (0.032) | 1.0003 (0.004) | 2 Knee schedule | 81.38 (0.02) | 88.66 (0.045) | 1.003 (0.002) | 2 #### 5.3.6 State of the Art Result To further demonstrate the effectiveness of Knee schedule, we took a recent high performing model, Cutoff (Shen et al., 2020)777We used the code available at https://github.com/dinghanshen/Cutoff, which had reported state-of-the-art accuracy on the IWSLT’14 (DE-EN) dataset. They reported a BLEU score of 37.6 when trained with an inverse square root learning rate schedule for 100 epochs, with the first 6000 steps allocated for warmup. We simply retrained the model with our Knee schedule, and achieved a new SOTA BLEU score of 37.78 (an absolute increase of 0.18). See Table 13 for the BLEU scores, training and validation perplexities. We also show that Knee schedule can train the model in 30% less training time (70 epochs), while achieving slightly better accuracy of 37.66 BLUE score compared to the 100 epoch baseline. The baseline schedule when run for 70 epochs achieves a much worse accuracy of 37.31. For both the full budget (100 epochs) and the short budget (70 epochs) Knee runs, we choose 50% of the total training epochs as explore epochs. We also perform warmup for the same number of steps as baseline. For all runs (Knee and baseline), we report the BLEU score obtained by averaging the last 5 checkpoints and computing on the test set. See Figure 12 and 13 in Appendix B for training curves. Table 13: Training, validation perplexity and test BLEU scores for IWSLT’14 DE-EN on Cutoff. The test BLEU scores are computed by averaging the last 5 checkpoints LR Schedule | Test BLEU | Train | Validation | Training ---|---|---|---|--- Score | Perplexity | Perplexity | Epochs Inv. Sqrt | 37.60 | 3.46 | 4.24 | 100 Knee | 37.78 | 3.29 | 4.13 | 100 Inv. Sqrt (short budget) | 37.31 | 3.76 | 4.29 | 70 Knee (short budget) | 37.66 | 3.48 | 4.18 | 70 ### 5.4 Hypothesis Validation with Knee schedule on Language Tasks For validating our hypothesis on the density of wide minima vs narrow minima, we did multiple experiments on vision tasks, most of which were discussed in Section 3. To summarize, in Figures 4 and 4, we showed that for Cifar-10 on Resnet-18, as the number of explore steps increase, the distribution of minima width and test accuracy sharpens and shifts towards wider minima and better accuracy, respectively. Table 14: IWSLT’14 (DE-EN) on the Transformer network trained with the Knee schedule. The explore duration is varied, while keeping the total training budget fixed at 50 epochs. We report averages over 3 runs. Explore Epochs | Test BLEU score | Training Perplexity ---|---|--- 5 | 34.93 | 3.29 10 | 35.02 | 3.22 15 | 35.08 | 3.11 20 | 35.10 | 3.08 25 | 35.23 | 3.02 30 | 35.28 | 2.99 40 | 35.53 | 3.00 We now perform similar experiments on the IWSLT’14 German to English dataset (Cettolo et al., 2014) trained on Transformer networks (Vaswani et al., 2017) to demonstrate that our hypothesis holds even on a completely different NLP dataset and network architecture. We train with the Knee schedule for a total budget of 50 epochs with explore lr as $3e^{-4}$, but keep varying the number of explore epochs. As shown in Table 14, the test BLEU score increases as we increase the number of explore epochs. Further, we found that among multiple trials, a 20 epoch explore run had a high BLEU score of 35.29, suggesting that the run got lucky. Thus, these results on the IWSLT’14 (DE-EN) dataset add more evidence to the wide-minima density hypothesis. ### 5.5 Learning Rate Sensitivity for Knee schedule We performed sensitivity analysis of the starting learning rate, referred to as the seed learning rate, for Knee schedule. We trained the Cifar-10 dataset on Resnet-18 with the Knee schedule for a shortened budget of 150 epochs, starting at different seed learning rates. For each experiment, we do a simple linear search to find the best explore duration. The test accuracies and optimal explore duration for the different seed learning rate choices is shown in Table 15. As shown, the seed learning rate can impact the final accuracy, but Knee schedule is not highly sensitive to it. In fact, we can achieve the target accuracy of 95.1 with multiple seed learning rates of 0.05, 0.075, 0.0875 and 0.115, as compared to the original seed learning rate of 0.1, by tuning the number of explore epochs. Another interesting observation is that the optimal explore duration varies inversely with the seed learning rate. Since a bigger learning rate has higher probability of escaping narrow minima compared to a lower learning rate, it would, on an average, require fewer steps to land in a wide minima. Thus, larger learning rates can explore faster, and spend more time in the exploit phase to go deeper in the wide minimum. This observation is thus consistent with our hypothesis and further corroborates it. We also note that by tuning both seed learning rate and explore duration, we can achieve the twin objectives of achieving a higher accuracy, as well as a shorter training time – e.g. here we are able to achieve an accuracy of 95.34 in 150 epochs (seed learning rate 0.075), compared to 95.1 achieved by the baseline schedule in 200 epochs. Table 15: Seed learning rate sensitivity analysis. Cifar-10 on Resnet-18 trained for 150 epochs with Knee schedule. We vary the seed learning rate and explore epochs to get the best test accuracy for the particular setting. We report averages over 3 runs. Seed LR | Test Accuracy | Optimal Explore Epochs ---|---|--- 0.03 | 95.07 | 120 0.05 | 95.12 | 120 0.0625 | 95.15 | 120 0.075 | 95.34 | 100 0.0875 | 95.22 | 100 0.1 | 95.14 | 100 0.115 | 95.20 | 60 0.125 | 95.06 | 60 0.15 | 95.04 | 30 ## 6 Conclusions In this paper, we make an observation that an initial explore phase with a high learning rate is essential for good generalization of DNNs. Further, we find that a minimum explore duration is required even if the training loss stops improving much earlier. We explain this observation via our hypothesis that in the DNN loss landscape, the density of wide minima is significantly lower than that of narrow minima. Motivated by this hypothesis, we present an Explore-Exploit based learning rate schedule, called the Knee schedule. We do extensive evaluation of Knee schedule on multiple models and datasets. 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Large batch optimization for deep learning: Training bert in 76 minutes. In _International Conference on Learning Representations_ , 2019. ## A Comparisons with Other Baseline Learning Rate Schedules In this section we compare Knee schedule against several other learning rate schedules – one-cycle, linear decay and cosine decay. One-Cycle: The one-cycle learning rate schedule was proposed in Smith (2018) (also see Smith (2017)). This schedule first chooses a maximum learning rate based on an learning rate range test. The learning rate range test starts from a small learning rate and keeps increasing the learning rate until the loss starts exploding (see figure 5). Smith (2018) suggests that the maximum learning rate should be chosen to be bit before the minima, in a region where the loss is still decreasing. There is some subjectivity in making this choice, although some blogs and libraries888See e.g. https://towardsdatascience.com/finding-good-learning-rate-and-the-one-cycle- policy-7159fe1db5d6 and https://sgugger.github.io/how-do-you-find-a-good- learning-rate.html. Also see https://docs.fast.ai/callbacks.lr_finder.html and https://docs.fast.ai/callbacks.one_cycle.html suggest using a learning rate one order lower than the one at minima. We go with this choice for all our runs. Once the maximum learning rate is chosen, the one-cycle schedule proceeds as follows. The learning rate starts at a specified fraction999See div_factor in https://docs.fast.ai/callbacks.one_cycle.html. We chose the fraction to be 0.1 in our experiments. of the maximum learning rate and is increased linearly to the maximum learning rate for 45 percent of the training budget and then decreased linearly for the remaining 45. For the final 10 percent, the learning rate is reduced by a large factor (we chose a factor of 10). We used an opensource implementation 101010https://github.com/nachiket273/One_Cycle_Policy for our experiments. (a) LR range test for CIFAR-10 (b) LR range test for IWSLT’14 DE-EN (c) LR range test for WMT’14 EN-DE (d) LR range test for ImageNet Figure 5: learning rate range test for selecting the maximum learning rate. A good choice is the learning rate is a bit before the minima in a region where the loss is still decreasing. Linear Decay: The linear decay learning rate schedule simply decays the learning rate linearly to zero starting from a seed learning rate. Cosine Decay: The cosine decay learning rate schedule decays the learning rate to zero following a cosine curve, starting from a seed learning rate. ### A.1 Cifar-10 Figure 5(a) shows the learning rate range test for Cifar-10 with the Resnet-18 network. The minima occurs around learning rate of 0.09, and we choose $9e^{-3}$ as the maximum learning rate for the One-Cycle runs. For linear, cosine decay schedules we start with a seed learning rate of 0.1 as used in the standard baselines. The training loss and test accuracy for the various schedules are shown in Table 16 for the full budget runs (200 epochs), and in Table 17 for the short budget runs (150 epochs). Table 16: Cifar-10 on Resnet-18 full budget training (200 epochs): Training loss and Test accuracy for more learning rate schedules. We report the mean and standard deviation over 7 runs. LR Schedule | Test Accuracy | Train Loss ---|---|--- One-Cycle | 94.08 (0.07) | 0.0041 (6e-5) Cosine Decay | 95.23 (0.11) | 0.0023 (9e-5) Linear Decay | 95.18 (0.15) | 0.0018 (7e-5) Knee schedule | 95.26 (0.11) | 0.0023 (1e-4) Table 17: Cifar-10 on Resnet-18 short budget training (150 epochs): Training loss and Test accuracy for more learning rate schedules. We report the mean and standard deviation over 7 runs. LR Schedule | Test Accuracy | Train Loss ---|---|--- One-Cycle | 93.84 (0.082) | 0.0052 (7e-5) Cosine Decay | 95.06 (0.16) | 0.0030 (2e-4) Linear Decay | 95.02 (0.10) | 0.0021 (1e-4) Knee schedule | 95.14 (0.18) | 0.0044 (3e-4) ### A.2 ImageNet Figure 5(d) shows the learning rate range test for ImageNet with the Resnet-50 network. The minima occurs around learning rate of 2.16, and we choose $0.216$ as the maximum learning rate for One-Cycle runs. For linear, cosine decay schedules we start with a seed learning rate of 0.1 as used in the standard baselines. The training loss and test accuracy for the various schedules are shown in Table 18 for the full budget runs (90 epochs), and in Table 19 for the short budget runs (50 epochs). Table 18: ImageNet with ResNet-50 full budget training (90 epochs): Training loss, Test Top-1 and Test Top-5 for more learning rate schedules. We report the mean and standard deviation over 3 runs. LR Schedule | Test Top-1 | Test Top-5 | Train Loss (av) ---|---|---|--- One Cycle | 75.39 (0.137) | 92.56 (0.040) | 0.96 (0.003) Cosine Decay | 76.41 (0.212) | 93.28 (0.066) | 0.80 (0.002) Linear decay | 76.54 (0.155) | 93.21 (0.051) | 0.75 (0.001) Knee schedule | 76.71 (0.097) | 93.32 (0.031) | 0.79 (0.001) Table 19: ImageNet with ResNet-50 short budget training (50 epochs): Training loss, Test Top-1 and Test Top-5 for more learning rate schedules. We report the mean and standard deviation over 3 runs. LR Schedule | Test Top-1 | Test Top-5 | Train Loss (av) ---|---|---|--- One Cycle | 75.36 (0.096) | 92.53 (0.079) | 1.033 (0.004) Cosine Decay | 75.71 (0.116) | 92.81 (0.033) | 0.96 (0.002) Linear decay | 75.82 (0.080) | 92.84 (0.036) | 0.91 (0.002) Knee schedule | 75.92 (0.11) | 92.90 (0.085) | 0.90 (0.003) ### A.3 WMT’14 EN-DE Figure 5(c) shows the learning rate range test for WMT’14 EN-DE on the transformer networks. The minima occurs near $1.25e^{-3}$. For the maximum learning rate, we choose $2.5e^{-4}$ for the default one-cycle policy. For linear, cosine decay schedules we start with a seed learning rate of $3e^{-4}$ as used in the standard baselines The training, validation perplexity and BLEU scores for the various schedules are shown in Table 20 for the full budget runs (70 epochs), and in Table 21 for the short budget runs (30 epochs). Table 20: WMT’14 (EN-DE) on Transformer networks full budget training (70 epochs): Training, validation perplexity and test BLEU scores for more learning rate schedules. The test BLEU scores are computed on the checkpoint with the best validation perplexity. We report the mean and standard deviation over 3 runs. LR Schedule | Test BLEU Score | Train ppl | Validation ppl ---|---|---|--- One-Cycle | 27.19 (0.081) | 3.96 (0.014) | 4.95 (0.013) Cosine Decay | 27.35 (0.09) | 3.87 (0.011) | 4.91 (0.008) Linear Decay | 27.29 (0.06) | 3.87 (0.017) | 4.89 (0.02) Knee schedule | 27.53 (0.12) | 3.89 (0.017) | 4.87 (0.006) Table 21: WMT’14 (EN-DE) on Transformer networks short budget training (30 epochs): Training, validation perplexity and test BLEU scores for more learning rate schedules. The test BLEU scores are computed on the checkpoint with the best validation perplexity. We report the mean and standard deviation over 3 runs. LR Schedule | Test BLEU Score | Train ppl | Validation ppl ---|---|---|--- One-Cycle | 26.80 (0.2) | 4.38 (0.017) | 5.02 (0.007) Cosine Decay | 26.95 (0.23) | 4.32 (0.013) | 4.99 (0.011) Linear Decay | 26.77 (0.12) | 4.36 (0.092) | 5.02 (0.01) Knee schedule | 27.28 (0.17) | 4.31 (0.02) | 4.92 (0.007) ### A.4 IWSLT’14 DE-EN Figure 5(b) shows the learning rate range test for IWSLT on the transformer networks. The minima occurs near $2.5e^{-3}$. For the maximum learning rate, we choose $2.5e^{-4}$ for the default one-cycle policy. For linear, cosine decay schedules we start with a seed learning rate of $3e^{-4}$ as used in the standard baselines The training, validation perplexity and BLEU scores for the various schedules are shown in Table 22 for the full budget runs (50 epochs), and in Table 23 for the short budget runs (35 epochs). Table 22: IWSLT’14 (DE-EN) on Transformer networks full budget training (50 epochs): Training, validation perplexity and test BLEU scores for more learning rate schedules. The test BLEU scores are computed on the checkpoint with the best validation perplexity. We report the mean and standard deviation over 3 runs. LR Schedule | Test BLEU Score | Train ppl | Validation ppl ---|---|---|--- One-Cycle | 34.77 (0.064) | 3.68 (0.009) | 4.97 (0.010) Cosine Decay | 35.21 (0.063) | 3.08 (0.004) | 4.88 (0.014) Linear Decay | 34.97 (0.035) | 3.36 (0.001) | 4.92 (0.035) Knee schedule | 35.53 (0.06) | 3.00 (0.044) | 4.86 (0.02) Table 23: IWSLT’14 (DE-EN) on Transformer networks short budget training (35 epochs): Training, validation perplexity and test BLEU scores for more learning rate schedules. The test BLEU scores are computed on the checkpoint with the best validation perplexity. We report the mean and standard deviation over 3 runs. LR Schedule | Test BLEU Score | Train ppl | Validation ppl ---|---|---|--- One-Cycle | 34.43 (0.26) | 3.98 (0.028) | 5.09 (0.017) Cosine Decay | 34.46 (0.33) | 3.86 (0.131) | 5.06 (0.106) Linear Decay | 34.16 (0.28) | 4.11 (0.092) | 5.14 (0.066) Knee schedule | 35.08 (0.12) | 3.58 (0.063) | 4.90 (0.049) ### A.5 SQuAD-v1.1 finetuning with BERTBASE We choose $1e^{-5}$ as the maximum learning rate for One-Cycle runs as the minima occurs close to $1e^{-4}$ . For linear, cosine decays we start with a seed learning rate of $3e^{-5}$ as used in standard baselines. Table 24 show the average training loss, average test EM and F1 scores for the various schedules. We did not do a short budget training for this dataset, as the full budget is just 2 epochs. Table 24: SQuAD-v1.1 fine-tuning on BERTBASE for more learning rate schedules. We report the average training loss, average test EM, F1 scores over 3 runs. LR Schedule | EM (av) | F1 (av) | Train Loss (av) ---|---|---|--- One Cycle | 79.9 (0.17) | 87.8 (0.091) | 1.062 (0.003) Cosine Decay | 81.31 (0.07) | 88.61 (0.040) | 0.999 (0.003) Linear decay | 80.89 (0.15) | 88.38 (0.042) | 1.0003 (0.004) Knee schedule | 81.38 (0.02) | 88.66 (0.045) | 1.003 (0.002) ## B Detailed Plots Figure 6: ImageNet on Resnet-50 trained with Momentum. Shown are the training loss, top-1/top-5 test accuracy and learning rate as a function of epochs, for the baseline scheme (orange) vs the Knee schedule scheme (blue). The plot is split into 3 parts to permit higher fidelity in the y-axis range. Figure 7: Cifar-10 on Resnet-18 trained with Momentum. Shown are the training loss, test accuracy and learning rate as a function of epochs, for the baseline scheme (orange) vs the Knee schedule scheme (blue). The plot is split into 3 parts to permit higher fidelity in the y-axis range. Figure 8: BERTLARGE pretraining for batch size of 16k with LAMB optimizer for the short budget runs. Shown are the training loss and learning rate as a function of steps, for the baseline scheme short budget (orange) vs the Knee schedule scheme short budget (blue). The plot is split into 2 parts to give a clear picture of the two phases of training Devlin et al. (2018). Note that even though the training loss curves look similar for the two runs, we see a significant gap in F1 score obtained when we fine-tune the model checkpoints on SQuAD-v1.1 Rajpurkar et al. (2016). See Table 9 for details. LR Schedule | F1 - Trial 1 | F1 - Trial 2 | F1 - Trial 3 | F1 avg. | F1 max ---|---|---|---|---|--- Baseline (short budget) | 90.39 | 90.64 | 90.53 | 90.52 | 90.64 Knee schedule ( short budget ) | 91.22 | 91.29 | 91.18 | 91.23 | 91.29 Knee schedule ( full budget ) | 91.45 | 91.41 | 91.51 | 91.46 | 91.51 Figure 9: SQuAD fine-tuning on BERTLARGE. We report F1 scores for 3 different trials as well as the maximum and average values. Figure 10: WMT’14 (EN-DE) on TransformerBASE network trained with RAdam. Shown are the training perplexity, validation perplexity and learning rate as a function of epochs, for the baseline scheme (orange) vs the Knee schedule scheme (blue). The plot is split into 3 parts to permit higher fidelity in the y-axis range. Figure 11: IWSLT’14 (DE-EN) on TransformerBASE network trained with RAdam. Shown are the training perplexity, validation perplexity and learning rate as a function of epochs, for the baseline scheme (orange) vs the Knee schedule scheme (blue). The plot is split into 3 parts to permit higher fidelity in the y-axis range. Figure 12: IWSLT’14 (DE-EN) on the SOTA model Cutoff(Shen et al., 2020), trained with Adam. Shown are the training perplexity, validation perplexity and learning rate as a function of epochs, for the baseline scheme (orange) vs the Knee schedule scheme (blue). Figure 13: IWSLT’14 (DE-EN) on the SOTA model Cutoff(Shen et al., 2020), trained with Adam with a reduced training budget of 70 epochs. Shown are the training perplexity, validation perplexity and learning rate as a function of epochs, for the baseline scheme (orange) vs the Knee schedule scheme (blue). Figure 14: SQuAD-v1.1 fine-tuning on BERTBASE trained with Adam. Shown are the training loss, test EM score, and learning rate as a function of epochs, for the baseline scheme (orange) vs the Knee schedule scheme (blue). The plot is split into 2 parts to permit higher fidelity in the y-axis range. It is clear that with Knee schedule the network starts to overfit after the 2nd epoch, where the testing loss continues to go down, but generalization suffers. We saw similar behavior with different seeds, and thus need to train with Knee schedule for only 2 epochs.

2024-09-04T02:54:58.282887

2003.03988

arxiv-papers

Original Article Journal Section STDWI, Spike-Timing-Dependent Weight Inference; LIF, Leaky Integrate and Fire; RDD, Regression Discontinuity Design. Nasir Ahmad<EMAIL_ADDRESS> # Overcoming the Weight Transport Problem via Spike-Timing-Dependent Weight Inference Nasir Ahmad Department of Artificial Intelligence, Donders Institute for Brain, Cognition and Behaviour, Radboud University, Nijmegen, the Netherlands Luca Ambrogioni Department of Artificial Intelligence, Donders Institute for Brain, Cognition and Behaviour, Radboud University, Nijmegen, the Netherlands Marcel van Gerven Department of Artificial Intelligence, Donders Institute for Brain, Cognition and Behaviour, Radboud University, Nijmegen, the Netherlands ###### Abstract We propose a solution to the weight transport problem, which questions the biological plausibility of the backpropagation algorithm. We derive our method based upon a theoretical analysis of the (approximate) dynamics of leaky integrate-and-fire neurons. Our results demonstrate that the use of spike timing alone outcompetes existing biologically plausible methods for synaptic weight inference in spiking neural network models. Furthermore, our proposed method is more flexible, being applicable to any spiking neuron model, is conservative in how many parameters are required for implementation and can be deployed in an online-fashion with minimal computational overhead. These features, together with its biological plausibility, make it an attractive mechanism for weight inference at single synapses. ###### keywords: Weight Transport Problem, Biologically Plausible Learning, Spiking Neural Network ## 1 Introduction Backpropagation of error is a successful approach for training rate-based neural network models [1, 2]. However, since its inception it has been criticised for its lack of biological plausibility [3, 4]. In particular, in order to update individual synaptic connections weights within a network, information is required about distant error signals and the weights of other synaptic connections of the network – information which is not available locally to the synapse. However, backpropagation’s flexibility, unrelenting success in application-based research, and most significantly its capacity for modelling and reproducing neural response statistics has contributed to a recent re-examination of its potential role and plausibility in neural systems [5, 6, 7, 8, 9]. A number of attempts have been made to explain mechanisms by which backpropagation’s implausibilities can be addressed. These can be divided into methods which propose alternative implementations of backpropagation, namely energy-based and dynamical systems methods which converge to backpropagation of error [10, 11, 12], for an overview see [6], and methods which show that components which are considered implausible can be approximated using alternative and plausible computations [13, 14, 15, 16, 17]. We focus on the latter approaches in this study. Figure 1: The weight transport problem in backpropagation of error. A. The computations involved in the forward-pass of an example feedforward neural network model. B. The backpropagation of error method. Specifically, the derivative of the loss function can be computed with respect to each weight matrix in our example network. Observe that the derivative of the loss function with respect to a weight matrix ($W_{1}$) deep in the network depends upon the weight matrices in the higher layers ($W_{2}$). C. Backpropagation of error requires a copy of the weights of the forward network. One particularly difficult-to-reconcile component of backpropagation is the need to propagate error signals backwards through a network (see Fig. 1). This requires that the backward propagating error signals between layers of neurons is weighted according to the forward synaptic connection weights, leading to a situation in which feedback weight matrices are copies of the feedforward matrices. This duplication of weights has been identified as particularly troubling in terms of a plausible biological implementation and is known as the weight transport problem [3]. Early attempts to address the weight transport problem included proposals that the feedback weights can converge to the values of the feedforward weights by applying the same weight changes to both matrices during training (see [13]). This explanation was criticised for simply shifting the problem from transporting weights to transporting weight changes in a network. More recently, feedback-alignment was proposed as a method to completely sidestep the need for weight symmetry [14]. It was empirically shown that by having fixed random feedback weight matrices between the layers of a network, the feedforward weight matrices are modified by backpropagation such they they come into alignment with the feedback matrices. This approach can also be implemented with a randomly weighted direct feedback error to every layer (direct feedback alignment, [18]), a method which has also been applied in spiking neural networks [19]. Though such an error distribution process is biologically plausible, the effectiveness of the approach is limited to shallow networks and the accuracy of deep networks appears to suffer severely under such a protocol [20]. Beyond static feedback matrices, matrices with arbitrary magnitudes but alignment of the signs of weights (i.e. positive feedforward weights are mirrored with positive feedback weights and vice versa) show greatly improved performance over feedback alignment [21, 22]. However, propagating the sign of feedback weights is itself a transport problem and performance with this method is less than optimal. Recently, methods have been proposed by which the symmetric feedback weight matrices could be learned by biologically plausible methods (using local only information). Specifically, methods have emerged which carry out a process of synaptic weight inference [15, 16]. In essence the backwards synaptic connections (which would propagate the error) attempt to infer the feedforward weight between two neurons by observation of their activity alone. This is a process in which, based upon the activity patterns of a pair of neurons, a feedback synapse can infer (and thereby copy) the strength of the feedforward synapse. Such a method was successfully applied in a rate-based neural network by Akrout et al. [15] (hereafter referred to as the Akrout method). This method makes use of inference phases during which neurons are randomly stimulated and their activation is correlated in order to infer synaptic weights. Alternative rate-based methods are available though we do not consider them given their non-locality [17]. A more recent proposal [16] considers a spiking neural network model and makes use of the spiking threshold of neurons to implement a quasi-experimental, causal inference method known as regression discontinuity design (we hereafter refer to this method as RDD, also see [23]). This method similarly uses inference phases in between training epochs in order to infer the backward synaptic weight matrices. These inference methods have proven successful in inferring the feedforward synaptic weights for use in the feedback weight matrices but also suffer from a number of drawbacks. First, the Akrout method operates on firing rates and requires a demeaning process which is carried out in batches. This demeaning and batching process is particularly troublesome when applied to spiking networks where the learning must therefore be carried out offline and firing rates measured by aggregating spikes at specific intervals. In the RDD method, weight inference requires a piece-wise linear fitting process in order to infer the synaptic weights. This procedure requires the storage of four times more parameters per synapse (than just the synaptic weight), a second state variable per neuron and a high computational complexity per update. Though these components and the calculation protocols might be possible for a neuron to compute, they incur a significant computational cost. To overcome these issues, we propose a spike-timing-dependent weight inference (STDWI) mechanism for solving the weight transport problem in spiking neural networks. Our method is motivated by analysis of the time-to-spike of various neuron models under the influence of an incident spikes. In order to estimate this in a biologically plausible and computationally efficient manner, we make use of local information for this computation, in particular just the spike times of the pre- and post-synaptic neurons. We show that under a number of conditions our method outperforms both the Akrout and RDD methods when applied to weight estimation in spiking neural network models. We also compare our method to an optimal Bayesian update rule for an integrate-and-fire neuron with stochastic input. Our rule proves effective as an approximation of this update rule. Furthermore, for networks in which the neurons emit action potentials at random times (i.e. without a correlation structure), our learning rule can analytically be shown to approximate a rate-based learning rule similar to the Akrout method. Finally, the update rule we propose is computationally cheap and can be applied in an online fashion. ## 2 Methods To address the weight transport problem, it has been proposed that network weights can be inferred from activity [15, 16]. We can formulate this problem as follows: Consider two neurons, labelled “A” and “B”, embedded in a larger network structure. Amongst other connections, there exists a ‘forward’ synaptic connection from neuron A to neuron B. Therefore, the activity of neuron B is dependent upon some internal dynamics as well as the network activity as a whole, including the activity of neuron A, via incoming synaptic connections. Let us also now consider a pseudo synaptic connection from B to A, a connection meant to carry error information backward through the network (note that this work and prior work do not describe this synapse as having an impact upon the network activity during inference). According to the backpropagation of error algorithm, the optimal value of this synaptic connection weight should be equivalent to the weight of the forward synapse from A to B. How the forward synaptic weight can be copied to the backward synaptic connection is the problem at hand. Here we address how to infer the forward synaptic weight value at the backwards synapse given knowledge of the spike times (indexed $k$) of the neurons A and B ($t_{A}^{k}$ and $t_{B}^{k}$ respectively) and by accounting for some average impact from all other synapses. We derive a computationally simple and biologically plausible method which, by use of appropriate approximations, achieves this aim and could be employed at the synaptic level to learn feedback weights for error propagation. ### 2.1 Derivation of the weight inference method In order to derive our proposed weight inference rule, we analyse a simplified deterministic leaky integrate-and-fire (LIF) neuron with instantaneous synaptic inputs from a single source and drift (where drift is a placeholder for the unknown impact of all other incident synapses) and then consider the impact of noise upon this model. A deterministic LIF neuron with drift $\mu$ has voltage dynamics $\tau_{m}\frac{dv(t)}{dt}=v_{r}-v(t)+\mu\,.$ (1) In the absence of any input spikes, this equation can be solved, for an arbitrary initial condition $v_{0}$, at time $t_{0}$, yielding $v(t)=(v_{r}+\mu)\left(1-e^{-(t-t_{0})/\tau_{m}}\right)+v_{0}e^{-(t-t_{0})/\tau_{m}}\,.$ (2) With this expression we can now consider two cases, one in which the neuron is not stimulated by any incoming spikes from neuron $j$ and, beginning at voltage $v_{0}$ at time $t_{0}$, it spikes with some time delay $\hat{T}$ (purely under the influence of drift). The other case is one in which the neuron received an additional instantaneous voltage injection of magnitude $w$ at time $t_{0}$ (i.e. a spike arrives and stimulates the neuron) and it spikes with a different time delay, $T$ (such that the second case involves replacement of $v_{0}$ with $v_{0}+w$). These cases can be subtracted at threshold in order to give an expression for $w$, the stimulation magnitude, of the form $w={e^{T/\tau_{m}}}(v_{r}+\mu- v_{0})\left(e^{-T/\tau_{m}}-e^{-\hat{T}/\tau_{m}}\right).$ (3) Equation (3) provides an exact solution for determining the amount of instantaneous voltage ($w$) injected to a neuron at some time $t_{0}$ given that its spike time was modified from an expected time $\hat{T}$ to the time $T$. This is under the assumption that other than the instantaneous voltage injection and a background drift, there are no other inputs to the neuron during this time. We wish to make use of this deterministic solution for application to noisy conditions. In particular, when the background drift is considered as due to input from many other neurons it would inherently be noisy (unlike our derivation above). However, the current expression includes a number of terms which are highly susceptible to noise. First, the exponential term, $e^{T/\tau_{m}}$ is a strictly positive function which is exponential in the time that the neuron took to spike. If we consider a case in which $T$ is noisy, this term scales our noise exponentially but never changes sign. Second, the expected time to spike, $\hat{T}$ is difficult to estimate in a noisy neuron. However, this term is crucial for our ability to accurately identify positive and negative weights and it must, therefore, be approximated. First we consider the exponential term $e^{T/\tau_{m}}$. Though this term might (in the noiseless case) aid in producing a highly accurate weight estimation, in the face of noise it introduces a significant error. Furthermore, in the noiseless case (where a single estimate of the weight is exact), its biggest function is to scale the estimated weight based upon the time taken to spike. This, in essence, reweighs cases in which the neuron dynamics take excess time to reach threshold – due to beginning far from threshold (low $v_{0}$), having a small drift, and/or when the incident spike arrives from a synapse with a small/negative weight. This is therefore a mechanism to ensure that, for cases in which our system setup results in dynamics which take some time to reach threshold, the weight scale is treated sensibly. However, in the coming derivation we intend to sample over multiple instances of such an estimation in a noisy system such that there is an unreliable signal of ‘time to spike’. And given that this term is heavily influenced by noise we wish to ignore it. Therefore, given its function, our intention to sample, and its susceptibility to noise, we test in this work the removal of this term from our weight estimation and instead propose weight estimation without this scaling. We empirically find this approach successful. Thus, our approach to (approximate) weight estimation can be described as $\tilde{w}=C(v_{r}+\mu- v_{0})\left(e^{-T/\tau_{m}}-e^{-\hat{T}/\tau_{m}}\right)$ (4) where $\tilde{w}$ is an approximate estimation of the weight (ignoring a rescaling based upon time to spike) and we have introduced a general constant $C$ to allow linear rescaling of weight estimates. Next, we wish to approximate $\hat{T}$ in the face of noisy samples. For this purpose, let us average our estimate of the weight over $K$ observations. In particular, let us consider a set of samples $T^{k}$, indexed by $k$, each of which correspond to the time to spike given that the ‘output’ neuron started from some initial voltage $v_{0}^{k}$ at the moment of an incident spike. For each of these samples, there exists an “expected” time from incident spike to neuron spike, $\hat{T}^{k}$, which corresponds to when the neuron would have spiked if not for this incident spike. Taking an average of the weight estimate over these $K$ samples yields an estimated weight $\tilde{w}^{K}=\frac{C}{K}\sum_{k=0}^{K}\left(v_{r}+\mu- v_{0}^{k}\right)\left(e^{-T^{k}/\tau_{m}}-e^{-\hat{T}^{k}/\tau_{m}}\right)$ (5) with $K$ indicating the number of observations/samples taken. If we assume that our $K$ samples are chosen independently of the incident activity (i.e. the incident spikes are random), then the values of the initial voltage, $v_{0}^{k}$, and expected times to spike, $\hat{T}^{k}$, are both independent of the sampling process (and of $w^{k}$ and $T^{k}$). Therefore, these can be independently averaged and, hence, replaced with $\langle v_{0}\rangle$ and $\langle\hat{T}\rangle$. Thus, we arrive at an expression $\tilde{w}^{K}=\frac{D}{K}\sum_{k=0}^{K}\left(e^{-T^{k}/\tau_{m}}-e^{-\langle\hat{T}\rangle/\tau_{m}}\right)\,,$ (6) where $D=C(v_{r}+\mu-\langle v_{0}\rangle)$ combines the various constants and scales our estimate of the weights. If we now finally consider how we ought to update our estimate of $w$ when we receive an additional $(K+1)$-th sample, we arrive at $\Delta\tilde{w}=\tilde{w}^{K+1}-\tilde{w}^{K}=\frac{1}{K+1}\left(D\left(e^{-T^{K+1}/\tau_{m}}-e^{-\langle\hat{T}\rangle/\tau_{m}}\right)-w^{K}\right).$ (7) Inspecting our derived update rule, the first exponential term in Eq. (7) is exponential in the time since an incident spike arrived. Given this, it is equivalent to sampling a trace which continuously measures the (fast exponential) instantaneous firing rate of the neuron from which the incident spike is arriving. The second exponential term is exponential in the average time since incident spikes ‘should’ arrive if the weight had been zero, $\langle\hat{T}\rangle$, an measure of the incident spike-rate. This term can be approximated as a sampling of a slow exponential measure of the average rate of the neuron from which incident spikes arrive. Finally, the constant term $D=C(v_{r}+\mu-\langle v_{0}\rangle)$, has a factor of the drift term $\mu$. In our model assumption, this drift is background input aside from the synapse under inference and affects the baseline time to spike of our output unit. This drift therefore scales up with the output neuron’s average firing rate. With these observations, we can make appropriate replacements in order to describe a local spike-timing-dependent weight inference rule. ### 2.2 Spike-timing-dependent weight inference We propose a spike-timing-dependent rule for the purpose of weight inference (STDWI) which can be deployed for parallel and online updates with minimal computational complexity. Our method maintains multiple online estimates of neuron firing rates through eligibility traces [24, 25] and makes use of these for synaptic weight estimation. In particular, each neuron (indexed $j$) maintains a fast trace $\epsilon_{j}^{f}(t)$ and a slow trace $\epsilon_{j}^{s}(t)$. The dynamics of the fast and slow traces are calculated for each neuron as $\tau_{f}\frac{d\epsilon_{j}^{f}(t)}{dt}=-\epsilon_{j}^{f}(t)+S_{j}(t)\quad\text{and}\quad\tau_{s}\frac{d\epsilon_{j}^{s}(t)}{dt}=-\epsilon_{j}^{s}(t)+\frac{\tau_{f}}{\tau_{s}}S_{j}(t)\,,$ (8) where $\tau_{f}$ and $\tau_{s}$ are the decay constants of the fast and slow traces respectively, and $S_{j}(t)$ is the spike train of the $j$th neuron. These traces are computed during simulation in a time-stepping method with the analytic (exponential) solution to these traces computed at every timestep. This spike train is computed from the set of $k$ spike times of the $j$th neuron, $t^{k}_{j}$, such that $S_{j}(t)=\sum_{k}\delta(t-t^{k}_{j})$, where $\delta(\cdot)$ is the Dirac delta function. Note that these two traces have an equal area (across time) when they both start with an initial value of zero due to the normalization of the slow trace spike-train, scaling factor $\tau_{f}/\tau_{s}$. This property ensures that both eligibility traces act to measure the firing rate of the neurons with the same scale. Having defined these eligibility traces, we define our weight inference rule as $\frac{dw_{ji}}{dt}=\alpha S_{i}(t)\left(\epsilon_{i}^{s}(t)\left(\epsilon_{j}^{f}(t)-\epsilon_{j}^{s}(t)\right)-\eta w_{ji}\right)\,,$ (9) where this rule describes inference of the weight of the forward synapse at the backward synapse (from neuron $i$ to neuron $j$), $w_{ji}$, with $\alpha$ as the learning rate and $\eta$ as the relative level of weight decay (both constant hyper-parameters). This learning rule and the fast and slow measures of the neuron’s firing rates are inspired by the synaptic inference rule derived in Section 2.1. Note that though this rule is given as a differential equation, since updates are gated by neuron $i$’s spike times, it is implemented as updating the synaptic weights such that $w_{ji}\leftarrow w_{ji}+dw_{ji}/dt$ at every timepoint where neuron $i$ spikes. Figure 2: A) Illustration of the difference between our derived method for weight inference by analysis of a deterministic LIF neuron (left) versus our proposed STDWI method (right) which uses a fast trace to measure the instantaneous firing rate (the first exponential term in Eq. (7)) and a slow trace to measure the average firing rate (second exponential term in Eq. (7)). B) Assuming regular neuron firing conditions, our method can be interpreted as an STDP rule of the form shown inset, where $T$ is the post minus pre-synaptic neuron spike time. Note pre and post-synaptic are termed relative to the backward synaptic connection. The formulation for the weight update given in Eq. (7) and our proposed STDWI rule given in Eq. (9) have corresponding terms, see Figure 2A. Both of these formulations include updates which occur only upon our pre-synaptic neuron spikes. Note we use the terms post-synaptic/pre-synaptic relative to the backward synaptic connection for application to the weight transport problem. In our approximation, we replace the first exponential term of Eq. (7) (an exponential measure of the time since the post-synaptic neuron’s last spike) with a fast timescale measure of the post-synaptic neuron’s firing rate (the fast trace) and we use a slow timescale measure of the post-synaptic neuron’s firing rate (the slow trace) to approximate the second exponential term (which computes a trace tracking the average post-synaptic neuron’s firing rate). Finally, we include a slow measure of the pre-synaptic neuron’s firing rate as a multiplicative factor, which is intended to capture the dependence of the weight estimate upon the pre-synaptic neuron drift. Figure 2A depicts how updates are calculated upon pre-synaptic neuron spikes for both the deterministic LIF and STDWI update, highlighting both the similarities and key differences between these implementations. Note that the learning rule being proposed here relates in a curious form to traditional Spike-Timing Dependent Plasticity (STDP) rules. In particular, the sign of the weight update is determined by the spike-timings and firing-rates of the pre and post-synaptic units. In general, if we assume some fixed regular firing rate of the post-synaptic neuron, then depending upon the spike-time of the pre-synaptic neuron relative to this regular firing, we encounter positive or negative weight estimation. This rule therefore appears in a mirrored-form to the commonly cited STDP observations [26], see Figure 2B. ### 2.3 Spiking neuron model For simulations in this study, we consider neurons with membrane leakage and conductance-based synaptic kernels whose membrane voltage dynamics can be described by $\tau_{m}\frac{dv_{i}(t)}{dt}=(v_{r}-v_{i}(t))+\frac{g_{D}}{g_{L}}\bigg{(}\sum_{j}w_{ij}\kappa_{j}(t)-v_{i}(t)\bigg{)}\,,$ (10) where $\tau_{m}$ is the leakage time constant, $v_{r}$ is the rest voltage, $g_{D}$ and $g_{L}$ are the dendritic and somatic leakage conductances, respectively, $w_{ij}$ is the weighting of the forward synaptic connection from the $j$th neuron to the $i$th neuron and $\kappa_{j}$ describes a filtered form of the $j$th neuron’s spike train. The form of the synaptic filtering kernel is taken as a double exponential with a fast rise and slow decay, such that $\kappa_{j}(t)=\frac{1}{\tau_{2}-\tau_{1}}\sum_{k}H(t-t^{k}_{j})\left(e^{-\frac{t-t^{k}_{j}}{\tau_{2}}}-e^{-\frac{t-t^{k}_{j}}{\tau_{1}}}\right)\,,$ (11) where $\tau_{1}$ and $\tau_{2}$ are the timescales of the fast rise and slow decay, taken to be 3ms and 10ms respectively, and $H(\cdot)$ is the Heaviside step function. When the membrane voltage, $v_{i}(t)$, reaches a threshold, $\theta$, an action potential is recorded and propagated. The membrane voltage is thereafter reset to a reset voltage $v_{\text{reset}}$. For the simulations in this study, we do not implement a refractory period explicitly. This is not expected to cause much deviation of the analysis in our low firing-rate regime. ### 2.4 Comparison against alternative weight inference methods In Section 3.2, we compare our method (STDWI) to alternative methods (RDD and Akrout methods) proposed for the local inference of weights in a network. The inference is carried out in networks composed of a group of spiking input neurons connected via a single forward weight matrix to a group of spiking output neuron. The spiking neuron dynamics are equivalent to those used in the work which introduced RDD as a causal weight inference method [16], see Section 2.3. The network is stimulated by selectively exciting input neurons, collecting the responses of input and output neurons, and applying the set of techniques to these neural data. During simulation, some percent of the input neurons are randomly sampled every 100ms and these are excited with background random Poisson distributed spike trains (with a fixed positive synaptic connection weight from stimulation nodes to the input neurons). Every 100ms the input neurons being stimulated are re-sampled. During this stimulation, non-selected neurons are left unstimulated with zero input. The STDWI and RDD methods are applied in a continuous form, paying no attention to the 100ms stimulation periods. The Akrout method was proposed for rate-based neural networks and makes use of a batch-wise de-meaning of firing rates. Therefore, the 100ms stimulation periods are considered as individual ’stimuli’ for the Akrout method, and the firing rates computed for each of these ‘stimuli’. These individual stimuli are then grouped into batches (batch-size chosen by grid search) and used to update the inferred weight according to the Akrout method. The spiking dynamics with stimulation, as described above, were simulated for 2500s. During weight inference, these 2500s of dynamics were then looped ten times in order to produce a total 25,000s training time. This looping was necessary due to memory and storage constraints and can be interpreted as ten epochs of training. All methods were trained with a learning rate of $5\times 10^{-5}$. This learning rate was chosen by iteratively reducing the learning rate until stable weight inference was clear for all methods. Conclusions are (and should be) drawn based upon asymptotic performance and not speed given that this hyperparameter was not tuned on a method-by-method basis. Free parameters were optimized for by measurement of sign-accuracy and Pearson correlation (best average performance) using a grid search carried out with a single seed of the network simulation. Selected parameters were then applied for inference to five other network seeds, and results collected. See Appendix B for the grid search results. ## 3 Results To validate our approach, we compare it against a Bayes-optimal method for a simple neuron model that affords an analytical solution. Furthermore, we compare it to two state-of-the-art synaptic weight inference methods for estimation of the connectivity of simulated spiking neural networks (see models described in Section 2.3). Code to reproduce results is available at https://github.com/nasiryahm/STDWI. ### 3.1 Comparison of STDWI to a Bayesian optimal method To verify the validity of our proposed STDWI rule and demonstrate its flexibility, we compare it against a Bayes-optimal method for inferring synaptic inputs to a neuron with internal state modelled by a Wiener process (Figure 3). Unlike a stochastic LIF neuron model, this model has a tractable hitting-time analysis and thereby we can form a Bayesian update rule for estimating the size of a synaptic input given a subsequent output neuron spike time. A detailed derivation of the Bayes-optimal method is provided in Appendix A. Figure 3: Weight inference accuracy of Bayesian and STDWI approaches applied to a pure Wiener process with jumps. Panels A and B show scatter plots of the true and inferred weights for the Bayesian and STDWI approach, respectively, at the end of the training time ($t=50s$). Panels C and D show how Pearson correlation and sign alignment between the true and inferred weights evolve through the training process. The standard deviation of the measures across 10 random network seeds are shown as envelopes about the curves. The Bayesian update rule occurs upon every pre-synaptic neuron spike and is based upon knowledge of when the last post-synaptic spike occurred (rather than knowledge of all past post-synaptic spikes), it would be an improper comparison to test the optimal Bayesian method against our full STDWI rule (which makes use of all previous spikes in its eligibility traces). Therefore, to ensure a fair comparison we modify our STDWI rule (Eq. 9) to use only single spikes. To do this we replaced the slow eligibility traces, $\epsilon_{j}^{s}(t)$, with a constant (optimally set as the average of the fast traces), and replaced the fast trace, $\epsilon_{j}^{f}(t)$, with a term which is exponential in the time since the last spike alone (rather than a decaying trace of all past post-synaptic spikes). This modification is equivalent to Eq. 7 if we treat the second exponential terms as a constant and use an arbitrary learning rate. We repeatedly simulated stochastic neurons, each with a single forward synaptic input connection but with varying synaptic connection strengths across simulations. We simulated the systems for 50s and thereafter used the network activity in this time period for synaptic weight inference. We repeated this analysis for synaptic weight strengths over a wide range to attempt inference of many different synaptic strengths. Figure 3 shows various measures of the similarity between the true and inferred jump widths for this simulation when using either the Bayesian or our derived method for weight inference. Both the scatter plots and learning curves show that the STDWI method closely matches the Bayes-optimal results, supporting the theoretical soundness of our approach. ### 3.2 Comparison of STDWI to alternative weight inference methods Figure 4: Weight inference accuracy comparison between the RDD, Akrout, and STDWI approaches for a network of LIF neurons with conductance-based synapses. Panels A, B and C show scatter plots of the true and inferred weights for each method at the end of training for a single network. Panels D and E show Pearson correlation and sign alignment between the inferred and true weights. Solid lines show the mean of these measures across ten randomly seeded networks and the shaded areas show the standard deviation across these networks. Panel F shows the convergence of the inferred weights for each method. The inferred weights for the 75% largest inferred weight (by magnitude) were collected, individually normalized to their final value and their average plot with standard deviation shown, as before, by the shaded area. We also compared our proposed STDWI approach to two existing methods for synaptic weight inference. In particular, we compare against the RDD and Akrout methods. Details of both methods are provided in Appendix B. To simulate a neural network model which is amenable to all of these weight inference methods, we use the same neural network models and setup as that described in [16]. This network is composed of LIF neurons with kernel- filtered, conductance-based synaptic inputs. We simulate two-layer network models with an input layer of 100 LIF neurons fully connected to an output layer of 10 LIF neurons. The synaptic weight matrix connecting these is drawn from a normal distribution with a small but positive mean. It is this weight matrix which must be inferred by the range of methods. The network is stimulated by selectively exciting input neurons. Some percentage of the input neurons are randomly sampled every 100ms and these are excited with background Poisson distributed input spike trains (with a fixed positive synaptic connection weight from stimulation nodes to the neurons). Every 100ms the input neurons being stimulated are re-sampled. During this stimulation process, non-selected neurons are left unstimulated with zero input. Figure 4 shows the result of weight inference with the range of methods discussed above for networks in which 20% of input neurons are simultaneously stimulated (sparse random stimulation). Scatter plots of the inferred vs true weights (see Panels 4A-C) show the strength of the STDWI method, which produces a tighter distribution of weights than competitors. Note that the scale of the synaptic weights inferred differs from the true weights for all methods, relating to the approximate nature of the methods. In practice, a rescaling could be applied to improve the correspondence to the scale of the true weights though none of our measures were sensitive to inferred weight scale and therefore this was disregarded. Panels 4D and E show the evolution of the Pearson correlation and sign alignment between the true and inferred weights through training for the range of methods. As can be seen, our proposed STDWI method outperforms both the RDD and Akrout methods, though the difference in the Pearson correlation of all three methods is small. Note that RDD outperforms Akrout in terms of sign accuracy. Finally, Panel 4F shows the successful convergence of inferred weights for all three methods. This plot shows the normalized weights (normalised through a division by the final converged weight value) of the top 75% largest magnitude network weights. These weights had a relatively unambiguous sign, hence their selection. This plot is provided to support the argument that the hyperparameter selections made were sufficient for stable inference by these methods. ### 3.3 The impact of stimulation protocol on weight inference It is also instructive to investigate how different stimulation protocols affect weight inference. To this end, and in contrast to the sparse stimulation in the previous section, we assume that all input neurons are stimulated (dense stimulation). Furthermore, we investigate how input timing correlations affect weight inference. Since input neurons are stimulated by random Poisson spike trains, we can create correlation between individual Poisson spike trains by a thinning process (via a Single Process Interaction Model, see [27]). Figure 5 shows results for this dense stimulation regime. Figure 5: Weight inference accuracy comparison between the RDD, Akrout, and STDWI approaches for a network of LIF neurons with conductance-based synapses when all input neurons are stimulated. Panels A, B and C show scatter plots of the true and inferred weights for each method at the end of training for a single network. Panels D and E show Pearson correlation and sign alignment between the inferred and true weights in the uncorrelated spiking case during training. Panels F and G show the final results (post-training) under varying input spike-time correlation. Solid lines (points) show the mean of these measures across five randomly seeded networks and the shaded areas (error bars) show the standard deviation across these networks. Scatter plots of the true vs inferred weights (see Panels 5A-C) again show that STDWI produces a tighter distribution of weights than its competitors. This highlights the smaller impact of stimulation density upon the STDWI inference method compared with the Akrout or RDD methods. These scatter plots show inferred weights for the dense stimulation case with zero correlation in timing between the various input stimulation spike-trains. Panels 5D and E show that the STDWI method remains most successful (as measured by the Pearson correlation and sign alignment mean) when compared with RDD and Akrout methods under dense stimulation. However, the Akrout method benefits significantly from dense stimulation (whereas the RDD method appears to suffer somewhat). Thus, the RDD method does not systematically outperform the Akrout method as previously reported (cf. Panels 4D and 5E). Panels 5F and G demonstrate how weight inference is affected by input timing correlations. STDWI remains largely successful, however as input spike timing correlation increases, the RDD method performs favourably. This may be expected as unlike the STDWI and Akrout methods, the RDD method compares only events which are near-threshold to establish synaptic weights. This filtering of events by which inference is done may be favourable in the regime of high input spike timing correlation, though the benefit only exists for some parameter range. ## 4 Discussion Our results demonstrate the efficacy of STDWI for synaptic weight inference across a range of network models and stimulation protocols. We have shown that our approach successfully approximates Bayes-optimal results in a simple neuron model and outperforms existing methods for weight inference. Our results also highlight the attention that must be paid to the employed stimulation protocols since the efficacy of different synaptic weight inference method has been shown to crucially depend on these. Existing methods cannot be so indiscriminately applied to arbitrary neuron models. For example, the RDD method requires a neuron model which has a second state variable mimicking the membrane voltage. This state variable should relax to the same value as the membrane voltage when the neuron is not spiking and otherwise should reflect how “driven” the neuron is when it spikes. However, such a state variable is not necessarily constructable for an arbitrary neuron model. In contrast, STDWI makes use of spike timing alone and is therefore agnostic to the neuron dynamics being simulated. In our analyses, we reported both on Pearson correlation and sign accuracy. STDWI systematically outperformed the alternative approaches on both measures for a range of parameters. One exception is in the investigation of timing- based correlations (Figure 5F and G), in which RDD outperformed the other methods for inference in the case of the medium correlation regime. This suggests a particular regime might favour the RDD method, however its failure in other regimes suggests that the current formulation of RDD for analysis about a spike-threshold may not be most effective. It is also important to realize that the number of variables stored per synaptic connection is greater for RDD compared to either the STDWI or Akrout methods. RDD requires a fitting process using data-points corresponding to events within which the neuron’s membrane voltage was near the spiking threshold. Aside from the selection of events close to threshold, the RDD method also uses four variables per synaptic connection to characterise a piece-wise linear function (with linear functions fit above and below the spiking threshold). By comparison, STDWI uses two variables for the fast and slow eligibility traces of each neuron and the Akrout method uses two variables storing the firing rate and mean firing rate of each unit within a batch. To derive our learning rule, we made use of a deterministic analysis of a LIF neuron and considered the spike times of a single input neuron. Our deterministic analyses later required approximations in order to remove terms which are highly affected by noise. Ideally we would instead have carried out a stochastic process analysis for a LIF neuron. The particular stochastic process to which our leaky neuron model corresponds is known as an Ornstein- Uhlenbeck (OU) process. Unfortunately a general analysis of the OU process that describes when we ought to expect such a neuron to spike (the hitting time) is non-trivial [28]. Nonetheless, the efficacy of our assumptions is validated by the quality of our results. Furthermore, under a rate-based analysis of our proposed STDWI rule, we can show a correspondence to the Akrout rule (see Appendix C). A limitation of our approach is that the inference process considers the spike times of a single input neuron. Instead, a multivariate approach which would take into account the spike-times of all input neurons to infer synaptic weights could prove even more powerful and accurate for weight inference. Indeed multivariate analyses, often making use of such multi-neuron spiking information along with cross-correlation measures and statistical significance testing, have been applied previously in approaches which aim to infer neural circuit connectivity from neural data [29, 30, 31, 32]. These approaches, however, make use of globally available network information and are not concerned with whether this information is locally available at the synapse. 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Biol Cybern 2008 Jun;98(6):459–478. ## Appendix A Bayesian weight estimation for a stochastic neuron model As a method of verification of our proposed STDWI rule and an exhibition of its flexibility, we compare it against an optimal Bayesian method for inferring a single synaptic input to a neuron with internal state modelled by Brownian motion with drift and diffusion (a Wiener process). Unlike a stochastic leaky integrate and fire neuron model, this model has a tractable hitting-time analysis and thereby we can form an optimal Bayesian update rule for estimating the size of a synaptic input given a subsequent output neuron spike time. This synaptic weight inference analysis for this simple neuron model and its similarity to our STDWI rule is described in the following section. ### A.1 Bayesian estimation of synaptic weights We wish to estimate the weight of synaptic connection given local-only information. In particular, this involves estimating the weight of a synaptic connection given the spike times of an input and output neuron (input and output relative to the forward synaptic connection) as well as the output neuron’s membrane voltages. Constraining this further, let us estimate a synaptic connection weight, $w$, between two neurons given a single input spike time, $t_{\text{in}}$, and the first output spike time which follows this input spike, $t_{\text{out}}$ where $t_{\text{out}}>t_{\text{in}}$. If we carry out all analysis relative to the input spike time, $t_{\text{in}}$, we can define the key dependent factors. First, the output neuron’s time to spike (the hitting time), following the input neuron spike, is a key measure which we define as $T=t_{\text{out}}-t_{\text{in}}$. The initial state of the output neuron is also a determining factor in this analysis as it defines the distance to threshold $\Delta$, which we elaborate on below. Given this setup and by Bayes’ rule, we aim to compute $p(w\mid T,\Delta)\propto p(T\mid w,\Delta)p(w)\,.$ (12) The likelihood term $p(T\mid w,\Delta)$ can be computed through analysis of the neural dynamics. To compute it, we must account for the impact of spikes from all other input neurons. In general this is non-trivial. To simplify this analysis, we consider the case of a non-leaky integrate-and-fire neuron driven by random input. ### A.2 Stochastic neuron model We consider a spiking neural network of neurons with membrane voltage under the effect of Brownian motion. As such, changes in the membrane voltage, $v(t)$, can be described by $\frac{dv(t)}{dt}=I(t)\,,$ (13) where $I(t)$ is the total input to the cell at time $t$. Notably, this change in membrane voltage is agnostic to the current voltage $v(t)$ (meaning there is no leakage effect). When this membrane voltage meets the threshold, $\theta$, an action potential is emitted and the membrane voltage is directly reset to the reset voltage $v_{\text{reset}}$. Let us consider the input, $I(t)$, as composed of input from the single synaptic connection and some background stochastic process. The synaptic connection is modelled as a producing instantaneous voltage injections which occur upon the spike times of the input neurons. The amplitudes of the instantaneous voltage injections induced by input spikes are equal to the weight of the synaptic connection from input to output neuron, $w$. Aside from these synaptic inputs, we also consider some background input which is a stochastic process. Assuming that there are a large number of randomly spiking input neurons, we can approximate their impact as a random Gaussian input with some mean and variance. This describes a stochastic process, known as a Wiener process, with some drift (mean input) and a diffusion constant (variance). This approximation for a neuron’s membrane voltage is valid in the limit of a large number of synaptic connections with small synaptic weight magnitudes. The above details are all approximations but provide us with a simple description of the neural dynamics such that $dv(t)=w\delta(t-t_{\text{in}})dt+\sqrt{D}dX_{i}(t)\,,$ (14) where $\delta(\cdot)$ is the Dirac-delta function and $X(t)$ is a Wiener process with drift $\mu$ and variance scaled by $D$. ### A.3 The hitting time of a non-leaky neuron We can now attempt to determine the “hitting time” of this system, i. e., the time $T$ at which it makes contact with our neuron membrane voltage threshold. The hitting-time density for a Wiener process with drift (by which we are approximating our non-leaky neuron) can be calculated as; $f(T\mid\Delta)=\frac{\Delta}{\sqrt{2D\pi T^{3}}}\exp\left(-\frac{(\Delta-\mu T)^{2}}{2DT}\right)\,,$ (15) where $\Delta=\theta-v_{0}$ is the membrane voltage distance to threshold ( where $v_{0}=v(t_{\text{in}})$), $T=t_{\text{out}}-t_{\text{in}}$ is defined as above, $\mu$ is the drift of our Wiener process, and $D$ is the variance of our Wiener process. In our neuron model, $\Delta$ corresponds to the difference between some initial membrane voltage $v_{0}$ and the threshold $\theta$, whereas $\mu$ corresponds to the average input to the output neuron from all input synapses in volts. The description assumes that the membrane voltage starts at some value $v_{0}$ and is under constant drift. However, instead we wish to assume that at the initial time, $t_{0}=t_{\text{in}}$, our input neuron fired and added some unknown voltage $w$ to the membrane voltage. Furthermore, rather than computing a probability distribution over the possible times at which the output neuron might spike, we instead know the next spike time of the output neuron, $t_{\textrm{out}}$, and wish to use it to infer the weight, $w$. We can therefore assume that for a given pair of input and output spikes, we have a fixed hitting time, $T$, as described above. Furthermore, under our synapse description for the non-leaky neuron (where synaptic inputs cause an instantaneous change in output neuron membrane voltage of size proportional to the synaptic weight) our initial membrane voltage, $v_{0}$, can be represented as the membrane voltage just prior to the input spike, plus the synaptic weight. That is, we take the limit of $v_{0}$ from below, i.e., $v_{0}=\lim_{t\to t_{0}^{-}}v(t)+w.$ This allows us to augment our first- passage density in terms of $w$ such that $f(T\mid w,\Delta)=\frac{(\Delta-w)}{\sqrt{2D\pi T^{3}}}\exp\bigg{(}-\frac{(\Delta-w-\mu T)^{2}}{2DT}\bigg{)}\,,$ (16) where we now define $\Delta=\theta-\lim_{t\to t_{0}^{-}}\>v(t)$. With this formulation of the hitting-time density, we can compute an estimate of the weight $w$ given a particular set of input and output neuron spike times. Thereafter we can update our estimate of the synaptic weight of interest through Eq. (12). To make our inference of $w$ tractable, we first take a Laplace approximation of Eq. (16). This produces a Gaussian with mean weight $\hat{w}=\Delta-\frac{\mu T+\sqrt{(\mu T)^{2}+4DT}}{2}\,,$ (17) calculated as the maximum of our likelihood $f(T\mid w,\Delta)$, and a variance $\hat{\sigma}=1/((\Delta-\hat{w})^{-2}+(DT)^{-1})\,.$ (18) Since we have Gaussian distributions for our likelihood, we can take a Gaussian conjugate prior with mean $\mu_{0}$ and variance $\sigma^{2}_{0}$ and obtain a closed-form solution to our posterior weight when given a single input-output spike pair as ${w}_{p}=\frac{1}{\sigma_{0}^{-2}+\hat{\sigma}^{-2}}\bigg{(}\frac{w_{0}}{\sigma_{0}^{2}}+\frac{\hat{w}}{\hat{\sigma}^{2}}\bigg{)}\,.$ (19) Similarly, we can compute the posterior variance as ${\sigma}_{p}^{2}=\bigg{(}\sigma_{0}^{-2}+\sigma^{-2}\bigg{)}^{-1}\,.$ (20) ### A.4 Weight estimation under negligible drift Let us assume that the diffusion term, $D$, is sufficiently small compared to the drift $\mu$ (such that $\mu\gg D$). This allows us to ignore the diffusion term in the numerator of Eq. (17). Having assumed this small diffusion scale, we can then describe the maximum likelihood estimate of the weight as $\hat{w}\approx\Delta-\mu T\,.$ (21) Furthermore, recall that $\Delta$ is the distance to threshold when the input neuron spikes, $\Delta=\theta-v(t_{\textrm{in}})$. By dividing this distance, $\Delta$, by the drift, $\mu$, we can calculate the expected time of the output spike under drift alone, $\hat{T}$, such that $\frac{\Delta}{\mu}=\hat{T}\implies\Delta=\mu\hat{T}\,.$ (22) Given these assumptions, we can approximate Eq. (17) as $\hat{w}\approx\mu\hat{T}-\mu T=\mu(\hat{T}-T).$ (23) This formulation can be understood well if we consider a non-leaky neuron under the effect of drift alone (without any stochastic input) and a single input neuron providing instantaneous voltage injections. In such a case, with knowledge of the initial membrane voltage and drift of the output neuron, we have a deterministic system which will spike at a specific time, $\hat{T}$. If we perturb this system with a spike from an input neuron (which causes a jump in the membrane voltage), we can decode the synaptic weight by simply measuring the effect on the timing of the output neuron spike time. The induced change in the output spike time is linearly proportional to the synaptic weight. ## Appendix B Details on baseline methods The STDWI method is compared to existing methods for synaptic weight inference. We provide more details on these methods below. ### B.1 The Akrout method In our simulations of LIF neurons, we compare against the Akrout method [15]. This rate-based method makes use of an inference phase in which neurons are stimulated (with mean zero) and then the levels of activity of input and output neurons are correlated to form a weight estimate. This approach was shown to be highly successful for weight inference and thereby training of rate-based neural network models. However, since we simulate spiking neurons, which cannot have a negative firing rate, we instead demean the neuron firing ratesand randomly stimulate the input neurons (post-synaptic from the perspective of the backward synapse). In particular, we use an update rule of the form $\Delta w_{ji}=\eta(r_{i}-\langle r_{i}\rangle)(r_{j}-\langle r_{j}\rangle)-\eta\lambda w_{ji}\,,$ (24) where $\Delta w_{ji}$ is the update to backward synaptic weight, from a neuron indexed $j$ to a neuron indexed $i$, which is attempting to estimate the weight of the forward synaptic connection, $w_{ij}$. $r_{i}$ and $r_{j}$ denote the firing rates of the $i$th and $j$th neurons, and $\langle\cdot\rangle$ indicates an average of these over time. Parameters $\eta$ and $\lambda$ are the learning rate and the weight decay respectively. The learning rate is fixed at with value $\eta=0.0001$ and the weight decay determined by grid search, see below. The firing rates $r_{i}$ and $r_{j}$ are calculated by computing the firing rates within the non-overlapping 100ms stimulation periods of the network. These stimulation periods are then grouped into batches (of size again determined by grid search) for calculation of the mean firing rates for this batch ($\langle r_{j}\rangle$ and $\langle r_{i}\rangle$ respectively) according to the weight-mirror gradient descent method described in [15]. ### B.2 Regression discontinuity design We also compare against the regression discontinuity design (RDD) method, which was proposed for application in spiking neural networks [16]. It makes use of all times at which a neuron spiked or almost spiked (i.e. its membrane voltage came within some margin of the spiking threshold but never reached it). It thereafter separately fits the almost-spiked and spiked events linearly against the membrane voltage. Notably, for the spiking events, a non- reset version of the membrane voltage is used for the linear fitting. Following a fitting process, the discontinuity of these linear fits at the spiking threshold is used as a measure of the synaptic weight. For full details of the RDD implementation, see [16]. ### B.3 Grid-based optimization of free parameters The methods compared have a number of free parameters that can be optimized for. In case of STDWI these are the time constants of the fast ($\tau_{f}$) and slow ($\tau_{s})$ traces. In case of RDD these are the distance to threshold at which samples are initiated and the window duration of a sample. For the Akrout method, the weight decay scaling and the batch-size are hyperparameters. These parameters are chosen from a grid-search using a single test network’s spike trains. The parameters producing highest average sign accuracy and Pearson correlation between the inferred and true weights are then chosen for analysis of a further four networks (each with a different random seed for input stimulation and the synaptic weight matrix). Figure 6: Variation in the performance of the STDWI, RDD, and Akrout methods with changes in the method parameters. The best parameter sets are highlighted with a black box. These were the parameter used to analyse all other seeded networks and produce the main results. Figure 6 shows the parameters maps for the grid searches carried out to select parameters for Figure 4. The same grid search parameter sweeps were repeated in order to choose parameters for Figure 5. ## Appendix C Rate-based analysis of the STDWI rule To appreciate the effect of STDWI rule, we can consider its approximate rate- based form under the assumption of random Poisson process sampled neuron spikes (for a review of the rationale of such methods see [33]). This produces an update rule based upon the firing rates of the neurons. Note that below, as in Section 2.2, we refer to pre/post-synaptic relative to a ‘backward’ synapse. In our case, the dependence upon the post-synaptic firing rate has two forms which correspond to a quickly-adapting exponential average, ${\lambda}_{\textrm{j}}^{\textrm{f}}$, and a slowly-adapting exponential average, ${\lambda}_{\textrm{j}}^{\text{s}}$. Similarly there is a dependence upon the pre-synaptic firing rate as a slowly-adapting exponential average, ${\lambda}_{\textrm{i}}^{\text{s}}$. Taking the assumption of Poisson random spiking, we can describe our weight update in a rate-based form as $\frac{d\hat{w}_{ji}}{dt}=\alpha S_{i}(t)\left(\lambda_{\textrm{i}}^{\textrm{s}}({\lambda}_{\textrm{j}}^{\textrm{f}}-{\lambda}_{\textrm{j}}^{\textrm{s}})-\eta\hat{w}_{ji}\right).$ (25) We can solve this equation for its fixed point ($\frac{d\hat{w_{ji}}}{dt}=0$), producing an expression for the fixed-point weight as $\hat{w}_{ji}^{*}=\frac{1}{\eta}{\lambda}_{\textrm{i}}^{\textrm{s}}\left({\lambda}_{\textrm{j}}^{\textrm{f}}-{\lambda}_{\textrm{j}}^{\textrm{s}}\right)$ (26) when $S_{i}(t)$ is non-zero. For networks with solely positive firing rates, Akrout et al. [15] proposed correlating the demeaned firing rates of pre and post-synaptic neurons in order to estimate synaptic weights. If we here interpret the slow firing rate measure of the input neuron activity as an approximation of its average value, then our method similarly correlates pre-synaptic firing rate with the demeaned post-synaptic neuron firing rate. Though this rate-based analysis shows similarities to the Akrout method, our spike timing implementation is unique in that it makes use of asymmetric causal kernels and has a demeaning process which slowly tracks the firing rates of neurons (rather than making use of batches). We attribute our performance gains to these features. Furthermore, given the spike-timing-dependent feature of the rule, weight updates can be computed in an event-driven fashion and with minimal communication between neurons (weight updates requiring communication only upon spike times). If we compare Eqs. (26) and (23) then we can also appreciate the correspondence of the STDWI rule and the Bayesian estimate. The STDWI update, instead of making use of an estimate of the a drift, $\mu$, makes use of the pre-synaptic neuron firing rate as a proxy. This is appropriate given the linear relationship between drift and firing rate for a non-leaky neuron. Furthermore, rather than directly comparing the expected and true time to spike, $\hat{T}$ and $T$ respectively, the STDWI rule keeps track of a slow and fast estimate of the post-synaptic neuron firing rate, through $\lambda_{\textrm{j}}^{\text{s}}$ and $\lambda_{\textrm{j}}^{\text{f}}$ respectively. The subtraction of these firing rate estimates in Eq. (26) provides a measure with a similar form to the subtraction of expected and true spike times ($\hat{T}-T$). Specifically, an earlier than average spike time induces a positive weight estimate and a later than average spike time induces a negative weight estimate.

2024-09-04T02:54:58.299400

2003.03994

arxiv-papers

WGMwgm QEqe EPep PMSpms BECbec DEde [orcid=0000-0003-0740-780X] [orcid=0000-0001-7809-7744] [1] [orcid=0000-0001-6768-1478] [orcid=0000-0002-7342-2090] [cor1]Corresponding author; Email Address<EMAIL_ADDRESS>Postal Address: Room No.-231, East Block, MIED, IIT Roorkee, Roorkee, Uttarakhand-247667, India; Phone: +91-8218612326 # Airline Crew Pairing Optimization Framework for Large Networks with Multiple Crew Bases and Hub-and-Spoke Subnetworks Divyam Aggarwal<EMAIL_ADDRESS>Department of Mechanical & Industrial Engineering (MIED), Indian Institute of Technology Roorkee, Roorkee, Uttarakhand-247667, India Dhish Kumar Saxena<EMAIL_ADDRESS>Thomas Bäck<EMAIL_ADDRESS>Leiden Institute of Advanced Computer Science (LIACS), Leiden University, Niels Bohrweg 1, 2333 CA Leiden, the Netherlands Michael Emmerich<EMAIL_ADDRESS> ###### Abstract Crew Pairing Optimization aims at generating a set of flight sequences (crew pairings), covering all flights in an airlines’ flight schedule, at minimum cost, while satisfying several legality constraints. CPO is critically important for airlines’ business viability considering that the crew operating cost is second only to the fuel cost. It poses an NP-hard combinatorial optimization problem, to tackle which, the state-of-the-art relies on relaxing the underlying Integer Programming Problem (IPP) into a Linear Programming Problem (LPP), solving the latter through Column Generation (CG) technique, and integerization of the resulting LPP solution. However, with the growing scale and complexity of the airlines’ networks (those with large number of flights, multiple crew bases and/or multiple hub-and-spoke subnetworks), the efficacy of the conventionally used exact CG-implementations is severely marred, and their utility has become questionable. This paper proposes an Airline Crew Pairing Optimization Framework, $AirCROP$, whose constitutive modules include the Legal Crew Pairing Generator, Initial Feasible Solution Generator, and an Optimization Engine built on heuristic-based CG- implementation. $AirCROP$’s novelty lies in not just the design of its constitutive modules but also in how these modules interact. In that, insights in to several important questions which the literature is otherwise silent on, have been shared. These relate to sensitivity analysis of $AirCROP^{\prime}s$ performance in terms of final solutions’ cost quality and run-time, with respect to - sources of variability over multiple runs for a given problem; cost quality of the initial solution and the run-time spent to obtain it; and termination parameters for LPP-solutioning and IPP-solutioning. In addition, the efficacy of the $AirCROP$ has been: (a) demonstrated on real-world airline flight networks with an unprecedented conjunct scale-and-complexity, marked by over 4200 flights, 15 crew bases, and billion-plus pairings, and (b) validated by the research consortium’s industrial sponsor. It is hoped that with the emergent trend of conjunct scale and complexity of airline networks, this paper shall serve as an important milestone for affiliated research and applications. ###### keywords: Airline Crew Scheduling Crew Pairing Combinatorial Optimization Column Generation Mathematical Programming Heuristics ## 1 Introduction Airline scheduling poses some of the most challenging optimization problems encountered in the entire Operations Research (OR) domain. For a large-scale airline, the crew operating cost constitutes the second-largest cost component, next to the fuel cost, and even its marginal improvements may translate to annual savings worth millions of dollars. Given the potential for huge cost-savings, Airline Crew Scheduling is recognized as a critical planning activity. It has received an unprecedented attention from the researchers of the OR community over the last three decades. Conventionally, it is tackled by solving two problems, namely, Crew Pairing Optimization Problem (CPOP) and Crew Assignment Problem, in a sequential manner. The former problem is aimed at generating a set of flight sequences (each called a crew pairing) that covers all flights from an airlines’ flight schedule, at minimum cost, while satisfying several legality constraints linked to federations’ rules, labor laws, airline-specific regulations, etc. These optimally-derived crew pairings are then fed as input to the latter problem, which is aimed to generate a set of pairing sequences (each sequence is a schedule for an individual crew member), while satisfying the corresponding crew requirements. Being the foremost step of the airline crew scheduling, CPOP is the main focus of this paper, and interested readers are referred to Barnhart . (2003) for a comprehensive review of the airline crew scheduling. CPOP is an NP-hard111For NP-hard (NP-complete) problems, no polynomial time algorithms on sequential computers are known up to now. However, verification of a solution might be (can be) accomplished efficiently, i.e., in polynomial time. combinatorial optimization problem (Garey Johnson, 1979). It is modeled as either a set partitioning problem (SPP) in which each flight is allowed to be covered by only one pairing, or a set covering problem (SCP) in which each flight is allowed to be covered by more than one pairing. In CPOP, a crew pairing has to satisfy hundreds of legality constraints (Section 2.2) to be classified as legal, and it is imperative to generate legal pairings in a time-efficient manner to assist optimization search. Several legal pairing generation approaches, based on either a flight- or a duty-network, have been proposed in the literature (Aggarwal ., 2018). Depending upon how the legal pairing generation module is invoked, two CPOP solution-architectures are possible. In the first architecture, all possible legal pairings are enumerated a priori the CPOP-solutioning. However, this is computationally- tractable only for small-scale CPOPs (with $\approx$¡1000 flights). Alternatively, legal pairings are generated during each iteration of the CPOP- solutioning, but only for a subset of flights, so the CPOP solution could be partially improved before triggering the next iteration. Such an architecture mostly suits medium- to large-scale CPOPs (with $\approx\geq$1000 flights) involving millions/billions of legal pairings, whose complete-enumeration is computationally-intractable. In terms of solution-methodologies, heuristic-based optimization techniques and mathematical programming techniques, are commonly employed (Section 2.3). In the former category, Genetic Algorithms (GAs) which are population-based randomized-search heuristics (Goldberg, 2006) are most commonly used. However, they are found to be efficient only for tackling very small-scale CPOPs (Ozdemir Mohan, 2001). Alternatively, several mathematical programming based approaches do exist to solve CPOPs of varying-scales. CPOP is inherently an Integer Programming Problem (IPP), and some approaches have used standard Integer Programming (IP) techniques to find a best-cost pairing subset from a pre-enumerated pairings’ set (Hoffman Padberg, 1993). However, these approaches have proven effective only with small-scale CPOPs with up to a million pairings. This perhaps explains the prevalence of an altogether different strategy, in which the original CPOP/IPP is relaxed into a Linear Programming Problem (LPP); the LPP is solved iteratively by invoking a LP solver and relying on Column Generation (CG) technique to generate new pairings as part of the pricing sub-problem; and finally, the resulting LPP solution is integerized using IP techniques and/or some special connection- fixing heuristics. The challenge associated with this strategy is that even though the LPP solver may lead to a near-optimal LPP solution, the scope of finding a good-cost IPP solution is limited to the pairings available in the LPP solution. To counter this challenge, heuristic implementations of branch- and-price framework (Barnhart . (1998)) in which CG is utilized during the integerization phase too, have been employed to generate new legal pairings at nodes of the IP-search tree. However, the efficacy of such heuristic implementations depends on a significant number of algorithmic-design choices (say, which branching scheme to adopt, or how many CG-iterations to perform at the nodes). Furthermore, it is noteworthy that the scale and complexity of flight networks have grown alarmingly over the past decades. As a result, an inestimably large number of new pairings are possible under the pricing sub- problem, given which most existing solution methodologies are rendered computationally-inefficient. Recognition of such challenges have paved the way towards domain-knowledge driven CG strategies to generate a manageable, yet crucial part of the overall pairings’ space under the pricing sub-problem (Zeren Özkol, 2016). Though rich in promise, the efficacy of this approach is yet to be explored vis-$\grave{a}$-vis the emergent large-scale and complex flight networks characterized by multiple crew bases and/or multiple hub-and- spoke subnetworks where billions of legal pairings are possible. In an endeavor to address airline networks with conjunct scale and complexity, this paper proposes an Airline Crew Pairing Optimization Framework ($AirCROP$) based on domain-knowledge driven CG strategies, and: * • presents not just the design of its constitutive modules (including Legal Crew Pairing Generator, Initial Feasible Solution Generator, and Optimization Engine powered by CG-driven LPP-solutioning and IPP-solutioning), but also how these modules interact * • discusses how sensitive its performance is to - sources of variability over multiple runs for a given problem; cost quality of the initial solution and the run-time spent to obtain it; and termination parameters for LPP- solutioning and IPP-solutioning. Such an investigation promises important insights for researchers and practitioners on critical issues which are otherwise not discussed in the existing literature. * • presents empirical results for real-world, large-scale (over 4200 flights), complex flight network (over 15 crew bases and multiple hub-and-spoke subnetworks) for a US-based airline, the data for which has been provided by the research consortium’s industrial partner. The outline of the remaining paper is as follows. Section 2 discusses the underlying concepts, related work, and problem formulation; Section 3 entails the details of the proposed $AirCROP$; Section 4 presents the results of the computational experiments along with the corresponding observations; and Section 5 concludes the paper as well as briefly describes the potential future directions. ## 2 Crew Pairing Optimization: Preliminaries, Related Work and Problem Formulation This section first describes the preliminaries, including the associated terminology, pairings’ legality constraints, and pairings’ costing criterion. Subsequently, the related work is presented in which the existing CPOP solution approaches are discussed. Lastly, the airline CPOP formulation is presented. ### 2.1 Associated Terminology In airline crew operations, each crew member is assigned a fixed (home) airport, called a crew base. A crew pairing (or a pairing) is a flight sequence operated by a crew, that begins and ends at the same crew base, and satisfies the given pairing legality constraints (detailed in Section 2.2). An example of a crew pairing with the Dallas (DAL) airport as the crew base is illustrated in Figure 1. In a crew pairing, the legal sequence of flights operated by a crew in a single working day (not necessarily equivalent to a calendar day) is called a crew duty or a duty. A sit-time or a connection-time is a small rest-period, provided between any two consecutive flights within a duty for facilitating operational requirements such as aircraft changes by the crew, turn-around operation for the aircraft, etc. An overnight-rest is a longer rest-period, provided between any two consecutive duties within a pairing. Moreover, two short-periods, provided in the beginning and ending of any duty within a pairing, are called briefing and de-briefing time, respectively. The total time elapsed in a crew pairing, i.e., the time for which a crew is away from its crew base is called the time away from base (TAFB). Sometimes, it is required for a crew to be transported at an airport to fly their next flight. For this, the crew travels as passenger in another flight, flown by another crew, to arrive at the required airport. Such a flight is called a deadhead flight or a deadhead for the crew traveling as passenger. It is desired by an airline to minimize the number of deadheads (ideally zero), as it affects the airline’s profit in two-folds. Firstly, the airline suffers a loss of the revenue on the passenger seat being occupied by the deadhead-ing crew, and secondly, the airline has to pay the hourly wages to the deadhead-ing crew even when it is not operating the flight. Figure 1: An example of a crew pairing starting from Dallas (DAL) crew base ### 2.2 Crew Pairing: Legality Constraints and Costing Criterion To govern the safety of crew members, airline federations such as European Aviation Safety Agency, Federal Aviation Administration, and others, have laid down several rules and regulations, which in addition to the airline-specific regulations, labor laws, etc. are required to be satisfied by a pairing to be “legal”. These legality constraints could be broadly categorized as follows: * • Connection-city constraint ($\mathcal{C}_{connect}$): this constraint requires the arrival airport of a flight (or the last flight of a duty) within a pairing to be same as the departure airport of its next flight (or the first flight of its next duty). * • Sit-time ($C_{sit}$) and Overnight-rest ($\mathcal{C}_{night}$) constraints: these constraints imposes the respective maximum and minimum limits on the duration of sit-times and overnight-rests, where these limits are governed by airlines and federations’ regulations. * • Duty constraints ($\mathcal{C}_{duty}$): these constraints govern the regulations linked to the crew duties. For instance, they impose maximum limits on the– number of flights allowed in a duty of a pairing; duty elapsed- time and the corresponding flying-time; number of duties allowed in a pairing, etc. * • Start- and end-city constraint ($\mathcal{C}_{base}$): this constraint requires the beginning airport (departure airport of the first flight) and ending airport (arrival airport of the last flight) of a pairing, to be the same crew base. * • Other constraints ($\mathcal{C}_{other}$): Airlines formulate some specific constraints, according to their operational requirements, so as to maximize their crew utilization. For example, a pairing is refrained from involving overnight-rests at the airports that belong to the same city as the crew base from which the pairing started, etc. Considering the multiplicity of the above constraints, it is critical to develop a time-efficient legal crew pairing generation approach, enabling their prompt availability, when their requirement arises during the optimization. In general, a pairing’s cost could be split into the flying cost and non- flying (variable) cost. The flying cost is the cost incurred in actually flying all the given flights, and is computed on hourly-basis. The variable cost is the cost incurred during the non-flying hours of the pairing, and is made up of two sub-components, namely, hard cost and soft cost. The hard cost involves the pairing’s hotel cost, meal cost, and excess pay– the cost associated with the difference between the guaranteed hours of pay and the actual flying hours. Here, the pairing’s hotel cost is the lodging cost incurred during its overnight-rests, and its meal cost is computed as a fraction of its TAFB. The soft cost is the undesirable cost associated with the number of aircraft changes (during flight-connections) in the pairing, etc. ### 2.3 Related Work As mentioned in Section 1, the existing CPOP solution approaches are based on either heuristic or mathematical programming techniques. Among the heuristic- based approaches, GA is the most widely adopted technique, and Beasley Chu (1996) is the first instance to customize a GA (using guided GA-operators) for solving a general class of SCPs. In that, the authors validated their proposed approach on small-scale synthetic test cases (with over 1,000 rows and just 10,000 columns). The important details of the GA-based CPOP solution approaches, available in the literature, are reported in Table 1. Table 1: Key facts around the GA-based CPOP solution approaches, available in the literature Literature Studies | Modeling | Timetable | Airline Test Cases* | Airlines ---|---|---|---|--- Levine (1996) | Set Partitioning | - | 40R; 823F; 43,749P | - Ozdemir Mohan (2001) | Set Covering | Daily | 28R; 380F; 21,308P | Multiple Airlines Kornilakis Stamatopoulos (2002) | Set Covering | Monthly | 1R; 2,100F; 11,981P | Olympic Airways Zeren Özkol (2012) | Set Covering | Monthly | 1R; 710F; 3,308P | Turkish Airlines Deveci Demirel (20181) | Set Covering | - | 12R; 714F; 43,091P | Turkish Airlines R represents the number of real-world test cases considered; F and P represents the maximum number of flights and pairings covered, therein. Notably, the utility of the studies reported in the table, have been demonstrated on CPOPs with reasonably small number of flights, leading to relatively smaller number of pairings. Though, CPOPs with 2,100 and 710 flights have been tackled by Kornilakis Stamatopoulos (2002) and Zeren Özkol (2012) respectively, only a subset of all possible legal pairings has been considered by them for finding the reported solutions. Zeren Özkol (2012) proposed a GA with highly-customized operators, which efficiently solved small-scale CPOPs but failed to solve large-scale CPOPs with the same search- efficiency. Furthermore, Aggarwal, Saxena . (20202) tackled a small-scale CPOP (with 839 flights and multiple hub-and-spoke sub-networks) using a customized- GA (with guided operators) as well as mathematical programming techniques. The authors concluded that customized-GAs are inefficient in solving complex versions of even small-scale flight networks, compared to a mathematical programming-based solution approach. Several mathematical programming-based CPOP solution approaches have been proposed in the literature over past few decades, and based on the size and characteristics of the flight network being tackled, these approaches have been categorized into either of the three general classes. In the first class of approaches, all legal pairings or a subset of good pairings are enumerated prior to the CPOP-solutioning, and the corresponding CPOP/IPP model is solved using standard IP techniques (such as branch-and-bound algorithm (Land Doig, 1960)). Gershkoff (1989) proposed an iterative solution approach, which is initialized using a set of artificial pairings (each covering a single flight at a high pseudo-cost). In that, each iteration involves selection of very few pairings (5 to 10); enumeration of all legal pairings using the flights covered in the selected pairings; optimization of the resulting SPP to find the optimal pairings; and lastly, replacement of the originally selected pairings with the optimal pairings, only if the latter offers a better cost. The search-efficiency of such an approach is highly dependent on the sub- problem-size (handled up to 100 flights and 5,000 pairings), as the length and breadth of the branching tree increases drastically with an increase in sub- problem-size. Hoffman Padberg (1993) proposed an alternative approach to tackle SPPs with up to 825 flights and 1.05 million pairings in which all possible pairings are enumerated a priori, and the resulting SPP is solved to optimality using a branch-and-cut algorithm222The branch-and-cut algorithm was first proposed by Padberg Rinaldi (1991) to solve Mixed Integer Programs (MIP), by integrating the standard branch-and-bound and cutting-plane algorithms. For comprehensive details of the MIP solvers, interested readers are referred to Lodi (2009); Linderoth Lodi (2011); Achterberg Wunderling (2013).. Such approaches are efficient only in tackling small-scale CPOPs, that too with up to a million pairings. However, even small-scale CPOPs may involve large number of pairings (an instance reported in Vance . (1997) had 250 flights and over five million pairings), rendering it computationally- intractable to use such approaches. The second class of approaches relies on relaxing the integer constraints in the original CPOP/IPP to form an LPP, which is then solved iteratively by– invoking an LP solver and generating new pairings using CG; and integerizing the resulting LPP solution. In any iteration of the LPP-solutioning (referred to as an LPP iteration), an LP solver (based on either a simplex method or an interior-point method333The class of interior-point methods was first introduced by Karmarkar (1984). In that, a polynomial-time algorithm, called Karmarkar’s algorithm, was proposed, which, in contrary to simplex method, searches for the best solution by traversing the interior of the feasible region of the search space.) is invoked on the input pairing set to find the LPP solution and its corresponding dual information (shadow price corresponding to each flight-coverage constraint), which are then utilized to generate new pairings as part of the pricing sub-problem, promising the corresponding cost-improvements. For the first LPP iteration, any set of pairings covering all the flights becomes the input to the LP solver, and for any subsequent LPP iteration, the current LPP solution and the set of new pairings (from the pricing sub-problem) constitute the new input. For more details on how new pairings are generated under the pricing sub-problem in the CG technique, interested readers are referred to Vance . (1997); Lübbecke Desrosiers (2005). As cited in Zeren Özkol (2016), the CG technique has several limitations, out of which the prominent ones are– heading-in effect (poor dual information in initial LPP iteration leads to generation of irrelevant columns), bang-bang effect (dual variables oscillate from one extreme point to another, leading to poor or slower convergence), and tailing- off effect (the cost-improvements in the later LPP iterations taper-off). While, different stabilization techniques are available for CG in the literature Du Merle . (1999); Lübbecke (2010), the use of interior point methods is gaining prominence. Anbil . (1991) presented the advancements at the American Airlines, and enhanced the approach proposed by Gershkoff (1989) (discussed above), by leveraging the knowledge of dual variables to screen- out/price-out the pairings from the enumerated set at each iteration, enabling it to solve larger sub-problems (up to 25 flights and 100,000 pairings). As an outcome of a collaboration between IBM and American Airlines, Anbil . (1992) proposed an iterative global solution approach (though falling short of global optimization) in which an exhaustive set of pairings ($\approx$5.5 million) is enumerated a priori. Several thousands of these pairings are used to initialize the iterative procedure, and in each of its iterations, a sub- problem is solved to obtain optimal dual variables, which are then used to price-out all 5.5 million pairings to select a sufficiently-sized set of good pairings ($\approx$5,000 pairings). For integerization of the LPP solution, the literature points to two prominent strategies. The first strategy is based on utilizing either a branch-and-bound, or a branch-and-cut algorithm. The other strategy utilizes some special “connection-fixing” heuristics either solely for integerization (Anbil ., 1992; Marsten, 1994), or during the iterations of the LPP-solutioning (Zeren Özkol, 2016) to boost the performance of the subsequent MIP solver (in some cases, may even get integer solution without using the MIP solver). These heuristics eliminate some irrelevant pairings by exploiting the knowledge of their linear variables and fixing some specific flight-connections. The limitation in this class of approaches is that even though, a good IPP solution to the original CPOP may exist and the LPP-solutioning leads to a near-optimal LPP solution, the pairings available in it may not fit well together to constitute a good-cost IPP solution. The third class of approaches share a similar solution-architecture as of the preceding class, however, differs in terms of the integerization of the LPP solution. In that, a heuristic branch-and-price framework444The branch-and- price algorithm was originally proposed by Barnhart . (1998) as an exact algorithm to solve then-known large-scale IPPs, and has been utilized to solve a variety of combinatorial optimization problems in transportation such as Desrosiers . (1984); Desrochers Soumis (1989); Desrochers . (1992). is adopted, wherein, CG is utilized during the integerization phase too, to generate new legal pairings at nodes of the MIP-search tree. Desrosiers . (1991) is the first instance that solved CPOP using a branch-and-price framework. However, given the inestimable number of legal pairings possible for even medium-scale CPOPs, numerous branch-and-price based heuristic- approaches have been proposed over the last three decades (Desaulniers ., 1997; Vance ., 1997; Anbil ., 1998; Desaulniers Soumis, 2010). Notably, the development of these approaches, being heuristic in nature, require a significant number of algorithmic-design choices to be taken empirically, which may vary with the characteristics of the flight networks being solved for. To name a few such decisions, which branching scheme should be employed (branching on linear variables, branching on flight-connections, or others), should CG be performed on each node of the MIP-search tree, how many CG iterations to be performed each time, etc. Furthermore, the commercial LP and MIP solvers are not much open to modifications, making it difficult for the new researchers to implement a computationally- and time-efficient branch-and- price framework from scratch. For further details of the existing CPOP solution approaches, interested readers are referred to recent survey articles– Kasirzadeh . (2017); Deveci Demirel (20182). In addition to the above classification of solution approaches, the literature differs on the notion of how the pricing sub-problem is modeled and solved to generate new legal pairings during the LPP iterations. However, the focus of this paper is not on the solution to the pricing sub-problem step, but on the interactions between different modules of a CG-based CPOP solution approach. Hence, for details on the existing work related to the pricing sub-problem step, interested readers are referred to Vance . (1997); Aggarwal, Saxena . (20201). ### 2.4 Integer Programming Problem Formulation As mentioned earlier, a CPOP is intrinsically an IPP, modeled either as a SCP or a SPP. Notably, the SCP formulation provides higher flexibility during its solutioning compared to the SPP formulation by accommodating deadhead flights in the model, possibly resulting in faster convergence (Gustafsson, 1999). For a given flight set $\mathcal{F}$ (including $F$ flights) that could be covered in numerous ways by a set of legal pairings $\mathcal{P}$ (including $P$ pairings), the set covering problem is aimed to find a subset of pairings ($\in\mathcal{P}$), say $\mathcal{P}_{IP}^{*}$, which not only covers each flight ($\in\mathcal{F}$) at least once, but does it at a cost lower than any alternative subset of pairings in $\mathcal{P}$. In that, while finding $\mathcal{P}_{IP}^{*}$ ($\subseteq\mathcal{P}$), each pairing $p_{j}\in\mathcal{P}$ corresponds to a binary variable $x_{j}$, which represents whether the pairing $p_{j}$ is included in $\mathcal{P}_{IP}^{*}$ (marked by $x_{j}=1$) or not ($x_{j}=0$). Here, $p_{j}$ is a $F$-dimensional vector, whose each element, say $a_{ij}$, represents whether the flight $f_{i}$ is covered by pairing $p_{j}$ (marked by $a_{ij}=1$) or not ($a_{ij}=0$). In this background, the IPP formulation, as used in this paper, is as follows. $\displaystyle\text{Minimize}~{}Z_{IP}=\sum_{j=1}^{P}c_{j}x_{j}+\psi_{D}\cdot\left(\sum_{i=1}^{F}\left(\sum_{j=1}^{P}a_{ij}x_{j}-1\right)\right),$ (1) $\displaystyle\text{subject to}\quad\sum_{j=1}^{P}a_{ij}x_{j}\geq 1,\quad~{}~{}~{}~{}~{}\forall i\in\\{1,2,...,F\\}$ (2) $\displaystyle\qquad\qquad\quad x_{j}\in\mathbb{Z}=\\{0,1\\},~{}~{}~{}~{}\forall j\in\\{1,2,...,P\\}$ (3) $\displaystyle\text{where},c_{j}$ $\displaystyle:~{}\text{the cost of a legal pairing }p_{j},$ $\displaystyle\psi_{D}$ $\displaystyle:~{}\text{an airline- defined penalty cost against each deadhead in the solution},$ $\displaystyle\quad a_{ij}$ $\displaystyle=~{}1,~{}\text{if flight}~{}f_{i}~{}\text{is covered in pairing}~{}p_{j};\text{ else }0$ $\displaystyle x_{j}$ $\displaystyle=~{}1,~{}\text{if pairing}~{}p_{j}~{}\text{contributes to Minimum}~{}Z;\text{ else }0$ In the objective function (Equation 1), the first component gives the sum of the individual costs of the pairings selected in the solution, while the other component gives the penalty cost for the deadheads incurred in the solution (note, $(\sum_{j=1}^{P}a_{ij}x_{j}-1)$ gives the number of deadheads, corresponding to the flight $f_{i}$). Notably, in the above formulation, it is assumed that the set of all possible legal pairings, namely, $\mathcal{P}$, are available a priori, and the task is to determine $\mathcal{P}_{IP}^{*}$. However, the generation of $\mathcal{P}$ a priori is computationally- intractable for large-scale CPOPs, as mentioned in Section 2.3. Hence, the solution to the CPOP/IPP is pursued in conjunction with the corresponding LPP (formulation deferred till Section 3.3.1) assisted by the CG technique. ## 3 Proposed Airline Crew Pairing Optimization Framework (AirCROP) This section presents the constitutive modules of the proposed optimization framework - $AirCROP$, their working, and their interactions. As per the schematic in Figure 2, $AirCROP$ accepts a set of given flights $\mathcal{F}$ along with the pairings’ legality constraints and costing criterion as input, and outputs a minimal-cost set of legal pairings $\mathcal{P}_{IP}^{\star}$, that covers all given flights. This transition from the input to output is enabled by the constitutive modules, namely, the Legal Crew Pairing Generator, the Initial Feasible Solution Generator, and an Optimization Engine in turn enabled by CG-driven LPP-solutioning and IPP-solutioning submodules and their intermittent interactions. While parts of these modules have been presented elsewhere (Aggarwal ., 2018; Aggarwal, Saxena ., 20201) in isolation, these are being detailed below towards a holistic view on the experimental results presented later. Figure 2: A schematic of $AirCROP$ illustrating the interactions between its constitutive modules– Legal Crew Pairing Generator, Initial Feasible Solution Generator, Optimization Engine (CG-driven LPP-solutioning interacting with IPP-solutioning). The CG heuristic in LPP-solutioning generates a set of fresh pairings $\mathcal{P}_{CG}^{t}$ at any LPP iteration $t$ using the following CG strategies: Deadhead reduction ($CGD$, generating $\mathcal{P}_{CGD}^{t}$), Crew Utilization enhancement ($CGU$, generating $\mathcal{P}_{CGU}^{t}$), Archiving ($CGA$, generating $\mathcal{P}_{CGA}^{t}$), and Random exploration ($CGR$, generating $\mathcal{P}_{CGR}^{t}$). The interactions between LPP- solutioning and IPP-solutioning are tracked by the counter $T$. ### 3.1 Legal Crew Pairing Generator This module enables generation of the legal pairings in a time-efficient manner, so they could feed real-time into the other modules - Initial Feasible Solution Generator and the optimization engine. For time-efficiency, it employs a parallel, duty-network based legal pairing generation approach, whose distinctive contributions are two-folds. Firstly, a crew base centric parallel architecture is adopted considering that several duty- and pairing- constitutive constraints do vary with crew bases. In that, for an input flight set, the legal pairing generation process is decomposed into independent sub- processes (one for each crew base), running in parallel on idle-cores of the central processing unit (CPU). This leads to a significant reduction in the pairing generation time ($\approx$10 folds for a CPOP with 15 crew bases, as demonstrated in Aggarwal . (2018)). Secondly, the set of all possible legal duties and the corresponding duty overnight-connection graph with-respect-to each crew base are enumerated and stored a priori the CPOP-solutioning. In a duty overnight-connection graph, a node represents a legal duty, and an edge between any two nodes represents a legal overnight-rest connection between the respective duties. Such a preprocessing ensures that all the connection-city, sit-time, duty, and overnight-rest constraints get naturally satisfied, eliminating the need for their re-evaluation during the generation of legal pairings, and leading to a significant reduction in the legal pairing generation time. The implementation of this module, formalized in Algorithms 1 & 2, is elaborated below. Input: $\mathcal{F}$; $\mathcal{B}$; and constraints: $\mathcal{C}_{connect},~{}\mathcal{C}_{sit},~{}\mathcal{C}_{duty}~{}\&~{}\mathcal{C}_{night}$ Output: $\mathcal{D}_{b}$ & $\mathcal{G}^{d}_{b}~{}\forall b\in\mathcal{B}$ $\mathcal{G}^{f}\leftarrow$ Generate the flight-connection graph by evaluating $\mathcal{C}_{connect}~{}\&~{}\mathcal{C}_{sit}$ between each pair of flights $\in\mathcal{F}$ $\triangleright$ $\mathcal{G}^{f}\equiv\left(\mathcal{F},\mathcal{E}^{f}\right)$ 1 for _each crew base $b\in\mathcal{B}$ in parallel_ do 2 for _each flight $f\in\mathcal{F}$_ do 3 Push $f$ into an empty $duty$ 4 if _updated flight-sequence in $duty$ satisfies constraints in $\mathcal{C}_{duty}$_ then 5 Add $duty$ to $\mathcal{D}_{b}$ 6 if _$f$ has at least one flight-connection in $\mathcal{G}^{f}$_ then 7 $\texttt{DFS(}duty,f,\mathcal{G}^{f},\mathcal{C}_{duty}\texttt{)}$, and add the enumerated duties to $\mathcal{D}_{b}$ 8 9 end if 10 11 end if 12 Pop out $f$ from $duty$ 13 14 end for 15 $\mathcal{G}^{d}_{b}\leftarrow$ Generate the duty overnight-connection graph by evaluating $\mathcal{C}_{night}$ between each pair of duties $\in\mathcal{D}_{b}$ 16 end for 17return $\mathcal{D}_{b}$ & $\mathcal{G}^{d}_{b}~{}\forall b\in\mathcal{B}$ $\triangleright$ DFS($duty,parent,\mathcal{G}^{f},\mathcal{C}_{duty}$) 18 for _each $child$ of $parent$ in $\mathcal{G}^{f}$_ do 19 Push $child$ into $duty$ 20 if _updated flight-sequence in $duty$ satisfies $\mathcal{C}_{duty}$_ then 21 yield $duty$ to $\mathcal{D}_{b}$ 22 if _$child$ has at least one connection in $\mathcal{G}^{f}$_ then 23 DFS($duty,child,\mathcal{G}^{f},\mathcal{C}_{duty}$) 24 25 end if 26 27 end if 28 Pop out $child$ from $duty$ 29 30 end for Algorithm 1 Procedure for enumeration of legal duties and duty overnight- connection graphs Input: $\mathcal{F}_{*}\text{ or }\mathcal{D}_{*};~{}\mathcal{B};~{}\mathcal{D}_{b}~{}\&~{}\mathcal{G}^{d}_{b}~{}\forall b\in\mathcal{B}$; and constraints: $\mathcal{C}_{base}~{}\&~{}\mathcal{C}_{other}$ Output: $\mathcal{P}_{*}$ 1 for _each crew base $b\in\mathcal{B}$ in parallel_ do 2 Update $\mathcal{D}_{b}~{}\&~{}\mathcal{G}^{d}_{b}$ by removing duties $\notin\mathcal{D}_{*}$ if $\mathcal{D}_{*}$ is input, or by removing those duties which cover flights $\notin\mathcal{F}_{*}$ if $\mathcal{F}_{*}$ is input 3 for _each $duty\in\mathcal{D}_{b}$_ do 4 if _departure airport of $duty$ is $b$_ then 5 Push $duty$ into an empty $pairing$ 6 if _updated duty-sequence in $pairing$ satisfies $\mathcal{C}_{other}$_ then 7 if _updated duty-sequence in $pairing$ satisfies $\mathcal{C}_{base}$_ then 8 Add $pairing$ to $\mathcal{P}_{*}$ 9 10 else if _$duty$ has at least one duty overnight-connection in $\mathcal{G}^{d}_{b}$_ then 11 $\texttt{DFS(}pairing,duty,\mathcal{G}^{d}_{b},\mathcal{C}_{base}\cup\mathcal{C}_{other}\texttt{)}$, and add enumerated pairings to $\mathcal{P}_{*}$ 12 13 end if 14 15 end if 16 Pop out $duty$ from $pairing$ 17 18 end if 19 20 end for 21 22 end for 23return $\mathcal{P}_{*}$ $\triangleright$ DFS($pairing,parent,\mathcal{G}^{d}_{b},\mathcal{C}_{base}\cup\mathcal{C}_{other}$) 24 for _each $child$ of $parent$ in $\mathcal{G}^{d}_{b}$_ do 25 Push $child$ into $pairing$ 26 if _updated duty-sequence in $pairing$ satisfies $\mathcal{C}_{other}$_ then 27 if _updated duty-sequence in $pairing$ satisfies $\mathcal{C}_{base}$_ then 28 yield $pairing$ to $\mathcal{P}_{*}$ 29 30 else if _$child$ has at least one duty overnight-connection in $\mathcal{G}^{d}_{b}$_ then 31 $\texttt{DFS(}pairing,child,\mathcal{G}^{d}_{b},\mathcal{C}_{base}\cup\mathcal{C}_{other}\texttt{)}$ 32 33 end if 34 35 end if 36 Pop out $child$ from $pairing$ 37 38 end for Algorithm 2 Procedure for enumeration of legal pairings from an input flight set $\mathcal{F}_{*}\text{ or a duty set }\mathcal{D}_{*}$ For solving any CPOP, the foremost step of the $AirCROP$ is to preprocess the entire duty-connection network– set of legal duties $\mathcal{D}_{b}$ and duty overnight-connection graph $\mathcal{G}^{d}_{b}\left(\equiv\left(\mathcal{D}_{b},~{}\mathcal{E}^{d}_{b}\right)\right)$ for each crew base $b$ in the given set of crew bases $\mathcal{B}$, where $\mathcal{E}^{d}_{b}$ is the set of legal overnight-rest connections between duty-pairs $\in\mathcal{D}_{b}$. The procedure for the above preprocessing is presented in Algorithm 1. In that, the first step is the generation of a flight-connection graph (denoted by $\mathcal{G}^{f}$) by evaluating the legality of connection-city ($\mathcal{C}_{connect}$) and sit-time ($\mathcal{C}_{sit}$) constraints between every flight-pair in the given flight schedule $\mathcal{F}$ (line 1). Here, in $\mathcal{G}^{f}~{}\left(\equiv\left(\mathcal{F},\mathcal{E}^{f}\right)\right)$, $\mathcal{F}$ is the set of nodes (flights) and $\mathcal{E}^{f}$ is the set of edges (legal flight connections). Subsequently, $\mathcal{G}^{f}$ is used for legal duty enumeration, by decomposing the process into independent sub- processes, one for each crew base $b\in\mathcal{B}$, and executing them in parallel (lines 2-12). In each of these sub-processes, enumeration of legal duties, starting from each flight $f\in\mathcal{F}$, is explored. In that: * • flight $f$ is added to an empty candidate duty stack, given by $duty$ (line 4). * • the flight-sequence in $duty$ is checked for satisfaction of duty constraints $\mathcal{C}_{duty}$, and if satisfied, $duty$ is added to the desired legal duty set $\mathcal{D}_{b}$ (lines 5-6). Notably, if $f$ has at least one connection with another flight in $\mathcal{G}^{f}$, and if the duty constraints permit, then more flights could be accommodated in $duty$, leading to enumeration of other legal duties (lines 7-9). * • a Depth-first Search (DFS) algorithm (Tarjan, 1972) is adapted, which is called recursively to enumerate legal duties, starting from a parent flight node ($parent$), by exploring its all successive paths in $\mathcal{G}^{f}$ in a depth-first manner (lines 16-25). In each recursion, a child flight node ($child$) is pushed into $duty$, the updated flight-sequence is checked for satisfaction of $\mathcal{C}_{duty}$, and if satisfied, $duty$ is yielded to $\mathcal{D}_{b}$, followed by another recursion of DFS() with $child$ as the new $parent$. In this way, all legal duties, starting from flight $f$, are enumerated. Subsequently, $f$ is popped out from $duty$, and duty enumeration using other flights in $\mathcal{F}$ is explored (lines 3 & 11). The resulting set $\mathcal{D}_{b}$ is then used to generate the duty overnight-connection graph $\mathcal{G}^{d}_{b}$), by evaluating the legality of connection-city ($\mathcal{C}_{connect}$) and overnight-rest ($\mathcal{C}_{night}$) constraints between every duty-pair $\in\mathcal{D}_{b}$ (line 13). Here, in $\mathcal{G}^{d}_{b}~{}\left(\equiv\left(\mathcal{D}_{b},\mathcal{E}^{d}_{b}\right)\right)$, $\mathcal{D}_{b}$ is the set of nodes (legal duties), and $\mathcal{E}^{d}_{b}$ is the set of edges (legal overnight-rest connections). The preprocessed sets of legal duties and the corresponding duty overnight- connection graphs are utilized to enumerate legal pairings for any input flight set (say $\mathcal{F}_{*}$) or a duty set (say $\mathcal{D}_{*}$), when required in real-time in other modules of the $AirCROP$. Its procedure, formalized in Algorithm 2, is elaborated below. For legal pairing enumeration, the same crew base driven parallel architecture is utilized in which the process is decomposed into independent sub-processes, one for each crew base $b\in\mathcal{B}$, running in parallel on idle-cores of the CPU (line 1). In each of these sub-processes, the first step is to update $\mathcal{D}_{b}$ and $\mathcal{G}^{d}_{b}$, by removing duties $\notin\mathcal{D}_{*}$ if $\mathcal{D}_{*}$ is input, or those duties that cover flights $\notin\mathcal{F}_{*}$ if $\mathcal{F}_{*}$ is input (line 2). Subsequently, the enumeration of legal pairings, starting from each duty ($duty$) $\in\mathcal{D}_{b}$, is explored (line 3). In that: * • the $duty$ is pushed into an empty candidate pairing stack, given by $pairing$, only if the departure airport of $duty$ is same as the crew base $b$ (lines 4-5). * • the $pairing$ is checked for satisfaction of pairing constraints $\mathcal{C}_{other}$, and if satisfied, $pairing$ is further checked for satisfaction of end-city constraint $\mathcal{C}_{base}$, which ensures that the arrival airport of the $pairing$’s last duty is same as the crew base $b$. * – If $pairing$ satisfies $\mathcal{C}_{base}$, it is classified as legal, and is added to the desired pairing set $\mathcal{P}_{*}$ (lines 7-8). * – If $pairing$ does not satisfy $\mathcal{C}_{base}$, it is not complete, and more duties are required to be covered in it to complete the legal duty- sequence. This is only possible if $duty$ has at least one overnight-rest connection in $\mathcal{G}^{d}_{b}$. And if it does, the DFS() sub-routine, similar to the one used in legal duty enumeration, is called recursively to enumerate legal pairings, starting from a parent duty node ($parent$), by exploring its all successive paths in $\mathcal{G}^{d}_{b}$ in a depth-first manner (lines 18-28). In each recursion: * $\circ$ a child duty node ($child$) is pushed into the $pairing$ (line 19). * $\circ$ the updated duty-sequence in $pairing$ is checked for satisfaction of first $\mathcal{C}_{other}$ and then $\mathcal{C}_{base}$ (lines 20-21). * $\circ$ if it satisfies both constraints, then $pairing$ is complete (legal), and is yielded to the desired pairing set $\mathcal{P}_{*}$ (line 22). * $\circ$ if it satisfies $\mathcal{C}_{other}$ but not $\mathcal{C}_{base}$, then another recursion of DFS() with $child$ as new $parent$ is called, only if $child$ has at least one duty overnight-rest connection in $\mathcal{G}^{d}_{b}$ (lines 23-25). In the above way, all legal pairings, starting from $duty$, are enumerated using the DFS() sub-routine. Subsequently, $duty$ is popped out of $pairing$ (line 13), and the legal pairing enumeration using other duties $\in\mathcal{D}_{b}$ is explored (line 3). Once, all the sub-processes are complete, the desired pairing set $\mathcal{P}_{*}$ is returned (line 17). ### 3.2 Initial Feasible Solution Generator An initial feasible solution (IFS) is any set of pairings, covering all flights in the given flight schedule, which is used to initialize a CPOP solution approach. For large-scale CPOPs, generation of an IFS standalone is a computationally-challenging task. This module is designed to generate a reasonably-sized IFS in a time-efficient manner for large and complex flight networks, which is then used to initialize the Optimization Engine of $AirCROP$. For this, it employs a novel Integer Programming based Divide-and- cover Heuristic (IPDCH), which relies on: (a) a divide-and-cover strategy to decompose the input flight schedule into sufficiently-small flight subsets, and (b) integer programming to find a lowest-cost pairing set, covering the maximum possible flights for each of the decomposed flight subsets. The procedure of the proposed IPDCH, formalized in Algorithm 3, is elaborated below. Input: $\mathcal{F},~{}K,~{}\texttt{Pairing\\_Gen()}$ Output: $\mathcal{P}_{IFS}$ 1 while _all flights $\in\mathcal{F}$ are not covered in $\mathcal{P}_{IFS}$_ do $\mathcal{F}_{K}\leftarrow$ Select $K$ random flights from $\mathcal{F}$ without replacement $\triangleright$ $K<F$ 2 $\mathcal{P}_{K}\leftarrow~{}\texttt{Pairing\\_Gen(}\mathcal{F}_{K}\texttt{)}$ $\mathcal{F}_{K^{\prime}}\leftarrow$ Flights covered in $\mathcal{P}_{K}$ $\triangleright$ $K^{\prime}\leq K$ 3 Add remaining flights $\left(\mathcal{F}_{K}\backslash\mathcal{F}_{K^{\prime}}\right)$ back to $\mathcal{F}_{K}$ 4 Formulate the IPP using flights in $\mathcal{F}_{K^{\prime}}$ and pairings in $\mathcal{P}_{K}$ 5 $\mathcal{P}_{IP}\leftarrow$ Solve the IPP using an MIP solver, and select pairings corresponding to non-zero variables 6 Add pairings from $\mathcal{P}_{IP}$ to $\mathcal{P}_{IFS}$ 7 Replace flights in $\mathcal{F}$ if it becomes empty 8 9 end while 10return $\mathcal{P}_{IFS}$ Algorithm 3 Procedure for IFS generation using the proposed IPDCH Being an iterative heuristic, IPDCH terminates when all flights in the input set are covered by pairings in the desired IFS, notated as $\mathcal{P}_{IFS}$ (lines 1). The input to the heuristic involves the given flight schedule $\mathcal{F}$ (with $F$ number of flights), the pairing generation sub-routine Pairing_Gen() (presented in Section 3.1), and a pre-defined decomposition parameter $K$, which regulates the number of flights to be selected from $\mathcal{F}$ in each IPDCH-iteration. The setting of $K$ largely depends upon the available computational resources, and the characteristics of the input flight dataset (as highlighted in Section 4.3.3). In each IPDCH-iteration, first a flight subset, say $\mathcal{F}_{K}$ $\left(K<F\right)$, is formed by randomly selecting $K$ number of flights from $\mathcal{F}$ without replacement (line 2). Subsequently, $\mathcal{F}_{K}$ is fed as input to the Pairing_Gen() sub-routine to enumerate the set of all possible legal pairings, say $\mathcal{P}_{K}$ (line 3). Notably, all flights in $\mathcal{F}_{K}$ may not get covered by pairings in $\mathcal{P}_{K}$, as random selection of flights does not guarantee legal connections for all selected flights. Let $\mathcal{F}_{K^{\prime}}$ $\left(K^{\prime}\leq K\right)$ be the set of flights covered in $\mathcal{P}_{K}$ (line 4). The remaining flights, given by $\mathcal{F}_{K}\backslash\mathcal{F}_{K^{\prime}}$, are added back to $\mathcal{F}$ (line 5). Subsequently, $\mathcal{F}_{K^{\prime}}$ and $\mathcal{P}_{K}$ are used to formulate the corresponding IPP (line 6), which is then solved using a commercial off-the-shelf MIP solver to find the optimal IPP solution, say $\mathcal{P}_{IP}$, constituted by pairings corresponding to only non-zero variables (line 7). The pairings in $\mathcal{P}_{IP}$ are then added to the desired set $\mathcal{P}_{IFS}$ (line 8). Lastly, the flights in $\mathcal{F}$ are replaced if it becomes empty (line 9). As soon as $\mathcal{P}_{IFS}$ covers all the required flights, IPDCH is terminated, and $\mathcal{P}_{IFS}$ is passed over to the Optimization Engine for its initialization. ### 3.3 Optimization Engine: Interactions between CG-driven LPP-solutioning and IPP-solutioning The search for minimal cost, full flight-coverage CPOP solution is enabled by an optimization engine. It tackles the underlying LPP and IPP through intermittent interactions of two submodules, namely, CG-driven LPP-solutioning and IPP-solutioning, tracked by a counter $T$. These submodules are presented below. #### 3.3.1 CG-driven LPP-solutioning As illustrated in Figure 2, this submodule entails several iterations (each referred to as an LPP iteration, and is tracked by $t$) in each of which: (a) an LP solver is invoked on the input pairing set, leading to the current LPP solution $\mathcal{P}_{LP}^{t}$, (b) the corresponding dual od the LPP is formulated using $\mathcal{P}_{LP}^{t}$, which is then solved to fetch dual variables (given by vector $Y^{t}$), and (c) a fresh set of pairings $\mathcal{P}_{CG}^{t}$, that promises associated cost-improvement, is generated using a domain-knowledge driven CG heuristic. For the first LPP iteration ($t=1$), the input to the LP solver is either $\mathcal{P}_{IFS}$ if $T=1$, or $\mathcal{P}_{IP}^{T-1}$ if $T>1$. For any subsequent LPP iteration ($t>1$), the input comprises of the current $\mathcal{P}_{CG}^{t}$ and $\mathcal{P}_{LP}^{t}$. In this background, each of these LPP iterations are implemented in the following three phases555For ease of reference, the notations introduced in these phases are kept independent of the LPP iteration counter $t$. However, these notations are super-scripted by $t$ in the corresponding discussions and pseudocodes with reference to a particular LPP iteration.: * • In the first phase, a primal of the LPP (Equations 4 to 6) is formulated from the input pairing set, and is solved using an interior-point method based commercial off-the-shelf LP solver (Gurobi Optimization, 2019). In the resulting LPP solution, a primal variable $x_{j}$, varying from $0$ to $1$, is assigned to each pairing $p_{j}$ in the input pairing set. These $x_{j}$s together constitute the primal vector, notated as $X~{}\left(=[x_{1}~{}x_{2}~{}x_{3}~{}...~{}x_{P}]^{\mathsf{T}}\right)$. The set of $x_{j}$s with non-zero values ($x_{j}\neq 0$) and the set of corresponding pairings are notated as $X_{LP}$ and $\mathcal{P}_{LP}$, respectively. $\displaystyle\text{Minimize}~{}Z_{LP}^{p}=\sum_{j=1}^{P}c_{j}x_{j}+\psi_{D}\cdot\left(\sum_{i=1}^{F}\left(\sum_{j=1}^{P}a_{ij}x_{j}-1\right)\right)=\sum_{j=1}^{P}\left(c_{j}+\psi_{D}\cdot\sum_{i=1}^{F}a_{ij}\right)x_{j}-F\cdot\psi_{D},$ (4) $\displaystyle\text{subject to}\quad\sum_{j=1}^{P}a_{ij}x_{j}\geq 1,\qquad\quad\forall i\in\\{1,2,...,F\\}$ (5) $\displaystyle\qquad\qquad\quad x_{j}\in\mathbb{R}=[0,1],\qquad\forall j\in\\{1,2,...,P\\}$ (6) It is to be noted that the minimization of $Z_{LP}^{p}$ will always lead to a solution with all primal variables $x_{j}\leq 1$, even without explicitly involving the corresponding constraint– Equation 6 (Vazirani, 2003). Hence, the contribution of each pairing in the LPP solution, given by its $x_{j}$, could be effectively treated as $x_{j}\in\mathbb{R}_{\geq 0}$ instead of Equation 6. * • In the second phase, dual variables are extracted from the current LPP solution. For this, the dual of the LPP (Equations 7 to 9) is formulated using the pairing set $\mathcal{P}_{LP}$, and is solved using an interior-point method (Andersen Andersen, 2000) based non-commercial LP solver (Virtanen ., 2020), to fetch the optimal dual solution. In that, a dual variable $y_{i}$ represents a shadow price corresponding to an $i^{th}$ flight-coverage constraint in the primal. The optimal dual vector, constituted by all $y_{i}$s in the optimal dual solution, is notated as $Y~{}\left(=[y_{1}~{}y_{2}~{}y_{3}~{}...~{}y_{F}]^{\mathsf{T}}\right)$, whose dimension is equal to $F$. $\displaystyle\text{Maximize}~{}Z_{LP}^{d}=\sum_{i=1}^{F}y_{i}-F\cdot\psi_{D},$ (7) $\displaystyle\text{subject to}\quad\sum_{i=1}^{F}a_{ij}y_{i}\leq\left(c_{j}+\psi_{D}\cdot\sum_{i=1}^{F}a_{ij}\right),~{}~{}~{}~{}\forall j\in\\{1,2,...,P_{LP}\\}$ (8) $\displaystyle\qquad\qquad\qquad\quad~{}~{}y_{i}\in\mathbb{R}\geq 0,\qquad\qquad\qquad~{}~{}~{}~{}\forall i\in\\{1,2,...,F\\}$ (9) $\displaystyle\text{where},\qquad P_{LP}:~{}\text{is the number of pairings in the set}~{}\mathcal{P}_{LP}$ $\displaystyle\qquad\qquad\quad~{}~{}~{}y_{i}:~{}\text{dual variable, corresponding to an $i^{th}$ flight-coverage constraint},$ Notably, in a conventional approach, the optimal $Y$ is directly computed from the optimal basis of the primal solution (obtained in the first phase), using the principles of duality theory, particularly the theorem of complementary slackness (Bertsimas Tsitsiklis, 1997), without explicitly solving the corresponding dual. However, in the second phase, solving the dual explicitly using the interior-point method (Andersen Andersen, 2000), in a sense, helps in stabilizing the oscillating behavior of dual variables over the successive LPP iterations (bang-bang effect, as discussed in Section 2.3). Moreover, this interior-point method is available via only a non-commercial LP solver (Virtanen ., 2020), and to ensure a time-efficient search, the above dual is formulated using the pairings $\in\mathcal{P}_{LP}$, instead of pairings from the large-sized input pairing set. * • In the last phase, the availability of dual variables from the second phase paves the way for solution to the pricing sub-problem. It is aimed to generate those legal pairings (non-basic), which if included as part of the input to the next LPP iteration, promise a better-cost (at least a similar-cost) LPP solution compared to the current solution. Such non-basic pairings are identified using a reduced cost metric, given by $\mu_{j}$ (Equation 10), which if negative (as CPOP is a minimization problem) indicates the potential in the pairing to further reduce the cost of the current LPP solution $Z_{LP}^{p}$, when included in the current basis (Bertsimas Tsitsiklis, 1997). Moreover, the potential of such a pairing to further reduce the current $Z_{LP}^{p}$, is in proportion to the magnitude of its $\mu_{j}$ value. $\displaystyle\mu_{j}=c_{j}-\mu d_{j},~{}\text{where,}~{}\mu d_{j}=\sum_{i=1}^{F}\left(a_{ij}\cdot y_{i}\right)~{}=\sum_{f_{i}\in p_{j}}y_{i}~{}~{}(~{}\text{represents the dual cost component of}~{}\mu_{j})$ (10) As mentioned in Section 2.3, the standard CG practices generate a complete pricing network and solves it as a resource-constrained shortest-path optimization problem, to identify only the pairing(s) with negative reduced cost(s). However, generation of a complete pricing network for CPOPs with large-scale and complex flight networks is computationally-intractable. To overcome this challenge, a domain-knowledge driven CG heuristic (Aggarwal, Saxena ., 20201) is employed here to generate a set of promising pairings (of pre-defined size, criterion for which is discussed in Section 4.2). Notably, the merit of this CG heuristic lies in the fact that from within the larger pool of pairings with negative $\mu_{j}$, besides selecting pairings randomly, it also selects pairings in a guided manner. In that, the selection of such pairings is guided by optimal solution features at a set level and an individual pairing level, and re-utilization of the past computational efforts. These optimal solution features are related to the minimization of deadheads and maximization of the crew utilization, respectively. In essence, while the standard CG practices present equal opportunity for any pairing with a negative $\mu_{j}$ to qualify as an input for the next LPP iteration, this CG heuristic, besides ensuring that the pairings have negative $\mu_{j}$, prioritizes some pairings over the others via its two-pronged strategy– exploration of the new pairings’ space and re-utilization of pairings from the past LPP iterations. In that: * – the exploration of the new pairings’ space is guided by three CG strategies, which are elaborated below. * $\circ$ Deadhead Reduction strategy ($CGD$): this strategy prioritizes a set of legal pairings that is characterized by low deadheads, a feature which domain knowledge recommends for optimality at a set level. To exploit this optimality feature, $CGD$ generates a new paring set $\mathcal{P}_{CGD}$, which not only provides an alternative way to cover the flights involved in a subset of the current $\mathcal{P}_{LP}$, but also ensures that some of these flights get covered with zero deadheads. It promises propagation of the zero deadhead feature over successive LPP iterations, as: (a) $\mathcal{P}_{CGD}$ alongside the current $\mathcal{P}_{LP}$ forms a part of the input for the next LPP iteration; (b) $\mathcal{P}_{CGD}$ provides a scope for better coverage (zero deadhead) of some flights, compared to the current $\mathcal{P}_{LP}$; and (c) $\mathcal{P}_{CGD}$ may focus on zero deadhead coverage for different flights in different LPP iterations. * $\circ$ Crew Utilization enhancement strategy ($CGU$): this strategy prioritizes a set of legal pairings each member of which is characterized by high crew utilization, a feature which domain knowledge recommends for optimality at an individual pairing level. To exploit this optimality feature, $CGU$: (a) introduces a new measure, namely, crew utilization ratio, given by $\gamma_{j}$ (Equation 11), to quantify the degree of crew utilization in a pairing $p_{j}$ at any instant; (b) identifies pairings from the current $\mathcal{P}_{LP}$, which are characterized by high dual cost component ($\mu d_{j}$, Equation 10), reflecting in turn on those constitutive flights that have high value of dual variables $y_{i}$, and hence, on the potential of these flights to generate new pairings with more negative $\mu_{j}$; and (c) utilizes these flights to generate promising pairings from which only the ones with high $\gamma_{j}$ are picked to constitute the new pairing set $\mathcal{P}_{CGU}$. $\displaystyle\gamma_{j}=\frac{1}{\text{Number of duties in }p_{j}}\cdot\sum_{d\in p_{j}}\frac{\text{Working hours in duty}~{}d}{\text{Permissible hours of duty }d}$ (11) In doing so, $CGD$ promises propagation of the higher crew utilization ratio over successive LPP iterations, given that in each LPP iteration, $\mathcal{P}_{CGU}$ alongside the current $\mathcal{P}_{LP}$ forms a part of the input for the next LPP iteration. * $\circ$ Random exploration strategy ($CGR$): this strategy, unlike $CGU$ and $CGD$ which are guided by optimal solution features, pursues random and unbiased exploration of the new pairings’ space, independent of the current LPP solution. It involves generation of new pairings for a random selected set of legal duties from which only the pairings with negative reduced cost are selected to constitute the new pairing set $\mathcal{P}_{CGR}$. Here, a random set of legal duties is used instead of a random set of flights, as the former has a higher probability of generating legal pairings, given that a majority of pairing legality constraints get satisfied with the preprocessing of legal duties. * – the re-utilization of pairings from the past LPP iterations is guided by an Archiving strategy ($CGA$), that prioritizes a set of legal pairings comprising of those flight-pairs, which as per the existing LPP solution, bear better potential for improvement in the objective function. Such a pairing set, originating from the flight-pair level information, is extracted from an archive (denoted by $\mathcal{A}$) of the previously generated pairings. In doing so, $CGA$ facilitates re-utilization of the past computational efforts, by providing an opportunity for a previously generated pairing to be re- inducted in the current pairing pool. For this, $CGA$: * $\circ$ updates the archive $\mathcal{A}$ in each LPP iteration such that any pairing is stored/retrieved with reference to a unique index $(f_{m},f_{n})$ reserved for any legal flight-pair in that pairing. * $\circ$ introduces a new measure, namely, reduced cost estimator, given by $\eta_{mn}$ (Equation 12), for a flight-pair $(f_{m},f_{n})$ in $\mathcal{A}$. In each LPP iteration, this estimator is computed for all the flight-pairs present in $\mathcal{A}$, by fetching $f_{m}$, $f_{n}$, $y_{m}$ and $y_{n}$. $\displaystyle\eta_{mn}$ $\displaystyle=\texttt{flying\\_cost($f_{m}$)}+\texttt{flying\\_cost($f_{n}$)}-y_{m}-y_{n}=\sum_{i\in\\{m,n\\}}\left(\texttt{flying\\_cost($f_{i}$)}-y_{i}\right)$ (12) Notably, this formulation is analogous to Equation 10, just that instead of the complete cost of a pairing, only the flying costs corresponding to the flights in a legal flight-pair are accounted for. Given this, $\eta_{mn}$ may be seen as an indicator of $\mu_{j}$ at the flight-pair level. * $\circ$ recognizes that towards further improvement in the current LPP solution, it may be prudent to include as a part of the input for the next LPP iteration– the new pairing set $\mathcal{P}_{CGA}$, constituted by preferentially picking pairings from $\mathcal{A}$, that cover flight-pairs with lower $\eta_{mn}$ value. In doing so, $CGA$ pursues the goal of continual improvement in the objective function, while relying on the flight-pair level information embedded in the LPP solution of current LPP iteration, and re-utilizing the computational efforts spent till that LPP iteration. For further details and associated nitty-gritty of the above domain-knowledge driven CG heuristic, interested readers are referred to the authors’ previous work– Aggarwal, Saxena . (20201). Once this CG heuristic generates a set of promising pairings $\mathcal{P}_{CG}$ of pre-defined size, it is merged with the current $\mathcal{P}_{LP}$, and fed as the input to the next LPP iteration ($t\mathrel{+}=1$). These LPP iterations are repeated until the cost-improvements over a pre- specified number of successive LPP iterations falls below a pre-specified cost-threshold (settings given in Section 4.2). In this submodule, these LPP iterations are repeated, until its termination criterion is not met. In that, the cost-improvement over LPP iterations is observed, and if it falls below a pre-specified cost-threshold, say $Th_{cost}$, over a pre-specified number of successive LPP iterations, say $Th_{t}$, then it is terminated. The settings of these pre-specified limits– $Th_{cost}$ and $Th_{t}$, are highlighted in Section 4.2. After termination, the final LPP solution $\mathcal{P}_{LP}^{T}$ is then passed over to the IPP-solutioning submodule for its integerization. #### 3.3.2 IPP-solutioning This submodule receives as input, the LPP solution $\mathcal{P}^{T}_{LP}$, and aims to find therein a full-coverage integer solution, notated as $\mathcal{P}^{T}_{IP}$. Towards it, an IPP (Equations 1 to 3) is formulated using $\mathcal{P}^{T}_{LP}$ and $\mathcal{F}$, and solved using a branch-and- cut algorithm based off-the-shelf commercial MIP solver (Gurobi Optimization, 2019). At each node of the MIP-search tree, this solver maintains a valid lower bound (cost of the LPP solution) and a best upper bound (cost of the IPP solution), and it self-terminates if the gap between these two bounds becomes zero, or all branches in the MIP-search tree have been explored. Considering that the MIP-search for large-scale CPOPs is extremely time-consuming, a pre- defined time limit, notated as $Th_{ipt}$ (setting highlighted in Section 4.2), is used to terminate this MIP solver, if it does not terminate by itself a priori. Once the $\mathcal{P}^{T}_{IP}$ is obtained, it is passed back to the previous submodule for the next LPP-IPP interaction ($T\mathrel{+}=1$), only if the termination criterion of the Optimization Engine is not satisfied. Overarching Optimization Engine In the wake of the above, the procedure of the overarching Optimization Engine, formalized in Algorithm 4, is elaborated below. Input: $\mathcal{F},~{}\mathcal{P}_{IFS},~{}Th_{cost},~{}Th_{t},~{}Th_{ipt},~{}\texttt{Pairing\\_Gen()},~{}\texttt{CGD()},~{}\texttt{CGU()},~{}\texttt{CGR()},~{}\texttt{CGA()}$ Output: $\mathcal{P}^{\star}_{IP}$ 1 $T\leftarrow 1$ while _termination criterion of Optimization Engine is not met_ do 2 $\triangleright$ CG-driven LPP-solutioning: $t\leftarrow 1$ while _termiantion criterion of CG-driven LPP-solutioning is not met_ do 3 if _$t=1$ and $T=1$_ then 4 Formulate the primal of the LPP using $\mathcal{P}_{IFS}$ and $\mathcal{F}$ 5 else if _$t=1$ and $T>1$_ then 6 Formulate the primal of the LPP using $\mathcal{P}^{T-1}_{IP}$ and $\mathcal{F}$ 7 else 8 Formulate the primal of the LPP using $\mathcal{P}^{t-1}_{CG}\cup\mathcal{P}^{t-1}_{LP}$ and $\mathcal{F}$ 9 end if 10 $\mathcal{P}^{t}_{LP},~{}X^{t}_{LP}\leftarrow$ Solve the primal using the interior-point method based LP solver $\triangleright$ Termination of the CG- driven LPP-solutioning: if _cost-improvements $\leq Th_{cost}$ over last $Th_{t}$ number of successive LPP iterations_ then 11 $\mathcal{P}^{T}_{LP}\leftarrow\mathcal{P}^{t}_{LP}$ Break 12 end if 13 Formulate the dual of the LPP using $\mathcal{F}$ and $\mathcal{P}^{t}_{LP}$ $Y^{t}\leftarrow$ Solve the dual using the interior- point method based LP solver $\triangleright$ Solution to pricing sub-problem using the CG heuristic: $\mathcal{P}^{t}_{CGD}\leftarrow\texttt{CGD($\mathcal{P}^{t}_{LP},X^{t}_{LP},Y^{t},\ldots$)}$ $\mathcal{P}^{t}_{CGU}\leftarrow\texttt{CGU($\mathcal{P}^{t}_{LP},X^{t}_{LP},Y^{t},\ldots$)}$ $\mathcal{P}^{t}_{CGR}\leftarrow\texttt{CGR($Y^{t},\ldots$)}$ $\mathcal{P}^{t}_{CGA}\leftarrow\texttt{CGA($\mathcal{P}^{t}_{LP},X^{t}_{LP},Y^{t},\ldots$)}$ $\mathcal{P}^{t}_{CG}\leftarrow\mathcal{P}^{t}_{CGD}\cup\mathcal{P}^{t}_{CGU}\cup\mathcal{P}^{t}_{CGR}\cup\mathcal{P}^{t}_{CGA}$ $t\mathrel{+}=1$ 14 end while 15 $\triangleright$ IPP-solutioning: Formulate the IPP using $\mathcal{P}^{T}_{LP}$ and $\mathcal{F}$ $\mathcal{P}^{T}_{IP}\leftarrow$ Solve the IPP using a branch-and-cut algorithm based MIP solver until its run- time becomes $\geq Th_{ipt}$ $\triangleright$ Termination of the Optimization Engine: if _$Z^{T}_{IP}\left(\text{cost of }\mathcal{P}^{T}_{IP}\right)=Z^{T}_{LP}\left(\text{cost of }\mathcal{P}^{T}_{LP}\right)$_ then 16 $\mathcal{P}^{\star}_{IP}\leftarrow\mathcal{P}^{T}_{IP}$ Break 17 end if 18 $T\mathrel{+}=1$ 19 end while return $\mathcal{P}_{IP}^{\star}$ Algorithm 4 Procedure for the Optimization Engine Its input involves the given flight set $\mathcal{F}$; the generated IFS $\mathcal{P}_{IFS}$; the pre-defined termination parameters– $Th_{cost}$ & $Th_{t}$ (for CG-driven LPP-solutioning) and $Th_{ipt}$ (for IPP-solutioning); and the sub-routines for Legal Crew Pairing Generator (Pairing_Gen()) and the four CG strategies ($\texttt{CGD()},~{}\texttt{CGU()},~{}\texttt{CGR()}~{}$and CGA()) in the proposed CG heuristic. In each LPP-IPP interaction of the Optimization Engine, first, the CG-driven LPP-solutioning is executed (lines 3-25). It entails several LPP iterations (tracked by $t$), in each of which the first step is to formulate the primal using $\mathcal{F}$ and the respective input pairing set. This input pairing set is: * • $\mathcal{P}_{IFS}$, if the first LPP iteration ($t=1$) of the first LPP-IPP interaction ($T=1$) is being executed (lines 5-6). * • $\mathcal{P}^{T-1}_{IP}$, if the first LPP iteration ($t=1$) of any subsequent LPP-IPP interaction ($T>1$) is being executed (lines 7-8). * • $\mathcal{P}^{t-1}_{CG}\cup\mathcal{P}^{t-1}_{LP}$, if any subsequent LPP iteration ($t>1$) of any LPP-IPP interaction ($T\geq 1$) is being executed (lines 9-11). Once the primal is formulated, it is solved using the corresponding LP solver to obtain the current optimal LPP solution, constituted by $\mathcal{P}^{t}_{LP}~{}$and$~{}X^{t}_{LP}$ (line 12). Subsequently, the termination criterion of CG-driven LPP-solutioning is checked (lines 13-16). If it is terminated, then the current LPP solution $\mathcal{P}^{t}_{LP}$ is fetched as the final LPP solution $\mathcal{P}^{T}_{LP}$ of this LPP-IPP interaction. If not, then a dual is formulated using $\mathcal{P}^{t}_{LP}$ and $\mathcal{F}$ (line 17), which is then solved using the corresponding LP solver to obtain the current optimal dual vector $Y^{t}$ (line 18). Using the current $\mathcal{P}^{t}_{LP}$, $X^{t}_{LP}$ and $Y^{t}$, a fresh set of pairings $\mathcal{P}^{t}_{CG}$ is obtained using the CG heuristic, which is constituted by the new pairing sets from the four underlying CG strategies (lines 19-23). At the end of the LPP iteration $t$, the fresh set of pairings $\mathcal{P}^{t}_{CG}$ is combined with the current $\mathcal{P}^{t}_{LP}$ to serve as input pairing set for the subsequent LPP iteration ($t\mathrel{+}=1$). Once this submodule is terminated, the resulting $\mathcal{P}^{T}_{LP}$ is passed over to the IPP-solutioning for its integerization, wherein, the MIP solver is used to obtain the IPP solution $\mathcal{P}^{T}_{IP}$ (lines 26 and 27). In that, the pre-defined $Th_{ipt}$ time-limit is used to terminate the MIP-search, if it does not self-terminate a priori. Subsequently, the resulting $\mathcal{P}^{T}_{IP}$ is passed back to the CG-driven LPP-solutioning for the next LPP-IPP interaction ($T\mathrel{+}=1$), or returned as the final integer solution $\mathcal{P}^{\star}_{IP}$, depending upon the termination condition of the Optimization Engine (lines 28-32). In that, if the cost of $\mathcal{P}^{T}_{IP}~{}\left(Z_{IP}^{T}\right)$, matches the cost of $\mathcal{P}^{T}_{LP}~{}\left(Z_{LP}^{p,T}\right)$, then the Optimization Engine is terminated. ## 4 Computational Experiments This section first presents the test cases and the computational setup, used to investigate the utility of $AirCROP$, its modules, and their interactions. Subsequently, the settings of parameters involved in different modules of $AirCROP$ are presented. Lastly, the experimental results are discussed. ### 4.1 Test Cases and Computational Setup The real-world airline test cases, used for experimentation, are detailed in Table 2. Each of these test cases involves a weekly flight schedule, and have been provided by the research consortium’s industrial sponsor (from the networks of US-based airlines). Table 2: Real-world airline test cases used in this research work Test Cases | $\\#$Flights | $\\#$Crew Bases | $\\#$Airports | $\\#$Legal Duties ---|---|---|---|--- TC-1 | 3202 | 15 | 88 | 454205 TC-2 | 3228 | 15 | 88 | 464092 TC-3 | 3229 | 15 | 88 | 506272 TC-4 | 3265 | 15 | 90 | 446937 TC-5 | 4212 | 15 | 88 | 737184 (a) (b) Figure 3: (a) Geographical representation of TC-5 flight network, where the red nodes, green edges and yellow nodes represent the airports, scheduled flights and crew bases, respectively, and (b) legal flight-connections, each represented by a point in the plot, where for a flight marked on the y-axis, the connecting flight is marked on the x-axis. The columns in Table 2, in order of their occurrence, highlight the notations for the different test cases; the number of its constituent flights; the number of constituent crew bases; and the total number of legal duties involved, respectively. It is critical to recognize that the challenge associated with solutioning of these test cases, depends not just on the number of flights involved but also to the fact that these flights are part of complex flight networks, characterized by a multiplicity of hubs as opposed to a single hub, and multiplicity of crew bases as opposed to a single crew base. In that, the number of legal pairings possible, grow exponentially with the number of hubs and crew bases. As a sample instance, the geographical representation of the flight network associated with TC-5, and the legal flight connections involved in it, are portrayed in Figure 3. Notably, in Figure 3(a), the presence of multiple hub-and-spoke subnetworks and multiple crew bases (highlighted in yellow color) is evident. Furthermore, the pattern visible in Figure 3(b) could be attributed to the (minimum and maximum) limits on the sit-time and overnight-rest constraints. For instance, a flight, say $f_{500}$, has legal connections only with those flights that depart from the arrival airport of $f_{500}$, and whose departure-time gap (difference between its departure-time and the arrival time of $f_{500}$) lies within the minimum and maximum allowable limits, of the sit-time or the overnight-rest. All the experiments in this research have been performed on an HP Z640 Workstation, which is powered by two Intel® Xeon® E5-2630v3 processors, each with 16 cores at 2.40 GHz, and 96 GBs of RAM. All codes related to the $AirCROP$ have been developed using the Python scripting language in alignment with the Industrial sponsor’s larger vision and preference. Furthermore: * • the interior-point method from Gurobi Optimizer 8.1.1 (Gurobi Optimization, 2019) is used to solve the primal in the CG- driven LPP-solutioning submodule. * • the interior-point method (Andersen Andersen, 2000) from SciPy’s linprog library (Virtanen ., 2020) is used to solve the dual in the CG-driven LPP- solutioning submodule. * • the branch-and-cut algorithm based MIP solver from Gurobi Optimizer 8.1.1 is used to solve the IPP in the Initial Feasible Solution Generator and the IPP- solutioning submodule. * • an $AirCROP$-run, in principle, terminates when the cost of the IPP solution matches the cost of its input LPP solution in a particular LPP-IPP interaction. However, for practical considerations on the time-limit, an $AirCROP$-run is allowed to terminate if the IPP and LPP costs do not conform with each other even after 30 LPP-IPP interactions are over, or 30 hours of total run-time is elapsed. ### 4.2 Parameter Settings The settings of the parameters associated with different modules and submodules of the $AirCROP$ are, as highlighted below. * • Initial Feasible Solution Generator: here, the proposed IPDCH involves the decomposition parameter $K$, which regulates the size of flight subsets formed in each of IPDCH-iteration. As mentioned before, the setting of $K$ is dependent on the characteristics of input flight dataset and the configuration of available computational resources. Here, the aim is to cover all given flights in a time-efficient manner. Hence, it is important to understand the effect of setting of $K$ on the time-performance of IPDCH, which is highlighted below. * – For a relatively lower value of $K$, smaller flight subsets with lesser number of legal flight-connections would be formed in each IPDCH-iteration, leading to coverage of relatively lesser number of unique flights in each of them. Though, this by itself is not a challenge, but this would necessitate a significant number of additional IPDCH-iterations (and the respective run- time), since the number of unique flights covered per IPDCH-iteration, which by construct reduces with the iterations, would get further reduced with relatively smaller flight subsets. * – On the flip side, for a relatively higher value of $K$, bigger flight subsets would be formed that would lead to coverage of higher number of unique flights per IPDCH-iteration. Though, this may reduce the total number of IPDCH- iterations required to generate the desired IFS, the overall run-time of the IPDCH may increase drastically. The rationale being that with bigger flight subset in each IPDCH-iteration, the number of possible legal pairings would increase drastically, leading to huge run-time for their generation as well as for the subsequent MIP-search. The above considerations suggest that $K$ should be reasonably-sized. Considering the given computational resources and the results of initial exploration around the possible number of pairings for differently-sized flight sets, the value of $K$ in each IPDCH-iteration is guided by a random integer between one-eighth and one-fourth of the size of the input flight set $\mathcal{F}$. It may be noted that this setting of $K$ has been selected considering the scale and complexity of the given test cases, and it needs to be re-visited if the scale and complexity of the flight network changes drastically. * • CG-driven LPP-solutioning: The parameters involved in the termination criterion for this submodule– $Th_{cost}$ & $Th_{t}$, are set as 100 USD & 10 iterations respectively, to achieve an LPP solution with a sufficiently good cost in a reasonably good time. Moreover, the sensitivity of these parameters towards the $AirCROP$’s performance is discussed in Section 4.3.4. Moreover, the effect of the parameter– size of $\mathcal{P}^{t}_{CG}$, on the performance of this submodule (the final LPP solution’s cost and required run- time), and the demand on the computational resources (dominantly, RAM) is highlighted below. * – for a relatively small-sized $\mathcal{P}^{t}_{CG}$, the alternative pairings available to foster further cost improvement shall be quite limited, amounting to smaller cost benefits in each phase of the CG-driven LPP-solutioning. This would necessitate far more LPP-IPP interactions, to reach the near-optimal cost. This pre se is not a challenge, however, significant amount of additional run-time may be required, since: (a) each call for CG-driven LPP- solutioning demands a minimum of 10 LPP iterations, before it could be terminated, (b) such calls when invoked repeatedly, may consume significant run-time, yet, without reasonable cost benefit. * – On the other hand, for a very large-sized $\mathcal{P}_{CG}^{t}$, though the potential for significant cost benefits may exist, the demand on the RAM may become overwhelming for any CG-driven LPP-solutioning phase to proceed. The above considerations suggest that the size of $\mathcal{P}_{CG}^{t}$ may neither be too small nor too large. Factoring these, the experiments here aim at $\mathcal{P}_{CG}^{t}$ sized approximately of a million pairings (significant size, yet, not overwhelming for 96 GB RAM). Furthermore, for a search that is not biased in favor of any particular CG strategy, the number of pairings from each CG strategy towards the overall CG heuristic are kept equable. * • IPP-solutioning: As mentioned before, the MIP-search on a large-scale IPP is time-intensive. Hence, the termination parameter– $Th_{ipt}$, that restricts the run-time of any IPP-solutioning phase if not self-terminated a priori, is reasonably set as 20 minutes, and its sensitivity on the $AirCROP$’s performance is discussed in Section 4.3.4. ### 4.3 Results & Observations This section presents the experimental results and associated inferences, in the order highlighted below. 1. 1. The performance of the proposed $AirCROP$ on the given test cases with the aforementioned parameter settings is discussed. 2. 2. The phenomenon referred to as performance variability (Lodi Tramontani, 2013) is discussed in the context of $AirCROP$. This aspect is pertinent since some variability in performance (even for the same random seed) is inevitable owing to $AirCROP$’s reliance on the mathematical programming solvers, which over the different runs may pick different permutations of the rows (flight- coverage) or columns (pairings). 3. 3. The impact of the initialization methods: (a) the proposed IPDCH, (b) an Enhanced-DFS heuristic, earlier proposed by the authors (Aggarwal ., 2018), and (c) a commonly adopted Artificial Pairings method (Hoffman Padberg, 1993; Vance ., 1997), on the final performance of $AirCROP$ is investigated. 4. 4. The sensitivity of $AirCROP$’s performance to the termination parameters in the Optimization Engine’s submodules (CG-driven LPP-solutioning and IPP- solutioning) has been discussed. #### 4.3.1 AirCROP’s Performance The results of the $AirCROP$-runs on the given test cases (TC-1 to TC-5) with the aforementioned parameter settings are reported in Table 3. In that, for each test case: * • the first row marked by “$\mathcal{P}_{IFS}$” highlights the cost associated with the IFS that initializes the $AirCROP$-run and the run-time consumed in its generation. * • the subsequent rows present the results of the LPP-IPP interactions (marked by the counter $T$). In that, for a particular $T$, the cost of the LP-solution passed on for its integerization and the associated time are highlighted. Also the cost of the IP-solution returned and the associated time are highlighted. Here, the unit of cost is USD, and the time corresponds to the HH:MM format. * • the final crew pairing solution ($\mathcal{P}^{\star}_{IP}$) is highlighted in the last row (emboldened) marked by “Final Solution”. It may be noted that the experimental results in the subsequent sections are presented in the same format, unless any digression is specifically highlighted. Table 3: $AirCROP$’s performance∗ on the given test cases LPP-IPP | TC-1 | TC-2 | TC-3 | TC-4 | TC-5 ---|---|---|---|---|--- Interactions $T$ | $\mathcal{P}_{LP}^{T}/\mathcal{P}_{IP}^{T}$ | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time $\mathcal{P}_{IFS}$ | 85893202 | 00:05 | 81950079 | 00:05 | 51552744 | 00:03 | 131716653 | 00:08 | 89690776 | 00:06 1 | $\mathcal{P}_{LP}^{1}$ | 3468349 | 03:56 | 3493986 | 03:56 | 3483057 | 05:18 | 3595565 | 03:27 | 4583484 | 07:48 $\mathcal{P}_{IP}^{1}$ | 3689420 | 00:20 | 3715798 | 00:20 | 3697204 | 00:20 | 3807233 | 00:20 | 4930789 | 00:20 2 | $\mathcal{P}_{LP}^{2}$ | 3467837 | 02:18 | 3494675 | 01:19 | 3484645 | 02:42 | 3600195 | 01:17 | 4588740 | 02:49 $\mathcal{P}_{IP}^{2}$ | 3557615 | 00:20 | 3587139 | 00:20 | 3590336 | 00:20 | 3679138 | 00:20 | 4734553 | 00:20 3 | $\mathcal{P}_{LP}^{3}$ | 3469591 | 00:47 | 3495254 | 01:22 | 3486614 | 01:59 | 3600813 | 01:16 | 4592143 | 01:46 $\mathcal{P}_{IP}^{3}$ | 3518161 | 00:02 | 3546777 | 00:02 | 3523538 | 00:02 | 3639313 | 00:01 | 4654258 | 00:20 4 | $\mathcal{P}_{LP}^{4}$ | 3471619 | 01:13 | 3496797 | 00:57 | 3491000 | 01:13 | 3601168 | 01:27 | 4593422 | 02:17 $\mathcal{P}_{IP}^{4}$ | 3489534 | 00:01 | 3505941 | 00:01 | 3496142 | 00:01 | 3621723 | 00:01 | 4634187 | 00:01 5 | $\mathcal{P}_{LP}^{5}$ | 3472403 | 00:31 | 3497106 | 00:23 | 3490420 | 00:56 | 3604082 | 00:37 | 4594282 | 02:14 $\mathcal{P}_{IP}^{5}$ | 3484783 | 00:01 | 3497106 | 00:01 | 3490420 | 00:01 | 3612845 | 00:01 | 4617838 | 00:01 6 | $\mathcal{P}_{LP}^{6}$ | 3473238 | 00:30 | | | | | 3604753 | 00:28 | 4595481 | 01:53 $\mathcal{P}_{IP}^{6}$ | 3473238 | 00:01 | | | | | 3604753 | 00:01 | 4615272 | 00:01 7 | $\mathcal{P}_{LP}^{7}$ | | | | | | | | | 4596466 | 01:12 $\mathcal{P}_{IP}^{7}$ | | | | | | | | | 4600428 | 00:01 8 | $\mathcal{P}_{LP}^{8}$ | | | | | | | | | 4595613 | 01:42 $\mathcal{P}_{IP}^{8}$ | | | | | | | | | 4595613 | 00:01 Final Solution | 3473238 | 10:05 | 3497106 | 08:46 | 3490420 | 12:55 | 3604753 | 09:24 | 4595613 | 22:52 ∗All values in the “Cost” columns are in USD, and all corresponding real values are rounded-off to the next integer values. All values in the “Time” columns are in HH:MM format, and all corresponding seconds’ values are rounded-off to the next minute values. The above results have been tested by the research consortium’s industrial sponsor, and verified to be highly-competitive compared to the best practice solutions known, for different test cases. In general, the obtained solutions have been found to be superior by about 1.5 to 3.0% in terms of the hard cost, which reportedly is one of the most important solution quality indicator. For reference, a comparison of the obtained solution vis-$\grave{a}$-vis the best known solution has been drawn for TC-5, in Table 4, where a significant difference in terms of the size of pairings can be observed. Notably, the key features contributing to lower hard cost relate to presence of pairings with relatively lower - TAFB, overnight rests and meal cost. However, the obtained solution also entails more crew changes, some of which (involving aircraft change) negatively impact the soft cost. Hence, there appears to be a trade- off between the hard cost and the soft cost. Table 4: Salient features of $\mathcal{P}^{\star}_{IP}$ for TC-5: $AirCROP$’s solution vis-$\grave{a}$-vis the best practice solution Features | $\bm{AirCROP}$’s solution | Best practice solution ---|---|--- $\\#$ pairings | 926 | 783 $\\#$ unique flights covered | 4,212 | 4,212 $\\#$ deadhead flights | 3 | 3 $\\#$ overnight-rests | 1,203 | 1,279 $\\#$ crew changes | 1,002 | 825 $\\#$ average crew changes per pairing | 1.082 | 1.054 Total TAFB (HH:MM) | 37444:54 | 38189:39 $\\#$ pairings covering 2 flights | 303 | 205 $\\#$ pairings covering 3 flights | 17 | 31 $\\#$ pairings covering 4 flights | 170 | 95 $\\#$ pairings covering 5 flights | 63 | 37 $\\#$ pairings covering 6 flights | 202 | 153 $\\#$ pairings covering 7 flights | 59 | 62 $\\#$ pairings covering 8 flights | 83 | 90 $\\#$ pairings covering 9 flights | 19 | 49 $\\#$ pairings covering 10 flights | 8 | 45 $\\#$ pairings covering 11 flights | 1 | 10 $\\#$ pairings covering 12 flights | 1 | 5 $\\#$ pairings covering 13 flights | 0 | 0 $\\#$ pairings covering 14 flights | 0 | 1 Hotel cost (USD) | 166,240 | 176,170 Meal cost (USD) | 157,269 | 160,397 Hard cost (USD) | 340,671 | 350,818 Soft cost (USD) | 51,600 | 42,750 Actual flying cost (USD) | 4,203,342 | 4,203,342 Total cost (USD) | 4,595,613 | 4,596,910 #### 4.3.2 Performance Variability in AirCROP These section investigates the sensitivity of $AirCROP$ with respect to the sources of variability over multiple runs, even for the same problem. This study assumes importance, considering that performance variability is rather inevitable when the mathematical programming based solution approaches are employed (Koch ., 2011). As cited by Lodi Tramontani (2013), variability in the performance of LP & MIP solvers may be observed on – changing the computing platform (which may change the floating-point arithmetic), permuting the constraints/variables of the respective mathematical models, or changing the pseudo-random numbers’ seed. These changes/permutations may lead to an entirely different outcome of the respective search algorithms (LP & MIP), as highlighted below. * • The root source for the performance variability in MIP is the imperfect tie- breaking. A majority of the decisions to be taken during an MIP-search are dependent on– the ordering of the candidates according to an interim score as well as the selection of the best candidate (one with the best score value). A perfect score that could fully-distinguish between the candidates is not-known mostly due to the lack of theoretical knowledge, and even if it is known, it may be too expensive to compute666For instance, in a strong branching scheme, the best variable to branch at each node is decided after simulating one-level of branching for each fractional variable, however, it is performed heuristically to make it a computationally-affordable task for MIP solvers (Linderoth Lodi, 2011). Furthermore, additional ties or tiebreaks could be induced by changing the floating-point operations, which inherently may change when the computing platform is changed. Amidst such an imperfect tie-breaking, the permutation of the variables/constraints changes the path within the MIP- search tree, leading to a completely different evolution of the algorithm with rather severe consequences. * • Depending upon the floating-point arithmetic or the sequence of variables loaded in an LPP, the performance of the simplex and interior-point methods may vary. * • The performance of the LP and MIP solvers is also affected by the choice of pseudo-random numbers’ seed, wherever the decisions are made heuristically. For instance, an interior-point method in the LP solvers performs a (random) crossover to one of the vertices of the optimal face when the search reaches its (unique) center. Table 5: Performance variability assessment for $AirCROP$ on two test instances∗ (TC-2 and TC-5) Test | LPP-IPP | Runs with performance variability | Runs without performance variability ---|---|---|--- Case | Interactions | Run-1 | Run-2 | Run (Seed-$\alpha$) | Run (Seed-$\beta$) | Run (Seed-$\gamma$) $T$ | $\mathcal{P}_{LP}^{T}/\mathcal{P}_{IP}^{T}$ | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time TC-2 | $\mathcal{P}_{IFS}$ | 81950079 | 00:05 | 74533686 | 00:04 | 129221508 | 00:08 | 114054265 | 00:07 | 52515476 | 00:04 1 | $\mathcal{P}_{LP}^{1}$ | 3493986 | 03:56 | 3494580 | 03:57 | 3495054 | 03:51 | 3493757 | 03:52 | 3493909 | 03:52 $\mathcal{P}_{IP}^{1}$ | 3715798 | 00:20 | 3746847 | 00:20 | 3769811 | 00:20 | 3711267 | 00:20 | 3722248 | 00:20 2 | $\mathcal{P}_{LP}^{2}$ | 3494675 | 01:19 | 3494540 | 02:18 | 3495311 | 01:57 | 3496733 | 01:57 | 3494657 | 03:02 $\mathcal{P}_{IP}^{2}$ | 3587139 | 00:20 | 3621066 | 00:20 | 3628514 | 00:20 | 3581978 | 00:20 | 3620745 | 00:20 3 | $\mathcal{P}_{LP}^{3}$ | 3495254 | 01:22 | 3496475 | 01:41 | 3497558 | 00:52 | 3499651 | 00:54 | 3496398 | 01:23 $\mathcal{P}_{IP}^{3}$ | 3546777 | 00:02 | 3555152 | 00:06 | 3566092 | 00:11 | 3536050 | 00:01 | 3551149 | 00:03 4 | $\mathcal{P}_{LP}^{4}$ | 3496797 | 00:57 | 3497750 | 01:36 | 3499237 | 01:26 | 3500818 | 01:03 | 3496069 | 01:37 $\mathcal{P}_{IP}^{4}$ | 3505941 | 00:01 | 3525600 | 00:01 | 3516807 | 00:01 | 3520552 | 00:01 | 3543236 | 00:02 5 | $\mathcal{P}_{LP}^{5}$ | 3497106 | 00:23 | 3498588 | 01:40 | 3500169 | 00:42 | 3500504 | 01:02 | 3496706 | 01:01 $\mathcal{P}_{IP}^{5}$ | 3497106 | 00:01 | 3498588 | 00:01 | 3517585 | 00:01 | 3500504 | 00:01 | 3501210 | 00:01 6 | $\mathcal{P}_{LP}^{6}$ | | | | | 3501523 | 00:43 | | | 3499063 | 00:41 $\mathcal{P}_{IP}^{6}$ | | | | | 3504085 | 00:01 | | | 3499063 | 00:01 7 | $\mathcal{P}_{LP}^{7}$ | | | | | 3502118 | 00:31 | | | | $\mathcal{P}_{IP}^{7}$ | | | | | 3502118 | 00:01 | | | | Final Solution | 3497106 | 08:46 | 3498588 | 12:05 | 3502118 | 11:05 | 3500504 | 09:38 | 3499063 | 12:27 TC-5 | $\mathcal{P}_{IFS}$ | 89690776 | 00:06 | 92080420 | 00:05 | 131443284 | 00:09 | 847887053 | 00:56 | 470430395 | 00:29 1 | $\mathcal{P}_{LP}^{1}$ | 4583484 | 07:48 | 4583476 | 08:00 | 4584525 | 07:28 | 4581988 | 08:47 | 4580130 | 07:36 $\mathcal{P}_{IP}^{1}$ | 4930789 | 00:20 | 4973580 | 00:20 | 4974341 | 00:20 | 4925863 | 00:20 | 4949616 | 00:20 2 | $\mathcal{P}_{LP}^{2}$ | 4588740 | 02:49 | 4588938 | 05:59 | 4589091 | 02:25 | 4584956 | 04:51 | 4584273 | 03:22 $\mathcal{P}_{IP}^{2}$ | 4734553 | 00:20 | 4765453 | 00:20 | 4782657 | 00:20 | 4749664 | 00:20 | 4753133 | 00:20 3 | $\mathcal{P}_{LP}^{3}$ | 4592143 | 01:46 | 4591571 | 02:35 | 4589952 | 02:14 | 4587812 | 03:02 | 4585046 | 03:40 $\mathcal{P}_{IP}^{3}$ | 4654258 | 00:20 | 4661078 | 00:20 | 4736313 | 00:20 | 4653279 | 00:20 | 4666390 | 00:20 4 | $\mathcal{P}_{LP}^{4}$ | 4593422 | 02:17 | 4595741 | 01:49 | 4591145 | 02:36 | 4589247 | 02:00 | 4588952 | 02:56 $\mathcal{P}_{IP}^{4}$ | 4634187 | 00:01 | 4624039 | 00:01 | 4654627 | 00:20 | 4614651 | 00:01 | 4628239 | 00:01 5 | $\mathcal{P}_{LP}^{5}$ | 4594282 | 02:14 | 4599006 | 01:14 | 4592463 | 02:03 | 4590573 | 01:05 | 4589577 | 02:02 $\mathcal{P}_{IP}^{5}$ | 4617838 | 00:01 | 4613385 | 00:01 | 4632708 | 00:02 | 4603938 | 00:01 | 4618710 | 00:01 6 | $\mathcal{P}_{LP}^{6}$ | 4595481 | 01:53 | 4598727 | 01:11 | 4593094 | 02:00 | 4591176 | 01:15 | 4589874 | 01:48 $\mathcal{P}_{IP}^{6}$ | 4615272 | 00:01 | 4605126 | 00:01 | 4625993 | 00:01 | 4591176 | 00:01 | 4607590 | 00:01 7 | $\mathcal{P}_{LP}^{7}$ | 4596466 | 01:12 | 4598412 | 01:39 | 4593431 | 01:04 | | | 4590674 | 01:24 $\mathcal{P}_{IP}^{7}$ | 4600428 | 00:01 | 4598412 | 00:01 | 4619643 | 00:01 | | | 4605058 | 00:01 8 | $\mathcal{P}_{LP}^{8}$ | 4595613 | 01:42 | | | 4594146 | 01:03 | | | 4591065 | 02:10 $\mathcal{P}_{IP}^{8}$ | 4595613 | 00:01 | | | 4594146 | 00:01 | | | 4591065 | 00:01 Final Solution | 4595613 | 22:52 | 4598412 | 23:37 | 4594146 | 22:27 | 4591176 | 22:59 | 4591065 | 26:32 ∗All values in the “Cost” columns are in USD, and all the corresponding real values are rounded-off to the next integer values. All values in the “Time” columns are in HH:MM, and all the corresponding seconds’ values are rounded- off to the next minute values. In the above background, the plausible reasons for variability in $AirCROP$’s performance are elaborated below. * • Generation of new legal pairings using a parallel architecture: in any LPP iteration $t$, new legal pairings are generated in parallel, by allocating the sub-processes to the idle-cores of the CPU. These sub-processes return their respective pairing sets as soon as they are terminated. This by itself is not a challenge, however, when the $AirCROP$ is re-run, the order in which these sub-processes terminate may not be same as before (as it depends on the state of the CPU), permuting the pairings in the cumulative pairing set $\mathcal{P}_{CG}^{t}$. This permuted pairing set, when fed as part of the input to the LP solver in the next LPP iteration, may lead to a different LPP solution, leading to a different outcome of the subsequent $AirCROP$’s search. To curb this, the pairings in the set that trigger the LP solver are sorted in lexicographical order of their representative strings. These strings are constructed from the indices of the flights covered in the corresponding pairings. For instance, the string corresponding to a pairing that covers flights $f_{1}$, $f_{10}$, $f_{100}$ & $f_{200}$ is $1\\_10\\_100\\_200$. Given that the pairings are distinct, the resulting strings are distinct too, allowing for a crisp sorting criterion and ensuring a fixed pairing sequence in each $AirCROP$-run. * • Numerical seed for generation of pseudo-random numbers: variability may also be introduced if the numerical seed employed to generate pseudo-random numbers for use in the proposed modules or the utilized LP & MIP solvers, varies. For instance, use of the default seed method of Python (i.e., the current time of the computing system) across different $AirCROP$ runs may lead to different pseudo-random numbers, each time. This in turn would trigger variability in the IFS generated by IPDCH (since the random selection of flights in each of its iterations, is impacted), and the pairing set resulting from the CG heuristic (since each of the underlying CG strategy is impacted). Such variability could be negated by use of a fixed numerical seed, instead of a time dependent one. The intriguing questions for researchers could relate to the impact that presence or absence of causes of variability may have on the quality of $AirCROP$’s solutions, in terms of both cost and run-time. Table 5 attempts to shed light on these questions through empirical evidence for two test cases involving 3228 flights (TC-2) and 4212 flights (TC-5), respectively. In each of these test cases, the effect of variability is revealed through: * • two independent runs (Run-1 and Run-2), in each of which the causes of variability exist, that is: (a) the permutations of pairings generated using the parallel architecture is possible, and (b) the default seed method of Python, based on the time of the computing system applies. * • three independent runs, in each of which the causes of variability have been eliminated, that is: (a) the lexicographical order of the pairings is imposed, and (b) a fixed numerical seed has been fed for random number generation. For these runs, the numerical seeds are given by $\alpha=0$, $\beta=1$, and $\gamma=2$, respectively. The key observations and inferences that could be drawn from each test case in Table 5 are highlighted below. * • understandably, the Run-1 and Run-2 (corresponding to the same numerical seed), yield different cost solutions over different run-time. Importantly, the variation in cost (despite the presence of causes of variability) is not alarming, though significantly different run-times may be required. * • each run (corresponding to Seed-$\alpha$, Seed-$\beta$, and Seed-$\gamma$, respectively) where the causes of variability have been negated, if repeated, yield the same cost solution in the same run-time though it has not been shown in the table for paucity of space. * • the runs corresponding to the numerical seeds given by $\alpha$, $\beta$, and $\gamma$, respectively, differ solely due to the difference in the corresponding random numbers generated, and subsequently utilized. It can be observed that the change in numerical seed does not significantly affect the cost-quality of the final $AirCROP$ solution though the associated run-time may vary significantly. The fact that $AirCROP$ can offer final solutions with comparable cost quality, regardless of the presence or absence of causes of variability, endorses the robustness of the constitutive modules of the $AirCROP$. Also, the variation in run-time could be attributed to different search trajectories corresponding to different permutations of variables or different random numbers. It may be noted that for the subsequent runs the lexicographical order of the pairings and a fixed numerical seed (Seed-$\alpha=0$) have been utilized. #### 4.3.3 Impact of Initialization on AirCROP’s Performance This section investigates the sensitivity of $AirCROP$ with respect to the cost quality of the initial solution and the run-time spent to obtain it. Towards it, the initial solution is obtained using three different methods (offering three input alternatives with varying cost and run-time) and the cost quality of $AirCROP^{\prime}s$ final solution alongside the necessary run-time is noted. Notably, in an initial attempt to generate IFS for large-scale CPOPs, the authors proposed a DFS algorithm based heuristic, namely, Enhanced-DFS heuristic (Aggarwal ., 2018). Its performance across the five test cases has been highlighted in Table 6. In that, TC-1 emerges as an outlier owing to alarmingly high run-time, when compared to all other test cases. Table 6: Performance of Enhanced-DFS heuristic (Aggarwal ., 2018) for IFS generation. Here, the real valued “Cost” is rounded-off to the next integer value, and the seconds’ in the “Time” column are rounded-off to the next minute values. Test Cases | Time (HH:MM) | Cost (USD) | # Pairings ---|---|---|--- TC-1 | 01:48 | 3863669070 | 477617 TC-2 | 00:02 | 167405376 | 26678 TC-3 | 00:03 | 167967482 | 26871 TC-4 | 00:13 | 1072078483 | 135269 TC-5 | 00:04 | 325922318 | 51920 A plausible explanation behind this aberration is that TC-1 involves some flights with very few legal flight connections, and a DFS based algorithm may have to exhaustively explore several flight connections, to be able to generate an IFS with full flight coverage. The need to do away with reliance on DFS so as to have equable run-time across different data sets explains the motivation for: * • proposition of IPDCH in this paper, which as highlighted in Section 3.2, relies on: (a) a divide-and-cover strategy to decompose the input flight schedule into sufficiently-small flight subsets, and (b) IP to find a lowest- cost pairing set that covers the maximum-possible flights for each of the decomposed flight subsets. * • consideration of a commonly adopted Artificial Pairings method (Vance ., 1997), that constructs a pairing set which covers all the flights, though some/all the pairings may not be legal. Hence, for this method the initial solution would be referred as $\mathcal{P}_{IS}$ instead of $\mathcal{P}_{IFS}$. Table 7: Performance assessment of $AirCROP$ on TC-1 and TC-5 when initialized using the proposed IPDCH, the Artificial Pairings method, and the Enhanced-DFS heuristic. LPP-IPP | TC-1 | TC-5 ---|---|--- Interactions | Enhanced-DFS | IPDCH | Artificial Pairings | Enhanced-DFS | IPDCH | Artificial Pairings $T$ | $\mathcal{P}_{LP}^{T}/\mathcal{P}_{IP}^{T}$ | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time $\mathcal{P}_{IFS}/\mathcal{P}_{IS}$ | 3863669070 | 01:48 | 74945982 | 00:05 | 14604919138 | $\approx$00:00 | 325922318 | 00:04 | 131443284 | 00:09 | 25409939785 | $\approx$00:00 1 | $\mathcal{P}_{LP}^{1}$ | 3463560 | 04:14 | 3465379 | 04:19 | 17589714 | 03:10 | 4583664 | 06:57 | 4584525 | 07:28 | 4585380 | 07:29 $\mathcal{P}_{IP}^{1}$ | 3650828 | 00:20 | 3689312 | 00:20 | 17833718 | 00:20 | 4943531 | 00:20 | 4974341 | 00:20 | 4960813 | 00:20 2 | $\mathcal{P}_{LP}^{2}$ | 3464678 | 01:51 | 3466567 | 01:32 | 17589851 | 01:29 | 4586675 | 03:41 | 4589091 | 02:25 | 4589470 | 03:34 $\mathcal{P}_{IP}^{2}$ | 17731125 | 00:20 | 3566415 | 00:20 | 3578030 | 00:20 | 4773342 | 00:20 | 4809348 | 00:20 | 4782657 | 00:20 3 | $\mathcal{P}_{LP}^{3}$ | 3466217 | 01:38 | 3467848 | 01:33 | 3466868 | 02:02 | 4586581 | 04:54 | 4589952 | 02:14 | 4593117 | 02:05 $\mathcal{P}_{IP}^{3}$ | 3531694 | 00:13 | 3556499 | 00:20 | 3553432 | 00:20 | 4701607 | 00:20 | 4736313 | 00:20 | 4672696 | 00:20 4 | $\mathcal{P}_{LP}^{4}$ | 3467672 | 01:19 | 3468777 | 01:26 | 3467935 | 00:47 | 4589568 | 01:51 | 4591145 | 02:36 | 4593938 | 02:18 $\mathcal{P}_{IP}^{4}$ | 3507987 | 00:01 | 3517901 | 00:01 | 3516376 | 00:02 | 4651824 | 00:20 | 4654627 | 00:20 | 4650449 | 00:06 5 | $\mathcal{P}_{LP}^{5}$ | 3469533 | 00:44 | 3468894 | 00:49 | 3468332 | 00:40 | 4591698 | 02:03 | 4592463 | 02:03 | 4596256 | 02:24 $\mathcal{P}_{IP}^{5}$ | 3483690 | 00:01 | 3499531 | 00:01 | 3496156 | 00:01 | 4616605 | 00:01 | 4632708 | 00:02 | 4620903 | 00:01 6 | $\mathcal{P}_{LP}^{6}$ | 3469276 | 00:52 | 3469352 | 00:48 | 3469095 | 00:47 | 4591969 | 01:05 | 4593094 | 02:00 | 4597203 | 00:49 $\mathcal{P}_{IP}^{6}$ | 3469276 | 00:01 | 3477354 | 00:01 | 3491947 | 00:01 | 4606253 | 00:01 | 4625993 | 00:01 | 4612164 | 00:01 7 | $\mathcal{P}_{LP}^{7}$ | | | 3469950 | 00:42 | 3469543 | 00:52 | 4592860 | 01:15 | 4593431 | 01:04 | 4597913 | 01:17 $\mathcal{P}_{IP}^{7}$ | | | 3469950 | 00:01 | 3487562 | 00:01 | 4592860 | 00:01 | 4619643 | 00:01 | 4606368 | 00:01 8 | $\mathcal{P}_{LP}^{8}$ | | | | | 3470100 | 00:38 | | | 4594146 | 01:03 | 4597730 | 02:00 $\mathcal{P}_{IP}^{8}$ | | | | | 3478057 | 00:01 | | | 4594146 | 00:01 | 4604551 | 00:01 9 | $\mathcal{P}_{LP}^{9}$ | | | | | 3470355 | 00:28 | | | | | 4597929 | 00:50 $\mathcal{P}_{IP}^{9}$ | | | | | 3470355 | 00:01 | | | | | 4597929 | 00:01 Final Solution | 3469276 | 13:22 | 3469950 | 12:18 | 3470355 | 12:00 | 4592860 | 23:13 | 4594146 | 22:27 | 4597929 | 23:57 ∗All values in the “Cost” columns are in USD, where the real values are rounded-off to the next integer values. All values in the “Time” columns are in HH:MM, where the seconds’ values are rounded-off to the next minute values. A comparison of the above three methods has been drawn in Table 7, for TC-1 (posing challenge to Enhanced-DFS) and TC-5 (largest flight set). In that, besides the cost and run-time of the initial solution for each test case, the results of all the iterations of $AirCROP$ leading up to the final solution have been presented. The latter is done to shed light on whether $AirCROP^{\prime}s$ final solution cost quality strongly depends on the cost of the initial solution. The prominent observations from the Table 7 include: * • In terms of run-time: IPDCH could outperform the Enhanced-DFS, as its run-time happened to be less than ten minutes in both the test cases. The Artificial pairing method even out performs IPDCH, since its run-time happened to be in milliseconds (formatted to 0 minutes in the table). * • In terms of initial cost: IPDCH could again outperform the Enhanced-DFS. This could be attributed to the use of IP to find a lowest-cost pairing set that covers the maximum-possible flights for each of the decomposed flight subsets. In contrast, the cost associated with the Artificial pairing method, is the worst. This is owing to a very high pseudo-cost attached to the pairings to offset their non-legality. Critically, regardless of the significantly varying run-time and the initial cost associated with the three methods, the variation in the cost of the final solution offered by $AirCROP$ is not significant. This endorses the robustness of its constitutive modules. #### 4.3.4 Impact of Termination Settings of Optimization Engine’s Submodules on AirCROP’s Performance This section investigates the sensitivity of $AirCROP$ to the termination parameter settings of the Optimization Engine’s submodules, namely, LPP- solutioning and IPP-solutioning. The parameters involved in LPP-solutioning are $Th_{cost}$ and $Th_{t}$, while $Th_{ipt}$ is involved in IPP-solutioning. To assess their impact on $AirCROP^{\prime}s$ performance, experiments are performed with three different sets of parameter settings each, for both the submodules. Impact of Termination Settings of CG-driven LPP-solutioning: As mentioned earlier, the CG-driven LPP-solutioning is terminated if the cost- improvement per LPP iteration falls below the pre-specified threshold $Th_{cost}$ (in USD) over $Th_{t}$ number of successive LPP iterations. To achieve a reasonable balance between $AirCROP^{\prime}s$ run time on the one hand and the cost reduction of the crew pairing solution on the other hand, three different sets of parameter settings are chosen, and experimented with. These settings of $\\{Th_{cost},Th_{t}\\}$ including $\\{500,5\\}$, $\\{100,10\\}$, and $\\{50,15\\}$ symbolize relaxed, moderate and strict settings, respectively, since the criterion for $AirCROP^{\prime}s$ termination gets more and more difficult as the settings change from $\\{500,5\\}$ to $\\{50,15\\}$. The results of the $AirCROP$-runs corresponding to these termination settings are reported in Table 8, and the key observations are as highlighted below. * • As the termination settings transition through relaxed, moderate and strict settings, the run-time to obtain the final solution increases, while the cost of the final solution decreases. An apparent exception to this trend is observed in TC-5 with the strict setting, but this could be explained by the fact that the upper limit of 30 hours set for $AirCROP^{\prime}s$ run time under practical considerations was exceeded during the fourth LPP-IPP interaction ($T=4$). It implies that due to the enforced termination in this particular case, $AirCROP$ could not fully utilize the potential for cost reduction. * • Despite the variation in the termination settings, the cost quality of $AirCROP^{\prime}s$ final solution does not vary as drastically, as its run time. For instance, as the settings switched from relaxed to moderate: an additional saving of 6384 USD could be achieved at the expense of additional 5:20 run time in the case of TC-2, while these indicators stand at 13388 USD and 10:25, respectively, in the case of TC-5. It can also be inferred that $\\{Th_{cost},Th_{t}\\}$ set as $\\{100,10\\}$ possibly offers a fair balance between solution’s cost quality and run time, and this explains why these settings have been used as the base settings for the experimental results presented in this paper, beginning with Table 3 and ending with Table 9. It is important to recognize that as the termination settings for LPP- solutioning are made stricter, its run time is bound to increase. It is also fair to expect that the cost quality of the final solution may be better, though it cannot be guaranteed. Any such departures from the expected trend may be due to the dependence of the quality of the final solution on the quality of the IPP-solution for each $T$. In that, if an IPP-solution for a particular $T$ may largely fail to approach the lower bound set by the corresponding LPP-solution, it may negatively influence the cost quality obtained in subsequent LPP- and IPP-solutioning phases. While such a possibility remains, it did not surface in the experiments above. Table 8: Performance assessment of $AirCROP$ on TC-2 and TC-5, against three different termination settings (Relaxed, Moderate and Strict Settings) of the CG-driven LPP-solutioning∗ LPP-IPP | TC-2 | TC-5 ---|---|--- Interactions | Relaxed Setting | Moderate Setting | Strict Setting | Relaxed Setting | Moderate Setting | Strict Setting | | $\bm{Th_{cost}=}$ 500, $\bm{Th_{t}=}$ 5 | $\bm{Th_{cost}=}$ 100, $\bm{Th_{t}=}$ 10 | $\bm{Th_{cost}=}$ 50, $\bm{Th_{t}=}$ 15 | $\bm{Th_{cost}=}$ 500, $\bm{Th_{t}=}$ 5 | $\bm{Th_{cost}=}$ 100, $\bm{Th_{t}=}$ 10 | $\bm{Th_{cost}=}$ 50, $\bm{Th_{t}=}$ 15 $T$ | $\mathcal{P}_{LP}^{T}/\mathcal{P}_{IP}^{T}$ | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time $\mathcal{P}_{IFS}$ | 129221508 | 00:08 | 129221508 | 00:08 | 129221508 | 00:08 | 131443284 | 00:09 | 131443284 | 00:09 | 131443284 | 00:09 1 | $\mathcal{P}_{LP}^{1}$ | 3510231 | 01:12 | 3495054 | 03:51 | 3489337 | 08:21 | 4603984 | 02:43 | 4584525 | 07:28 | 4581711 | 10:05 $\mathcal{P}_{IP}^{1}$ | 3844119 | 00:20 | 3769811 | 00:20 | 3698316 | 00:20 | 5165821 | 00:20 | 4974341 | 00:20 | 4946510 | 00:20 2 | $\mathcal{P}_{LP}^{2}$ | 3510105 | 00:26 | 3495311 | 01:57 | 3491725 | 03:36 | 4605049 | 01:12 | 4589091 | 02:25 | 4582977 | 09:32 $\mathcal{P}_{IP}^{2}$ | 3729820 | 00:20 | 3628514 | 00:20 | 3607470 | 00:20 | 4962643 | 00:20 | 4782657 | 00:20 | 4780005 | 00:20 3 | $\mathcal{P}_{LP}^{3}$ | 3506864 | 00:35 | 3497558 | 00:52 | 3491685 | 04:15 | 4602283 | 01:09 | 4589952 | 02:14 | 4585457 | 05:56 $\mathcal{P}_{IP}^{3}$ | 3659201 | 00:20 | 3566092 | 00:11 | 3578774 | 00:20 | 4818918 | 00:20 | 4736313 | 00:20 | 4678596 | 00:20 4 | $\mathcal{P}_{LP}^{4}$ | 3506644 | 00:32 | 3499237 | 01:26 | 3494201 | 02:29 | 4604535 | 00:52 | 4591145 | 02:36 | 4595692 | 03:34 $\mathcal{P}_{IP}^{4}$ | 3606381 | 00:20 | 3516807 | 00:01 | 3540972 | 00:01 | 4727106 | 00:20 | 4654627 | 00:20 | 4624747 | 00:01 5 | $\mathcal{P}_{LP}^{5}$ | 3507647 | 00:29 | 3500169 | 00:42 | 3494409 | 02:36 | 4603253 | 00:47 | 4592463 | 02:03 | | $\mathcal{P}_{IP}^{5}$ | 3559484 | 00:04 | 3517585 | 00:01 | 3527254 | 00:01 | 4683130 | 00:20 | 4632708 | 00:02 | | 6 | $\mathcal{P}_{LP}^{6}$ | 3507101 | 00:20 | 3501523 | 00:43 | 3496498 | 01:00 | 4603093 | 00:45 | 4593094 | 02:00 | | $\mathcal{P}_{IP}^{6}$ | 3547304 | 00:02 | 3504085 | 00:01 | 3496498 | 00:01 | 4681335 | 00:20 | 4625993 | 00:01 | | 7 | $\mathcal{P}_{LP}^{7}$ | 3508166 | 00:18 | 3502118 | 00:31 | | | 4603638 | 00:46 | 4593431 | 01:04 | | $\mathcal{P}_{IP}^{7}$ | 3517436 | 00:01 | 3502118 | 00:01 | | | 4651002 | 00:06 | 4619643 | 00:01 | | 8 | $\mathcal{P}_{LP}^{8}$ | 3508502 | 00:17 | | | | | 4604073 | 00:44 | 4594146 | 01:03 | | $\mathcal{P}_{IP}^{8}$ | 3508502 | 00:01 | | | | | 4634316 | 00:02 | 4594146 | 00:01 | | 9 | $\mathcal{P}_{LP}^{9}$ | | | | | | | 4606250 | 00:28 | | | | $\mathcal{P}_{IP}^{9}$ | | | | | | | 4614420 | 00:01 | | | | 10 | $\mathcal{P}_{LP}^{10}$ | | | | | | | 4607534 | 00:17 | | | | $\mathcal{P}_{IP}^{10}$ | | | | | | | 4607534 | 00:01 | | | | Final Solution | 3508502 | 05:45 | 3502118 | 11:05 | 3496498 | 23:28 | 4607534 | 12:02 | 4594146 | 22:27 | 4624747 | 30:17 ∗All values in the “Cost” columns are in USD, and all the corresponding real values are rounded-off to the next integer values. All values in the “Time” columns are in HH:MM, and all the corresponding seconds’ values are rounded- off to the next minute values. Impact of Termination Settings of IPP-solutioning: As mentioned before, integerization of an LPP solution using an MIP solver is extremely time-consuming, particularly for large-scale CPOPs, and more so those involving complex flight networks. Hence, from a practical perspective, the $AirCROP$ framework imposes a threshold on the upper time limit for IPP- solutioning (for any given $T$), namely $Th_{ipt}$, in case it does not self- terminate a priori. To investigate the impact of $Th_{ipt}$ on $AirCROP^{\prime}s$ performance, experiments are performed with three different settings, including, 00:20 (one-third of an hour), 00:40 (two-third of an hour), and 01:00 (an hour). The results are presented in Table 9, and the key observations are as follows. In the case of TC-2, as the $Th_{ipt}$ is raised, the run-time to obtain the final solution increases, while the cost of the final solution decreases. However, there are exceptions to this trend in the case of TC-5. Notably, the cost quality of the final solution corresponding to $Th_{ipt}=$ 00:20 remains superior to that obtained for both $Th_{ipt}=$ 00:40 and 01:00. For these two settings, the quality of LPP- solution at $T=8$ turned worse compared to the case of $Th_{ipt}=$ 00:20, and the gap could not be bridged even in the subsequent LPP-IPP interaction ($T=9$). The worsening of LPP-solution could be attributed to the fact that LPP-solutioning relies on random number based heuristics, and the resulting pairing combinations may not necessarily offer lower cost within the pre- specified termination settings. Based on the above, it may be inferred that despite the changes in the termination parameter settings, $AirCROP$ is able to offer solutions with reasonably close cost quality, though significant variations in run time may be observed. It is also evident that even the lowest setting (desired from a practical perspective) for $Th_{ipt}=$ 00:20 offers a good balance between solution’s cost quality and run time, and this explains why it has been used as the base setting for the experimental results presented in this paper. Table 9: Performance assessment of $AirCROP$ on TC-2 and TC-5, against three different termination settings ($Th_{ipt}=$ 00:20, 00:40 & 01:00) of the IPP- solutioning∗ LPP-IPP | TC-2 | TC-5 ---|---|--- Interactions | $\bm{Th_{ipt}=}$ 00:20 | $\bm{Th_{ipt}=}$ 00:40 | $\bm{Th_{ipt}=}$ 01:00 | $\bm{Th_{ipt}=}$ 00:20 | $\bm{Th_{ipt}=}$ 00:40 | $\bm{Th_{ipt}=}$ 01:00 $T$ | $\mathcal{P}_{LP}^{T}/\mathcal{P}_{IP}^{T}$ | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time | Cost | Time $\mathcal{P}_{IFS}$ | 129221508 | 00:08 | 129221508 | 00:08 | 129221508 | 00:08 | 131443284 | 00:09 | 131443284 | 00:09 | 131443284 | 00:09 1 | $\mathcal{P}_{LP}^{1}$ | 3495054 | 03:51 | 3495054 | 04:21 | 3495054 | 03:57 | 4584525 | 07:28 | 4584525 | 07:46 | 4584525 | 07:49 $\mathcal{P}_{IP}^{1}$ | 3769811 | 00:20 | 3744301 | 00:40 | 3760028 | 01:00 | 4974341 | 00:20 | 4958532 | 00:40 | 4987497 | 01:00 2 | $\mathcal{P}_{LP}^{2}$ | 3495311 | 01:57 | 3497483 | 01:49 | 3495562 | 02:19 | 4589091 | 02:25 | 4585347 | 05:00 | 4588371 | 03:56 $\mathcal{P}_{IP}^{2}$ | 3628514 | 00:20 | 3629401 | 00:40 | 3632875 | 01:00 | 4782657 | 00:20 | 4778465 | 00:40 | 4766924 | 01:00 3 | $\mathcal{P}_{LP}^{3}$ | 3497558 | 00:52 | 3497473 | 01:33 | 3494305 | 01:50 | 4589952 | 02:14 | 4589481 | 02:56 | 4588911 | 04:25 $\mathcal{P}_{IP}^{3}$ | 3566092 | 00:11 | 3566247 | 00:40 | 3579899 | 01:00 | 4736313 | 00:20 | 4699845 | 00:40 | 4713402 | 01:00 4 | $\mathcal{P}_{LP}^{4}$ | 3499237 | 01:26 | 3500607 | 01:06 | 3495273 | 01:13 | 4591145 | 02:36 | 4590618 | 01:56 | 4591028 | 02:09 $\mathcal{P}_{IP}^{4}$ | 3516807 | 00:01 | 3524672 | 00:01 | 3551863 | 00:05 | 4654627 | 00:20 | 4656611 | 00:40 | 4681015 | 01:00 5 | $\mathcal{P}_{LP}^{5}$ | 3500169 | 00:42 | 3501809 | 00:49 | 3496754 | 00:52 | 4592463 | 02:03 | 4591826 | 01:18 | 4591448 | 02:03 $\mathcal{P}_{IP}^{5}$ | 3517585 | 00:01 | 3501809 | 00:01 | 3528564 | 00:01 | 4632708 | 00:02 | 4644467 | 00:15 | 4639287 | 00:29 6 | $\mathcal{P}_{LP}^{6}$ | 3501523 | 00:43 | | | 3496342 | 00:53 | 4593094 | 02:00 | 4592492 | 02:21 | 4591372 | 02:06 $\mathcal{P}_{IP}^{6}$ | 3504085 | 00:01 | | | 3512692 | 00:01 | 4625993 | 00:01 | 4617694 | 00:01 | 4616944 | 00:01 7 | $\mathcal{P}_{LP}^{7}$ | 3502118 | 00:31 | | | 3497967 | 00:59 | 4593431 | 01:04 | 4594599 | 01:30 | 4594479 | 01:23 $\mathcal{P}_{IP}^{7}$ | 3502118 | 00:01 | | | 3519996 | 00:01 | 4619643 | 00:01 | 4607261 | 00:01 | 4608085 | 00:01 8 | $\mathcal{P}_{LP}^{8}$ | | | | | 3498726 | 01:24 | 4594146 | 01:03 | 4595739 | 01:08 | 4595424 | 01:03 $\mathcal{P}_{IP}^{8}$ | | | | | 3518299 | 00:01 | 4594146 | 00:01 | 4598624 | 00:01 | 4603634 | 00:01 9 | $\mathcal{P}_{LP}^{9}$ | | | | | 3499104 | 00:40 | | | 4595703 | 00:45 | 4596929 | 00:59 $\mathcal{P}_{IP}^{9}$ | | | | | 3504258 | 00:01 | | | 4595703 | 00:01 | 4596929 | 00:01 10 | $\mathcal{P}_{LP}^{10}$ | | | | | 3499117 | 01:10 | | | | | | $\mathcal{P}_{IP}^{10}$ | | | | | 3509608 | 00:01 | | | | | | 11 | $\mathcal{P}_{LP}^{11}$ | | | | | 3499609 | 00:45 | | | | | | $\mathcal{P}_{IP}^{11}$ | | | | | 3499609 | 00:01 | | | | | | Final solution | 3502118 | 11:05 | 3501809 | 12:24 | 3499609 | 19:22 | 4594146 | 22:27 | 4595703 | 27:55 | 4596929 | 30:35 ∗All values in the “Cost” columns are in USD, and all the corresponding real values are rounded-off to the next integer values. All values in the “Time” columns are in HH:MM, and all the corresponding seconds’ values are rounded- off to the next minute values. ## 5 Conclusion and Future Research For an airline, crew operating cost is the second largest expense, after the fuel cost, making the crew pairing optimization critical for business viability. Over the last three decades, CPOP has received an unprecedented attention from the OR community, as a result of which numerous CPOP solution approaches have been proposed. Yet, the emergent flight networks with conjunct scale and complexity largely remain unaddressed in the available literature. Such a scenario is all the more alarming, considering that the air traffic is expected to scale up to double over the next 20 years, wherein, most airlines may need to cater to multiple crew bases and multiple hub-and-spoke subnetworks. This research has proposed an Airline Crew Pairing Optimization Framework ($AirCROP$) based on domain-knowledge driven CG strategies for efficiently tackling real-world, large-scale and complex flight networks. This paper has presented not just the design of the $AirCROP$’s constitutive modules, but has also shared insights on how these modules interact and how sensitive the $AirCROP^{\prime}s$ performance is to the sources of variability, choice of different methods and parameter settings. Given a CPOP, $AirCROP$ first preprocesses the entire duty overnight- connection network via its Legal Crew Pairing Generator777This module is utilized again to facilitate legal crew pairings when required in real-time in other modules of $AirCROP$ Subsequently, $AirCROP$ is initialized using an IFS generated by the proposed method (IPDCH). Next, the $AirCROP$’s Optimization Engine attempts to find a good-quality CPOP solution via intermittent interactions of its submodules, namely, CG-driven LPP-solutioning and IPP- solutioning. The efficacy of $AirCROP$ has been demonstrated on real-world airline flight network characterized by an unprecedented (in reference to available literature) conjunct scale-and-complexity, marked by over 4200 flights, 15 crew bases, multiple hub-and-spoke subnetworks, and billion-plus pairings. The distinctive contribution of this paper is also embedded in its empirical investigation of critically important questions relating to variability and sensitivity, which the literature is otherwise silent on. In that: * • first, the sensitivity analysis of $AirCROP$ is performed in the presence and absence of sources of variability. It is empirically highlighted that $AirCROP$ is capable of offering comparable cost solutions, both in the presence or absence of the sources of variability. This endorses the robustness of its constitutive modules. * • second, the sensitivity of $AirCROP$ with respect to the cost quality of the initial solution and the associated run-time is investigated vis-$\grave{a}$-vis three different initialization methods. Again, the robustness of $AirCROP$ is endorsed, considering that it is found to be capable of offering similar cost solutions, despite the significantly varying cost and run-time of the initial solutions. * • last, the sensitivity of $AirCROP$ to the termination parameter settings associated with the Optimization Engine’s submodules, is investigated. The fact that with the variation in termination settings of both LPP-solutioning and IPP-solutioning (independent of each other)- the $AirCROP$’s performance strongly aligns with the logically expected trends, is a testimony to the robustness of its constitutive modules. Notably, $AirCROP$ has been implemented using Python scripting language, aligned with the industrial sponsor’s preferences. However, a significant reduction in run-time could be achieved by the use of compiled programming languages such as C++, Java, etc. Moreover, employing the domain-knowledge driven CG strategies during the IPP-solutioning phase too, may augment the overall cost- and time-efficiency of the $AirCROP$. Furthermore, the emerging trend of utilizing the Machine Learning capabilities for assisting combinatorial optimization tasks, may also hold promise for the airline crew pairing optimization, towards which an exploratory attempt has been made by the authors (Aggarwal, Singh Saxena, 2020). Despite the scope for improvement, the authors hope that with the emergent trend of evolving scale and complexity of airline flight networks, this paper shall serve as an important milestone for the affiliated research and applications. ## Acknowledgment This research work is a part of an Indo-Dutch joint research project, supported by the Ministry of Electronics and Information Technology (MEITY), India [grant number 13(4)/2015-CC&BT]; Netherlands Organization for Scientific Research (NWO), the Netherlands; and General Electric (GE) Aviation, India. The authors thank GE Aviation, particularly, Saaju Paulose (Senior Manager), Arioli Arumugam (Senior Director- Data & Analytics), and Alla Rajesh (Senior Staff Data & Analytics Scientist) for providing real-world test cases, and sharing their domain knowledge which has helped the authors significantly in successfully completing this research work. ## References * Achterberg Wunderling (2013) achterberg2013mixedAchterberg, T. 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2024-09-04T02:54:58.320151

2003.03999

arxiv-papers

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) ​​​ CERN-EP-2020-023 LHCb-PAPER-2020-001 May 28, 2020 Search for the rare decays $B^{0}_{s}\rightarrow e^{+}e^{-}$ and $B^{0}\rightarrow e^{+}e^{-}$ LHCb collaboration†††Authors are listed at the end of this Letter. A search for the decays $B^{0}_{s}\rightarrow e^{+}e^{-}$ and $B^{0}\rightarrow e^{+}e^{-}$ is performed using data collected with the LHCb experiment in proton-proton collisions at center-of-mass energies of $7$, $8$ and $13\,\text{TeV}$, corresponding to integrated luminosities of $1$, $2$ and $2\,\text{fb}^{-1}$, respectively. No signal is observed. Assuming no contribution from $B^{0}\rightarrow e^{+}e^{-}$ decays, an upper limit on the branching fraction $\mathcal{B}(B^{0}_{s}\rightarrow e^{+}e^{-})<9.4\,(11.2)\times 10^{-9}$ is obtained at $90\,(95)\,\%$ confidence level. If no $B^{0}_{s}\rightarrow e^{+}e^{-}$ contribution is assumed, a limit of $\mathcal{B}(B^{0}\rightarrow e^{+}e^{-})<2.5\,(3.0)\times 10^{-9}$ is determined at $90\,(95)\,\%$ confidence level. These upper limits are more than one order of magnitude lower than the previous values. Published in Phys. Rev. Lett. 124 (2020) 211802 © 2020 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence. Searches for rare particle decays provide ideal probes for contributions from physics processes beyond the Standard Model (SM). Recent measurements of decays involving $b\\!\rightarrow s\ell^{+}\ell^{-}$ transitions (the inclusion of charge-conjugated processes is implied throughout this Letter) hint at deviations from SM predictions in lepton-flavor universality tests [1, 2, 3, 4, 5, 6] and thus motivate measurements of decay rates into final states involving leptons. Following the observation of the decay ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ [7, 8], the search for ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ decays provides an independent test of lepton-flavor universality. According to SM predictions (calculated from Ref. [9], neglecting QED corrections that are expected to be at the percent level), ${{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}}$ decays have branching fractions of $\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})=$(8.60\pm 0.36)\text{\times}{10}^{-14}$$ and $\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})=$(2.41\pm 0.13)\text{\times}{10}^{-15}$$. With contributions beyond the SM, these branching fractions could be significantly larger, reaching values of $\mathcal{O}(10^{-8})$ for $\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})$ and $\mathcal{O}(10^{-10})$ for $\mathcal{B}\mbox{(${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$)}$ [10]. These values are close to the current experimental bounds of $\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})<$2.8\text{\times}{10}^{-7}$$ and $\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})<$8.3\text{\times}{10}^{-8}$$ at $90\text{\,}\mathrm{\char 37\relax}$ confidence level (CL) [11], set by the CDF collaboration. In this Letter a search for ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ decays is presented using data collected with the LHCb experiment in proton-proton collisions at center-of-mass energies of $7\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ in 2011, $8\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ in 2012 and $13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ in 2015 and 2016, corresponding to integrated luminosities of $1$, $2$ and $2\text{\,}\mathrm{f}\mathrm{b}^{-1}$, respectively. The signal yields are determined from a fit to the data and normalized to those of the ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$ decay, where the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson decays to $e^{+}e^{-}$, which has a precisely measured branching fraction [12] and a similar dielectron signature in the detector. The LHCb detector [13, 14] is a single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the study of particles containing $b$ or $c$ quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the $pp$ interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three stations of silicon-strip detectors and straw drift tubes placed downstream of the magnet. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors. Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers. The online event selection is performed by a trigger [15], which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. At the hardware trigger stage, events are required to have a high-energy deposit in the calorimeters associated with a signal electron candidate, or a muon candidate with high transverse momentum $p_{\mathrm{T}}$, or a photon, electron or hadron candidate with high transverse energy from the decays of other particles from the $pp$ collision. The software trigger requires a two- track secondary vertex with a significant displacement from any primary $pp$ interaction vertex (PV). At least one charged particle must have high $p_{\mathrm{T}}$ and be inconsistent with originating from a PV. A multivariate algorithm [16, 17] is used in the trigger for the identification of secondary vertices consistent with the decay of a $b$ hadron. Simulated samples are used to optimize the candidate selection, estimate selection efficiencies and describe the expected invariant-mass shapes of the signal candidates and background decays. In the simulation, $pp$ collisions are generated using Pythia [18, *Sjostrand:2007gs] with a specific LHCb configuration [20]. Decays of unstable particles are described by EvtGen [21], in which final-state radiation is generated using Photos [22]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [23, *Agostinelli:2002hh] as described in Ref. [25]. The simulation is corrected for data-simulation differences in $B$-meson production kinematics, detector occupancy and isolation criteria [26] using ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$ and ${{B}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ decays, with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\\!\rightarrow{e^{+}e^{-}}$ and $\phi\\!\rightarrow{{K}^{+}}{{K}^{-}}$. Particle identification variables are calibrated using data from ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$ and ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays [27]. The calibration data are binned in momentum and pseudorapidity of the particle as well as detector occupancy to account for possible differences in kinematics between the investigated decay and the calibration data. The ${{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}}$ candidates are selected in events passing the trigger requirements by combining two tracks that are inconsistent with originating from any PV in the event and which form a good- quality secondary vertex. The tracks are also required to have a momentum larger than 3 $\text{\,Ge\kern-1.00006ptV\\!/}c$ and $p_{\mathrm{T}}$ greater than 500 $\text{\,Me\kern-1.00006ptV\\!/}c$, and must be identified as electrons using information from the Cherenkov detectors and calorimeters. The dielectron candidate’s momentum must be aligned with the vector pointing from a PV (the associated PV) to the two-track vertex and have a considerable transverse component. The candidate must also have an invariant mass in the range $[4166,6566]\,\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. The measured electron momenta are corrected for losses due to bremsstrahlung radiation by adding the momentum of photons consistent with being emitted upstream of the magnet [28]. Candidates in data and simulation are separated into three categories with either zero, one, or both electrons having a bremsstrahlung correction applied. To avoid experimenters’ bias, the narrowest dielectron invariant-mass region containing $90\text{\,}\mathrm{\char 37\relax}$ of simulated ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ decays, corresponding to a range of [$4689$, $5588$] $\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, was removed from the data set until the analysis procedure was finalized. Candidates for the normalization mode, ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$, are constructed similarly, but require an additional track consistent with being a kaon and originating from the same vertex as the dielectron candidate. The dielectron candidate must have an invariant mass in the range $[2450,3176]\,\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, consistent with arising from a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson decay. In addition, the reconstructed ${B}^{+}$ candidate mass, when the dielectron candidate is constrained to the known ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [12], must be above $5175\,\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, suppressing partially reconstructed decays. A boosted decision tree (BDT) algorithm[29, 30, 31] is used to separate ${{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}}$ signal from random combinations of two electrons (combinatorial background). The BDT is trained separately for data taking periods 2011–2012 (Run 1) and 2015–2016 (Run 2) on simulated ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ decays as signal proxy and dielectron candidates from data with a mass above $5588\,\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ as background proxy. The split between the data taking periods is done to account for changes in the center- of-mass energies and trigger strategies, which significantly impact the data distributions and improve the BDT and the particle identification algorithms in Run 2. It is checked that the data behave consistently across the data- taking periods. The BDT input variables comprise of the following: kinematic information on the electron tracks and $B$ candidate, information on the displacement of the electrons and $B$ candidate from the associated PV, and isolation variables that quantify the compatibility of other tracks in the event with originating from the same decay as the $B$ candidate [26, 32]. Candidates with a BDT response compatible with that of the background are discarded, with the threshold chosen by maximizing the figure of merit ${\epsilon_{\text{signal}}}/{(\sqrt{N_{\text{background}}}+3/2)}$ [33], where $\epsilon_{\text{signal}}$ is the signal efficiency and the expected background yield in the signal region is $N_{\text{background}}$. The final selected data set is separated by data-taking period and by category of bremsstrahlung correction. The branching fraction $\mathcal{B}({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})$ is measured relative to that of the normalization channel via $\displaystyle\mathcal{B}({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})$ $\displaystyle=N({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})\times\alpha\times\mathcal{B}({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})\times\left(\frac{f_{d(s)}}{f_{u}}\right)^{-1},$ (1) where $\displaystyle\alpha$ $\displaystyle\equiv\frac{\varepsilon({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})}{\varepsilon({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})}\times\frac{1}{N({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})},$ (2) $\varepsilon({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})$ and $\varepsilon({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})$ denote the efficiencies of the signal and normalization modes, and $N$(${{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}}$) and $N$(${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$) their yields. The normalization mode branching fraction (including that for the decay ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\\!\rightarrow{e^{+}e^{-}}$) is $\mathcal{B}({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})=$(6.03\pm 0.17)\text{\times}{10}^{-5}$$, taken from Ref.[12]. The $b$-hadron fragmentation fraction ratio $f_{d}/f_{u}$ is assumed to be unity, while $f_{s}/f_{u}=$0.259\pm 0.015$$ [34] is used for the Run 1 data and is scaled by $1.068\pm 0.016$ for the Run 2 data, according to Ref. [35], to account for center-of-mass energy differences. A measurement of $f_{s}/f_{u}$ from Run 2 yields a consistent, but less precise, result [36]. The yield of the normalization mode is determined using an unbinned maximum- likelihood fit to the ${K}^{+}$ $e^{+}e^{-}$ invariant mass separately for each year of data taking and bremsstrahlung category. The fit model comprises a Gaussian function with power-law tails [37] for the signal component, where the tail parameters are fixed from simulation, and an exponential function to describe combinatorial background. Summed over the bremsstrahlung categories, the yield of the normalization mode is $20\,480\pm 140$ in the Run 1 data and $33\,080\pm 180$ in the Run 2 data. The selection efficiencies $\varepsilon({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})$ and $\varepsilon({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})$ are determined separately for each year of data taking and bremsstrahlung category using simulated decays that are weighted to better represent the data. Calibration data are used to evaluate particle- identification efficiencies [27]. Trigger efficiencies are also estimated from data, using the technique described in Ref. [38]. For simulated ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ decays, the mean ${B}^{0}_{s}$ lifetime [39] is assumed. The selection efficiency is assumed to be the same for both ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ and ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ decays, which is consistent with results from simulation. The normalization factors, $\alpha$, are combined across the data-taking periods and given in Table 1, split by bremsstrahlung category (for the selection efficiency ratio between normalization and signal mode, see the Supplemental Material [40]). Table 1: Normalization factors $\alpha$ for ${{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}}$. The bremsstrahlung category denotes whether zero, one or both electrons are corrected for bremsstrahlung losses. The uncertainties include statistical uncertainties and uncertainties due to limited size of the simulated samples. Bremsstrahlung category | 2011–2012 $[10^{-5}]$ | 2015–2016 $[10^{-5}]$ ---|---|--- No correction | $2.85\pm 0.24$ | $1.84\pm 0.08$ One electron corrected | $1.13\pm 0.08$ | $0.70\pm 0.03$ Both electrons corrected | $1.73\pm 0.20$ | $1.04\pm 0.06$ In addition to the combinatorial background, backgrounds due to misidentification and partial reconstruction are present in the data. These backgrounds differ significantly between the categories of bremsstrahlung correction. Their invariant-mass shapes and relative contributions are evaluated using simulation. In the lower mass region, partially reconstructed backgrounds of the types ${B}\\!\rightarrow X{e^{+}e^{-}}$ and ${{B}^{+}}\\!\rightarrow{{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(\rightarrow Y^{+}{e^{-}}{{\overline{\nu}}_{e}}){e^{+}}{{\nu}_{e}}$ dominate, where $X$ and $Y$ represent hadronic systems. The main source of background in the $B$-mass region, however, stems from misidentified particles in the decays ${{B}^{0}}\\!\rightarrow{{\pi}^{-}}{e^{+}}{{\nu}_{e}}$ and ${B}\\!\rightarrow{h}^{+}{h}^{\prime-}$, where $h$ and $h^{\prime}$ are hadrons. The latter has a peaking structure in the $B$-mass region. Backgrounds involving misidentified particles contribute mostly to categories in which at most one of the electrons has a bremsstrahlung correction applied. The contribution from combinatorial background is evaluated from same-sign lepton pairs in data and found to be small. The yields of the backgrounds are Gaussian constrained to their expected values, estimated from simulation using their known branching fractions [12]. The shape of the invariant mass of the ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ components is modeled using a Gaussian function with power-law tails, where the parameters are obtained from simulation and differ between each bremsstrahlung category and year of data taking. The peak values and the widths of the functions are corrected for data-simulation differences by a factor determined from the normalization mode. The parameters of the ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ line shapes are fixed to the same values with the exception of the peak value, which is shifted according to the known ${B}^{0}_{s}$–${B}^{0}$ mass difference [12]. Due to the limited mass resolution, arising from imperfect bremsstrahlung recovery, the line shapes from ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ are highly overlapping. Therefore the branching fraction of ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ is obtained by performing a simultaneous fit to the dielectron invariant-mass distribution of all six data sets while neglecting the contribution from ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$, and vice versa. In these fits, the only shared parameters between categories are the branching fractions $\mathcal{B}({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})$ and $\mathcal{B}({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})$, and the ratio of the fragmentation fractions $f_{s}/f_{u}$. Systematic uncertainties are estimated separately for each data set. Dominant sources of systematic uncertainties in the normalization arise from the uncertainty on the fragmentation fraction ratio, the technique used to evaluate the trigger efficiencies, and the determination of particle- identification efficiencies; the systematic uncertainties from these sources extend to $5.8\text{\,}\mathrm{\char 37\relax}$, $5.3\text{\,}\mathrm{\char 37\relax}$, and $5.3\text{\,}\mathrm{\char 37\relax}$ on the branching fractions, respectively. The uncertainty on $\mathcal{B}({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})$ of $2.8\text{\,}\mathrm{\char 37\relax}$ [12] is taken into account. A difference of up to $4.1\text{\,}\mathrm{\char 37\relax}$ is found between the efficiency of the BDT selection on simulated ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$ decays and ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$ decays in data, which is assigned as a systematic uncertainty. The fraction of candidates in each bremsstrahlung-correction category of the signal modes is taken from simulation. The difference between simulation and data is investigated using ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$ decays and its effect on the normalization, up to $4.0\text{\,}\mathrm{\char 37\relax}$, is taken as a systematic uncertainty. Systematic uncertainties on the invariant-mass resolution corrections are determined by repeating the correction procedure with pseudoexperiments obtained with the bootstrapping method [41], yielding up to $1.1\text{\,}\mathrm{\char 37\relax}$. A difference between the total selection efficiencies in the ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ channels of up to $2.5\text{\,}\mathrm{\char 37\relax}$ is assigned as a systematic uncertainty on the ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ normalization factor. Due to the presence of an additional kaon in the final state of the normalization mode, the track-reconstruction efficiency is different between the signal and normalization modes. An uncertainty of $1.1\text{\,}\mathrm{\char 37\relax}$ is assigned to the branching fraction as a systematic uncertainty on the kaon reconstruction efficiency arising from the limited knowledge of the interactions in the detector material [42]. Finally, an uncertainty of $1.0\text{\,}\mathrm{\char 37\relax}$ is assigned to account for small differences in detector occupancy between the signal and normalization mode arising from the trigger selection. The dominant sources of systematic uncertainties on the background composition are due to the imprecise knowledge of the branching fractions of the background components. The largest uncertainty of this type on the expected background yield in the $B$-mass region is $14\text{\,}\mathrm{\char 37\relax}$, determined from refitting the mass sidebands while varying the background components according to their uncertainties. Taking all correlations into account, overall single event sensitivities of $[4.71\pm 0.12\text{(stat.)}\pm 0.33\text{(syst.)}]\times 10^{-10}$ for ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and $[1.271\pm 0.034\text{(stat.)}\pm 0.063\text{(syst.)}]\times 10^{-10}$ for ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ are obtained. The dielectron invariant-mass spectrum, summed over bremsstrahlung categories, is shown in Fig. 1, with the result of the ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ fit. The individual categories are shown in the Supplemental Material [40], as well as the distributions with the result of the ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ fit. The measured branching fractions are $\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})=$(2.4\pm 4.4)\text{\times}{10}^{-9}$$ and $\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})=$(0.30\pm 1.29)\text{\times}{10}^{-9}$$, where the uncertainties include both statistical and systematic components. The results are in agreement with the background-only hypothesis. Figure 1: Simultaneous fit to the dielectron invariant-mass distribution, with $\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})$ fixed to zero. The sum of bremsstrahlung categories is shown for (left) Run 1 and (right) Run 2. The relative proportions of background contributions change between Run 1 and Run 2 due to different performances of the particle identification algorithms and BDT selections. Upper limits on the branching fractions are set using the CLs method [43], as implemented in the GammaCombo framework[44, 45] with a one-sided profile likelihood ratio [46] as test statistic. The likelihoods are computed from fits to the invariant-mass distributions. In the fits, the normalization factor, normalization mode branching fraction, fragmentation fraction ratio, and background yields are Gaussian constrained to their expected values within statistical and systematic uncertainties. Pseudoexperiments, in which the nuisance parameters are set to their fitted values from data, are used for the evaluation of the test statistic. The expected and observed CLs distributions are shown in Fig. 2. The upper observed limits are $\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})<9.4\,(11.2)\times 10^{-9}$ and $\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})<2.5\,(3.0)\times 10^{-9}$ at $90\,(95)\,\%$ confidence level. These are consistent with the expected upper limits of $\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})<7.0\,(8.6)\times 10^{-9}$ and $\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})<2.0\,(2.5)\times 10^{-9}$ at $90\,(95)\,\%$ confidence level, obtained as the median of limits determined on background-only pseudoexperiments. Figure 2: CLs values as a function of the branching fractions of the decays (left) ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ and (right) ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$. The red solid line (black solid line with data points) corresponds to the distribution of the expected (observed) upper limits, and the light blue (dark blue) band contains the $1\sigma$ $(2\sigma)$ uncertainties on the expected upper limits. Thresholds corresponding to $90\,\%$ and $95\,\%$ confidence level are indicated with dashed lines. The observed values are plotted for branching fractions greater than the measured branching fraction in the data; the test statistic is defined to be nonzero only in that region. In conclusion, a search for the rare decays ${{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}}$ is performed using data from proton-proton collisions recorded with the LHCb experiment, corresponding to a total integrated luminosity of $5\text{\,}\text{\,fb}^{-1}$. No excess of events is observed over the background. The resulting limits on the branching fractions are $\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})<9.4\,(11.2)\times 10^{-9}$ and $\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})<2.5\,(3.0)\times 10^{-9}$ at $90\,(95)\,\%$ confidence level, when neglecting the contribution from the other decay. The mean ${B}^{0}_{s}$ lifetime is assumed for ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ decays. Assuming SM-like $C\\!P$-odd ($C\\!P$-even) ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ decays, an increase (decrease) of $2.4\text{\,}\mathrm{\char 37\relax}$ with respect to the quoted limit is found. The results improve the limits on these branching fractions [11] by more than one order of magnitude and constrain contributions beyond the SM, for example from scalar and pseudoscalar currents [10]. ## Acknowledgements We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL- GRID (Poland) and OSC (USA). We are indebted to the communities behind the multiple open-source software packages on which we depend. Individual groups or members have received support from AvH Foundation (Germany); EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU, and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society and the Leverhulme Trust (United Kingdom). ## Supplemental Material for LHCb-PAPER-2020-001 The individual categories of the simultaneous fit to the dielectron invariant- mass using the ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ hypothesis are presented in Fig. 3. The fit to the invariant dielectron mass including the ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ hypothesis instead of the ${{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}}$ hypothesis is shown in Fig. 4, where the bremsstrahlung categories are summed. The individual categories of the simultaneous fit to the dielectron invariant-mass using the ${{B}^{0}}\\!\rightarrow{e^{+}e^{-}}$ hypothesis are presented in Fig. 5. Table 2 lists the inputs to the normalization factors: the ratio of normalization and signal efficiencies and the normalization yield. The efficiency of the normalization mode differs from the signal and causes the efficiency ratio to decrease with bremsstrahlung category due to the slightly different reconstruction and preselection and a different impact of the BDT selection, where the differences mainly originate from the additional track in the normalization mode. Figure 3: Simultaneous fit to the dielectron invariant-mass distribution in all categories, with $\mathcal{B}({{B}^{0}}\\!\rightarrow{e^{+}e^{-}})$ fixed to zero. The top figures show the three bremsstrahlung categories in the Run 1 data set and the bottom figures show the Run 2 data set. From left to right, the data sets correspond to the bremsstrahlung correction category with no correction, correcting one electron and correcting both electrons. The relative proportions of background contributions change between Run 1 and Run 2 due to different performances of the particle-identification algorithms and BDT selections. Their relative fractions between bremsstrahlung categories follow the expectation from simulation. Figure 4: Simultaneous fit to the dielectron invariant-mass distribution, with $\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})$ fixed to zero. The bremsstrahlung categories are summed over the (left) Run 1 and (right) Run 2 data sets. The relative proportions of background contributions change between Run 1 and Run 2 due to different performances of the particle-identification algorithms and BDT selections. Figure 5: Simultaneous fit to the dielectron invariant-mass distribution in all categories, with $\mathcal{B}({{B}^{0}_{s}}\\!\rightarrow{e^{+}e^{-}})$ fixed to zero. The top figures show the three bremsstrahlung categories in the Run 1 data set and the bottom figures show the Run 2 data set. From left to right, the data sets correspond to the bremsstrahlung correction category with no correction, correcting one electron and correcting both electrons. The relative proportions of background contributions change between Run 1 and Run 2 due to different performances of the particle-identification algorithms and BDT selections. Their relative fractions between bremsstrahlung categories follow the expectation from simulation. Table 2: Inputs for the normalization factors, the efficiency ratio $\varepsilon({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})/\varepsilon({{B}_{({s})}^{0}}\\!\rightarrow{e^{+}e^{-}})$ and normalization yield $N({{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}})$. The bremsstrahlung category (Brem. cat.) denotes whether zero, one or both electrons are corrected for bremsstrahlung losses. The uncertainties on the efficiency ratios include statistical uncertainties from the calibration and uncertainties due to limited size of the simulated samples. 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Zucchelli19,e. 1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil 2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 3Center for High Energy Physics, Tsinghua University, Beijing, China 4School of Physics State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing, China 5University of Chinese Academy of Sciences, Beijing, China 6Institute Of High Energy Physics (IHEP), Beijing, China 7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, China 8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy, France 9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France 10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France 11Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France 12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3, Paris, France 13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany 14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany 15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany 16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 17School of Physics, University College Dublin, Dublin, Ireland 18INFN Sezione di Bari, Bari, Italy 19INFN Sezione di Bologna, Bologna, Italy 20INFN Sezione di Ferrara, Ferrara, Italy 21INFN Sezione di Firenze, Firenze, Italy 22INFN Laboratori Nazionali di Frascati, Frascati, Italy 23INFN Sezione di Genova, Genova, Italy 24INFN Sezione di Milano-Bicocca, Milano, Italy 25INFN Sezione di Milano, Milano, Italy 26INFN Sezione di Cagliari, Monserrato, Italy 27INFN Sezione di Padova, Padova, Italy 28INFN Sezione di Pisa, Pisa, Italy 29INFN Sezione di Roma Tor Vergata, Roma, Italy 30INFN Sezione di Roma La Sapienza, Roma, Italy 31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands 32Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, Netherlands 33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Kraków, Poland 34AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science, Kraków, Poland 35National Center for Nuclear Research (NCBJ), Warsaw, Poland 36Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania 37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI), Gatchina, Russia 38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute (ITEP NRC KI), Moscow, Russia, Moscow, Russia 39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia 40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS), Moscow, Russia 41Yandex School of Data Analysis, Moscow, Russia 42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia 43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI), Protvino, Russia, Protvino, Russia 44ICCUB, Universitat de Barcelona, Barcelona, Spain 45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela, Santiago de Compostela, Spain 46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia - CSIC, Valencia, Spain 47European Organization for Nuclear Research (CERN), Geneva, Switzerland 48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland 49Physik-Institut, Universität Zürich, Zürich, Switzerland 50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine 51Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine 52University of Birmingham, Birmingham, United Kingdom 53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom 54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom 55Department of Physics, University of Warwick, Coventry, United Kingdom 56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom 57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 58School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom 59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom 60Imperial College London, London, United Kingdom 61Department of Physics and Astronomy, University of Manchester, Manchester, United Kingdom 62Department of Physics, University of Oxford, Oxford, United Kingdom 63Massachusetts Institute of Technology, Cambridge, MA, United States 64University of Cincinnati, Cincinnati, OH, United States 65University of Maryland, College Park, MD, United States 66Los Alamos National Laboratory (LANL), Los Alamos, United States 67Syracuse University, Syracuse, NY, United States 68Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria, associated to 2 69School of Physics and Astronomy, Monash University, Melbourne, Australia, associated to 55 70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2 71Guangdong Provencial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou, China, associated to 3 72School of Physics and Technology, Wuhan University, Wuhan, China, associated to 3 73Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia, associated to 12 74Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16 75Van Swinderen Institute, University of Groningen, Groningen, Netherlands, associated to 31 76Universiteit Maastricht, Maastricht, Netherlands, associated to 31 77National Research Centre Kurchatov Institute, Moscow, Russia, associated to 38 78National University of Science and Technology “MISIS”, Moscow, Russia, associated to 38 79National Research University Higher School of Economics, Moscow, Russia, associated to 41 80National Research Tomsk Polytechnic University, Tomsk, Russia, associated to 38 81University of Michigan, Ann Arbor, United States, associated to 67 aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil bLaboratoire Leprince-Ringuet, Palaiseau, France cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia dUniversità di Bari, Bari, Italy eUniversità di Bologna, Bologna, Italy fUniversità di Cagliari, Cagliari, Italy gUniversità di Ferrara, Ferrara, Italy hUniversità di Genova, Genova, Italy iUniversità di Milano Bicocca, Milano, Italy jUniversità di Roma Tor Vergata, Roma, Italy kAGH - University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Kraków, Poland lDS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain mHanoi University of Science, Hanoi, Vietnam nUniversità di Padova, Padova, Italy oUniversità di Pisa, Pisa, Italy pUniversità degli Studi di Milano, Milano, Italy qUniversità di Urbino, Urbino, Italy rUniversità della Basilicata, Potenza, Italy sScuola Normale Superiore, Pisa, Italy tUniversità di Modena e Reggio Emilia, Modena, Italy uUniversità di Siena, Siena, Italy vMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines wNovosibirsk State University, Novosibirsk, Russia xINFN Sezione di Trieste, Trieste, Italy ySchool of Physics and Information Technology, Shaanxi Normal University (SNNU), Xi’an, China zUniversidad Nacional Autonoma de Honduras, Tegucigalpa, Honduras

2024-09-04T02:54:58.337551

2003.04013

arxiv-papers

11institutetext: ASTRON - the Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, the Netherlands 22institutetext: Department of Electronic and Electrical Engineering, University of Bath, Claverton Down, Bath, BA2 7AY, UK 33institutetext: School of Science and Technology, Nottingham Trent University, Clifton Lane, Nottingham, NG11 8NS, UK 44institutetext: Space Environment and Radio Engineering, School of Engineering, The University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK 55institutetext: Space Research Centre, Polish Academy of Sciences, Bartycka 18A, 00-716 Warsaw, Poland 66institutetext: Space Radio-Diagnostics Research Centre, University of Warmia and Mazury, ul. Romana Prawocheskiego 9, 10-719 Olsztyn, Poland 77institutetext: Technische Universität Berlin, Institut für Geodäsie und Geoinformationstechnik, Fakultät VI, Sekr. H 12, Hauptgebäude Raum H 5121, Straße des 17. Juni 135, 10623 Berlin, Germany 88institutetext: GFZ German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, Germany 99institutetext: Shell Technology Center, Bangalore, India 1010institutetext: Science and Technology B.V., Delft, the Netherlands 1111institutetext: RAL Space, UKRI STFC, Rutherford Appleton Laboratory, Harwell Campus, Oxfordshire, OX11 0QX, UK 1212institutetext: Mt Stromlo Observatory, Research School of Astronomy and Astrophysics, Australian National University, Cotter Road, Weston Creek, ACT 2611, Australia 1313institutetext: Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany 1414institutetext: Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029, Hamburg, Germany 1515institutetext: Thüringer Landessternwarte, Sternwarte 4, D-07778 Tautenburg, Germany 1616institutetext: Jodrell Bank Centre for Astrophysics (JBCA), Department of Physics & Astronomy, Alan Turing Building, Oxford Road, University of Manchester, Manchester M139PL, UK 1717institutetext: Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, The Netherlands 1818institutetext: LPC2E - Université d’Orléans / CNRS, 45071 Orléans cedex 2, France 1919institutetext: Station de Radioastronomie de Nançay, Observatoire de Paris, PSL Research University, CNRS, Univ. Orléans, OSUC, 18330 Nançay, France 2020institutetext: Radboud University, Department of Astrophysics/IMAPP, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands 2121institutetext: Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands 2222institutetext: Vrije Universiteit Brussel, Astronomy and Astrophysics Research Group, Pleinlaan 2, 1050 Brussel, Belgium 2323institutetext: Kapteyn Astronomical Institute, University of Groningen, P.O.Box 800, 9700AV Groningen, the Netherlands 2424institutetext: Leibniz- Institut für Astrophysik Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany 2525institutetext: ECAP, Friedrich-Alexander-Universität Erlangen- Nürnberg, Erwin-Rommel-Str. 1, 91054 Erlangen, Germany 2626institutetext: DESY, Platanenallee 6, 15738 Zeuthen, Germany 2727institutetext: CIT, Rijksuniversiteit Groningen, Nettelbosje 1, 9747 AJ Groningen, The Netherlands 2828institutetext: Max- Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany 2929institutetext: Anton Pannekoek Institute, University of Amsterdam, Postbus 94249, 1090 GE Amsterdam, The Netherlands 3030institutetext: Fakultät für Physik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany 3131institutetext: South African Radio Astronomy Observatory, 2 Fir Street, Black River Park, Observatory, Cape Town, 7925, South Africa 3232institutetext: Department of Physics and Astronomy, University of the Western Cape, Cape Town 7535, South Africa 3333institutetext: Department of Physics and Electronics, Rhodes University, PO Box 94, Makhanda, 6140, South Africa 3434institutetext: Jagiellonian University in Kraków, Astronomical Observatory, ul. Orla 171, PL 30-244 Kraków, Poland 3535institutetext: Department of Physics, Khalifa University, PO Box 127788, Abu Dhabi, United Arab Emirates 3636institutetext: Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-439 92 Onsala, Sweden 3737institutetext: JIVE, Joint Institute for VLBI-ERIC, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, the Netherlands 3838institutetext: LESIA & USN, Observatoire de Paris, CNRS, PSL, SU/UP/UO, 92195 Meudon, France # A LOFAR Observation of Ionospheric Scintillation from Two Simultaneous Travelling Ionospheric Disturbances R.A. Fallows111Corresponding author<EMAIL_ADDRESS>11 B. Forte 22 I. Astin 22 T. Allbrook222Now at BAE Systems (operation) Ltd. 22 A. Arnold333Now an independent researcher 22 A. Wood 33 G. Dorrian 44 M. Mevius 11 H. Rothkaehl 55 B. Matyjasiak 55 A. Krankowski 66 J.M. Anderson 7788 A. Asgekar 99 I.M. Avruch 1010 M.J.Bentum 11 M.M. Bisi 1111 H.R. Butcher 1212 B. Ciardi 1313 B. Dabrowski 66 S. Damstra 11 F. de Gasperin 1414 S. Duscha 11 J. Eislöffel 1515 T.M.O Franzen 11 M.A. Garrett 16161717 J.-M. Grießmeier 18181919 A.W. Gunst 11 M. Hoeft 1515 J.R. Hörandel 202021212222 M. Iacobelli 11 H.T. Intema 1717 L.V.E. Koopmans 2323 P. Maat 11 G. Mann 2424 A. Nelles 25252626 H. Paas 2727 V.N. Pandey 112323 W. Reich 2828 A. Rowlinson 112929 M. Ruiter 11 D.J. Schwarz 3030 M. Serylak 31313232 A. Shulevski 2929 O.M. Smirnov 33333131 M. Soida 3434 M. Steinmetz 2424 S. Thoudam 3535 M.C. Toribio 3636 A. van Ardenne 11 I.M. van Bemmel 3737 M.H.D. van der Wiel 11 M.P. van Haarlem 11 R.C. Vermeulen 11 C. Vocks 2424 R.A.M.J. Wijers 2929 O. Wucknitz 2828 P. Zarka 3838 P. Zucca 11 (Received November 30, 2019) ###### Abstract This paper presents the results from one of the first observations of ionospheric scintillation taken using the Low-Frequency Array (LOFAR). The observation was of the strong natural radio source Cassiopeia A, taken overnight on 18-19 August 2013, and exhibited moderately strong scattering effects in dynamic spectra of intensity received across an observing bandwidth of 10-80 MHz. Delay-Doppler spectra (the 2-D FFT of the dynamic spectrum) from the first hour of observation showed two discrete parabolic arcs, one with a steep curvature and the other shallow, which can be used to provide estimates of the distance to, and velocity of, the scattering plasma. A cross- correlation analysis of data received by the dense array of stations in the LOFAR “core” reveals two different velocities in the scintillation pattern: a primary velocity of $\sim$20-40 m s-1 with a north-west to south-east direction, associated with the steep parabolic arc and a scattering altitude in the F-region or higher, and a secondary velocity of $\sim$110 m s-1 with a north-east to south-west direction, associated with the shallow arc and a scattering altitude in the D-region. Geomagnetic activity was low in the mid- latitudes at the time, but a weak sub-storm at high latitudes reached its peak at the start of the observation. An analysis of Global Navigation Satellite Systems (GNSS) and ionosonde data from the time reveals a larger–scale travelling ionospheric disturbance (TID), possibly the result of the high–latitude activity, travelling in the north-west to south-east direction, and, simultaneously, a smaller–scale TID travelling in a north-east to south- west direction, which could be associated with atmospheric gravity wave activity. The LOFAR observation shows scattering from both TIDs, at different altitudes and propagating in different directions. To the best of our knowledge this is the first time that such a phenomenon has been reported. ###### keywords: ionospheric scintillation – travelling ionospheric disturbances – instability mechanisms ## 1 Introduction Radio waves from compact sources can be strongly affected by any ionised medium through which they pass. Refraction through large-scale density structures in the medium leads to strong lensing effects where the radio source appears, if imaged, to focus, de-focus and change shape as the density structures in the line of sight themselves move and change. Diffraction of the wavefront by small-scale density structures leads to variations building up in the intensity of the wavefront with distance from the scattering medium, due to interference between the scattered waves, an effect known as scintillation. Observations of all these effects thus contain a great deal of information on the medium through which the radio waves have passed, including the large- scale density, turbulence, and the movement of the medium across the line of sight. Since the second world war, a large number of studies have shown the effect of ionospheric density variations on radio signals, as reviewed by Aarons (1982), and this can lead to disruption for applications using Global Navigation Satellite Systems (GNSS, e.g., GPS), as thoroughly reviewed by, e.g., Hapgood (2017). The Low-Frequency Array (LOFAR - van Haarlem et al. (2013)) is Europe’s largest low-frequency radio telescope, operating across the frequency band 10–250 MHz, and with a dense array of stations in the Netherlands and, at the time of writing, 13 stations internationally from Ireland to Poland. It was conceived and designed for radio astronomy but, at these frequencies, the ionosphere can also have a strong effect on the radio astronomy measurement (de Gasperin et al., 2018). Ionospheric scintillation, which is rarely seen over the mid-latitudes on the high-frequency signals of GNSS, is seen almost continually in observations of strong natural radio sources by LOFAR. The wide bandwidth available with LOFAR enables an easy and direct assessment of scattering conditions and how they change in a given observation, including whether scattering is weak or strong, or refractive effects dominate, and enables further information to be gleaned from delay-Doppler spectra (the 2-D FFT of a dynamic spectrum, termed variously as the “scattering function”, “generalised power spectrum”, or “secondary spectrum” - here we use the term “delay-Doppler” spectrum as this clearly describes what the spectrum shows). In observations of interstellar scintillation these spectra can exhibit discrete parabolic arcs which can be modelled to give information on the distance to the scattering “screen” giving rise to the scintillation and its velocity across the line of sight (Stinebring et al., 2001; Cordes et al., 2006). Broadband observations of ionospheric scintillation are not common, but such arcs have been observed using the Kilpisjärvi Atmospheric Imaging Receiver Array (KAIRA, McKay-Bukowski et al. (2014) – an independent station built using LOFAR hardware in arctic Finland) in a study by Fallows et al. (2014). Model spectra produced by Knepp and Nickisch (2009) have also illustrated parabolic arc structures, particularly in the case of scattering from a thin scattering screen. The wide spatial distribution of LOFAR stations also enables scintillation conditions at these observing frequencies to be sampled over a large part of western Europe. A dense “core” of 24 stations, situated near Exloo in the north-east of the Netherlands, over an area with a diameter of $\sim$3.5 km further provides a more detailed spatial view of the scintillation pattern in its field of view. LOFAR thus enables detailed studies of ionospheric scintillation to be undertaken which can both reveal details which would be unavailable to discrete-frequency observations such as those taken using GNSS receivers, and act as a low-frequency complement to these observations to probe potentially different scattering scales. A number of different phenomena can lead to scattering effects in radio wave propagation through the mid-latitude ionosphere: Ionisation structures due to gradients in the spatial distribution of the plasma density can arise from a southward expansion of the auroral oval or from large- to small- scale travelling ionospheric disturbances (TIDs). Large-scale TIDs (LSTIDs) with wavelengths of about 200 km typically propagate southward after forming in the high-latitude ionosphere in response to magnetic disturbances (e.g. storms or sub-storms, Tsugawa et al. (2004)). On the other hand, medium-scale TIDs (MSTIDs) seem to form in response to phenomena occurring in the neutral atmosphere triggering atmospheric gravity waves (AGWs), which then propagate upwards to generate TIDs at ionospheric heights (Kelley, 2009). The morphology of MSTIDs varies with local time, season, and magnetic longitude. Their propagation shows irregular patterns that vary on a case-by-case basis, although they commonly seem to propagate mainly equatorward during winter daytime and westward during summer night-time (Hernández-Pajares et al., 2006, 2012; Tsugawa et al., 2007; Saito and Fukao, 1998; Emardson et al., 2013). Smaller-scale ionisation gradients, likely associated with the Perkins instability (Kelley, 2009, 2011), can then form as a consequence of the presence of MSTIDs, potentially leading to scintillation at LOFAR frequencies. In this paper, we perform an in-depth analysis of ionospheric scintillation seen in an observation of the strong natural radio source Cassiopeia A (Cas A) overnight on 18-19 August 2013. This observation was amongst the first of its kind taken with LOFAR and exhibited quite strong scattering effects across the 10-80 MHz band. The purpose of this paper is both technical and scientific: We first describe the observation itself, and then demonstrate several techniques to analyse LOFAR data and show how these can bring out the details of ionospheric structures. Finally, we use supporting data from GNSS and ionosondes to get a broader picture of conditions in the ionosphere at the time and how these give rise to the scintillation seen by LOFAR. ## 2 The LOFAR Observation LOFAR observed Cas A (Right Ascension 23h23m24s, Declination +58°48’54”) between 21:05 UT on 18 August 2013 and 04:05 UT on 19 August 2013, recording dynamic spectra from each individual station with a sampling time of 0.083 s over the band 2.24-97.55 MHz from each available station. The observing band was sampled with 7808 channels of 12.207 kHz each, but averaged over each successive 16-channel block to 488 subbands of 195.3125 kHz for the analyses described in this paper. At the time of observation the available stations were the 24 stations of the LOFAR “core”, 13 “remote” stations across the north-east of the Netherlands, and the international stations at Effelsburg, Unterweilenbach, Tautenburg, Potsdam, and Jülich (Germany), Nançay (France), Onsala (Sweden), and Chilbolton (UK). The reader is referred to van Haarlem et al. (2013) for full details of the LOFAR receiving system. The raw data for this observation can be obtained from the LOFAR long-term archive (lta.lofar.eu); observation ID L169059 under project “IPS”. We first illustrate the data in a more traditional sense. Figure 1 shows time series’ at three discrete observing frequencies of the data taken by LOFAR station CS002, at the centre of the core, and their associated power spectra. The power spectra show a fairly typical shape for intensity scintillation: An initial flat section at the lowest spectral frequencies represents scattering from larger-scale density structures which are close enough to the observer that the scattered waves have not had the space to fully interfere to develop a full intensity scintillation pattern; the turnover (often termed the “Fresnel Knee”) indicates the largest density scales for which the intensity scintillation pattern has fully formed; this is followed by a power-law in the spectra illustrating the cascade from larger to smaller density scales, which is cut off in these spectra by white noise due to the receiving system (the flat section covering high spectral frequencies). Figure 1: 1 Time series of intensity received at three discrete frequencies by LOFAR station CS002 during the observation of Cas A on 18-19 August 2013, plus, 1 and 1, power spectra of two 10-minute periods within these time series’. However, the advantage of observing a natural radio source with LOFAR is that full dynamic spectra can be produced covering the full observed band. Dynamic spectra of data taken by LOFAR station CS002 are presented in Figure 2, which includes a dynamic spectrum of the full observation, alongside more detailed views of three different single hours of the observation to illustrate the range of scattering conditions seen. The strength of the scattering can be seen much more clearly in this view, compared to time series’ from discrete observing frequencies. In general, scattering appears weak in this observation at the highest observing frequencies (where intensity remains highly correlated across the observing band) with a transition to strong scattering conditions as the observing frequency decreases. The frequency range displayed in these dynamic spectra is restricted to exclude the radio–frequency interference (RFI) which dominates below about 20 MHz and a fade in signal strength at the higher frequencies due to the imposition of a hard filter to exclude the FM waveband. Figure 2: Dynamic spectra of normalised intensity data taken by LOFAR station CS002 during the observation of Cas A on 18-19 August 2013. The dynamic spectrum of the entire observing period is given at the top, with zooms into three different hours of observation below to illustrate the range of conditions seen. White areas within the plots indicate where RFI was identified. RFI is still visible as white areas within the plots. These were identified by applying a median filter to the data using a window of (19.5 MHz $\times$ 4.2 s) to flatten out the scintillation pattern and then applying a threshold to identify the RFI. This method appears to be quite successful at identifying the RFI without also falsely identifying strong peaks in the scintillation as RFI. For subsequent analysis the RFI data points are replaced by an interpolation from nearby data, using the Python Astropy (Astropy Collaboration et al., 2013; Price-Whelan et al., 2018) library routine, “interpolate_replace_nans”. Normalisation of the data, to correct for long- period temporal variations in the system (e.g., gain variations resulting from the varying sensitivity of the receiving antenna array with source elevation), is carried out after RFI excision by dividing the intensities for each single frequency subband by a fitted 3rd-order polynomial. When analysing the data, a variety of scattering conditions are observed during the course of the observation, as indicated in Figure 2. Different conditions also naturally occurred over the various international stations compared to those observed over the Dutch part of LOFAR. In this paper we therefore focus our analysis on only the first hour of observation and the measurements taken by the 24 core stations. This allows us to demonstrate the analysis techniques and to investigate the reason for the scintillation seen over this interval. Observations from later in this dataset undoubtedly show other effects and may be discussed in a subsequent publication. ## 3 LOFAR Data Analysis Methods and Results ### 3.1 Delay-Doppler Spectra The first stage of analysis was the calculation of delay–Doppler spectra: These were created from the dynamic spectra using five-minute time slices, advancing every minute through the observation, following the methods described in Fallows et al. (2014). To avoid regions more heavily contaminated by RFI, the frequency band used was restricted to 28.5–64.1 MHz. Example spectra from the first hour are presented in Figure 3. Figure 3: Example delay-Doppler spectra from the first hour of observation, taken using five-minute chunks of the dynamic spectrum from CS103 over the frequency band 28.5-64.1 MHz. The spectra show two clear arcs: the first is a steeper arc which varies in curvature throughout the first hour (henceforth labelled for convenience as the “primary arc”); the second is a very shallow arc (henceforth labelled as the “secondary arc”) which remains stable for the first 40 minutes of the observation before fading away. By the end of the first hour of observation the primary arc also becomes less distinctive for a short while before the delay–Doppler spectra again show distinctive structure, including a return of the secondary arc. The variability of the curvature of the primary arc appears to follow a wave–like pattern during this part of the observation, as displayed in Figure 4. Here, simple parabolas involving only the square term ($y=Cx^{2}$ where $C$ is the curvature) were plotted with various curvatures until a reasonable eyeball fit was achieved, and the resulting curvatures plotted for every minute of observation for the first hour. It proved impossible to achieve reasonable fits using least-squares methods due to confusion from non–arc structure in the spectra: Fitting curvatures to these scintillation arcs is a well–known problem in the interstellar scintillation field and new methods of attempting this were presented at a recent workshop, but they are not easily described and have yet to be published. Hence, we do not attempt their application here. Figure 4: Curvatures of the steeper arc seen in delay-Doppler spectra calculated using data from CS103, from simple parabolas fitted by eye. The grey bounds represent an estimated error. The presence of two scintillation arcs likely indicates that scattering is dominated by two distinct layers in the ionosphere. A simple analysis, as described in Fallows et al. (2014), can be used to estimate the altitude of the scattering region with a basic formula relating arc curvature $C$ to velocity $V$ and distance $L$ along the line of sight to the scattering region (Cordes et al., 2006): $L=2CV^{2}$ (1) The square term for the velocity illustrates the importance of gaining a good estimate of velocity to be able to accurately estimate the altitude of the scattering region via this method. ### 3.2 Scintillation Pattern Flow The core area of LOFAR contains 24 stations within an area with a diameter of $\sim$3.5 km. When viewing dynamic spectra from each of these stations it is clear that the scintillation pattern is mobile over the core (i.e., temporal shifts in the scintillation pattern are clear between stations) but does not necessarily evolve significantly. Therefore, the flow of the scintillation pattern over the core stations may be viewed directly by simply plotting the intensity received, for a single subband, by each station on a map of geographical station locations, for data from successive time steps. A movie (CasA_20130818_NL.mp4) of the scintillation pattern flow through the observation is published as an online supplement to this article. The result, for 12 example time steps, is displayed in Figure 5, where a band of higher intensities can be seen to progress from north-west to south-east over the core. It should be noted that the data were integrated in time to 0.92 s for this purpose, to reduce both flicker due to noise and the duration of the movie. This does not average over any scintillation structure in this observation; structure with periodicities shorter than one second would be obvious in the delay–Doppler spectra as an extension of the arc(s) to greater than 0.5 Hz along the Doppler frequency axis. Figure 5: Normalised intensities received by all core stations at an observing frequency of 44.13 MHz, plotted on a geographical map of the stations. The intensities are colour-coded using a colour scale from yellow to purple with a range of 0.8 to 1.3 respectively. Times are at $\sim$10 s intervals from 21:22:25 UT at top left to 21:24:15 UT at bottom right, and each plot uses data samples with an integration time of 0.92 s. Plot diameter is $\sim$4.5 km. However, this is not the entire picture because the lines of sight from radio source to receivers are moving through the ionosphere as the Earth rotates, meaning that the scintillation pattern flow observed is a combination of flow due to the movement of density variations in the ionosphere and the movement of the lines of sight themselves through the ionosphere. Since the speed with which any single point on a line of sight passing through the ionosphere is dependent on the altitude of that point (the so-called ionosphere “pierce point”), this altitude needs to be either assumed or calculated to estimate a correction to the overall flow speed to obtain the natural ionospheric contribution. This introduces a natural uncertainty into estimates of velocity. Figure 6 shows the track of an ionospheric pierce-point at an assumed altitude of 200 km (an altitude chosen as representative of a typical F-region altitude where large-scale plasma structures are commonly observed) for the line of sight from core station CS002 to the radio source Cassiopeia A through the 7-hour course of the observation to illustrate this movement. Although not the subject of this paper, it is worth noting that an east to west flow seen later in the observation appears to be solely due to the lines of sight moving across a mostly static ionospheric structure (see the online movie), if the 200 km pierce point is assumed, further illustrating the necessity to take accurate care of the contribution from line of sight movement when assessing ionospheric speeds. Figure 6: Map showing the track of the 200 km pierce point of the line of sight from core station CS002 to Cassiopeia A from 2013-08-18T21:05:00 to 2013-08-19T04:05:00 UT. The thicker orange part of the track enhances the first hour of the observation. The black line winding a path across the centre of the image is the location of the border between the Netherlands and Germany. The location of CS002 is marked with a black star. The movie of the scintillation pattern flow, assuming a 200 km pierce point, shows a clear general north-west to south-east flow during the first hour of the observation, but also indicates some short (minutes) periods of confusion in which a north-east to south-west component might be just about discernable. Any second flow is likely to be associated with a second ionospheric layer and so warrants further investigation. ### 3.3 Estimating Velocities The representation of the scintillation pattern flow in movie form gives a direct and broad picture of the flow pattern and is very helpful in discovering short time-scale changes in speed and direction. However a cross- correlation analysis is still necessary to assess actual velocity(s). Correlation functions are calculated as follows:- * • Time series’ of intensity received by each station are calculated by averaging over the frequency band 55–65 MHz, with these frequencies chosen as the scintillation pattern remains highly correlated over this band; * • For each three-minute data slice, advancing the start time of each successive slice by 10 s:- * – Calculate auto- and cross- power spectra using intensities from every station pair within the LOFAR core; * – Apply low- and high-pass filters to exclude the DC-component and any slow system variation unlikely to be due to ionospheric effects, and white noise at the high spectral frequencies. The white noise is also subtracted using an average of spectral power over the high frequencies; * – Inverse–FFT the power spectra back to the time domain to give auto- and cross- correlation functions. In the analysis the high- and low-pass filter values were set to 0.01 Hz and 0.5 Hz respectively. This process results in a large set of cross-correlation functions for each time slice, each of which has an associated station-station baseline and a primary peak at, typically, a non-zero time delay from which a velocity can be calculated. However, the direction of the scintillation pattern flow still needs to be found for calculation of the actual velocity. For this, directions were assumed for each degree in the full 360–degree range of possible azimuth directions and the velocities re-calculated using the components of all baselines aligned with each assumed direction. This results, for each time slice, in 360 sets of velocities and from each set a median velocity and standard deviation about the median can be calculated (the median is used as this is less susceptible to rogue data points than the mean). The actual flow direction corresponds to the azimuth with the maximum median velocity and minimum standard deviation, as illustrated in Figure 7. From this analysis the primary velocity of $\sim$20–40 m s-1 travelling from north-west to south-east is found, illustrated in Figure 7, corresponding to the obvious scintillation pattern flow seen in the movie. However, the presence of a second flow is still not obvious, although a hint of it can be seen in, for example, the second peak in the median velocity seen in Figure 7. Figure 7: Plots for single 3-minute time slices of the median velocity and standard deviation of velocities about the median versus azimuth direction, calculated from the range of velocities found from all cross-correlation functions with the baselines within each station pair re-calculated for each assumed azimuth direction, in the usual form, counting clockwise from north. 7 Time slice commencing 21:05:00 UT using cross-correlations calculated after applying a high-pass filter at 0.01 Hz; 7 Time slice commencing 21:15:00 UT using cross-correlations calculated after applying a high-pass filter at 0.07 Hz. Note that the same y-axis is used for both velocity and standard deviation. A closer look at the auto- power spectra yielded the key to finding the second flow. Many spectra show a “bump” which can be viewed as being a second spectrum superposed on the main one. This is illustrated in Figure 8. To isolate this part of the spectrum, the spectra were re-filtered with a high- pass filter value of 0.07 Hz (the low-pass filter value remained the same), and correlation functions re-calculated. After following the same analysis as above to find median velocities and standard deviations, the second flow was found, as illustrated in Figure 7. Figure 8: Example power spectrum calculated from three minutes of intensity data received by CS003. The black curve is the raw spectrum, the blue curve is the filtered and noise-subtracted spectrum. The locations of the low-pass filter and both high-pass filters used are illustrated. The analysis, using both high-pass filter values, has been carried out for the full data set. The velocities and associated directions in degrees azimuth for the first hour of the observation are given in Figure 9. Error bounds in the velocities are calculated as the standard deviation about the median of all velocity values available for the calculated azimuth direction. Figure 9: Top: Velocities calculated for the first hour of observation from cross-correlations created after filtering using the two different high-pass filter values. Bottom: Directions of these velocities, in degrees azimuth. The higher velocity (henceforth labelled as the “secondary velocity”) shows some scatter: Periods where the secondary velocity drops to around the primary velocity values are due to the secondary velocity not being detected at these times; in these cases, it can still be detected in short-duration drops of velocity if correlation functions are re-calculated using an even higher high- pass filter value (the bump in these spectra appears shifted to slightly higher spectral frequencies). Values which decrease/increase towards/away from the primary velocity values likely represent a mix between the two velocities. The larger error bars seen in velocities may also be indicative in some instances of the standard deviation being broadened by some velocity values being more dominated by the other flow. The more extended period of scatter around 21:40 to 22:00 UT is a period where the secondary velocity is less apparent and the secondary scintillation arc fades from the delay-Doppler spectra. This indicates that the secondary structure is restricted in either space or time, either moving out of the field of view of the observation or ceasing for a period around 21:40 UT. It gives a first indication that the secondary velocity is associated with the secondary scintillation arc. ### 3.4 Estimating Scattering Altitudes The velocities can now be used to estimate scattering altitudes, using the curvatures of the scintillation arcs and the simple formula given in Equation 1. Initially the movement of the line of sight through the ionosphere is not accounted for, since this correction also requires an estimate of the pierce- point altitude to be reasonably calculated. Therefore an initial calculation of the scattering altitudes is made based on velocity values which are not corrected for this movement. Using the primary velocities and combining these with the curvatures of the primary arc (Figure 4) in Equation 1, a range of distances, L, along the line of sight to the scattering region are found. These distances are converted to altitudes by accounting for source elevation (Cas A increased in elevation from 55 ∘ to 64 ∘ during the first hour of observation). This process resulted in a range of altitudes to the scattering region of 200 to 900 km. Doing the same for the secondary velocities and applying an arc curvature of 3.2$\pm$0.3 for the secondary scintillation arc gives estimated scattering altitudes of only $\sim$70 km. If the primary/secondary velocities are combined vice-versa with the secondary/primary arc curvatures respectively, then the resulting scattering altitudes are clearly unreasonable (the secondary arc, primary velocity combination gives estimated altitudes of only $\sim$10 km for example), lending further credence to the secondary velocity being associated with the secondary arc. Velocity contributions from the line of sight movement are calculated as follows: For each time slice, t, the geographical locations beneath the pierce point of the line of sight through the ionosphere corresponding to the estimated scattering altitude at t are calculated, for both t and t + $\delta$t, where $\delta$t is taken as 3 minutes (the actual value is unimportant for this calculation). A velocity and its direction are found from the horizontal distance between these two locations and the direction of travel from one to the other. The general direction of the movement of the line of sight through the ionosphere is indicated by the orange line in Figure 6. Although the high scattering altitudes related to the primary scintillation arc and primary scintillation velocities lead to line-of-sight movements of up to $\sim$35 m s-1, this movement is almost perpendicular to the direction of the primary scintillation velocity, limiting the actual contribution to $\sim$ 5 m s-1. The line of sight movement is, however, in a very similar direction to the secondary velocities but the low corresponding scattering altitudes also limit the contribution in this case to $\sim$5 m s-1. An iterative procedure is then followed to correct the scintillation velocities for line-of-sight movement at the calculated scattering altitudes, re-calculate these altitudes, and re-calculate the line-of-sight movement. This procedure converges to a set of final scattering altitudes within 5 iterations. These are presented in Figure 10, with error bounds taken as the lowest and highest possible altitudes resulting from applying this procedure using the lower and upper limits of the arc curvature and scintillation velocity error bounds. Figure 10: Scattering altitudes estimated using Equation 1, the primary velocities and primary scintillation arc curvatures (blue curve) and the secondary velocities and the curvature of the secondary scintillation arc (red dashed curves). The range of scattering altitudes encompassed by the error bounds is quite large in some instances, particularly where the calculated altitudes are higher. Although the square term for the velocity in Equation 1 could lead to the natural conclusion that the error in the velocity dominates the error in scattering altitude, the errors in the velocity calculations are, for the most part, relatively small. Nevertheless, the error in the secondary velocity does appear to be the dominant error in the lower range of scattering altitudes (the red curves in Figure 10. However, the dominant error for the higher range of scattering altitudes appears to be the scintillation arc curvatures, illustrating the importance of developing accurate fitting methods for these curvatures. Despite these concerns, it is clear that scattering is seen from two layers in the ionosphere; the primary scintillation arc arises from scattering in the F-region and the secondary scintillation arc arises from scattering much lower down in the D-region. Plasma decays by recombination with neutral species. In the F-region these densities are lower and so plasma lifetimes are longer than in the D-region. Typical plasma lifetimes in the F-region are of the order of hours, while they are of the order of minutes in the D-region. Hence the structures seen in each level may have a different source and time history. ## 4 Conditions in the Ionosphere We now investigate what the overall ionospheric conditions were at the time and hence the possible cause(s) of the scintillation seen by LOFAR at the time. ### 4.1 Geomagnetic Conditions The overall geomagnetic conditions at the time are given in Figure 11, which shows 24–hour traces of the H–component of magnetic field for a representative set of magnetometers from the Norwegian magnetometer chain for 18 August 2013. Activity can be described as unsettled, with a minor substorm at high latitudes, peaking at the start of the LOFAR observation. However, geomagnetic activity remains quiet further south, and Kp took a value of 1 at 21 UT on 18th August 2013, indicating that this is unlikely to be a direct cause of the scintillation seen at LOFAR latitudes. We therefore investigate whether TIDs were present at the time and whether these could be consistent with the scintillation seen by LOFAR. Figure 11: Traces of the H-component of the geomagnetic field recorded on 18 August 2013 by a selection of magnetometer stations from the Norwegian chain. From top to bottom these are, along with their geomagnetic latitudes (2004, altitude 100 km): Longyearbyen (75.31∘N), Bjørnøya (71.52∘N), Nordkapp (67.87∘N), Tromsø(66.69∘N), Rørvik (62.28∘N), and Karmøy (56.43∘N). ### 4.2 Ionosonde Data The presence of TIDs can be detected through the simultaneous appearance of wave-like structures on multiple sounding frequencies recorded by an ionosonde. This method is generally limited to a single point of observation and detection. The spatial extent of TIDs can be attempted by comparing multiple traces from different ionosondes, but this is limited by the low density of ionosondes in a given region. Measurements from the ionosonde in Chilton (UK) do indeed suggest the presence of wave-like patterns which, in principle, could be due to a large-scale TID propagating southward and/or MSTID triggered by a local Atmospheric Gravity Wave (Figure 12). Figure 12: Multiple traces from the ionosonde in Chilton (UK) recorded between 20:00 18 August 2013 and 06:00 19 August 2013. ### 4.3 GNSS Data However, measurements from ground-based GNSS receivers offer a more comprehensive view of the characteristics of any MSTIDs present (Kelley, 2009). In the present study, we focus on perturbations in the slant Total Electron Content (STEC) observed over the evening of 18 August 2013 from a network of GNSS stations around the LOFAR core stations (see Figure 13). These stations are sufficient to infer the presence of TIDs and to infer the upper spatial scale-size limit of smaller-scale irregularities causing the intensity scintillation seen at LOFAR wavelengths. The presence of TID-induced perturbations can be deduced from the presence of wave-like residuals on the STEC calculated for each satellite-receiver pair. Figure 13: Map showing the locations of the GNSS stations used. STEC was calculated and detrended following the methods of Hernández-Pajares et al. (2006), with the detrending carried out according to: $\Delta STEC\left(t\right)=STEC\left(t\right)-\frac{STEC\left(t+\tau\right)+STEC\left(t-\tau\right)}{2}\left[TECu\right]$ (2) where $\tau=300s$. It is worth noting that the measured carrier phases $L_{1}$ and $L_{2}$ vary with time as a consequence of the motion of GNSS satellites relative to a given receiver on the Earth’s surface. As such, the spatial and temporal variabilities of ionisation gradients (such as those connected with TIDs and corresponding instabilities) become entangled. The various detrending methods (similar to equation 2) lead to an estimate of ionisation gradients by considering temporal gradients only, with spatial and temporal variabilities intrinsically entangled in the GNSS observations. Figure 14 shows examples of wave-like residuals on STEC for one pair of GNSS stations (Dentergem and Bruxelles in Belgium) observing the same GNSS satellite. The wave pattern is strongest over the first two hours shown (18:00 - 20:00 UT) but then weakens considerably by the start of the LOFAR observation, although it remains evident. STEC from the observations of both stations appears well–correlated, with the Bruxelles dataset lagging behind that of Dentergem. Since Dentergem lies to the WNW of Bruxelles, this suggests a strong westerly component in the direction of travel, which could correspond with the secondary velocity seen by LOFAR. Figure 14: Example of a satellite-station pair. 14 PRN01 as observed on 18 August 2013 from Dentergem (DENT, blue line) and Bruxelles (BRUX, red line), both in Belgium, with baseline oriented from WNW to ESE; 14 azimuth/elevation plot for PRN01 as observed from Dentergem. Figure 15 shows hourly plots of the overall geographical distribution of the STEC residuals calculated for all satellite passes seen within each hour by the GNSS stations used. The patterns shown in Figure 15 suggest a spatially and temporally varying propagation of MSTID wavefronts with components along the NE-SW as well as the NW-SE directions. Furthermore, the examples shown in Figure 15 also indicate the presence of smaller-scale ionisation structures in proximity to the wavefronts of the MSTIDs. This suggests that the scintillation seen by LOFAR is likely associated with the perpendicular propagation of two MSTIDs. However, the STEC variations here are also seen to fade by the start of the LOFAR observation. Figure 15: Hourly geographical distribution of all STEC perturbations in the evening of 18 August 2013: 15 18:00-19:00 UT, 15 19:00-20:00 UT, 15 20:00-21:00 UT, and 15 21:00-22:00 UT. A further illustration looks at the overall power spectral densities for the STEC residuals on all satellite–receiver pairs considered here over the hourly periods 20:00 UT to 21:00] UT and 21:00 UT to 22:00 UT (Figure 16). The earlier hour is chosen alongside the hour covering the LOFAR data period as this better displays the components seen in the spectra The temporal frequencies f can be converted into spatial scales L by assuming a given velocity VREL for the motion of the ionospheric structures across a GNSS raypath. That is: $L=\frac{V_{REL}}{f}$ (3) where VREL = VIONO-VSAT is the relative velocity between the velocity of the ionospheric structures and the scan velocity of a single raypath (at the same shell height). VSAT can be of the order of a few tens of m s-1 at 300 km. Figure 16: Power Spectral Densities of all the TEC residuals considered during the hours 16 20:00-21:00 UT and 16 21:00-22:00 UT. The arrows indicate the two components considered in the text. There appear to be two main components in the energy cascade from larger to smaller ionisation scales: one with a period of $\sim$1666 s, and another component with a period of $\sim$666 s. Taking VREL to be $\sim$100 m s-1 (the secondary velocity seen by LOFAR as this is in a south-westerly direction and the example GNSS data in Figure 14 indicate a westerly component), these periodicities correspond to spatial scales of the order of 166 km and 66 km respectively. Beyond these scales the STEC analysis is limited by the sensitivity of the technique (Tsugawa et al., 2007), as the Power Spectral Densities reach the noise floor (Figure 16). These orders of magnitudes suggest the presence of a larger–scale TID together with a smaller–scale TID (Kelley, 2009), while the energy cascade that can be observed through the Power Spectral Densities indicates that the large–scale structure breaks down into small–scale structures, likely owing to some instability mechanism. ### 4.4 Estimation of Scale Sizes of Plasma Structures The scale sizes of the plasma structures causing the scintillation seen by LOFAR can also be calculated. The variations in the intensity of the received signal are caused by irregularities with a spatial scale size ranging from the Fresnel dimension to an order of magnitude below this value (Basu et al., 1998). The Fresnel length DF is related to the wavelength of the radio wave $\lambda$ and the line of sight distance from the receiver to the scattering region L: $D_{F}=\sqrt{2\lambda L}$ (4) The Fresnel length was calculated for plasma structures at altitudes of 70 km, 200 km, 350 km and 700 km, elevations of 55∘ and 64∘, and at frequencies of 25.19 MHz, 35.15 MHz and 60.15 MHz, and the results are shown in Table 1. The altitudes were chosen to cover the range of altitudes identified for the primary and secondary features in the LOFAR analysis, with the addition of 350 km as this altitude is commonly used within studies using GNSS satellites. The elevations of the radio source at the start and the end of the first hour of observation were used to establish the range of Fresnel scales for each altitude. The frequencies were chosen to match Figure 1. Table 1 shows that the Fresnel length ranges between $\sim$1 km and $\sim$5 km and therefore the plasma structures causing the variations in signal intensity are likely to have a spatial scale size between $\sim$100 m and $\sim$5 km. The velocities calculated from the LOFAR data indicate that such structures would take tens of seconds to pass through the source-to-receiver line and the intensity variations in the observed signal occur on a similar timescale. Altitude | 70 km | 200 km | 350 km | 700 km ---|---|---|---|--- Frequency | | | | 25.19 MHz | 1.4 | 2.3–2.4 | 3.0–3.2 | 4.3–4.5 35.15 MHz | 1.2 | 1.9–2.0 | 2.6–2.7 | 3.6–3.8 60.15 MHz | 0.9 | 1.5–1.6 | 2.0–2.1 | 2.8–2.9 Table 1: The Fresnel length at altitudes of 70 km, 200 km, 350 km and 700 km for three different frequencies received by LOFAR station CS002. The ranges represent calculation using the source elevation for the start and for the end of the first hour of observation. Values are in km. ## 5 Further Discussion Geomagnetic activity was low in the mid-latitudes at the time, so enhanced activity was unlikely to be the direct cause of the scintillation observed. However, a weak sub-storm was seen at high latitudes and this reached its peak at the time of the start of the observation. An analysis of GNSS and ionosonde data reveals the presence of an MSTID travelling in the north-west to south- east direction. The larger-scale nature of this TID, and its direction of travel, are strongly consistent with the primary velocity and F-region scattering altitudes seen in the LOFAR observation. It is possible that this TID was caused by the geomagnetic activity at high latitude, but this is not confirmed. Simultaneously, an MSTID is also present travelling in a north-east to south-west direction which would most likely be associated with an atmospheric gravity wave propagating up from the neutral atmosphere. The smaller–scale nature of it, its direction of travel, and likely low-altitude source make it highly consistent with the secondary velocity and D–region scattering altitudes observed by LOFAR. The amplitude of TID activity observed through GNSS STEC residuals decreased after 20:00 UT (as visible from Figure 14 as well as from the comparison of hourly geographical maps in Figure 15). However, the LOFAR observation did not start until 21:05 UT and the presence of scintillation on the radio frequencies observed by LOFAR remained significant for much of the first hour of observation. Whilst the presence of MSTIDs seems evident from the ionosonde multiple traces and GNSS STEC residuals in the region considered, their signatures do not appear simultaneously above the LOFAR core stations between 21:00 UT and 22:00 UT. This can be explained by the inability of GNSS to detect smaller amplitudes in STEC residuals, as the noise floor is encountered for observations with pierce points above the core LOFAR stations (Figures 15 and 16). The scale sizes of plasma structures calculated for the LOFAR data indicate that these are an order of magnitude lower than those estimated from GNSS STEC. Smaller ionisation scales developing, for example, through the Perkins instability could induce scintillation on the VHF radio frequencies received by LOFAR but not on the L-band frequencies of GNSS. Hence, scintillation from these mid-latitude smaller-scale ionisation structures, formed through the Perkins instability in conjuction with the presence of TIDs, is likely to be what is detected through LOFAR. ## 6 Conclusions and Outlook This paper presents the results from one of the first observations of ionospheric scintillation taken using LOFAR, of the strong natural radio source Cassiopeia A taken overnight on 18–19 August 2013. The observation exhibited moderately strong scattering effects in dynamic spectra of intensity received across an observing bandwidth of 10–80 MHz. Delay–Doppler spectra from the first hour of observation showed two discrete parabolic arcs, one with a steep and variable curvature and the other with a shallow and static curvature, indicating that the scintillation was the result of scattering through two distinct layers in the ionosphere. A cross-correlation analysis of the data received by stations in the LOFAR core reveals two different velocities in the scintillation pattern: A primary velocity of $\sim$20-40 m s-1 is observed travelling in a north-west to south- east direction, which is associated with the primary parabolic arc and altitudes of the scattering layer varying in the range $\sim$200–700 km. A secondary velocity of $\sim$110 m s-1 is observed travelling in a north-east to south-west direction, which is associated with the secondary arc and a much lower scattering altitude of $\sim$60–70 km. The latter velocity is associated with a secondary “bump” seen at higher spectral frequencies in power spectra calculated from time series’ of intensities, indicating that it is more strongly associated with smaller–scale structure in the ionosphere. GNSS and ionosonde data from the time suggest the presence of two MSTIDs travelling in perpendicular directions. The F-region scattering altitudes calculated from the LOFAR primary scintillation arc and primary velocity, and the larger density scales associated with this, suggest that this is associated with a larger–scale TID seen in GNSS data potentially resulting from high–latitude geomagnetic activity. The D-region scattering altitudes of the secondary arc and secondary velocity suggest an atmospheric gravity wave source for a smaller-scale TID. These TIDs trigger an instability which leads to the breakdown of the large-scale density structure into smaller scales, giving rise to the scintillation observed. In the mid-latitude ionosphere the Perkins mechanism is the most likely instability and the features of the smaller-scale density variations observed seem consistent with this. To the best of our knowledge this is the first time that two TIDs have been directly observed simultaneously at different altitudes. This observation demonstrates that LOFAR can be a highly valuable tool for observing ionospheric scintillation in the mid–latitudes over Europe and enables methods of analysis to be used which give greater insight into the likely sources of scattering and could be used to improve modelling of them. With a far greater range of frequencies (multi–octave if the LOFAR high–band is also used) and fine sampling both across the frequency band and in time, LOFAR observations offer a wider sensitivity than that available to GNSS measurements. The analysis techniques shown in this paper also demonstrate that LOFAR can observe ionospheric structures at different altitudes simultaneously; a capability not commonly available for GNSS observations. It also complements these measurements by probing potentially different scintillation regimes to those observed by GNSS. Since this observation was taken, many more have been carried out under a number of projects, recording ionospheric scintillation data at times when the telescope would otherwise be idle. These demonstrate a wide range of scintillation conditions over LOFAR, some of which are seen only very occasionally and perhaps by only one or two of the international stations, illustrating the value to be had by monitoring the ionosphere at these frequencies. A Design Study, LOFAR4SpaceWeather (LOFAR4SW – funded from the European Community’s Horizon 2020 Programme H2020 INFRADEV-2017-1 under grant agreement 777442) currently underway will design a possible upgrade to LOFAR to enable, amongst other space weather observations, ionospheric monitoring in parallel with the regular radio astronomy observations. Such a design, if implemented, would enable a full statistical study of ionospheric scintillation at these frequencies, alongside the advances in scintillation modelling and our understanding of the ionospheric conditions causing it which can be gleaned in focussed studies such as that presented here. ###### Acknowledgements. This paper is based on data obtained with the International LOFAR Telescope (ILT) under project code “IPS”. LOFAR (van Haarlem et al., 2013) is the Low Frequency Array designed and constructed by ASTRON. It has observing, data processing, and data storage facilities in several countries, that are owned by various parties (each with their own funding sources), and that are collectively operated by the ILT foundation under a joint scientific policy. The ILT resources have benefitted from the following recent major funding sources: CNRS-INSU, Observatoire de Paris and Université d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science and Technology Facilities Council, UK; Ministry of Science and Higher Education, Poland. The work carried out at the University of Bath was supported by the Natural Environment Research Council [Grant Number NE/R009082/1] and by the European Space Agency/Thales Alenia Space Italy [H2020-MOM-TASI-016-00002]. We thank Tromsø Geophysical Observatory, UiT the Arctic University of Norway, for providing the lyr, bjn, nor, tro, rvk, and kar magnetometer data. The Kp index and the Chilton ionosonde data were obtained from the U.K. Solar System Data Centre at the Rutherford Appleton Laboratory. 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2024-09-04T02:54:58.353153

2003.04055

arxiv-papers

September 16, 2019 # Spin-Triplet Superconductivity in UTe2 and Ferromagnetic Superconductors Dai Aoki1,2 E-mail<EMAIL_ADDRESS>Ai Nakamura1 Fuminori Honda1 DeXin Li1 Yoshiya Homma1 Yusei Shimizu1 Yoshiki J. Sato1 Georg Knebel2 Jean-Pascal Brison2 Alexandre Pourret2 Daniel Braithwaite2 Gerard Lapertot2 Qun Niu2 Michal Vališka2 Hisatomo Harima3 and Jacques Flouquet2 1IMR1IMR Tohoku University Tohoku University Oarai Oarai Ibaraki Ibaraki 311-1313 311-1313 Japan 2University Grenoble Alpes Japan 2University Grenoble Alpes CEA CEA IRIG-PHELIQS IRIG-PHELIQS F-38000 Grenoble F-38000 Grenoble France 3Graduate School of Science France 3Graduate School of Science Kobe University Kobe University Kobe 657-8501 Kobe 657-8501 Japan Japan<EMAIL_ADDRESS> ###### Abstract The spin-triplet state is most likely realized in uranium ferromagnetic superconductors, UGe2, URhGe, UCoGe. The microscopic coexistence of ferromagnetism and superconductivity means that the Cooper pair should be realized under the strong internal field due the ferromagnetism. leading to the spin-triplet state with equal spin pairing. The field-reinforced superconductivity, which is observed in all three materials when the ferromagnetic fluctuations are enhanced, is one of the strong evidences for the spin-triplet superconductivity. We present here the results of a newly discovered spin-triplet superconductor, UTe2, and compare those with the results of ferromagnetic superconductors. Although no magnetic order is found in UTe2, there are similarities between UTe2 and ferromagnetic superconductors. For example, the huge upper critical field exceeding the Pauli limit and the field-reentrant superconductivity for $H\parallel b$-axis are observed in UTe2, URhGe and UCoGe. We also show the specific heat results on UTe2 in different quality samples, focusing on the residual density of states in the superconducting phase. ferromagnetism, superconductivity, metamagnetism, reentrant superconductivity, spin triplet, specific heat The coexistence of ferromagnetism and superconductivity attracts much attention because unconventional superconductivity with spin-triplet state is realized. [1, 2] In general, ferromagnetism and superconductivity are antagonistic because the large internal field due to the ferromagnetism easily destroys the superconducting Cooper pairs in conventional superconductors. Thus it is natural to consider that the spin-triplet superconductivity with equal spin-pairing is realized in ferromagnetic superconductors. The microscopic coexistence of ferromagnetism and superconductivity is established only in uranium compounds so far, namely UGe2 [3], URhGe [4] and UCoGe [5]. All of these materials have fairly small ordered moments ($1$–$0.05\,\mu_{\rm B}/{\rm U}$) in the ferromagnetic phase compared to that for the U free ion ($\sim 3\,\mu_{\rm B}$). Thus the $5f$ electrons for these compounds are believed to be itinerant in the first principle, although the magnetic anisotropy is rather strong, indicating the Ising properties. The superconductivity occurs well below the Curie temperature, $T_{\rm Curie}$, in the ferromagnetic state. One of the highlights in ferromagnetic superconductors is the field-reentrant or field-reinforced superconductivity. In URhGe, for example, the reentrant superconductivity appears when the field is applied along the hard-magnetization $b$-axis in the orthorhombic structure. [6] While the transition temperature $T_{\rm sc}$ is $0.25\,{\rm K}$ at zero field, the reentrant superconducting phase has a maximum $T_{\rm sc}$ of $0.4\,{\rm K}$ at $H_{\rm R}\sim 12\,{\rm T}$, indicating that the superconductivity is indeed reinforced under magnetic field. The similar field-reinforced superconductivity is also observed in UCoGe. [7] Recently a new spin-triplet superconductor, namely UTe2 was discovered [8, 9]. UTe2 has the body-centered orthorhombic structure with the space group $Immm$ (#71, $D_{2h}^{25}$). The distance of the first nearest neighbor for U atom is $3.78\,{\rm\AA}$, which is larger than the so-called Hill limit ($\sim 3.5\,{\rm\AA}$). Although no magnetic order was found down to $0.025\,{\rm K}$, UTe2 is believed to be at the verge of ferromagnetic order. In fact, the ferromagnetic fluctuations were observed in $\mu$SR [10] and NMR experiments [11]. By substituting Te with Se, the ferromagnetic order appears at $69$ and $33\,{\rm K}$ in UTe0.72Se1.28 and UTe0.24Se1.76, respectively, [12] although the space group for these materials is $Pnma$, which is different from that in UTe2. The superconducting transition occurs at $1.6\,{\rm K}$ with the sharp and large specific heat jump. The large residual density of states nearly $50\,{\%}$ may suggest the possibility for the spontaneous spin-polarization and the “partially-gapped” superconductivity similar to the A1 state with non- unitary state. However, it should be stressed that the direct transition from the paramagnetic state to the non-unitary state at zero field is forbidden from the symmetry restriction in this orthorhombic system. [13] Thus, the hidden feature in the superconducting state is expected. No other transition in the superconducting state is not observed yet at least at zero field. The pressure study is definitely important to solve this problem. One of the strongest support for the spin-triplet superconductivity in UTe2 is the huge upper critical field, $H_{\rm c2}$. In all the field directions, $H_{\rm c2}$ extremely exceeds the Pauli limit ($\sim 3\,{\rm T}$) expected for the weak- coupling BCS theory. The values of $H_{\rm c2}$ at $0\,{\rm K}$ are $7$ and $11\,{\rm T}$ for $H\parallel a$ and $c$-axis, respectively. For $H\parallel b$-axis, the spectacular field-reentrant superconductivity is observed. [14, 15] The transition temperature monotonously decreases with field down to $0.4\,{\rm K}$ at $16\,{\rm T}$, then increases with field up to $0.9\,{\rm K}$ at $35\,{\rm T}$. The first order metamagnetic transtion occurs at $H_{\rm m}=35\,{\rm T}$ [16, 17], and the superconductivity is abruptly collapsed above $H_{\rm m}$. The metamagnetic transition at $H_{\rm m}$ is connected to the so-called $T_{\chi,\rm max}$ at low fields, where the magnetic susceptibility shows a broad maximum for $H\parallel b$-axis. The magnetic susceptibility shows the Curie-Weiss behavior at high temperatures. At low temperatures, the anisotropic susceptibility is observed with the relation, $\chi_{a}>\chi_{c}>\chi_{b}$, which is consistent with the anisotropy of $H_{\rm c2}$. In order to study more details on superconducting properties in UTe2, we have grown single crystals of UTe2 with different quality, and measured the specific heat at low temperatures. We compare the ($H,T$) phase diagrams for $H\parallel b$-axis in UTe2, URhGe and UCoGe. Figure 1: (Color online) Photographs of UTe2 single crystals grown by (a) chemical vapor transport method and (b) Te-flux method. (c) Laue photograph of UTe2 single crystal along $c$-axis. (d) U-Te phase diagram cited from Ref.18 Single crystals of UTe2 were grown using chemical vapor transport method. The starting materials of U and Te were put into a quartz ampoule with the atomic ratio, U : Te = 2 : 3, together with iodine as the transport agent to be the density, $3\,{\rm mg/cm}^{3}$ in the inner volume of the quartz ampoule. The ampoule was slowly heated and was kept at the temperature gradient of $1060\,^{\circ}{\rm C}$/$1000\,^{\circ}{\rm C}$ for 10 days. Many single crystals were obtained at lower temperature side, as shown in Fig. 1(a). The obtained single crystals were checked by the single crystal X-ray analysis. The lattice parameters and the atomic coordinates are in good agreement with the values in the previous report. [19] The single crystals were oriented using the Laue photograph, as shown in Fig. 1(c). The clear superconducting transition was observed in resistivity and specific heat. The highest residual resistivity ratio (RRR) is about 40. Note that the we also obtained single crystals from the previous recipe, that is, a stoichiometric amount of starting materials and lower temperature gradient $950\,^{\circ}{\rm C}$/$850\,^{\circ}{\rm C}$. The single crystals were grown at high temperature side in this case. However, the quality of the single crystal is lower with the low RRR ($\sim 2$–$3$), and no superconductivity was observed down to $0.1\,{\rm K}$. As shown in Fig. 1(d), UTe2 is an incongruent melting compound in the U-Te phase diagram, and single crystals of UTe2 can be grown using the Te-flux method as well. The off-stoichiometric amounts of U and Te ($22$ and $78\,{\rm at\%}$, respectively) were put into an alumina crucible, which was sealed in a Ta-tube under Ar atmosphere gas. The Ta-tube was then sealed again in a quartz ampoule. The quartz ampoule was slowly heated up to $1050\,^{\circ}{\rm C}$ and was cooled down to $960\,^{\circ}{\rm C}$. The Te-flux was removed at $960\,^{\circ}{\rm C}$ in a centrifuge. The obtained single crystals were large, as shown in Fig. 1(b). However the residual resistivity ratio is not very large (${\rm RRR}\sim 3$). Although the superconductivity was confirmed by the resistivity, it was not a bulk property as no anomaly was detected in the specific heat. Hereafter, we show the results of single crystals obtained by the chemical vapor transport method with off-stoichiometric amounts of starting materials and the high temperature gradient. Figure 2(a) shows the temperature dependence of the electronic specific heat in UTe2. The part of the data is replotted from Ref. 8, 20. The data of sample #1 show the highest $T_{\rm sc}$ with a sharp and large jump at $T_{\rm sc}$, indicating the highest quality sample. The value of $T_{\rm sc}$ defined by the entropy balance in #1 is $1.65\,{\rm K}$, and the residual $\gamma$-value, $\gamma_{0}$, which is extrapolated from the fitting $C_{\rm e}/T=\gamma_{0}+\alpha T^{2}$ at low temperatures assuming a point node gap, is $\gamma_{0}=52\,{\rm mJ\,K^{-2}mol^{-1}}$. The residual $\gamma$-value is equal to $44\,{\%}$ of the $\gamma$-value in the normal state. The lower quality samples show the lower $T_{\rm sc}$ and the higher residual $\gamma$-value. For example, $T_{\rm sc}$ and $\gamma_{0}$ in sample #4 are $1.23\,{\rm K}$ and $89\,{\rm mJ\,K^{-2}mol^{-1}}$, respectively. In sample #5, no superconductivity was observed down to $0.4\,{\rm K}$ in specific heat, and it is confirmed by the resistivity measurement down $0.1\,{\rm K}$. Figure 2(b) shows $T_{\rm sc}$ as a function of residual $\gamma$-value normalized by the $\gamma$-value in the normal state. It is clear that $T_{\rm sc}$ decreases with increasing the residual $\gamma$-value. It is known that the decrease of $T_{\rm sc}$ can be described by the Abrikosov-Gor’kov pair- breaking theory. On the basis of this model, the relation between $T_{\rm sc}$ and the residual density of states had been studied theoretically [21] and experimentally [22] in high $T_{\rm c}$ cuprates and heavy fermion systems, where the rapid increase of the residual density of states is reported, compared to the decrease of $T_{\rm sc}$. This can be explained by the unitarity scattering in unconventional superconductivity. The present result in Fig. 2(b) supports the unconventional superconductivity in UTe2. An important question is whether the residual density of states exists in the ideal single crystal without impurities. In that case, the partial density of states would be gapped, and the so-called A1 state should be realized, where the time reversal symmetry must be broken at zero field. There is, however, no experimental evidence for the breaking of time reversal symmetry in UTe2. Figure 2: (Color online) (a) Electronic specific heat in the form of $C_{\rm e}/T$ vs $T$ of UTe2 in different samples. The phonon contribution is subtracted from the fitting at high temperature above $T_{\rm sc}$. Part of the data is replotted from Ref. 8, 20. (b) $T_{\rm sc}$ as a function of the normalized residual $\gamma$-value for different quality samples. Next we show in Fig. 3 the ($H,T$) phase diagrams of UTe2 and two ferromagnetic superconductors, URhGe and UCoGe for $H\parallel b$-axis, corresponding to the hard-magnetization axis. The field-reentrant or field- reinforced superconductivity is observed both in URhGe and in UCoGe. The enhancement of superconductivity is clearly related to the suppression of $T_{\rm Curie}$, where the ferromagnetic instabilities are realized. In URhGe, the suppression of $T_{\rm Curie}$ leads to the spin-reorientation at $H_{\rm R}$ in field sweep at low temperatures. The slope of magnetization curve for $H\parallel b$-axis is larger than that for $c$-axis. The moment gradually tilts from $c$ to $b$-axis, and finally it re-orients to $b$-axis at $H_{\rm R}\sim 12\,{\rm T}$. The $\gamma$-value is enhanced with increasing field, taking a maximum at $H_{\rm R}$. In NMR experiments, the spin-spin relaxation rate, $1/T_{2}$ shows the diverging behavior around $H_{\rm R}$, indicating the strong enhancement of the longitudinal ferromagnetic fluctuations [23]. The 2nd order transition of $T_{\rm Curie}$ at low fields changes into the weak 1st order transition at $H_{\rm R}$ through the tricritical point (TCP). The reentrant superconductivity appears with the maximum $T_{\rm sc}=0.4\,{\rm K}$ exactly at $H_{\rm R}$. In UCoGe, the suppression of $T_{\rm Curie}$ with field is similar to the case for URhGe. However, the spin reorientation is not observed in magnetization curve, indicating the strong Ising property compared to URhGe. The superconductivity shows an “S”-shaped curve, which is also connected to the suppression of $T_{\rm Curie}$. The enhancement of $\gamma$-value and the development of longitudinal fluctuation are observed in the field scan for $H\parallel b$-axis. In UTe2, the field reentrant superconductivity is also observed, while the temperature range and field range are much wider, compared to those in ferromagnetic superconductors, URhGe and UCoGe. The reentrant superconductivity is again linked to the metamagnetic transition at $H_{\rm m}$. The clear difference from ferromagnetic superconductors is that $H_{\rm m}$ at high fields originates from the broad maximum of magnetic susceptibility, $T_{\chi,\rm max}$, instead of $T_{\rm Curie}$. In the heavy fermion system, it is well known that $H_{\rm m}$ is scaled with $T_{\chi,\rm max}$ [24]. The value of $H_{\rm m}=35\,{\rm T}$ in UTe2 is consistent with $T_{\chi,\rm max}=35\,{\rm K}$. The mass enhancement around $H_{\rm m}$ is detected in the resistivity $A$ coefficient [16] and $\gamma$-value from the Maxwell relation in magnetization [17] and the direct specific heat measurements [25]. The crossover at $T_{\chi,\rm max}$ changes into the 1st order transition at $H_{\rm m}$ through the critical end point (CEP). It should be noted that the reentrant superconductivity is abruptly suppressed above $H_{\rm m}$ in UTe2. On the other hand, the reentrant superconductivity in URhGe still survives in the small field range above $H_{\rm m}$. This is probably due to the abrupt change of $T_{\rm sc}$ as it is inferred from the sharp increase of magnetoresistance at $H_{\rm m}$, implying the drastic change of the electronic state at $H_{\rm m}$. Figure 3: (Color online) ($H,T$) phase diagrams for $H\parallel b$-axis in URhGe, UCoGe and UTe2. The data are taken from Ref. 1, 14, 16, 17 In summary, we presented the single crystal growth of the novel spin-triplet superconductor, UTe2, and the results of specific heat in different quality samples. The higher quality sample shows the higher $T_{\rm sc}$ and the lower residual density of states. The rapid increase of the residual density of states compared to the decrease of $T_{\rm sc}$ supports the unconventional superconductivity in UTe2. The unusual field-reentrant superconductivity is a common feature in ferromagnetic superconductors and UTe2. The precise high field experiments from the microscopic point of views and pressure experiments using a high quality single crystal are desired for the future studies. ## Acknowledgements We thank Y. Tokunaga, S. Ikeda, Y. Ōnuki, K. Ishida, K. Izawa, K. Miyake, V. Mineev, S. Ran, J. Ishizuka, Y. Yanase, K. Machida, C. Paulsen and K. Miyake for fruitful discussion. 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Niu, M. Vališka, D. Braithwaite, G. Lapertot, J.-P. Brison, S. Mishra, I. Sheikin, G. Seyfarth, D. Aoki, and J. Flouquet: J. Phys. Soc. Jpn. 88 (2019) 063707. * [15] S. Ran, I.-L. Liu, Y. S. Eo, D. J. Campbell, P. Neves, W. T. Fuhrman, S. R. Saha, C. Eckberg, H. Kim, J. Paglione, D. Graf, J. Singleton, and N. P. Butch: arXiv:1905.04343 . * [16] W. Knafo, M. Vališka, D. Braithwaite, G. Lapertot, G. Knebel, A. Pourret, J.-P. Brison, J. Flouquet, and D. Aoki: J. Phys. Soc. Jpn. 88 (2019) 063705. * [17] A. Miyake, Y. Shimizu, Y. J. Sato, D. Li, A. Nakmura, Y. Homma, F. Honda, J. Flouquet, M. Tokunaga, and D. Aoki: J. Phys. Soc. Jpn. 88 (2019) 063706. * [18] Y. Xu, M. Yamazaki, and P. Villars: Jpn. J. Appl. Phys. 50 (2011) 11RH02. * [19] S. Ikeda, H. Sakai, D. Aoki, Y. Homma, E. Yamamoto, A. Nakamura, Y. Shiokawa, Y. Haga, and Y. Ōnuki: J. Phys. Soc. Jpn. Suppl 75 (2006) 116\. * [20] T. Metz, S. Bae, S. Ran, I.-L. Liu, Y. S. Eo, W. T. Fuhrman, D. F. Agterberg, S. Anlage, N. P. Butch, and J. Paglione: arXiv:1908.01069 . * [21] A. Okada and K. Miyake: J. Phys. Soc. Jpn. 80 (2011) 084708. * [22] Y. Kitaoka, K. Ishida, and K. Asayama: J. Phys. Soc. Jpn. 63 (1994) 2052\. * [23] Y. Tokunaga, D. Aoki, H. Mayaffre, S. Krämer, M.-H. Julien, C. Berthier, M. Horvatić, H. Sakai, S. Kambe, and S. Araki: Phys. Rev. Lett. 114 (2015) 216401. * [24] D. Aoki, W. Knafo, and I. Sheikin: C. R. Physique 14 (2013) 53. * [25] S. Imajo, Y. Kohama, A. Miyake, C. Dong, J. Flouquet, K. Kindo, and D. Aoki: J. Phys. Soc. Jpn. 88 (2019) 083705.

2024-09-04T02:54:58.365392

2003.04100

arxiv-papers

# Tunable Casimir equilibria with phase change materials: from quantum trapping to its release Lixin Ge<EMAIL_ADDRESS>School of Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000, China Xi Shi Department of physics, Shanghai Normal University, Shanghai, 200234, China Zijun Xu School of Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000, China Ke Gong School of Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000, China ###### Abstract A stable suspension of nanoscale particles due to the Casimir force is of great interest for many applications such as sensing, non-contract nano- machines. However, the suspension properties are difficult to change once the devices are fabricated. Vanadium dioxide (VO2) is a phase change material, which undergoes a transition from a low-temperature insulating phase to a high-temperature metallic phase around a temperature of 340 K. In this work, we study Casimir forces between a nanoplate (gold or Teflon) and a layered structure containing a VO2 film. It is found that stable Casimir suspensions of nanoplates can be realized in a liquid environment, and the equilibrium distances are determined, not only by the layer thicknesses but also by the matter phases of VO2. Under proper designs, a switch from quantum trapping of the gold nanoplate (“on” state) to its release (“off” state) as a result of the metal-to-insulator transition of VO2, is revealed. On the other hand, the quantum trapping and release of a Teflon nanoplate is found under the insulator-to-metal transition of VO2. Our findings offer the possibility of designing switchable devices for applications in micro-and nano- electromechanical systems. ## I Introduction Micro- and nano-electromechanical systems (MEMS and NEMS), which integrate electrical and mechanical functionality on the micro- and nano-scales, have attracted enormous attention Lyshevski (2018); Craighead (2000). Thanks to small sizes, the MEMS and NEMS exhibit low mass, high mechanical resonance frequencies and quantum effects, leading to a broad range of applications such as biological/chemical detections Eom et al. (2011), accelerometers Xu et al. (2011) and micro/nanomachines Wang (2013). One major problem in MEMS and NEMS is the $stiction$ which makes the systems collapse and permanent adhesion caused by the attractive Casimir forces Buks and Roukes (2001); Chan et al. (2001). The Casimir force is a macroscopic quantum effect which arises from quantum fluctuations of the electromagnetic field Casimir (1948). In most cases, two neutral, parallel plates consisted of the same materials are attractive to each other, and the magnitudes of the attraction depend on several parameters such as separations, geometric thicknesses, finite conductivities and temperatures (see, e.g., the review Klimchitskaya et al. (2009) and Refs.Yampol’skii et al. (2008, 2010). Therefore, repulsive Casimir forces are highly required for non-contact and low-friction MEMS and NEMS. The repulsive Casimir forces have been intensively studied in many systems Woods et al. (2016) including liquid-separated environments Munday et al. (2009); van Zwol and Palasantzas (2010); Phan and Viet (2011); Dou et al. (2014), meta-materials Rosa et al. (2008); Zhao et al. (2009, 2011); Song et al. (2018), topological insulators Grushin and Cortijo (2011); Chen and Wan (2012); Nie et al. (2013) and specific geometrics Tang et al. (2017); Levin et al. (2010). In addition, the concept of Casimir equilibria was also investigated, using the enclosed geometries Rodriguez et al. (2008); Rahi and Zaheer (2010) and dispersive materials Rodriguez et al. (2010a). Lately, stable Casimir equilibria of nanoplates above a Teflon-coated gold substrate were reported by Zhao et al Zhao et al. (2019). However, the Casimir equilibria of previous studies were mainly in passive systems. Once the devices are fabricated, the trapping properties are difficult to change. Thus, the tunable trapping or even the switching from the trapping to its release by external stimuli (e.g., heating, electric fields or optical waves) is highly desired in MEMS and NEMS. In order to active modulate the Casimir effect, one straight way is to change the dielectric properties of materials under external means Torricelli et al. (2012); Sedighi et al. (2013); Torricelli et al. (2010). Vanadium dioxide (VO2) Shao et al. (2018); Zylbersztejn and Mott (1975) is a phase change material(PCM), which undergoes a transition from a low-temperature insulating phase to a high-temperature metallic phase at critical temperature 340 K. The phase transition of VO2 is accompanied by a structural transformation from the monoclinic phase to the tetragonal one. Meanwhile, the dielectric function of VO2 changes dramatically during the phase transition, leading to many interesting applications Wu et al. (2017); Liu et al. (2017); Kats et al. (2012); van Zwol et al. (2012). In general, the phase transition of VO2 can be induced by changing the temperature of systems. Alternatively, the phase transition can be driven by optical lasers Cavalleri et al. (2001); Rini et al. (2008) or electrical gratings Qazilbash et al. (2008); Nakano et al. (2012) on a sub-picosecond timescale. Recently, VO2 has been employed to study the tunable Casimir effect in the vacuum Galkina et al. (2009); Pirozhenko and Lambrecht (2008); Castillo-Garza et al. (2007). For a large separation (e.g., $>$1 $\mu$m), the contrast of Casimir forces due to the phase-transition is quite large (e.g., over 2 times for two semi-infinite plates of VO2, this value could be even larger for the case of finite thickness Galkina et al. (2009); Pirozhenko and Lambrecht (2008)). As the separation is small (e.g., $\sim$100 nm), however, the modulation of Casimir forces owning to the phase transition and finite-thickness decreases greatly Pirozhenko and Lambrecht (2008); Castillo-Garza et al. (2007). Nonetheless, the Casimir forces are always attractive and only magnitude modulations have been reported in a vacuum-separated configuration. The influences of phase transition of VO2 on the sign modulation of Casimir forces (e.g., from attraction to repulsion) are yet less explored. In a liquid environment, the function of sign modulation and the related phenomena such as tunable Casimir equilibria are expected based on the phase transition of VO2. Here, the Casimir forces between a nanoplate and a layered structure separated by a liquid are investigated. The layered structure consists of two kinds of materials, i.e., Vanadium dioxide (VO2) and Teflon. It is found that stable Casimir equilibria of gold nanoplates can be realized when a VO2 film is buried under a semi-infinite Teflon. The properties of Casimir equilibria are determined, not only by the layer thicknesses but also by the matter phases of VO2. For thick-film VO2, the Casimir equilibria and quantum traps can be achieved for both the metallic and insulating phases. On the other hand, a switch from quantum trapping of the gold nanoplate(“on” state) to its release (“off” state) can be triggered by the metal-to-insulator phase transition when the thickness of VO2 is thin (e.g., 20 nm). Finally, stable suspensions of Teflon nanoplates are also proposed with a complementary design, where the Teflon substrate is coated by a VO2 film. Unlike the case of gold nanoplates, the quantum trapping of Teflon nanoplates and its release correspond to the insulating and metallic phases of VO2. Moreover, the switching phenomena can be realized only with a several-nanometers thickness of VO2. Figure 1: (color online) (a) Schematic view of a gold nanoplate suspended in a liquid environment. (b) The permittivity of different materials (gold, VO2, bromobenzene and Teflon) as a function of imaginary frequency. ## II Theoretical models The system in this work is schematically shown in Fig. 1(a), where a gold nanoplate with thickness $L_{g}$ is suspended in a liquid of bromobenzene. The separation between the nanoplate and the substrate is $d$. The substrate is composed of a VO2 film buried under a semi-infinite plate of Teflon. The thicknesses of the top-layer Teflon and VO2 are denoted as $L_{T}$ and $L_{\mathrm{V}}$, respectively. The in-plane dimension of the gold nanoplate is much larger than $L_{g}$ and $d$, and it is considered as a slab during our calculations. The Casimir force is calculated by $F_{c}=-\partial E_{c}(d)/\partial d$, where $E_{c}(d)$ is the Casimir energy between the gold nanoplate and the substrate, having the form Nie et al. (2013); Zhao et al. (2019) $E_{c}(d)=A\hbar\int_{0}^{\infty}\frac{d\xi}{2\pi}\int\frac{d^{2}\mathbf{k_{\|}}}{(2\pi)^{2}}\log\det\left[1-\mathbf{R_{1}}\cdot\mathbf{R_{2}}e^{-2k_{3}d}\right],$ (1) where $\hbar$ is the reduced Planck constant, $A$ is the in-plane area, $\mathbf{k_{\parallel}}$ is the parallel wavevector, $k_{3}=\sqrt{k_{\parallel}^{2}+\varepsilon_{liq}(i\xi)\xi^{2}/c^{2}}$ is the vertical wavevector, $c$ is the speed of light in vacuum, $\varepsilon_{liq}(i\xi)$ is the permittivity of the intervening liquid evaluated with imaginary frequency $\omega=i\xi$, $\mathbf{R}_{1,2}$ is the $2\times 2$ reflection matrix for layered structures, having the form $\mathbf{R_{j}}=\left(\begin{array}[]{cc}r_{j}^{s}&0\\\ 0&r_{j}^{p}\end{array}\right),$ (2) where $r_{j}$ with $j$=1 and $j$=2 are the reflection coefficients for the upper and lower layered structures, and the superscripts $s$ and $p$ correspond to the polarizations of transverse electric ($\mathbf{TE}$) and transverse magnetic ($\mathbf{TM}$) modes, respectively. Note that the temperature $T$ for Eq. (1) equals 0 K and it is an effective approximation as the separation $d$ is smaller than 1 $\mu m$ for finite temperatures Milton (2004). For a nanoplate suspended in a liquid, the reflection coefficients can be given analytically as follows Zhao et al. (2011) $r^{\alpha}=\frac{r_{0,j}^{\alpha}+r_{j,0}^{\alpha}e^{-2K_{j}L_{j}}}{1+r_{0,j}^{\alpha}r_{j,0}^{\alpha}e^{-2K_{j}L_{j}}},$ (3) where $\alpha=s$ and $p$, $L_{j}$ is the thickness of the nanoplate, $K_{j}=\sqrt{k_{\parallel}^{2}+\varepsilon_{j}(i\xi)\xi^{2}/c^{2}}$ is the vertical wavevector, $\varepsilon_{j}(i\xi)$ is the permittivity of the nanoplate. The subscripts of $r_{m,n}^{\alpha}$ represent the light is incident from the medium $m$ to $n$ (0 means the liquid). Alternatively, the reflection coefficients for layered structures can be calculated by a transfer matrix method. The general form is given as $r=M_{21}/M_{11}$, where $M_{21}$ and $M_{11}$ are the elements of the $M$ matrixZhan et al. (2013). The $M$ matrix is the multiplications of transmission matrices across different interfaces and propagation matrices in different layers. Considering an arbitrary $N$-layer system, the $M$-matrix is given as : $M=D_{0,1}P(L_{1})D_{1,2}P(L_{2})...D_{N-1,N}P(L_{N})D_{N,N+1},$ (4) where the transmission matrix $D_{j,j+1}$ is given as: $D_{j,j+1}=\frac{1}{2}\left[\begin{array}[]{cc}1+\eta&1-\eta\\\ 1-\eta&1+\eta\end{array}\right],$ (5) where $\eta=\varepsilon_{j}(i\xi)K_{j+1}/(\varepsilon_{j+1}(i\xi)K_{j})$ for p-polarization and $\eta=K_{j+1}/K_{j}$ for s-polarization. The propagation matric in the $j$-th layer (for both $s$ and $p$ polarizations) is written as: $P(L_{j})=\left[\begin{array}[]{cc}e^{K_{j}L_{j}}&0\\\ 0&e^{-K_{j}L_{j}}\end{array}\right].$ (6) For example, we have $N=2$ for the multilayered substrate in Fig. 1. The $M$ matrix is given by $M=D_{0,1}P(L_{1})D_{1,2}P(L_{2})D_{2,3}$, where the subscripts 0, 1, 2 and 3 represent the media of liquid, Teflon, VO2 and Teflon (from top to down); the thicknesses $L_{1}=L_{T}$, $L_{2}=L_{V}$. ## III Results and discussions Figure 1(b) shows the permittivity for different materials, where the used models and parameters are given in the Appendixes. The dielectric function of VO2 changes dramatically under different temperatures. For temperature $T>T_{c}$, VO2 is in the metallic phase and it acts as a poor metal. For $T<T_{c}$, it is in the insulating phase (or called semiconducting phase), and the corresponding dielectric function nearly matches that of intrinsic silicon at low frequency Pirozhenko and Lambrecht (2008). To create repulsive Casimir forces between two dissimilar plates separated by a liquid, the permittivity should satisfy $\varepsilon_{1}(i\xi)>\varepsilon_{liq}(i\xi)>\varepsilon_{2}(i\xi)$ for a vast range of frequency Munday et al. (2009). Clearly, the dielectric functions of gold and VO2 (either metallic or insulating phase) are larger than that of bromobenzene over a wide range of frequency. Therefore, the Casimir force is always attractive for the layered structure of gold/bromobenzene/VO2. While the Casimir force for the structure of gold/bromobenzene/Teflon is repulsive instead. Nonetheless, the Casimir equilibria can not be found for above two layered structures. Figure 2: (color online) Casimir pressure via different thicknesses of VO2, where the thickness $L_{T}$=45 nm and $L_{g}$=40 nm are fixed. (a) Thick films. The solid and dashed lines represent the pressure for the metallic and insulating phases of VO2, respectively. (b) Thin films. The positive (negative) sign of the pressure corresponds to the repulsive (attractive) force. ### III.1 Tunable Casimir equilibria for gold nanoplates Now we consider the Casimir forces as the substrate is composed of a VO2 film and Teflon (see Fig. 1(a)). The Casimir pressure ($P_{c}=F_{c}/A$) for the thick film of VO2 is given in Fig. 2(a). The results show that the curves are almost identical for $L_{\mathrm{V}}$=200, 500 and 1000 nm, indicating the weak impact of the thickness for thick-film configurations. The pressure is repulsive at small separation (e.g., $d<60$ nm), making the nanoplate stay away from the substrate. As the separation increases further, the Casimir equilibria (zero pressure) occur and quantum traps can be realized for both metallic (solid lines) and insulating phases (dashed lines). In addition, the equilibrium distance $d_{c}$ is shifted under the phase transition of VO2. On the other hand, the thin-film thickness and the phase transition of VO2 can play an important role in Casimir pressure as shown in Fig. 2(b). For the thickness $L_{\mathrm{V}}$ =10 and 20 nm, quantum traps can be realized for the metallic phase, whereas no trap is found for the insulating phase. Under such configurations, a switch from quantum trapping of the nanoplate(“on” state) to its release (“off” state) can be triggered by the metal-insulator transition of VO2. However, the quantum trapping occurs for both metallic and insulating phases as the thickness $L_{\mathrm{V}}$ increases to 30 nm, and the “off” state disappears. Compared with the vacuum-separated configuration Castillo-Garza et al. (2007), not only the magnitude of Casimir forces can be modified in a liquid environment, but also the sign could be switched (e.g., from attraction to repulsion for $d$=100 nm, $L_{V}$=30 nm), due to the phase- transition of VO2. Figure 3: (color online) Casimir pressure contributed from different frequencies and different parallel wavevectors. (a) and (b) $d$=30 nm; (c) and (d) $d$=85 nm (close to critical separation); (e) and (f) $d$=150 nm. (a), (c)and (e) VO2 in the metallic phase ($T>T_{c}$); (b), (d) and (f) VO2 in the insulating phase ($T<T_{c}$). The layer thicknesses are set as $L_{\mathrm{V}}$=20 nm and $L_{T}$=45 nm. To understand the switch transition from the “on” to the “off” state, the contour plots of Casimir pressure are shown in Fig. 3 under different separations. The sign of pressure is determined by the competition of VO2 film (attraction) and low-refractive-index Teflon (repulsion). For small separation $d$=30 nm, the pressure is dominant by the repulsive component as shown in Figs. 3(a) and 3(b). For the metallic phase, the attractive component increases and it compensates the repulsive one as the separation becomes 85 nm ($d\approx d_{c}$), resulting in Casimir equilibrium (see Fig. 3(c)). While the repulsion is still dominant for the insulating phase as shown in Fig. 3(d). As $d$ increases further to 150 nm, the Casimir pressure turns out to be dominantly attractive in Fig. 3(e) for the metallic phase, resulting in a restoring force for stable trapping. By contrast, the pressure is still dominant by repulsion for the insulating phase as shown in Fig. 3(f). The pressure maps between the metallic and insulating phases are almost identical for large energy (e.g., $>$2 eV), whereas the discrepancy manifests at low energy. The results indicate that the attractive component appears only at low frequency and small $k$ vector for metallic VO2, where the field cannot penetrate the metalZhao et al. (2019). Conversely, the field can penetrate the thin-film of insulating VO2 easily, leading to repulsive Casimir forces. Figure 4: (color online) (a)The equilibrium distances via the thicknesses of VO2 under three different configurations (see the inset on the right). The thickness $L_{T}$ is set as 45 nm. The solid (dashed) curves for type III represent stable (unstable) equilibria. Contour plots of Casimir pressure via the thicknesses of coating Teflon for (b) metallic VO2 and (c) insulating VO2, where the thickness $L_{\mathrm{V}}$=20 nm is fixed. In (b) and (c), the gray zones represent a strong repulsive pressure larger than 1 Pa. The colors of the curves denote the same meaning as those in (a). Practically, the influences of gravitation and buoyancy on the force balances should be taken into account. The condition for the force equilibrium is written as $\vec{n}\cdot(\mathbf{F}_{c}+\mathbf{F}_{\mathrm{GB}})$=0, where $\vec{n}$ is the unit vector normal to the surface, $F_{\mathrm{GB}}=(\rho_{g}-\rho_{liq})gL_{g}A$ is the sum of gravity and buoyancy, $g$ is the gravitational acceleration, $\rho_{g}\approx$19.3 g/cm3 and $\rho_{liq}\approx$1.50 g/cm3 is the density of gold and liquid bromobenzene, respectively. The magnitude of $F_{\mathrm{GB}}/A$ is about 7.0 mPa as the thickness $L_{g}$=40 nm. Three types of configurations are depicted in the inset of Fig. 4(a) for the cross-section views. The type I configuration corresponds to a zero-projection (or weightlessness in aerospace), where the switching from quantum trapping (metallic state) to its release (insulating state) can be obtained as $L_{\mathrm{V}}$ in a proper range, from about 2 to 22 nm. For type II configuration, the attractive $F_{\mathrm{GB}}$ can compensate the long-range repulsive Casimir force at large $d$, leading to stable suspensions for both $T>T_{c}$ and $T<T_{c}$. However, the equilibrium distances are different, and it can be inferred that the stiffness of trapping for metallic phase is stronger than that of the insulating phase. For type III configuration (a flipped down system), the switching between trapping and its release can also be realized. Interestingly, there are two equilibrium distances for this configuration. It is not difficult to know that the smaller equilibrium distance (solid lines) is stable, whereas the other one (dashed lines) with larger distance is unstable to small perturbations in position. For both type II and III configurations, the deviations from Type I become strong as $d_{c}$ is large. In addition to the thickness of VO2 film, the top-layer Teflon can also play a significant role in the Casimir effect. The plots of Casimir pressure via the thicknesses of the coating Teflon $L_{T}$ are shown in Figs. 4(b) and 4(c), where $L_{\mathrm{V}}$=20 nm is fixed. The results show that the switching between quantum trapping and it release occurs only when $L_{T}$ is larger than about 42 nm (no gravity). The larger the $L_{T}$, the larger of the position for the Casimir equilibrium. As $L_{T}$ is smaller than 42 nm, the equilibrium distance is also small, and quantum trappings can be realized for both metallic and insulating phases. For comparison, the gravitation and buoyancy are taken into account. Again, strong discrepancies among three configurations occur as the equilibrium positions larger than about 150 nm, resulting from the comparable magnitude of $F_{GB}$ and the Casimir force. The impact of $F_{GB}$ can be further reduced by decreasing the thickness $L_{g}$ near the skin depth (about 22 nm) Lisanti et al. (2005). Figure 5: (color online) Casimir pressure for a complementary design. A thin film of VO2 with thickness $L_{V}$ is deposited on a Teflon substrate. (a)The metallic VO2. (b)The insulating VO2. The thickness of the suspended nanoplate is set as 100 nm. Figure 6: (color online) Casimir pressure calculated for finite temperatures and 0 K approximation from Eq. (1). (a)The trapping and release of a gold nanoplate. The parameters for the substrate are $L_{T}$=45 nm and $L_{V}$=20 nm. (b)The trapping and release of a Teflon nanoplate. The thickness $L_{V}$ is set as 2 nm. ### III.2 Tunable Casimir equilibria for Teflon nanoplates The active control of the low-refractive-index nanoplates can also be significant in many applications. Inspiring by the work Zhao et al. (2019), a complementary design is schematically shown in the inset of Fig. 5(a). A Teflon nanoplate is suspended in a liquid of bromobenzene, and the substrate is a semi-infinite plate of Teflon coated by a VO2 film (high refractive index). Under such design, the Casimir force is repulsive at very short separation, due to the dominant interaction between Teflon/bromobenzene/VO2. As the separation increases, the attractive interaction from Teflon/bromobenzene/Teflon can be dominant instead, resulting in a stable Casimir trapping. To verify the design, the Casimir pressure is given quantitatively in Figs. 5(a) and 5(b) as a function of separation. Interestingly, the Casimir pressure shows a long-range repulsive behavior for the metallic VO2, which corresponds to the “off” state. The repulsion pressure becomes stronger as the thickness $L_{\mathrm{V}}$ enlarges from 2 to 6 nm. For $L_{\mathrm{V}}$= 2 nm, a Casimir equilibria and strong restoring forces can be found when VO2 is in the insulating phase. Therefore, the quantum trapping and release of a Teflon nanoplate can be achieved under the insulator-to-metal transition of VO2. As the thickness is 4 nm, the restoring force decreases and the trapping stiffness drops considerably. The calculation results indicate that the Casimir pressure is quite sensitive to the thickness of VO2. Due to the low density of Teflon (2.1 g/cm3), the pressure $F_{GB}/A$ for the Teflon nanoplate is about 0.6 mPa, which is reduced significantly compared with those of gold nanoplates. ### III.3 Finite temperatures effect To achieve the phase transition of VO2, the temperatures of the devices need to be changed. We assume that the dielectric functions of the gold and Teflon are temperature-independent. For organic liquids, the change of refractive index due to the temperature Li et al. (1994) is an order of $10^{-4}/$ K, and the permittivity of bromobenzene is also treated as temperature-independent. Nonetheless, it is interesting to check the finite temperature effect on Casimir forces. The integral over frequency $\xi$ in Eq. (1) now is replaced by a discrete summation Rahi et al. (2009): $\frac{\hbar}{2\pi}\int_{0}^{\infty}d\xi\leftrightarrow k_{b}T\overset{\infty}{\underset{n=0}{\sum}}^{\prime},$ (7) where $\xi$ is replaced by discrete Matsubara frequencies $\xi_{n}=2\pi\frac{k_{b}T}{\hbar}n(n=0,1,2,3\ldots),$ $k_{B}$ is the Boltzmann’s constant and the prime denotes a prefactor 1/2 for the term $n$=0. The Casimir pressures under different temperatures are shown in Figs. 6(a) and 6(b), where two different designs are demonstrated. It is found that the curves for temperature 320 K (insulating phase) overlap with those calculated from Eq. (1). For the temperature of 360 K, there is only a small deviation between 0 K and 360 K. Overall, the calculation results from 320 and 360 K confirm the accuracy of the 0 K approximation. Recently, the switching between repulsive and attractive Casimir forces based on PCM has also been reported Boström et al. (2018), where the equilibrium distances for switching occur only at several nanometers. The equilibrium distances in our work are more accessible to experiments, and it can be tuned by designing the geometric thickness of VO2 and the Teflon. Figure 7: (color online) The total energy of a suspended gold nanoplate (a) and a Teflon nanoplate (b) under different types of gravity projection. The solid and dashed lines represent the cases for the metallic VO2 ($T$=360 K) and insulating VO2 ($T$=320 K), respectively. The in-plane area $A$ is set as 10 $\mu m\times$10 $\mu m$. Other parameters are kept the same as those in Fig. 6. ### III.4 The effect of Brownian motion In a real configuration, the position of a nanoplate has a fluctuation around the equilibrium distances due to the Brownian motion. To evaluate the effect of Brownian motion, the total energy of the suspended nanoplate should be known, which are written as $U(d)=E_{c}+\Lambda\times(E_{g}+E_{b})$, where $E_{c}$ is the Casimir energy given by Eq. [1], $E_{g}=\rho_{p}gL_{p}Ad$ and $E_{b}=-\rho_{liq}gL_{p}Ad$ are respectively the energies caused by the gravity and buoyancy Phan et al. (2012), $\rho_{p}$ and $L_{p}$ represent the density and thickness of the suspended nanoplate. The coefficient $\Lambda$ is the parameter depending on the gravity projection. For type I configuration (see the inset of Fig. 4), $\Lambda$=0. While we have $\Lambda$=1 and -1 for type II and type III configurations. The total energy of a gold and Teflon nanoplate are shown in Figs. 7(a) and 7(b), respectively. The minimum of $U(d)/k_{B}T$ corresponds to the equilibrium distance $d_{c}$. Clearly, stable quantum trapping can be realized for a gold (Teflon) nanoplate when VO2 is in the metallic (insulating) phase. Due to the balance of repulsive Casimir force and gravity, stable trapping can also be realized for type II configuration. Theoretically, the transition rate from the equilibrium distance to another position due to the Brownian motion is proportional to $\exp(-\triangle U/k_{B}T)$ Phan et al. (2012); Rodriguez et al. (2010b), where $\triangle U$ represents the energy barrier between these two positions. The calculated results indicate that the transition rates from Casimir equilibria to stiction are negligible since the energy barriers $\triangle U//k_{B}T$ are quite large (e.g., over $10^{4}$) for the gold and Teflon nanoplates. For a flipped-down system (type III), quantum trapping can be realized for gold (Teflon) nanoplate when VO2 is in the metallic (insulating) phase. However, there is a nonzero possibility that the nanoplates can escape from the equilibrium distances to the free-liquid regime ($d\rightarrow\infty$). Fortunately, the energy barrier $\triangle U/k_{B}T$ for such a transition is the order of $10^{2}$ as shown in Figs. 7(a) and 7(b), and the transition rate of the escape is also negligible. ## IV Conclusions In summary, the Casimir forces between a nanoplate and a layered structure containing VO2 films are investigated. In a liquid-separated environment, not only the magnitude of Casimir forces can be modified, but also the sign could be switched (e.g., from attraction to repulsion), due to the phase-transition of VO2. Moreover, a stable Casimir suspension of nanoplates and its tunability are revealed. For a gold nanoplate, a switch from the quantum trapping to its release is obtained under the metal-to-insulator transition of VO2. In addition, the quantum trapping and release of a Teflon nanoplate are demonstrated with a complementary design. The switching performances due to the layer thicknesses, gravitation and temperatures are discussed as well. Theoretically, the bromobenzene can be substituted by other high-refractive- index liquids (e.g., glycerol and styrene van Zwol and Palasantzas (2010)) as long as the boiling points are larger than $T_{c}$. The Teflon can also be replaced by other low-refractive-index materials (e.g., mesoporous silica Dou et al. (2014)). This work offers the possibility of designing switchable devices in MEMS/NEMS, resulting from the quantum fluctuations of the electromagnetic field. ###### Acknowledgements. This work is supported by the National Natural Science Foundation of China (Grant No. 11804288, No. 11704254, No. 61571386 and No. 61974127), and the Innovation Scientists and Technicians Troop Construction Projects of Henan Province. The research of L.X. Ge is further supported by Nanhu Scholars Program for Young Scholars of XYNU. ## Appendix A The permittivity of gold Here, a generalized Drude-Lorentz model is applied for the permittivity of gold Sehmi et al. (2017): $\varepsilon(i\xi)=\varepsilon_{D}(i\xi)+\varepsilon_{L}(i\xi),$ (8) where the Drude term is given by: $\varepsilon_{D}(i\xi)=\varepsilon_{\infty}+\frac{\gamma\sigma}{\xi(\xi+\gamma)},$ (9) where $\varepsilon_{\infty}=$0.83409, $\sigma=$3134.5 eV, and $\gamma=$0.02334 eV. The Lorentz term is described by four pairs of poles: $\varepsilon_{L}(i\xi)=\overset{4}{\underset{j=1}{\sum}}\left(\frac{i\sigma_{j}}{i\xi-\Omega_{j}}+\frac{i\sigma_{j}^{\ast}}{i\xi+\Omega_{j}^{\ast}}\right)$ (10) where $\sigma_{j}$ and $\Omega_{j}$ are the generalized conductivity and resonant frequency of the $j$-th Lorentz pole. The star superscripts represent the operation of complex conjugation. The generalized Drude-Lorentz model respects causality, and it can represent the exact physical resonances in the material. The parameters for the model are listed in the Table I. Table 1: The fitted parameters for Lorentz poles of gold Sehmi et al. (2017). $j$-th | $\sigma_{j}(\mathrm{eV})$ | $\Omega_{j}(\mathrm{eV})$ ---|---|--- 1 | -0.01743+0.3059*I | 2.6905-0.16645*I 2 | 1.0349+1.2919*I | 2.8772-0.44473*I 3 | 1.2274+2.5605*I | 3.7911-0.81981*I 4 | 9.85+37.614*I | 4.8532-13.891*I ## Appendix B The permittivity of VO2 For temperature $T>T_{c}$, VO2 is in the metallic phase, and the permittivity is given by Pirozhenko and Lambrecht (2008); Castillo-Garza et al. (2007) $\displaystyle\varepsilon(i\xi)$ $\displaystyle=$ $\displaystyle 1+\frac{\omega_{p}^{2}}{\xi(\xi+\gamma)}+\frac{\varepsilon_{\infty}-1}{1+\xi^{2}/\omega_{\infty}^{2}}$ (11) $\displaystyle+\underset{j=1}{\overset{4}{\sum}}\frac{s_{j}}{1+(\xi/\omega_{j})^{2}+\Gamma_{j}\xi/\omega_{j}},$ where $\varepsilon_{\infty}=3.95,\omega_{p}=3.33$ eV, and $\gamma=0.66$ eV. The parameters $s_{j}$ and $\Gamma_{j}$ represent respectively the strength and linewidth of the $j$-th oscillator (resonant frequency $\omega_{j}$). For temperature $T<T_{c}$, VO2 is in the insulating phase, and the permittivity is described as $\varepsilon(i\xi)=1+\frac{\varepsilon_{\infty}-1}{1+\xi^{2}/\omega_{\infty}^{2}}+\underset{j=1}{\overset{7}{\sum}}\frac{s_{j}}{1+(\xi/\omega_{j})^{2}+\Gamma_{j}\xi/\omega_{j}},$ (12) where $\varepsilon_{\infty}=4.26$ and $\omega_{\infty}=15$ eV. The above equations for metallic and insulating VO2 are valid for a wide range of frequency (up to about 10 eV)Castillo-Garza et al. (2007), which are modified versions of Ref. Verleur et al. (1968). The parameters are listed in Table II. Table 2: The parameters for the metallic and insulating VO2 Castillo-Garza et al. (2007). $j$-th ($T>T_{c}$) | $S_{j}$ | $\omega_{j}(\mathrm{eV})$ | $\Gamma_{j}$ ---|---|---|--- 1 | 1.816 | 0.86 | 0.95 2 | 0.972 | 2.8 | 0.23 3 | 1.04 | 3.48 | 0.28 4 | 1.05 | 4.6 | 0.34 $j$-th ($T<T_{c}$) | $S_{j}$ | $\omega_{j}(\mathrm{eV})$ | $\Gamma_{j}$ 1 | 0.79 | 1.02 | 0.55 2 | 0.474 | 1.30 | 0.55 3 | 0.483 | 1.50 | 0.50 4 | 0.536 | 2.75 | 0.22 5 | 1.316 | 3.49 | 0.47 6 | 1.060 | 3.76 | 0.38 7 | 0.99 | 5.1 | 0.385 Table 3: The parameters for Teflon(left) and bromobenzene (right)van Zwol and Palasantzas (2010). $j$-th | $C_{j}$ | $\omega_{j}(\mathrm{eV})$ | $C_{j}$ | $\omega_{j}(\mathrm{eV})$ ---|---|---|---|--- 1 | 0.0093 | 0.0003 | 0.0544 | 0.00502 2 | 0.0183 | 0.0076 | 0.0184 | 0.0309 3 | 0.139 | 0.0557 | 0.0475 | 0.111 4 | 0.112 | 0.126 | 0.532 | 6.75 5 | 0.195 | 6.71 | 0.645 | 13.3 6 | 0.438 | 18.6 | 0.240 | 24.0 7 | 0.106 | 42.1 | 0.00927 | 99.9 8 | 0.0386 | 77.6 | | ## Appendix C The permittivity of Teflon and bromobenzene The permittivity for the Teflon and bromobenzene are given by the oscillator model van Zwol and Palasantzas (2010): $\varepsilon(i\xi)=1+\underset{j}{\overset{}{\sum}}\frac{C_{j}}{1+(\xi/\omega_{j})^{2}},$ (13) where $C_{j}$ corresponds to the oscillator strength for the $j$-th resonance, and $\omega_{j}$ is the corresponding resonant frequency. 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2024-09-04T02:54:58.377245

2003.04110

arxiv-papers

# MS-TP-20-17 Continuum extrapolation of Ward identities in $\mathbf{\mathcal{N}=1}$ supersymmetric SU(3) Yang-Mills theory Sajid Ali<EMAIL_ADDRESS>University of Münster, Institute for Theoretical Physics, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany Government College University Lahore, Department of Physics, Lahore 54000, Pakistan Georg Bergner<EMAIL_ADDRESS>University of Jena, Institute for Theoretical Physics, Max-Wien-Platz 1, D-07743 Jena, Germany University of Münster, Institute for Theoretical Physics, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany Henning Gerber<EMAIL_ADDRESS>University of Münster, Institute for Theoretical Physics, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany Istvan Montvay<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22607 Hamburg, Germany Gernot Münster11footnotemark: 1 University of Münster, Institute for Theoretical Physics, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany Stefano Piemonte <EMAIL_ADDRESS>University of Regensburg, Institute for Theoretical Physics, Universitätsstr. 31, D-93040 Regensburg, Germany Philipp Scior <EMAIL_ADDRESS>Universität Bielefeld, Fakultät für Physik, Universitätsstr. 25, D-33615 Bielefeld, Germany (14th May 2020) ###### Abstract Abstract: In $\mathcal{N}=1$ supersymmetric Yang-Mills theory, regularised on a space-time lattice, in addition to the breaking by the gluino mass term, supersymmetry is broken explicitly by the lattice regulator. In addition to the parameter tuning in the theory, the supersymmetric Ward identities can be used as a tool to investigate lattice artefacts as well as to check whether supersymmetry can be recovered in the chiral and continuum limits. In this paper we present the numerical results of an analysis of the supersymmetric Ward identities for our available gauge ensembles at different values of the inverse gauge coupling $\beta$ and of the hopping parameter $\kappa$. The results clearly indicate that the lattice artefacts vanish in the continuum limit, confirming the restoration of supersymmetry. ## 1 Introduction Supersymmetry (SUSY) is an elegant idea which relates fermions and bosons, whose spin differs by 1/2, through supercharges [1]. SUSY provides dark matter candidates, arising from the lightest supersymmetric particles [2]. In addition to that, supersymmetric extensions of the Standard Model would resolve the hierarchy problem [3]. $\mathcal{N}=1$ supersymmetric Yang-Mills (SYM) theory, which is being considered in this article, provides an extension of the pure gluonic part of the Standard Model [4]. It describes the strong interactions between gluons and gluinos, the superpartners of the gluons. Gluinos are Majorana particles that transform under the adjoint representation of the gauge group. The on-shell Lagrangian of $\mathcal{N}=1$ SYM theory, which consists of the gluon fields $A^{a}_{\mu}(x)$ and the gluino fields $\lambda^{a}(x)$, where $a=1,\ldots,N^{2}_{c}-1$, can be written in Minkowski space as $\mathcal{L}_{\text{SYM}}=-\frac{1}{4}F^{a}_{\mu\nu}F^{a,\mu\nu}+\frac{\mathrm{i}}{2}\bar{\lambda}^{a}\gamma^{\mu}\left(\mathcal{D}_{\mu}\lambda\right)^{a}-\frac{m_{\tilde{g}}}{2}\bar{\lambda}^{a}\lambda^{a},$ (1) where the first term, containing the field strength tensor $F^{a}_{\mu\nu}$, is the gauge part, and $\mathcal{D}_{\mu}$ in the second term is the covariant derivative in the adjoint representation of the gauge group SU($N_{c}$), $N_{c}$ being the number of colors. The last part of the above Lagrangian is a gluino mass term which breaks SUSY softly for $m_{\tilde{g}}\neq 0$, which means that it does not affect the renormalisation properties of the theory and that the spectrum of the theory depends on the gluino mass in a continuous way. The physical spectrum of this theory is expected to consist of bound states of gluons and gluinos, arranged in mass degenerate supermultiplets if SUSY is not broken [5, 6]. In order to perform Monte-Carlo simulations of the theory, we discretise the Euclidean action and put it onto a four-dimensional hypercubic lattice. We use the Curci-Veneziano version [7] of the lattice action $S=S_{g}+S_{f}$, where the gauge part $S_{g}$ is defined by the usual plaquette action $S_{g}=-\frac{\beta}{N_{c}}\sum_{p}\mathrm{Re}\left[\mathrm{tr}\left(U_{p}\right)\right],$ (2) with the inverse gauge coupling given by $\beta=2N_{c}/g^{2}$, and the fermionic part $S_{f}=\frac{1}{2}\sum_{x}\left\\{\bar{\lambda}^{a}_{x}\lambda_{x}^{a}-\kappa\sum_{\mu=1}^{4}\left[\bar{\lambda}^{a}_{x+\hat{\mu}}V_{ab,x\mu}(1+\gamma_{\mu})\lambda^{b}_{x}+\bar{\lambda}^{a}_{x}V^{T}_{ab,x\mu}(1-\gamma_{\mu})\lambda^{b}_{x+\hat{\mu}}\right]\right\\}$ (3) implements the gluinos as Wilson fermions. Here the adjoint link variables are defined by $V_{ab,x\mu}=2\,\mathrm{tr}\,(U_{x\mu}^{\dagger}T_{a}U_{x\mu}T_{b})$, where $T_{a}$ are the generators of the gauge group, and the hopping parameter $\kappa$ is related to the bare gluino mass $m_{\tilde{g}}$ by $\kappa=1/(2m_{\tilde{g}}+8)$. In order to approach the limit of vanishing gluino mass, the hopping parameter has to be tuned properly. In our numerical investigations the fermionic part is additionally $O(a)$ improved by adding the clover term $-(c_{sw}/4)\,\bar{\lambda}(x)\sigma_{\mu\nu}F^{\mu\nu}\lambda(x)$ [8]. In our previous investigations we have determined the low-lying mass spectrum of the theory with gauge group SU(2) and SU(3) non-perturbatively from first principles using Monte Carlo techniques [4, 9, 10, 11], and obtained mass degenerate supermultiplets [12]. ## 2 SUSY Ward identities In classical physics, Noether’s theorem provides a relation between symmetries and conservation laws. In the case of quantum field theories, symmetries are translated to Ward identities, representing quantum versions of Noether’s theorem. In $\mathcal{N}=1$ supersymmetric Yang-Mills theory a gluino mass term breaks SUSY softly. The soft breaking effects vanish in the chiral limit, a limit where theory is characterised by massless gluinos. In order to analyse this breaking of supersymmetry and to identify the chiral limit, we employ the Ward identities for supersymmetry. Moreover, on the lattice supersymmetry is broken explicitly due to the introduction of the discretisation of space-time lattice as a regulator of the theory. SUSY Ward identities can be used to check whether supersymmetry is restored in the continuum limit. In the Euclidean continuum, on-shell supersymmetry transformations of the gauge and gluino fields are given by $\delta A_{\mu}^{a}=-2\,\mathrm{i}\,\overline{\lambda}^{a}\gamma_{\mu}\,\varepsilon\,,\quad\delta\lambda^{a}=-\sigma_{\mu\nu}F_{\mu\nu}^{a}\,\varepsilon\,,$ (4) where the transformation parameter $\varepsilon$ is an anticommuting Majorana spinor. From the variation of the action under a supersymmetry transformation with a space-time-dependent parameter $\varepsilon(x)$ one derives the SUSY Ward identities. For any suitable gauge invariant local operator $Q(y)$, they read $\left\langle\partial^{\mu}S_{\mu}(x)Q(y)\right\rangle=m_{\tilde{g}}\left\langle\chi(x)Q(y)\right\rangle-\left\langle\frac{\delta Q(y)}{\delta\bar{\epsilon}(x)}\right\rangle,$ (5) where $S_{\mu}(x)=(S_{\mu}^{\alpha}(x))$ is the supercurrent of spin 3/2, and the term $m_{\tilde{g}}\left\langle\chi(x)Q(y)\right\rangle$ is due to the gluino mass in the action of the theory. In the continuum the supercurrent $S_{\mu}(x)$ and the operator $\chi(x)$ are given by $\displaystyle S_{\mu}(x)$ $\displaystyle=-\frac{2\,\mathrm{i}}{g}\mathrm{tr}\left[F^{\nu\rho}(x)\sigma_{\nu\rho}\gamma_{\mu}\lambda(x)\right],$ (6) $\displaystyle\chi(x)$ $\displaystyle=+\frac{2\,\mathrm{i}}{g}\mathrm{tr}\left[F^{\mu\nu}(x)\sigma_{\mu\nu}\lambda(x)\right].$ (7) The last term of Eq. (5) is a contact term, which contributes only if $x=y$, and it can be avoided if $Q(y)$ is not localised at $x$. Therefore the contact term is ignored in the following discussions. The four-dimensional space-time lattice breaks SUSY explicitly. As a consequence, the lattice versions of the Ward identities differ from their continuum counter parts by an additional term $\left\langle X_{S}(x)Q(y)\right\rangle$. The explicit form of this term is known, but need not be displayed here. At tree level this term is proportional to the lattice spacing $a$ and vanishes in the limit of zero lattice spacing. At higher orders in perturbation theory, nevertheless, the contribution of this term is finite in the continuum limit due to divergences proportional to 1/$a$ that multiply the factor $a$. This plays a role for the renormalisation of the supercurrent and of the gluino mass [7, 13]. In the renormalisation of SUSY Ward identities, operators of dimensions $\leq 11/2$ have to be taken into account. They lead to a modification of the gluino mass, and in addition a current $T_{\mu}$, mixing with the supercurrent, appears, corresponding to an operator of dimension $9/2$. Consequently, on the lattice the following Ward identities are obtained $Z_{S}\left\langle\nabla_{\mu}S_{\mu}(x)Q(y)\right\rangle+Z_{T}\left\langle\nabla_{\mu}T_{\mu}(x)Q(y)\right\rangle=m_{S}\left\langle\chi(x)Q(y)\right\rangle+O(a),$ (8) where $Z_{S}$ and $Z_{T}$ are renormalisation coefficients. The subtracted gluino mass is defined as $m_{S}=m_{\tilde{g}}-\bar{m}$, where $\bar{m}$ is the mass subtraction coming from the operators of dimension $7/2$. The mixing current is defined as $T_{\mu}(x)=\frac{2\,\mathrm{i}}{g}\mathrm{tr}\left[F_{\mu\nu}(x)\gamma_{\nu}\lambda(x)\right].$ (9) Regarding the local insertion operator $Q(y)$, our choice is the spinor $Q(y)=\chi^{(\mathrm{sp})}(y)$, with $\chi^{(\mathrm{sp})}(y)=\sum_{i<j}\mathrm{tr}\left[F_{ij}(y)\sigma_{ij}\lambda(y)\right],$ (10) where the indices $i,j\in\\{1,2,3\\}$. The reason behind this choice is that it gives the best signal [13]. ## 3 Numerical analysis of SUSY Ward identities We have analysed the SUSY Ward identities numerically, employing the configurations produced in our project on $\mathcal{N}=1$ supersymmetric Yang- Mills theory with gauge group SU(3). Numerically it is convenient to use integrated Ward identities where integration or sum is performed over all three spatial coordinates. The resulting identities will then hold for every time-slice distance $t$. In the analysis the data from all time-slice distances in an interval $t_{min}\leq t\leq t_{max}$ are included. The lower limit $t_{min}$ is always taken to be larger or equal than 3 in order to avoid contamination from contact terms. The choice of $t_{min}$ for the different ensembles of configurations is discussed below. Since the correlation functions are symmetric or antisymmetric in $t$, the upper limit $t_{max}$ is chosen to be half of the time extent of the lattice. Each term in Eq. (8) is a 4$\times$4 matrix in spin-space and can be expanded in the basis of 16 Dirac matrices, i. e. $\left\\{\boldsymbol{1},\gamma_{5},\gamma_{\mu},\gamma_{\mu}\gamma_{5},\mathrm{i}\sigma_{\mu\nu}\right\\}$. It can be shown, with the help of discrete symmetries, that only the following two contributions are non-zero [13]: $\hat{x}_{b,t,1}+A\hat{x}_{b,t,2}=B\hat{x}_{b,t,3},\qquad\text{with}\quad b=1,2\,,$ (11) where $A=Z_{T}Z^{-1}_{S}$, $B=am_{S}Z^{-1}_{S}$, and $\displaystyle\hat{x}_{1,t,1}$ $\displaystyle\equiv\sum_{\vec{x}}\left\langle\nabla_{4}S_{4}(x)Q(0)\right\rangle,$ $\displaystyle\hat{x}_{2,t,1}$ $\displaystyle\equiv\sum_{\vec{x}}\left\langle\nabla_{4}S_{4}(x)\gamma_{4}Q(0)\right\rangle,$ $\displaystyle\hat{x}_{1,t,2}$ $\displaystyle\equiv\sum_{\vec{x}}\left\langle\nabla_{4}T_{4}(x)Q(0)\right\rangle,$ $\displaystyle\hat{x}_{2,t,2}$ $\displaystyle\equiv\sum_{\vec{x}}\left\langle\nabla_{4}T_{4}(x)\gamma_{4}Q(0)\right\rangle,$ (12) $\displaystyle\hat{x}_{1,t,3}$ $\displaystyle\equiv\sum_{\vec{x}}\left\langle\chi(x)Q(0)\right\rangle,$ $\displaystyle\hat{x}_{2,t,3}$ $\displaystyle\equiv\sum_{\vec{x}}\left\langle\chi(x)\gamma_{4}Q(0)\right\rangle.$ In these equations the Dirac indices of $S_{4}(x)$, $T_{4}(x)$, $\chi(x)$ and of the insertion operator $Q(0)$ are not written, and sums over repeated (hidden) Dirac indices are implied. Also, $O(a)$ terms that vanish in the continuum limit are not written explicitly in these equations. Introducing a double index $i=(b,t)$, running over $2T$ values, where $T$ is the time extent of the lattice, and denoting $A_{1}=1,A_{2}=A,A_{3}=-B$, Eq. (11) is written compactly $\sum_{\alpha=1}^{3}A_{\alpha}\hat{x}_{i\alpha}=0\,.$ (13) In these equations the $\hat{x}_{i\alpha}=\langle x_{i\alpha}\rangle$ are the expectation values of random variables $x_{i\alpha}$, which themselves are considered to be the results of a finite Markov chain. We compute the estimators $x_{i\alpha}$ for the correlation functions $\hat{x}_{i\alpha}$ numerically using high performance facilities. The Eqs. (13), including all time-slice distances $t$ from $t_{min}$ to $t_{max}$, are solved simultaneously for $A_{\alpha}$ by means of minimal chi-squared methods. Two methods, namely the so-called Local Method and Global Method, have been used in the past by our collaboration [4, 13]. These methods, however, do not take properly into account correlations between the different quantities appearing in Eq. (13). For this purpose we have developed a new method based on a generalised least squares fit, the so-called GLS Method [14], based on the maximum likelihood. For fixed $A_{\alpha}$ ($\alpha=1,2,3$) and given numerical data $x_{i\alpha}$, the probability distribution $P\sim\exp(-L)$ of the quantities $\hat{x}_{i\alpha}$, subject to the constraints (13), has its maximum at a point where $L=L_{min}$, with $L_{min}=\frac{1}{2}\sum_{i,\alpha,j,\beta}(A_{\alpha}x_{i\alpha})(D^{-1})_{ij}(A_{\beta}x_{j\beta})\,,$ (14) where $D_{ij}=\sum_{\alpha,\beta}A_{\alpha}A_{\beta}(\langle x_{i\alpha}x_{j\beta}\rangle-\langle x_{i\alpha}\rangle\langle x_{j\beta}\rangle).$ (15) Next, the desired coefficients $A_{\alpha}$ have to be found such that $L_{min}$ as a function of $A_{2}$ and $A_{3}$ is minimised. This cannot be solved analytically, and we find $A_{\alpha}$ numerically such that the global minimum of $L_{min}(A_{2},A_{3})$ is reached; for details see Ref. [15]. In particular, owing to $A_{3}=-am_{S}Z^{-1}_{S}$ this provides us with the subtracted gluino mass $m_{S}$ up to the renormalisation factor. To estimate the statistical uncertainties we employ the standard Jackknife procedure. ### 3.1 Discretisation effects All terms in the Ward identity (8), including the $O(a)$ term $\left\langle X_{S}(x)Q(y)\right\rangle$, are correlation functions of gauge invariant operators. In the corresponding Eqs. (11) they are correlation functions of operators localised on time slices or pairs of adjacent time slices at distance $t$. As for any gauge invariant correlation function of this type, they decay exponentially in $t$, with a decay rate given by the mass gap of the theory. For very small $t$ the contributions of higher masses will affect the impact of the $O(a)$ term on the Ward identities. Therefore we expect that the value of the obtained gluino mass will depend on the minimal time slice distance $t_{min}$. This effect should become negligible at sufficiently large $t_{min}$. On the other hand, if $t_{min}$ is chosen too large, noise in the data will dominate. The behaviour that can be observed in Fig. 1 is compatible with these expectations. Figure 1: The subtracted gluino mass $am_{S}Z^{-1}_{S}$ as a function of $t_{min}$ calculated with the GLS Method at $\beta=5.6$. At small values of $t_{min}$ the subtracted gluino mass is affected by contact terms and by $O(a)$ terms. Data from $t_{min}=2$ and $t_{min}=3$ are shown, but do not enter our final analysis. An adequate choice of $t_{min}$ is therefore important for the quality of the results. We cope with this in two ways. In order to avoid perturbing effects at too small $t_{min}$ and a poor signal- to-noise ratio at too large $t_{min}$, for each hopping parameter and inverse gauge coupling, the value of $t_{min}$ is selected by finding an optimal starting point where a plateau in the subtracted gluino mass begins. The results are presented in Tab. 1. $\beta=5.4$ | $\beta=5.4$ | $\beta=5.45$ | $\beta=5.5$ | $\beta=5.6$ ---|---|---|---|--- ​​$V=12^{3}\times 24$​​ | ​​$V=16^{3}\times 32$​​ | ​​$V=16^{3}\times 32$​​ | ​​$V=16^{3}\times 32$​​ | ​​ $V=24^{3}\times 48$​​ $\kappa$ | ​​$t_{min}$​​ | $\kappa$ | ​​$t_{min}$​​ | $\kappa$ | ​​$t_{min}$ ​​ | $\kappa$ | ​​ $t_{min}$ ​​ | $\kappa$ | ​​$t_{min}$​​ 0.1695 | 4 | 0.1692 | 4 | 0.1685 | 5 | 0.1667 | 5 | 0.1645 | 7 0.1700 | 4 | 0.1695 | 4 | 0.1687 | 5 | 0.1673 | 5 | 0.1650 | 7 0.1703 | 4 | 0.1697 | 4 | 0.1690 | 5 | 0.1678 | 5 | 0.1655 | 6 0.1705 | 4 | 0.1700 | 4 | 0.1692 | 5 | 0.1680 | 5 | 0.1660 | 7 - | - | 0.1703 | 4 | 0.1693 | 4 | 0.1683 | 5 | - | - - | - | 0.1705 | 4 | - | - | - | - | - | - Table 1: The values of $t_{min}$ for all available gauge ensembles, chosen such that a plateau is formed. In the second approach, we consider that our simulations of the theory are done at different values of the lattice spacing $a$, which leads to different $O(a)$ terms in the Ward identities. A fixed value of $t_{min}$ in lattice units would mean a lower limit on the time-slice distances in physical units, that is on the cutoff-scale and shrinks to zero in the continuum limit. Instead it would be more appropriate to consider $t_{min}$ at constant physical distance for all gauge ensembles. This is done in the following way. At the coarsest lattice spacing, at inverse gauge coupling $\beta_{0}$, the value of $t_{min}$ is selected according to the plateau criterion explained above. For finer lattice spacings at inverse gauge couplings $\beta_{i}$ the corresponding $t_{min}$ are then obtained by scaling with a physical scale. In order to determine the physical scale we use the mass $m_{g\tilde{g}}$ of the gluino-glue particle and the Wilson flow parameter $w_{0}$. Correspondingly, $t_{min}$ is scaled according to $\displaystyle t_{min,{\beta_{i}}}$ $\displaystyle=t_{min,\beta_{0}}\frac{m_{g\tilde{g},\beta_{0}}}{m_{g\tilde{g},\beta_{i}}}\,,$ (16) $\displaystyle\text{or}\quad t_{min,{\beta_{i}}}$ $\displaystyle=t_{min,\beta_{0}}\frac{w_{0,\beta_{i}}}{w_{0,\beta_{0}}}\,,$ (17) where $\beta_{0}=5.4$, $\beta_{1}=5.45$, $\beta_{2}=5.5$, and $\beta_{3}=5.6$. The resulting $t_{min}$ is rounded to the nearest integer value. The values obtained by this method are collected in Tab. 2. In most points they are equal or almost equal to those in Tab. 1. $\beta$ | $t_{min}$ from $m_{g\tilde{g}}$ | $t_{min}$ from $w_{0}$ ---|---|--- 5.4 | 4 | 4 5.45 | 5 | 5 5.5 | 5 | 6 5.6 | 7 | 7 Table 2: The values of $t_{min}$ at fixed physical temporal distance from scaling with the gluino-glue mass $m_{g\tilde{g}}$ and with the Wilson flow parameter $w_{0}$. ### 3.2 Adjoint pion and remnant gluino mass The chiral limit is defined by the vanishing of the subtracted gluino mass. Its measured values can therefore be employed for the tuning of the hopping parameter $\kappa$ to the chiral limit. On the other hand, we can also use the vanishing of the adjoint pion mass $m_{\text{a-}\pi}$ for the tuning [16]. The adjoint pion $\text{a-}\pi$ is an unphysical particle in the SYM theory, that can be defined in partially quenched chiral perturbation theory [17]. In the numerical simulations its correlation function can be computed as the connected piece of the correlation function of the $\text{a-}\eta^{\prime}$ particle. Similar to the Gell-Mann-Oakes-Renner relation of QCD [5], in the continuum limit there is a linear relation between the adjoint pion mass squared and the gluino mass: $m^{2}_{\text{a-}\pi}\propto m_{\tilde{g}}$. The numerical results for the subtracted gluino mass from the Ward identities and the adjoint pion mass squared in lattice units are shown for $\beta=5.6$ in Fig. 2 together with their extrapolations towards the chiral limit. (a) The subtracted gluino mass $am_{S}Z^{-1}_{S}$ and the adjoint pion mass squared $(am_{\text{a-}\pi})^{2}$ as a function of $1/(2\kappa)$, and the corresponding extrapolations towards the chiral limit ($\kappa_{c}$). (b) The subtracted gluino mass $am_{S}Z^{-1}_{S}$ as a function of the adjoint pion mass squared $(am_{\text{a-}\pi})^{2}$ in order to obtain the remnant gluino mass $\Delta(am_{S}Z^{-1}_{S})$. Figure 2: Chiral limit and determination of the remnant gluino mass at $\beta=5.6$. All quantities are in lattice units. In the continuum the subtracted gluino mass and the adjoint pion mass should vanish at the same point. On the lattice, however, this is not the case due to lattice artefacts. As an estimate for this discrepancy we determine the value of the subtracted gluino mass at vanishing adjoint pion mass. This quantity is called the remnant gluino mass $\Delta(am_{S}Z^{-1}_{S})$, and it is expected to vanish in the continuum limit. The values of the remnant gluino mass, obtained by taking an average of the values calculated using the procedures explained above, are presented in Tab. 3. $\beta$ | 5.4 | 5.45 | 5.5 | 5.6 ---|---|---|---|--- $\Delta(am_{S}Z^{-1}_{S})$ | 0.0334(48) | 0.019(12) | 0.0099(88) | 0.0103(33) Table 3: The values of the remnant gluino mass $\Delta(am_{S}Z^{-1}_{S})$ obtained at four different values of the inverse gauge coupling. ### 3.3 Continuum limit The remnant gluino mass is a lattice artefact and should vanish in the continuum limit $a\rightarrow 0$. It is therefore a quantity to check on whether supersymmetry is recovered or not. Concerning the dependence of the remnant gluino mass on the lattice spacing, arguments based on partially quenched chiral perturbation theory suggest that the remnant gluino mass is of order $a^{2}$ at $m^{2}_{\text{a-}\pi}=0$ [13]. In order to investigate this relation, the remnant gluino mass has to be expressed in physical units. Our choice for the scale is the Wilson flow parameter $w_{0}$, which is defined through the gradient flow [10]. We use its values extrapolated to the chiral limit, $w_{0,\chi}$. Similarly the lattice spacing is represented by $a/w_{0,\chi}$. Our numerical results for the remnant gluino mass as a function of the lattice spacing and its extrapolation towards the continuum limit are shown in Fig. 3. The data points in Fig. 3(a) show the results from separate chiral extrapolations for each lattice spacing and the corresponding extrapolation to the continuum limit. The extrapolation to the continuum and the error of this extrapolation are obtained by means of parametric bootstrap with linear fits. On the other hand, Fig. 3(b) is obtained by means of a simultaneous fit of the dependence on the hopping parameter and the lattice spacing [18]. (a) The remnant gluino mass from separate extrapolations to the chiral limit where $m^{2}_{\text{a-}\pi}$ is zero, and the extrapolation to the continuum limit. (b) The remnant gluino mass from a simultaneous chiral and continuum extrapolation. By construction, in this method the data points coincide with the error band. Figure 3: The remnant gluino mass $\Delta{(w_{0}m_{S}Z^{-1}_{S})}$ in physical units $w_{0}$ as a function of the lattice spacing squared, and its linear extrapolation towards the continuum limit. The remnant gluino mass in the continuum limit is compatible with zero within one standard-deviation, confirming the preliminary results present in Ref. [15] with only two data points. Lattice artefacts vanish in the continuum limit as expected, and supersymmetry is recovered in the chiral and continuum limits, in agreement with our findings from the mass spectrum [12]. ## 4 Conclusion In this paper we have presented numerical results of an analysis of SUSY Ward identities in $\mathcal{N}=1$ supersymmetric Yang-Mills theory on the lattice with gauge group SU(3). Contact terms and $O(a)$ lattice artefacts in the Ward identities have been controlled by suitable choices of time-slice distances. Ensembles of gauge configurations at four different values of the lattice spacing and various hopping parameters have been analysed, allowing us for the first time to perform an extrapolation to the continuum limit, where the lattice artefacts vanish. The remnant gluino mass has been extrapolated in two alternative ways, on the one hand by extrapolating to the chiral limit at each lattice spacing separately and then to the continuum limit, and on the other hand by means of a simultaneous extrapolation to the chiral and continuum limit. With both extrapolations the lattice artefacts in the subtracted gluino mass appear to scale to zero as of order $a^{2}$ in agreement with the theoretical expectations. Our findings support the validity of SUSY Ward identities and the restoration of supersymmetry in the continuum limit. ## Acknowledgments The authors gratefully acknowledge the Gauss Centre for Supercomputing e. V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer JUQUEEN and JURECA at Jülich Supercomputing Centre (JSC) and SuperMUC at Leibniz Supercomputing Centre (LRZ). Further computing time has been provided on the compute cluster PALMA of the University of Münster. This work is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group “GRK 2149: Strong and Weak Interactions - from Hadrons to Dark Matter”. G. Bergner acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) Grant No. BE 5942/2-1. S. Ali acknowledges financial support from the Deutsche Akademische Austauschdienst (DAAD). ## References * [1] J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton University Press, 1992. * [2] G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 267 (1996) 195, [arXiv: hep-ph/9506380 ]. * [3] J. D. Lykken, [arXiv: 1005.1676[hep-ph]]. * [4] G. Bergner, P. Giudice, I. Montvay, G. Münster and S. Piemonte, JHEP 1603 (2016) 080, [arXiv: 1512.07014[hep-lat]]. * [5] G. Veneziano and S. Yankielowicz, Phys. Lett. B 113 (1982) 231. * [6] G. R. Farrar, G. Gabadadze and M. Schwetz, Phys. Rev. D 58 (1998) 015009, [arXiv: hep-th/9711166 ]. * [7] G. Curci and G. Veneziano, Nucl. Phys. B 292 (1987) 555. * [8] S. Musberg, G. Münster and S. Piemonte, JHEP 1305 (2013) 143, [arXiv: 1304.5741[hep-lat]]. * [9] S. Ali, G. Bergner, H. Gerber, P. Giudice, S. Kuberski, I. Montvay, G. Münster and S. Piemonte, EPJ Web Conf. 175 (2018) 08016, [arXiv: 1710.07464[hep-lat]]. * [10] S. Ali, G. Bergner, H. Gerber, P. Giudice, I. Montvay, G. Münster, S. Piemonte and P. Scior, JHEP 1803 (2018) 113, [arXiv: 1801.08062[hep-lat]]. * [11] S. Ali, G. Bergner, H. Gerber, S. Kuberski, I. Montvay, G. Münster, S. Piemonte and P. Scior, JHEP 1904 (2019) 150, [arXiv: 1901.02416[hep-lat]]. * [12] S. Ali, G. Bergner, H. Gerber, I. Montvay, G. Münster, S. Piemonte and P. Scior, Phys. Rev. Lett. 122 (2019) 2216011, [arXiv: 1902.11127[hep-lat]]. * [13] F. Farchioni, A. Feo, T. Galla, C. Gebert, R. Kirchner, I. Montvay, G. Münster and A. Vladikas, Eur. Phys. J. C 23 (2002) 719, [arXiv: hep-lat/0111008 ]. * [14] S. Ali, PhD thesis, University of Münster, June 2019. * [15] S. Ali, G. Bergner, H. Gerber, I. Montvay, G. Münster, S. Piemonte and P. Scior, Eur. Phys. J. C 78 (2018) 404, [arXiv: 1802.07067[hep-lat]]. * [16] K. Demmouche, F. Farchioni, A. Ferling, I. Montvay, G. Münster, E. E. Scholz and J. Wuilloud, Eur. Phys. J. C 69 (2010) 147, [arXiv: 1003.2073[hep-lat]]. * [17] G. Münster and H. Stüwe, JHEP 1405 (2014) 034, [arXiv: 1402.6616[hep-th]]. * [18] H. Gerber, PhD thesis, University of Münster, May 2019.

2024-09-04T02:54:58.388585

2003.04121

arxiv-papers

# The inverse theorem for the nonlinear Roth configuration: an exposition Sean Prendiville Department of Mathematics and Statistics Lancaster University UK<EMAIL_ADDRESS> ###### Abstract. We give an exposition of the inverse theorem for the cut-norm associated to the nonlinear Roth configuration, established by Peluse and the author in [6]. ###### Contents 1. 1 Introduction 2. 2 An outline of our argument 3. 3 PET induction 4. 4 An inverse theorem for the arithmetic box norm 5. 5 Quantitative concatenation 6. 6 Degree lowering 7. 7 Proof of the cut norm inverse theorem 8. A Basic theory of the Gowers norms ## 1\. Introduction Peluse and the author recently obtained an effective bound on the density of sets of integers lacking the configuration $x,\ x+y,\ x+y^{2}\qquad(y\neq 0).$ (1.1) We call this pattern the _nonlinear Roth configuration_ , after Bourgain and Chang [1]. ###### Theorem 1.1 (Peluse and Prendiville [6]). There exists an absolute constant $c>0$ such that if $A\subset\left\\{1,2,\dots,N\right\\}$ lacks the configuration (1.1), then $|A|\ll N(\log\log N)^{-c}.$ We have since removed a logarithm from this bound. ###### Theorem 1.2 (Peluse and Prendiville [7]). There exists an absolute constant $c>0$ such that if $A\subset\left\\{1,2,\dots,N\right\\}$ lacks the configuration (1.1), then $|A|\ll N(\log N)^{-c}.$ The main innovation behind both of these results is [6, Theorem 7.1], an inverse theorem for the counting operator associated to this configuration. It is the purpose of this note to give an exposition of this inverse theorem. The approach is essentially the same as that in [6]. We hope that having two distinct accounts is useful for those interested in utilising these ideas. ###### Definition 1.3 (Counting operator). For positive integers $q\leq N$ write $M:=\left\lfloor\sqrt{N/q}\right\rfloor.$ (1.2) Given this, define the _counting operator_ on the functions $f_{i}:\mathbb{Z}\to\mathbb{C}$ by $\Lambda_{q,N}(f_{0},f_{1},f_{2}):=\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[M]}f_{0}(x)f_{1}(x+y)f_{2}(x+qy^{2}).$ (1.3) When the $f_{i}$ all equal $f$ we simply write $\Lambda_{q,N}(f)$. ###### Definition 1.4 (Local function). We call a function $\phi:\mathbb{Z}\to\mathbb{C}$ a _local function of resolution $M$ and modulus $q$_ if there exists a partition of $\mathbb{R}$ into intervals of length $M$ such that $\phi$ is constant on the intersection of every such interval with every congruence class mod $q$. ###### Definition 1.5 (Cut norm). Define the _cut norm_ of $f:\mathbb{Z}\to\mathbb{C}$ by $\left\|f\right\|_{q,N}:=\sup\\{|\Lambda_{q,N}(f,g_{1},g_{2})|,\ |\Lambda_{q,N}(g_{1},f,g_{2})|,\ |\Lambda_{q,N}(g_{1},g_{2},f)|\\},$ (1.4) where the supremum is taken over all 1-bounded functions $g_{i}:[N]\to\mathbb{C}$. We note that, in spite of our nomenclature, this is not a norm but a seminorm. One could remedy this by summing over $y\geq 0$ in the counting operator (1.3) This seminorm is useful in [7]. However, it is too restrictive for the approach developed in [6], where we (implicitly) only work with the following quantities: $\left\|f\right\|^{\sharp}_{q}:=\sup\\{|\Lambda_{q,N}(g_{0},g_{1},f)|:|g_{i}|\leq 1\ \text{ and }\ \mathrm{supp}(g_{i})\subset[N]\\}$ (1.5) and $\left\|f\right\|^{\flat}_{q}:=\sup\\{|\Lambda_{q,N}(f,g_{1},g_{2})|,\ |\Lambda_{q,N}(g_{1},f,g_{2})|\ :\ |g_{i}|\leq 1\ \text{ and }\ \mathrm{supp}(g_{i})\subset[N]\\}.$ (1.6) Here then is a re-formulation and slight generalisation of [6, Theorem 7.1]. ###### Theorem 1.6 (Partial cut norm inverse theorem). Let $q\leq N$ be positive integers, $\delta>0$, and $f:\mathbb{Z}\to\mathbb{C}$ be a $1$-bounded function with support in $[N]$. Suppose that $\left\|f\right\|^{\flat}_{q,N}\geq\delta.$ Then either $N\ll(q/\delta)^{O(1)}$ or there exists a 1-bounded local function $\phi$ of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$, modulus $qq^{\prime}$ for some $q^{\prime}\ll\delta^{-O(1)}$, and such that $\sum_{x\in[N]}f(x)\phi(x)\gg\delta^{2^{66}}N.$ This exposition is organised as follows. In §2, we give a more detailed outline of the proof of Theorem 1.6. In §§3–5 we develop an effective approach to a (special case of a) so-called _concatenation_ theorem of Tao and Ziegler [10]. This allows us to show that if our counting operator is large, then the function weighting the nonlinear term must have large Gowers uniformity norm. The drawback is that the degree of the resulting Gowers norm is large (in our approach it is the $U^{5}$-norm). In §6 we give a _degree-lowering_ procedure, which utilises properties specific to our configuration to show that one may replace the $U^{5}$-norm with the $U^{1}$-norm. In §7 we combine the results of the previous sections in order to prove Theorem 1.6. ### 1.1. Notation #### 1.1.1. Standard conventions We use $\mathbb{N}$ to denote the positive integers. For a real $X\geq 1$, write $[X]=\\{1,2,\ldots,\left\lfloor X\right\rfloor\\}$. A complex-valued function is _1-bounded_ if the modulus of the function does not exceed 1. We use counting measure on $\mathbb{Z}$, so that for $f,g:\mathbb{Z}\to\mathbb{C}$ we have $\left\langle f,g\right\rangle:=\sum_{x}f(x)\overline{g(x)}\qquad\text{and}\qquad\left\|f\right\|_{L^{p}}:=\biggl{(}\sum_{x}|f(x)|^{p}\biggr{)}^{\frac{1}{p}}.$ Any sum of the form $\sum_{x}$ is to be interpreted as a sum over $\mathbb{Z}$. We use Haar probability measure on $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$, so that for measurable $F:\mathbb{T}\to\mathbb{C}$ we have $\left\|F\right\|_{L^{p}}:=\biggl{(}\int_{\mathbb{T}}|F(\alpha)|^{p}\mathrm{d}\alpha\biggr{)}^{\frac{1}{p}}=\biggl{(}\int_{0}^{1}|F(\alpha)|^{p}\mathrm{d}\alpha\biggr{)}^{\frac{1}{p}}$ For $\alpha\in\mathbb{T}$ we write $\left\|\alpha\right\|$ for the distance to the nearest integer. For a finite set $S$ and function $f:S\to\mathbb{C}$, denote the average of $f$ over $S$ by $\mathbb{E}_{s\in S}f(s):=\frac{1}{|S|}\sum_{s\in S}f(s).$ Given functions $f,g:G\to\mathbb{C}$ on an additive group with measure $\mu_{G}$ we define their convolution by $f*g(x):=\int_{G}f(x-y)g(y)\mathrm{d}\mu_{G},$ (1.7) when this makes sense. We define the Fourier transform of $f:\mathbb{Z}\to\mathbb{C}$ by $\hat{f}(\alpha):=\sum_{x}f(x)e(\alpha x)\qquad(\alpha\in\mathbb{T}),$ (1.8) again, when this makes sense. Here $e(\alpha)$ stands for $e^{2\pi i\alpha}$. The difference function of $f:\mathbb{Z}\to\mathbb{C}$ is the function $\Delta_{h}f:\mathbb{Z}\to\mathbb{C}$ given by $\Delta_{h}f(x)=f(x)\overline{f(x+h)}.$ Iterating gives $\Delta_{h_{1},\dots,h_{s}}f:=\Delta_{h_{1}}\dots\Delta_{h_{s}}f.$ This allows us to define the Gowers $U^{s}$-norm $\left\|f\right\|_{U^{s}}:=\left(\sum_{x,h_{1},\dots,h_{s}}\Delta_{h_{1},\dots,h_{s}}f(x)\right)^{1/2^{s}}.$ (1.9) If $\|\cdot\|$ is a seminorm on an inner product space, recall that its dual seminorm $\|\cdot\|^{*}$ is defined by $\|f\|^{*}:=\sup_{\|g\|\leq 1}|\langle f,g\rangle|.$ Hence $\left|\left\langle f,g\right\rangle\right|\leq\left\|f\right\|^{*}\left\|g\right\|.$ (1.10) For a function $f$ and positive-valued function $g$, write $f\ll g$ or $f=O(g)$ if there exists a constant $C$ such that $|f(x)|\leq Cg(x)$ for all $x$. We write $f=\Omega(g)$ if $f\gg g$. We sometimes opt for a more explicit approach, using $C$ to denote a large absolute constant, and $c$ to denote a small positive absolute constant. The values of $C$ and $c$ may change from line to line. #### 1.1.2. Local conventions Up to normalisation, all of the above are well-used in the literature. Next we list notation specific to our paper. We have tried to minimise this in order to aid the casual reader. For a real parameter $H\geq 1$, we use $\mu_{H}:\mathbb{Z}\to[0,1]$ to represent the following normalised Fejér kernel $\mu_{H}(h):=\frac{1}{\left\lfloor H\right\rfloor}\left(1-\frac{|h|}{\left\lfloor H\right\rfloor}\right)_{+}=\frac{(1_{[H]}*1_{[H]})(h)}{\left\lfloor H\right\rfloor^{2}}.$ (1.11) For a multidimensional vector $h\in\mathbb{Z}^{d}$ we write $\mu_{H}(h):=\mu_{H}(h_{1})\dotsm\mu_{H}(h_{d}).$ (1.12) We observe that this is a probability measure on $\mathbb{Z}^{d}$ with support in the interval $(-H,H)^{d}$. ## 2\. An outline of our argument In this section we describe the ideas behind Theorem 1.6. In the hope of making the ideas clearer, we make the simplification that $q=1$ in our counting operator (1.3). Hence, for finitely supported functions $f_{0},f_{1},f_{2}:\mathbb{Z}\to\mathbb{C}$, write $\Lambda(f_{0},f_{1},f_{2}):=\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[N^{1/2}]}f_{0}(x)f_{1}(x+y)f_{2}(x+y^{2}).$ (2.1) For this operator, Theorem 1.6 can be deduced from the following. ###### Lemma 2.1. Let $f_{0},f_{1},f_{2}:\mathbb{Z}\to\mathbb{C}$ be $1$-bounded functions supported in the interval $[N]$ and $\delta>0$. Suppose that $|\Lambda(f_{0},f_{1},f_{2})|\geq\delta.$ Then either $N\ll\delta^{-O(1)}$ or there exist positive integers $q\ll\delta^{-O(1)}$ and $N^{\prime}\gg\delta^{O(1)}N^{1/2}$ such that $\sum_{x}\left|\sum_{y\in[N^{\prime}]}f_{1}(x+qy)\right|\gg\delta^{O(1)}NN^{\prime}.$ (2.2) Using the notation (1.9), notice that the left-hand side of (2.2) is equal to $\sum_{x}\left\|f_{1}\right\|_{U^{1}(x+q\cdot[N^{\prime}])}.$ ### 2.1. Quantitative concatenation To prove Lemma 2.1, we first prove that our counting operator (2.1) is controlled by the $U^{5}$-norm of $f_{2}$. The purpose of this subsection is to sketch how we do this with polynomial bounds. By applying the Cauchy–Schwarz and van der Corput inequalities a number of times, we show in §3 that, when $f_{0},f_{1},f_{2}:\mathbb{Z}\to\mathbb{C}$ are $1$-bounded functions supported in the interval $[N]$, largeness of the counting operator (2.1) implies largeness of the sum $\sum_{a,b\in[N^{1/2}]}\sum_{h_{1},h_{2},h_{3}\in[N^{1/2}]}\sum_{x}\Delta_{ah_{1},bh_{2},(a+b)h_{3}}f_{2}(x).$ (2.3) This deduction is made following the PET induction scheme of Bergelson and Leibman [2]. The gain in working with the counting operator (2.3) over (2.1) is that univariate polynomials such as $y^{2}$, whose image constitute a sparse set, have been replaced by bilinear forms such as $ah_{1}$, whose image is much denser In §§4–5, we show that largeness of (2.3) implies largeness of $\|f_{2}\|_{U^{5}}$. If there were no dependence between the coefficients of the $h_{i}$ in (2.3), then we could in fact bound (2.3) in terms of $\|f_{2}\|_{U^{3}}$. Since the argument is informative, we illustrate why this is the case for the sum $\sum_{a,b,c\in[N^{1/2}]}\sum_{h_{1},h_{2},h_{3}\in[N^{1/2}]}\sum_{x}\Delta_{ah_{1},bh_{2},ch_{3}}f_{2}(x).$ (2.4) The following fact is key, the formal version of which is Lemma 5.3. ###### Claim 2.2. If $\displaystyle\sum_{a,h\in[N^{1/2}]}\sum_{x}\Delta_{ah}f(x)$ is large then so is $\displaystyle\sum_{k\in(-N,N)}\sum_{x}\Delta_{k}f(x)$. ###### Sketch proof. Apply the Cauchy–Schwarz inequality to double the $a$ and $h$ variables, yielding a bound in terms of $\sum_{a,a^{\prime}\in[N^{1/2}]}\sum_{h,h^{\prime}\in[N^{1/2}]}\sum_{x}\Delta_{ah-a^{\prime}h^{\prime}}f(x).$ (2.5) For a random choice of $a,a^{\prime}\in[N^{1/2}]$, the progression $a\cdot[N^{1/2}]-a^{\prime}\cdot[N^{1/2}]$ covers a large portion of the interval $(-N,N)$ relatively smoothly. One can make this intuition rigorous and thus deduce largeness of the sum $\sum_{k\in(-N,N)}\sum_{x}\Delta_{k}f(x).$∎ Applying Claim 2.2 three times allows us to replace each of $ah_{1}$, $bh_{2}$ and $ch_{3}$ in (2.4) with $k_{1},k_{2},k_{3}\in(-N,N)$, yielding largeness of $\left\|f_{2}\right\|_{U^{3}}$. Since the PET induction scheme outputs (2.3), and not (2.4), the problem remains of how to handle the dependency between the differencing parameters in (2.3). If we were not concerned with quantitative bounds, we could apply a ‘concatenation’ theorem of Tao and Ziegler [10, Theorem 1.24] to obtain largeness of the $U^{9}$-norm of $f_{2}$. However, the qualitative nature of this argument means that it cannot be used to obtain bounds in the nonlinear Roth theorem. In its place we prove Theorem 5.6, which is a special case of [10, Theorem 1.24], using a very different argument that gives polynomial bounds. We spend the remainder of this subsection sketching the argument. We begin by viewing (2.3) as the average $\sum_{a,h_{1}\in[N^{1/2}]}\left\|\Delta_{ah_{1}}f_{2}\right\|_{a},$ (2.6) where $\|f\|_{a}^{4}:=\sum_{b\in[N^{1/2}]}\sum_{h_{2},h_{3}\in[N^{1/2}]}\sum_{x}\Delta_{bh_{2},(a+b)h_{3}}f(x)$ (2.7) One can view this as an average of 2-dimensional Gowers box norms where, for fixed $b$, the inner sum corresponds to a box norm in the ‘directions’ $b$ and $a+b$. Note that if we could bound the quantity $\|\Delta_{ah_{1}}f_{2}\|_{a}$ in terms of the $U^{4}$-norm of $\Delta_{ah_{1}}f_{2}$ for many pairs $(a,h_{1})$, then by Claim 2.2 we deduce largeness of the $U^{5}$-norm of $f_{2}$. We show that, on average, one can indeed control $\|\cdot\|_{a}$ in terms of $\|\cdot\|_{U^{4}}$, with polynomial bounds. The following can be extracted from the proof of (the more general) Theorem 5.6. ###### Lemma 2.3. For each $a\in[N^{1/2}]$ let $f_{a}:\mathbb{Z}\to\mathbb{C}$ be a $1$-bounded function supported in the interval $[N]$. Suppose that $\mathbb{E}_{a\in[N^{1/2}]}\|f_{a}\|_{a}^{4}\geq\delta\left\|1_{[N]}\right\|_{a}^{4}.$ Then $\mathbb{E}_{a\in[N^{1/2}]}\|f_{a}\|_{U^{4}}^{16}\gg\delta^{O(1)}\left\|1_{[N]}\right\|_{U^{4}}^{16}.$ To finish this subsection, we briefly discuss the proof of this key lemma. For most choices of $a,b\in[N^{1/2}]$, the ‘directions’ $a$ and $a+b$ of the box norm $\sum_{h_{2},h_{3}\in[N^{1/2}]}\sum_{x}\Delta_{bh_{2},(a+b)h_{3}}f_{a}(x)$ (2.8) are close to ‘independent’, in the sense that at least one of the directions $a$ and $a+b$ is large and together they have small greatest common divisor. The proof of Lemma 2.3 thus begins by viewing $\|\cdot\|_{a}$ as an average of box norms $\|f\|_{\square(X,Y)}^{4}:=\sum_{x_{1},x_{2}\in X,y_{1},y_{2}\in Y}f(x_{1},y_{1})\overline{f(x_{1},y_{2})f(x_{2},y_{1})}f(x_{2},y_{2}).$ (2.9) It is easy to show that largeness of $\|f\|_{\square(X,Y)}$ implies that $f$ correlates with a function of the form $(x,y)\mapsto l(x)r(y)$. We show, analogously, that provided $b$ and $a+b$ are not too small and have greatest common divisor not too large, then largeness of the arithmetic box norm (2.8) implies that $f_{a}$ correlates with a product $g_{b}h_{a+b}$ of 1-bounded functions, where $g_{b}$ is $b$-periodic and $h_{a+b}$ is almost periodic under shifts by integer multiples of $a+b$. As a consequence, for most $a\in[N^{1/2}]$, largeness of $\|f_{a}\|_{a}$ implies largeness of $\sum_{b\in[N^{1/2}]}\sum_{x}f_{a}(x)g_{b}(x)h_{a+b}(x).$ (2.10) In fact, an application of Cauchy–Schwarz allows us give an explicit description of $h_{a+b}$ in terms of $f_{a}$, namely we may take it to be of the form $h_{a+b}(x)=\mathbb{E}_{k\in[N^{1/2}]}f_{a}(x+(a+b)k)g_{b}(x+(a+b)k).$ (2.11) This presentation makes apparent the almost periodicity of $h_{a+b}$. ###### Claim 2.4. Largeness of (2.10) implies that $\mathbb{E}_{b\in[N^{1/2}]}h_{a+b}$ has large $U^{3}$-norm. Let us first show why Claim 2.4 in turn implies that $f_{a}$ has large $U^{4}$-norm, completing our sketch proof of Lemma 2.3. The expression (2.11) and the triangle inequality for Gowers norms together imply that largeness of $\mathbb{E}_{b\in[N^{1/2}]}\left\|h_{a+b}\right\|_{U^{3}}$ implies largeness of $\mathbb{E}_{b\in[N^{1/2}]}\left\|f_{a}g_{b}\right\|_{U^{3}}$. Utilising the $b$-periodicity of $g_{b}$ we have $\left\|f_{a}g_{b}\right\|_{U^{3}}=\mathbb{E}_{k\in[N^{1/2}]}\left\|f_{a}(\cdot)g_{b}(\cdot+bk)\right\|_{U^{3}}.$ (2.12) The product $f_{a}(\cdot)g_{b}(\cdot+bk)$ resembles a difference function in the direction $b$. Indeed the Gowers–Cauchy–Schwarz inequality (see [9, Exercise 1.3.19]) shows that if (2.12) is large (on average over $b\in[N^{1/2}]$) then so is $\mathbb{E}_{b,k\in[N^{1/2}]}\left\|\Delta_{bk}f_{a}\right\|_{U^{3}}$ Largeness of $\left\|f_{a}\right\|_{U^{4}}$ then follows from Claim 2.2. Finally we sketch the proof of Claim 2.4. The Cauchy–Schwarz inequality allows us to remove the weight $f_{a}(x)$ from (2.10) and deduce largeness of $\sum_{x}\sum_{b,b^{\prime}\in[N^{1/2}]}\overline{g_{b}(x)h_{a+b}(x)}g_{b^{\prime}}(x)h_{a+b^{\prime}}(x).$ Using the periodicity properties of $g_{b}$, $g_{b^{\prime}}$ and $h_{a+b}$, this is approximately equal to $\sum_{x}\sum_{\begin{subarray}{c}b,b^{\prime}\in[N^{1/2}]\\\ k_{1},k_{2},k_{3}\in[N^{1/2}]\end{subarray}}\overline{g_{b}(x-bk_{1})h_{a+b}(x-(a+b)k_{2})}g_{b^{\prime}}(x-b^{\prime}k_{3})h_{a+b^{\prime}}(x).$ Changing variables in $x$, we obtain largeness of the sum $\sum_{x}\sum_{\begin{subarray}{c}b,b^{\prime}\in[N^{1/2}]\\\ k_{1},k_{2},k_{3}\in[N^{1/2}]\end{subarray}}\overline{g_{b}(x+(a+b)k_{2}+b^{\prime}k_{3})h_{a+b}(x+bk_{1}+b^{\prime}k_{3})}\\\ g_{b^{\prime}}(x+bk_{1}+(a+b)k_{2})h_{a+b^{\prime}}(x+bk_{1}+(a+b)k_{2}+b^{\prime}k_{3}).$ The point here is that all but the last function have arguments depending on at most two of the bilinear forms $bk_{1}$, $(a+b)k_{2}$ and $b^{\prime}k_{1}^{\prime}$. This enables us to employ the Gowers–Cauchy–Schwarz inequality (in the form of Lemma A.4) to deduce largeness of a sum similar to $\sum_{x}\sum_{\begin{subarray}{c}b,b^{\prime}\in[N^{1/2}]\\\ k_{1},k_{2},k_{3}\in[N^{1/2}]\end{subarray}}\Delta_{bk_{1},\,(a+b)k_{2},\,b^{\prime}k_{3}}h_{a+b^{\prime}}(x).$ The utility of this expression is that the directions of the differencing parameters are all ‘independent’ of the direction of periodicity of $h_{a+b^{\prime}}$. Indeed the approximate $(a+b^{\prime})$-periodicity of $h_{a+b^{\prime}}$ means that one can replace $\Delta_{y}h_{a+b^{\prime}}$ with $\mathbb{E}_{k}\Delta_{y+(a+b^{\prime})k}h_{a+b^{\prime}}$ at the cost of a small error. We thereby obtain largeness of $\sum_{x}\sum_{b,b^{\prime}\in[N^{1/2}]}\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3}\in[N^{1/2}]\\\ k_{1}^{\prime},k_{2}^{\prime},k_{3}^{\prime}\in[N^{1/2}]\end{subarray}}\Delta_{bk_{1}+(a+b^{\prime})k_{1}^{\prime},\,(a+b)k_{2}+(a+b^{\prime})k_{2}^{\prime},\,b^{\prime}k_{3}+(a+b^{\prime})k_{3}^{\prime}}h_{a+b^{\prime}}(x).$ (2.13) For a random triple $(a,b,b^{\prime})\in[N^{1/2}]$ the greatest common divisor of the pairs $(b,a+b^{\prime})$, $(a+b,a+b^{\prime})$ and $(b^{\prime},a+b^{\prime})$ are all small, and these are the pairs appearing in the differencing parameters of (2.13). The argument used to treat (2.5) may be therefore be employed to replace (2.13) with $\sum_{x}\sum_{b^{\prime}\in[N^{1/2}]}\sum_{k_{1},k_{2},k_{3}\in[N]}\Delta_{k_{1},k_{2},k_{3}}h_{a+b^{\prime}}(x),$ and thereby yield Claim 2.4. ### 2.2. Degree lowering After we have shown that $\Lambda(f_{0},f_{1},f_{2})$ is controlled by the $U^{5}$-norm of $f_{2}$, we carry out a ‘degree lowering’ argument. This technique originated in the work [5] in finite fields. The basic idea is that, under certain conditions, one can combine $U^{s}$-control with understanding of two-term progressions to deduce $U^{s-1}$-control. Repeating this gives a sequence of implications $U^{5}\text{-control}\implies U^{4}\text{-control}\implies U^{3}\text{-control}\implies U^{2}\text{-control}\implies U^{1}\text{-control}.$ Despite the appearance of the $U^{5}$-norm, $U^{4}$-norm, and $U^{3}$-norm, the degree lowering argument, both in [5] and here, does not require the $U^{s}$-inverse theorem for any $s\geq 3$. Instead it relies on Fourier analysis in the place of these inverse theorems. Adapting the degree lowering argument of [5] to the integer setting requires several significant modifications. The first modification is that the $U^{s}$-control described above is control in terms of the $U^{s}$-norm of the dual function $F(x):=\mathbb{E}_{y\in[N^{1/2}]}f_{0}(x-y^{2})f_{1}(x+y-y^{2}).$ (2.14) Thus, to begin the degree lowering argument, we must show that largeness of $\Lambda(f_{0},f_{1},f_{2})$ implies largeness of $\|F\|_{U^{5}}$. To do this, we use a simple Hahn–Banach decomposition as described in [3, Proposition 3.6], for details see §7. We conclude this section by sketching an instance of degree-lowering: how $U^{3}$-control of the dual (2.14) implies $U^{2}$-control, starting from the assumption that $\|F\|_{U^{3}}^{8}\geq\delta\left\|1_{[N]}\right\|_{U^{3}}^{8}.$ Using the fact that $\|F\|_{U^{3}}^{8}=\sum_{h}\|\Delta_{h}F\|_{U^{2}}^{4}$ and applying the $U^{2}$-inverse theorem, we deduce the existence of a function $\phi:\mathbb{Z}\to\mathbb{T}$ such that, for at least $\gg\delta N$ choices of differencing parameter $h$, we have $\left|\sum_{x\in[N]}\Delta_{h}F(x)e(\phi(h)x)\right|\gg\delta N.$ (2.15) Note that if, in the above inequality, we could replace the function $\phi(h)$ by a constant $\beta\in\mathbb{T}$ not depending on $h$, then we could easily deduce largeness of $\|F\|_{U^{2}}$. Indeed, writing $g(h)$ for the phase of the sum inside absolute values, this would give $\sum_{x,h}\overline{g(h)}\overline{F(x+h)}F(x)e(\beta x)\gg\delta^{O(1)}N^{3},$ and the usual argument111One can either use orthogonality and extraction of a large Fourier coefficient, as in the proof of Lemma A.1, or use two applications of Cauchy–Schwarz. showing $U^{2}$-control of the equation $x+y=z$ implies that $\|F\|_{U^{2}}^{4}\gg\delta^{O(1)}\left\|1_{[N]}\right\|_{U^{2}}$. It thus remains to show that such a $\beta$ exists. Expanding the definition of the difference and dual functions in (2.15), and using the Cauchy–Schwarz inequality (as is done in greater generality in the proof of Lemma 6.3), one can show that there exists $h^{\prime}$ such that for many $h$ satisfying (2.15) we have $\left|\sum_{x}\sum_{y\in[N^{1/2}]}\Delta_{h-h^{\prime}}f_{0}(x)\Delta_{h-h^{\prime}}f_{1}(x+y)e([\phi(h)-\phi(h^{\prime})][x+y^{2}])\right|\gg\delta^{O(1)}N^{3/2}$ Further application of Cauchy–Schwarz allows us to remove the difference functions from the above inequality and deduce largeness of the exponential sum $\sum_{z\in[N^{1/2}]}\left|\sum_{y\in[N^{1/2}]}e(2\left[\phi(h)-\phi(h^{\prime})\right]yz)\right|.$ Summing the inner geometric progression and using a Vinogradov-type lemma then shows that $\phi(h)-\phi(h^{\prime})$ is major arc. There are very few major arcs, so the pigeonhole principle gives the existence of $\beta_{0}\in\mathbb{T}$ such that $\phi(h)-\phi(h^{\prime})$ is very close to $\beta_{0}$ for many $h\in(-N,N)$ that also satisfy (2.15). We may therefore take $\beta=\beta_{0}+\phi(h^{\prime})$ in the argument following (2.15). ## 3\. PET induction We prove Theorem 1.6 over the course of §§3–7. We begin in §§3–5 by showing how our counting operator $\Lambda_{q,N}(f_{0},f_{1},f_{2})$, as defined in (1.3), is controlled by the $U^{5}$-norm of $f_{2}$. This argument starts with the PET induction scheme of Bergelson–Leibman [2], which in some sense ‘linearises’ a polynomial progression, replacing univariate polynomials such as $y^{2}$ with bilinear forms $ah$. The outcome of this procedure is Lemma 3.3. For the following, we recall our definition (1.11) of $\mu_{H}$. ###### Lemma 3.1 (van der Corput inequality). Let $f:\mathbb{Z}\to\mathbb{C}$ be 1-bounded and $M,H\geq 1$. Then we have the estimate $\biggl{|}\mathbb{E}_{y\in[M]}f(y)\biggr{|}^{2}\leq\frac{M+H}{M}\sum_{h}\mu_{H}(h)\mathbb{E}_{y\in[M]}\Delta_{h}f(y).$ ###### Proof. This is standard, see for instance [8, Lemma 3.1].∎ ###### Lemma 3.2 (Difference functions control linear configurations). Let $f_{i}:\mathbb{Z}\to\mathbb{C}$ be $1$-bounded functions with support in an interval $I_{i}$ of size $|I_{i}|=N$. Then for any $a,b\in\mathbb{Z}$ and $1\leq H\leq M$ we have $\biggl{|}\mathbb{E}_{x\in I_{0}}\mathbb{E}_{y\in[M]}f_{0}(x)f_{1}(x+ay)f_{2}(x+by)f_{3}(x+(a+b)y)\biggr{|}^{8}\\\ \ll\sum_{h}\mu_{H}(h)\mathbb{E}_{x\in I_{3}}\Delta_{ah_{1},bh_{2},(a+b)h_{3}}f_{3}(x).$ (3.1) ###### Proof. Applying Cauchy-Schwarz in the $x$ variable gives $\biggl{|}\mathbb{E}_{x\in I_{0}}\mathbb{E}_{y\in[M]}f_{0}(x)f_{1}(x+ay)f_{2}(x+by)f_{3}(x+(a+b)y)\biggr{|}^{2}\\\ \leq\frac{1}{N}\sum_{x}\bigg{|}\mathbb{E}_{y\in[M]}f_{1}(x+ay)f_{2}(x+by)f_{3}(x+(a+b)y)\bigg{|}^{2}.$ Bounding the inner sum using van der Corput’s inequality (Lemma 3.1) and making the change of variables $x\mapsto x-ay$ (valid since $x$ is ranging over $\mathbb{Z}$), the latter is at most $2\sum_{h_{1}}\mu_{H}(h_{1})\mathbb{E}_{x\in I_{1}}\mathbb{E}_{y\in[M]}\Delta_{ah_{1}}f_{1}(x)\Delta_{bh_{1}}f_{2}(x+(b-a)y)\Delta_{(a+b)h_{1}}f_{3}(x+by).$ Here we may restrict $x$ to $I_{1}$ on observing that the support of $\Delta_{ah_{1}}f_{1}$ is contained in the support of $f_{1}$. Making use of the fact that $\mu_{H}$ is a probability measure, we repeat the procedure of applying Cauchy–Schwarz, van der Corput then a change of variables, to deduce that $\biggl{|}\mathbb{E}_{x\in I_{0}}\mathbb{E}_{y\in[M]}f_{0}(x)f_{1}(x+ay)f_{2}(x+by)f_{3}(x+(a+b)y)\biggr{|}^{4}\\\ \leq 8\sum_{h_{1},h_{2}}\mu_{H}(h_{1})\mu_{H}(h_{2})\mathbb{E}_{x\in I_{2}}\mathbb{E}_{y\in[M]}\Delta_{bh_{1},(b-a)h_{2}}f_{2}(x)\Delta_{(a+b)h_{1},bh_{2}}f_{3}(x+ay).$ A final iteration of the same procedure then yields (3.1). ∎ Before embarking on the following, we remind the reader of our convention (1.2) regarding $M$. ###### Lemma 3.3 (Linearisation). Let $f_{i}:\mathbb{Z}\to\mathbb{C}$ be $1$-bounded functions, each with support in the interval $[N]$. Then for any $1\leq H\leq M$ we have $\left|\Lambda_{q,N}(f_{0},f_{1},f_{2})\right|^{32}\ll\sum_{a,b,h}\mu_{M}(a)\mu_{M}(b)\mu_{H}(h)\mathbb{E}_{x\in[N]}\Delta_{2q(a+b)h_{1},\,2qbh_{2},\,2qah_{3}}f_{2}(x).$ (3.2) ###### Proof. We repeat the procedure given in the proof of Lemma 3.2, applying Cauchy- Schwarz, followed by van der Corput’s inequality and a change of variables. A first application gives $\left|\Lambda_{q,N}(f_{0},f_{1},f_{2})\right|^{2}\leq\\\ 2\sum_{a}\mu_{M}(a)\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[M]}\Delta_{a}f_{1}(x)f_{2}\bigl{(}x+qy^{2}-y\bigr{)}\overline{f_{2}\bigl{(}x+q(y+a)^{2}-y\bigr{)}}.$ A second application then gives $\left|\Lambda_{q,N}(f_{0},f_{1},f_{2})\right|^{4}\ll\sum_{a,b}\mu_{M}(a)\mu_{M}(b)\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[M]}f_{2}(x)\overline{f_{2}\bigl{(}x+2qay+qa^{2}\bigr{)}}\\\ \overline{f_{2}\bigl{(}x+2qby+qb^{2}-b\bigr{)}}f_{2}\bigl{(}x+2q(a+b)y+q(a+b)^{2}-b\bigr{)}.$ Applying Lemma 3.2 to bound the inner sum over $x$ and $y$, we obtain (3.2) after a final change of variables ∎ ## 4\. An inverse theorem for the arithmetic box norm The objective in this section is to characterise those 1-bounded functions $f:\mathbb{Z}\to\mathbb{C}$ with support in $[N]$ for which the following quantity is large $\sum_{h,x}\mu_{H}(h)\Delta_{ah_{1},bh_{2}}f(x).$ (4.1) One can think of this as an arithmetic analogue of the two-dimensional ‘box norm’ (2.9). In our eventual application we are able to ensure that $a$ and $b$ are a generic pair of integers from the interval $[N^{1/2}]$. In particular, at least one of them has size proportional to $N^{1/2}$ and their highest common factor is small. One may think of this as a proxy for linear independence. We begin by characterising largeness of (4.1) when the directions are coprime. ###### Lemma 4.1 (Inverse theorem for the arithmetic box norm). Let $a,b$ be positive integers with $\gcd(a,b)=1$. Suppose that $f:\mathbb{Z}\to\mathbb{C}$ is $1$-bounded with support in the interval $[N]$ and satisfies $\sum_{h,x}\mu_{H}(h)\Delta_{ah_{1},bh_{2}}f(x)\geq\delta N.$ (4.2) Then there exist 1-bounded functions $g,h:\mathbb{Z}\to\mathbb{C}$ such that * • $g$ is $a$-periodic, in the sense that $g(x+a)=g(x)$ for all $x$; * • $h$ is approximately $b$-periodic, in the sense that for any $\varepsilon>0$ we have $\\#\left\\{x\in[N]:h(x+by)\neq h(x)\text{ for some }|y|\leq\varepsilon N/b\right\\}\leq\left(1+\tfrac{2\varepsilon N}{b}\right)\left(1+\tfrac{N}{a}\right);$ and furthermore $\biggl{|}\sum_{x}f(x)g(x)h(x)\biggr{|}\geq\delta\left\lfloor H\right\rfloor^{2}-2\left(\tfrac{H}{a}+\tfrac{Hb}{N}\right)\left\lfloor H\right\rfloor^{2}.$ (4.3) ###### Remark. In parsing the above inequalities, it may be helpful to keep in mind that in our application $a$, $b$ and $H$ are of order $\sqrt{N}$, with $H$ considerably smaller than $a$, in which case the lower bound in (4.3) becomes $\Omega(\delta H^{2})$. ###### Proof. The majority of our proof is concerned with manipulating (4.2) until we can interpret it as a genuine box norm (2.9), and thereby apply the box norm inverse theorem. The essential observation is that, since $\gcd(a,b)=1$, every integer $x$ can be uniquely represented in the form $x=ay+bz\qquad(y\in\mathbb{Z},\ z\in[a]).$ We note that if $x\in[N]$ then the constraint on $z$ forces $y$ to lie in the range $-b<y<N/a$. Defining $F:\mathbb{Z}\times\mathbb{Z}\to\mathbb{C}$ by $F(y,z):=f(ay+bz)$, the left-hand side of (4.2) becomes $\sum_{y,y^{\prime}\in\mathbb{Z}}\sum_{\begin{subarray}{c}z\in[a]\\\ z^{\prime}\in\mathbb{Z}\end{subarray}}F(y,z)\overline{F(y^{\prime},z)}\overline{F(y,z^{\prime})}F(y^{\prime},z^{\prime})\mu_{H}(y^{\prime}-y)\mu_{H}(z^{\prime}-z).$ If $z^{\prime}$ and $z$ contribute to the above sum then $z^{\prime}\in z+(-H,H)\subset(-H+1,a+H).$ Hence we can restrict the range of summation of $z^{\prime}$ to $[a]$, at the cost of perturbing the sum by at most $2\left\lfloor H\right\rfloor(\frac{N}{a}+b).$ It follows that $\biggl{|}\sum_{y,y^{\prime}}\sum_{z,z^{\prime}\in[a]}F(y,z)\overline{F(y^{\prime},z)}\overline{F(y,z^{\prime})}F(y^{\prime},z^{\prime})\mu_{H}(y^{\prime}-y)\mu_{H}(z^{\prime}-z)\biggr{|}\\\ \geq\delta N-2\left\lfloor H\right\rfloor\left(\tfrac{N}{a}+b\right).$ We remove the Fejér kernels by Fourier expansion: $\sum_{\begin{subarray}{c}y,y^{\prime}\\\ z,z^{\prime}\in[a]\end{subarray}}F(y,z)\overline{F(y^{\prime},z)F(y,z^{\prime})}F(y^{\prime},z^{\prime})\mu_{H}(y^{\prime}-y)\mu_{H}(z^{\prime}-z)=\\\ \int_{\mathbb{T}^{2}}\sum_{\begin{subarray}{c}y,y^{\prime}\\\ z,z^{\prime}\in[a]\end{subarray}}F(y,z)\overline{F(y^{\prime},z)F(y,z^{\prime})}F(y^{\prime},z^{\prime})\hat{\mu}_{H}(\alpha)\hat{\mu}_{H}(\beta)e(\alpha(y^{\prime}-y)+\beta(z^{\prime}-z))\mathrm{d}\alpha\mathrm{d}\beta\\\ \leq\left(\int_{\mathbb{T}}|\hat{\mu}_{H}(\alpha)|\mathrm{d}\alpha\right)^{2}\sup_{\alpha,\beta\in\mathbb{T}}\biggl{|}\sum_{\begin{subarray}{c}y,y^{\prime}\\\ z,z^{\prime}\in[a]\end{subarray}}F(y,z)F_{2}(y^{\prime},z)F_{3}(y,z^{\prime})F_{4}(y^{\prime},z^{\prime})\biggr{|},$ where $F_{2}(y^{\prime},z):=\overline{F(y^{\prime},z)}e(-\beta z)$, $F_{3}(y,z^{\prime}):=\overline{F(y,z^{\prime})}e(-\alpha y)$, and $F_{4}(y^{\prime},z^{\prime})$ $:=F(y^{\prime},z^{\prime})e(\alpha y^{\prime}+\beta z^{\prime})$. We observe that $\hat{\mu}_{H}(\alpha)=|\hat{1}_{[H]}(\alpha)|^{2}/\left\lfloor H\right\rfloor^{2}$, which implies that $\int_{\mathbb{T}}|\hat{\mu}(\alpha)|d\alpha=\left\lfloor H\right\rfloor^{-1}$. Therefore $\biggl{|}\sum_{\begin{subarray}{c}y,y^{\prime}\\\ z,z^{\prime}\in[a]\end{subarray}}F(y,z)F_{2}(y^{\prime},z)F_{3}(y,z^{\prime})F_{4}(y^{\prime},z^{\prime})\biggr{|}\geq\delta\left\lfloor H\right\rfloor^{2}N-2\left\lfloor H\right\rfloor^{3}\left(\tfrac{N}{a}+b\right),$ (4.4) for $1$-bounded functions $F_{i}:\mathbb{Z}\times[a]\to\mathbb{C}$ of the form $F_{i}(y,z)=f(ay+bz)e(\alpha_{1}y+\alpha_{2}z)$. Since $f$ is supported on $[N]$, there are exactly $N$ pairs $(y^{\prime},z^{\prime})\in\mathbb{Z}\times[a]$ for which $F(y^{\prime},z^{\prime})\neq 0$. Thus, by pigeonholing in $y^{\prime}$ and $z^{\prime}$ in (4.4) and setting $L(y):=F_{3}(y,z^{\prime})$ and $R(z):=F_{2}(y^{\prime},z)F_{4}(y^{\prime},z^{\prime})$, we get that $\biggl{|}\sum_{y}\sum_{z\in[a]}F(y,z)L(y)R(z)\biggr{|}\geq\delta\left\lfloor H\right\rfloor^{2}-2\left\lfloor H\right\rfloor^{3}\left(\tfrac{1}{a}+\tfrac{b}{N}\right).$ For each $x\in\mathbb{Z}$, define $l(x)\in\mathbb{Z}$ and $r(x)\in[a]$ by $x=al(x)+br(x)$, and set $g(x):=R\circ r(x)$ and $h(x):=L\circ l(x)$. Then it remains to check the invariance properties of $g$ and $h$. To see that $g(x)=g(x+ay)$ for all $x,y\in\mathbb{Z}$, just note that $r(x)=r(x+ay)$ for every $x,y\in\mathbb{Z}$. Finally we establish that, for most $x\in[N]$, we have $h(x)=h(x+bz)$ for all $|z|\leq\varepsilon N/b$. First note that $l(x)=l(x+bz)$ whenever $\varepsilon N/b<r(x)\leq a-\varepsilon N/b$. Hence for this to fail, $x$ must lie in one of at most $1+2\varepsilon N/b$ congruence classes modulo $a$. The number of such $x$ lying in the interval $[N]$ is at most $\left(1+\frac{2\varepsilon N}{b}\right)\left(1+\frac{N}{a}\right).$ ∎ The lemma also yields a result in the situation in which $\gcd(a,b)>1$. In proving this we take the opportunity to smooth out the $b$-invariance of $h$ slightly, whilst also giving an explicit description of $h$ in terms of $f$. More concretely, we replace $h$ with a projection of $fg$ onto cosets of $b\cdot\mathbb{Z}$. ###### Lemma 4.2. There exists an absolute constant $c>0$ such that on assuming $1\leq H\leq c\delta^{3}N^{1/2}$ and $1\leq K\leq c\delta^{2}H^{2}N^{-1/2}$ the following holds. Let $a,b\in[N^{1/2}]$ with $\gcd(a,b)\leq\delta^{-1}$ and $a,b\geq\delta N^{1/2}$. Suppose that $f:\mathbb{Z}\to\mathbb{C}$ is $1$-bounded, supported on the interval $[N]$, and satisfies $\biggl{|}\sum_{h,x}\mu_{H}(h)\Delta_{ah_{1},bh_{2}}f(x)\biggr{|}\geq\delta N.$ Then there exists a 1-bounded $a$-periodic function $g$ such that $\sum_{x}f(x)g(x)\sum_{k}\mu_{K}(k)\overline{f(x+bk)g(x+bk)}\gg\delta^{2}H^{4}/N.$ (4.5) ###### Proof. Set $q:=\gcd(a,b)\leq\delta^{-1}$. For each $u\in[q]$, define a $1$-bounded function $f_{u}:\mathbb{Z}\to\mathbb{C}$ by $f_{u}(x):=f(u+qx)$, and let $I_{u}:=\left\\{x:u+qx\in[N]\right\\}$ denote the interval on which $f_{u}$ is supported. By the pigeon-hole principle, for some $u$ we have $\sum_{x,h_{1},h_{2}}\mu_{H}(h_{1})\mu_{H}(h_{2})\Delta_{\frac{a}{q}h_{1},\frac{b}{q}h_{2}}f_{u}(x)\geq\delta|I_{u}|.$ Note that $\gcd(a/q,b/q)=1$, so by the previous lemma, there exist 1-bounded functions $g_{u},h_{u}:\mathbb{Z}\to\mathbb{C}$ such that $\biggl{|}\sum_{x}f_{u}(x)g_{u}(x)h_{u}(x)\biggr{|}\geq\delta\left\lfloor H\right\rfloor^{2}-2\left(\tfrac{Hq}{a}+\tfrac{Hb}{q|I_{u}|}\right)\left\lfloor H\right\rfloor^{2}\gg\delta H^{2}.$ Furthermore, $g_{u}$ is $(a/q)$-periodic and $\\#\left\\{x\in I_{u}:h_{u}(x)\neq h_{u}(x+yb/q)\text{ for some }|y|\leq\varepsilon|I_{u}|q/b\right\\}\\\ \leq\left(1+\tfrac{2q\varepsilon|I_{u}|}{b}\right)\left(1+\tfrac{q|I_{u}|}{a}\right)\ll\tfrac{N}{a}+\tfrac{\varepsilon N^{2}}{ab}.$ Defining $g_{u^{\prime}}$ and $h_{u^{\prime}}$ to be identically zero when $u^{\prime}\neq u$, we set $g(u^{\prime}+qx):=g_{u^{\prime}}(x)$ and $h(u^{\prime}+qx):=h_{u^{\prime}}(x)$. One can then check that $g$ is $a$-invariant, that $\biggl{|}\sum_{x}f(x)g(x)h(x)\biggr{|}\gg\delta H^{2},$ and that $\\#\left\\{x\in[N]:h(x)\neq h(x+by)\text{ for some }|y|\leq\varepsilon N/b\right\\}\ll\tfrac{N}{a}+\tfrac{\varepsilon N^{2}}{ab}.$ We may use the latter property to show that, provided $K\geq 1$, we have $\biggl{|}\sum_{x}f(x)g(x)h(x)-\sum_{x}h(x)\mathbb{E}_{y\in[K]}g(x+by)f(x+by)\biggr{|}\ll\tfrac{NK}{a}.$ Provided that $K\leq c\delta^{2}H^{2}N^{-1/2}$ we deduce that $\biggl{|}\sum_{x}h(x)\mathbb{E}_{y\in[K]}g(x+bk)f(x+bk)\biggr{|}\gg\delta H^{2}.$ One can check that, as a function of $x$, the inner expectation is 1-bounded with support in $[-2N,2N]$. Applying the Cauchy–Schwarz inequality and changing variables then gives (4.5). ∎ Finally we observe that a function of the form $h(x):=\sum_{k}\mu_{K}(k)f(x+by)$ (4.6) has nice $b$-periodicity properties. ###### Lemma 4.3. If $h$ is defined as in (4.6) for some 1-bounded $f$, then $h$ is $O(K^{-1})$-Lipschitz along $b\cdot\mathbb{Z}$, in that for any $x,y\in\mathbb{Z}$ we have $h(x+by)=h(x)+O(|y|/K)$. ###### Proof. Recalling the definition (1.11), note that $\mu_{K}$ is $(2/\left\lfloor K\right\rfloor)$-Lipschitz, in that $|\mu_{K}(k+y)-\mu_{K}(k)|\leq 2|y|/\left\lfloor K\right\rfloor$ for all $k,y\in\mathbb{Z}$. Hence, for $|y|\leq K$, a change of variables gives $|h(x+by)-h(x)|\leq\sum_{k}|\mu_{K}(k-y)-\mu_{K}(k)|\ll\frac{|y|}{K}\sum_{|k|<2K}1.$ ∎ ## 5\. Quantitative concatenation The endpoint of this section is to show how our counting operator (1.3) is controlled by the $U^{5}$-norm. We begin with four technical lemmas. The first says that convolving Fejér kernels along progressions of coprime common difference covers a substantial portion of an interval in a somewhat regular manner, a fact that can be interpreted Fourier analytically in the following. ###### Lemma 5.1. Let $K,L\geq 1$ and let $a,b$ be integers satisfying $a\geq\delta L$, $b\geq\delta K$ and $\gcd(a,b)\leq\delta^{-1}$. Then $\int_{\mathbb{T}}\bigl{|}\widehat{\mu}_{K}(a\beta)\bigr{|}\bigl{|}\widehat{\mu}_{L}(b\beta)\bigr{|}\mathrm{d}\beta\ll\frac{\delta^{-4}}{\left\lfloor K\right\rfloor\left\lfloor L\right\rfloor}.$ ###### Proof. Expanding Fourier transforms, one can check that $\int_{\mathbb{T}}\bigl{|}\widehat{\mu}_{H}(a\beta)\bigr{|}\bigl{|}\widehat{\mu}_{K}(b\beta)\bigr{|}\mathrm{d}\beta\\\ =\left\lfloor K\right\rfloor^{-2}\left\lfloor L\right\rfloor^{-2}\\#\biggl{\\{}(x,y)\in[K]^{2}\times[L]^{2}:a(x_{1}-x_{2})=b(y_{1}-y_{2})\biggr{\\}}.$ Writing $d:=\gcd(a,b)$, the number of solutions to the equation is at most $\left\lfloor K\right\rfloor\left\lfloor L\right\rfloor\left(\tfrac{\left\lfloor K\right\rfloor}{b/d}+1\right)\left(\tfrac{\left\lfloor L\right\rfloor}{a/d}+1\right).$ ∎ Our next lemma allows us to discard pairs of integers $a,b$ which are not sufficiently coprime. We exploit this repeatedly. ###### Lemma 5.2. For fixed integers $0\leq a_{1},a_{2}\leq M$. The number of pairs $(b,c)$ of integers $0\leq b,c\leq M$ such that $\gcd(a_{1}+b,a_{2}+c)>\delta^{-1}$ is $\ll\delta M^{2}$. ###### Proof. Notice that if $d=\gcd(a_{1}+b,a_{2}+c)$ then $d\leq 2M$. Hence $\displaystyle\sum_{\begin{subarray}{c}0\leq b,c\leq M\\\ \gcd(a_{1}+b,a_{2}+c)>\delta^{-1}\end{subarray}}1\leq\sum_{\delta^{-1}<d\leq 2M}\ \biggl{(}\ \sum_{0\leq m\leq 2M,\ d\mid m}1\biggr{)}^{2}$ $\displaystyle\leq\sum_{\delta^{-1}<d\leq 2M}\left(\frac{2M}{d}+1\right)^{2}$ $\displaystyle\ll M^{2}\sum_{d>\delta^{-1}}\frac{1}{d^{2}}\ll\delta M^{2}.$ ∎ The following lemma says that, as $a$ and $h$ range over $[N^{1/2}]$, the difference function $\Delta_{ah}f$ behaves like $\Delta_{k}f$ with $k\in[N]$, at least on average. ###### Lemma 5.3. Let $f:\mathbb{Z}\to\mathbb{C}$ be a 1-bounded function with support in $[N]$. Suppose that $\delta N^{1/2}\leq H\leq N^{1/2}$ and $\mathbb{E}_{a\in[N^{1/2}]}\sum_{h}\mu_{H}(h)\left\|\Delta_{ah}f\right\|_{U^{s}}^{2^{s}}\geq\delta\left\|1_{[N]}\right\|_{U^{s}}^{2^{s}}.$ Then $\left\|f\right\|_{U^{s+1}}^{2^{s+1}}\gg\delta^{12}\left\|1_{[N]}\right\|_{U^{s+1}}^{2^{s+1}}$ ###### Proof. Expanding the definition of the $U^{s}$-norm $\mathbb{E}_{a\in[N^{1/2}]}\sum_{h}\mu_{H}(h)\left\|\Delta_{ah}f\right\|_{U^{s}}^{2^{s}}\\\ =\sum_{h_{1},\dots,h_{s},x}\overline{\Delta_{h_{1},\dots,h_{s}}f(x)}\mathbb{E}_{a\in[N^{1/2}]}\sum_{h}\mu_{H}(h)\Delta_{h_{1},\dots,h_{s}}f(x+ah).$ Employing the Cauchy–Schwarz inequality to double the $a$ and $h$ variables gives $\mathbb{E}_{a,a^{\prime}\in[N^{1/2}]}\sum_{h_{i}}\sum_{x}\sum_{h,h^{\prime}}\mu_{H}(h)\mu_{H}(h^{\prime})\Delta_{h_{1},\dots,h_{s},ah-a^{\prime}h^{\prime}}f(x)\gg\delta^{2}N^{s+1}.$ By Lemma 5.2 and the pigeon-hole principle, we deduce the existence of $a,a^{\prime}\gg\delta^{2}N^{1/2}$ with $\gcd(a,a^{\prime})\ll\delta^{-2}$ such that $\sum_{h_{i}}\sum_{x}\sum_{h,h^{\prime}}\mu_{H}(h)\mu_{H}(h^{\prime})\Delta_{h_{1},\dots,h_{s},ah-a^{\prime}h^{\prime}}f(x)\gg\delta^{2}N^{s+1}.$ By Fourier inversion and extraction of a large Fourier coefficient, there exists $\alpha\in\mathbb{T}$ such that the right-hand side above is at most $\int_{\mathbb{T}}\left|\widehat{\mu}_{H}(a\beta)\right|\left|\widehat{\mu}_{H}(a^{\prime}\beta)\right|\mathrm{d}\beta\biggl{|}\sum_{h_{i}}\sum_{x}\Delta_{h_{1},\dots,h_{s},h_{s+1}}f(x)e(\alpha h_{s+1})\biggr{|}.$ The result follows on employing Lemma 5.1 and Lemma A.3. ∎ We now prove a similar lemma, but with $\Delta_{ah}f$ replaced by $fg_{a}$ where $g_{a}$ is $a$-periodic. The moral is that these are similar quantities (on average). ###### Lemma 5.4. Let $f,g_{a}:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions such that $g_{a}$ is $a$-periodic and $\mathrm{supp}(f)\subset[N]$. Suppose that $\mathbb{E}_{a\in[N^{1/2}]}\left\|fg_{a}\right\|_{U^{s}}^{2^{s}}\geq\delta\left\|1_{[N]}\right\|_{U^{s}}^{2^{s}}.$ Then $\left\|f\right\|_{U^{s+1}}^{2^{s+1}}\gg\delta^{24}\left\|1_{[N]}\right\|_{U^{s+1}}^{2^{s+1}}$ ###### Proof. Fix $a\in[N^{1/2}]$. By the periodicity of $g_{a}$ and a change of variables, we have $\sum_{h_{i}}\sum_{x}\Delta_{h_{1},\dots,h_{s}}g_{a}(x)\Delta_{h_{1},\dots,h_{s}}f(x)=\sum_{h_{i}}\sum_{x}\Delta_{h_{1},\dots,h_{s}}g_{a}(x)\mathbb{E}_{y\in[N^{1/2}]}\Delta_{h_{1},\dots,h_{s}}f(x+ay).$ Notice that the sum over $x$ is non-zero only if $|x|,|h_{i}|<N$, hence by Cauchy–Schwarz and a change of variables $\displaystyle\biggl{(}\mathbb{E}_{a\in[N^{1/2}]}\left\|fg_{a}\right\|_{U^{s}}^{2^{s}}\biggr{)}^{2}$ $\displaystyle\ll N^{s+1}\mathbb{E}_{a\in[N^{1/2}]}\sum_{h_{i}}\sum_{x}\sum_{y}\mu_{N^{1/2}}(y)\Delta_{h_{1},\dots,h_{s},ay}f(x)$ $\displaystyle=N^{s+1}\mathbb{E}_{a\in[N^{1/2}]}\sum_{y}\mu_{N^{1/2}}(y)\left\|\Delta_{ay}f\right\|_{U^{s}}^{2^{s}}$ The result follows on employing Lemma 5.3. ∎ We are now ready to give the technical heart of this section. The (somewhat lengthy) assumptions come from our eventual application of Lemma 4.2. ###### Lemma 5.5. Fix $a\in\mathbb{N}$ and let $\delta N^{1/2}\leq K\leq N^{1/2}$. For each $b\in[N^{1/2}]$ let $f,g_{b},h_{b}:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions such that $\mathrm{supp}(f),\mathrm{supp}(h_{b})\subset[N]$ and where $g_{b}$ is $b$-periodic. Set $\tilde{h}_{b}(x):=\sum_{k}\mu_{K}(k)h_{b}(x+(a+b)k)$ and suppose that $\sum_{\begin{subarray}{c}\delta\sqrt{N}\leq b\leq\sqrt{N}\\\ \gcd(a,b)\leq\delta^{-1}\end{subarray}}\sum_{x}f(x)g_{b}(x)\tilde{h}_{b}(x)\geq\delta N^{3/2}.$ Then $\mathbb{E}_{b\in[N^{1/2}]}\big{\|}h_{b}\big{\|}_{U^{3}}^{8}\gg\delta^{208}\left\|1_{[N]}\right\|_{U^{3}}^{8}.$ ###### Proof. To ease notation, write $\tilde{h}_{b}(x):=\sum_{k}\mu_{K}(k)h_{b}(x+(a+b)k)$ We apply Cauchy–Schwarz to remove the weight $f(x)$ and double the $b$ variable, yielding $\sum_{\begin{subarray}{c}\delta\sqrt{N}\leq b,b^{\prime}\leq\sqrt{N}\\\ \gcd(a,b)\leq\delta^{-1}\end{subarray}}\sum_{x}g_{b}(x)\tilde{h}_{b}(x)\overline{g_{b^{\prime}}(x)\tilde{h}_{b^{\prime}}(x)}\geq\delta^{2}N^{2}.$ Employing Lemma 5.2, we may discard those $b,{b^{\prime}}$ for which one of $\gcd(b^{\prime},a+{b})$ or $\gcd(a+b^{\prime},a+{b})$ is greater than $C\delta^{-2}$. On combining this with the popularity principle, we deduce the existence of $\mathcal{B}\subset[\delta N^{1/2},N^{1/2}]$ of size $|\mathcal{B}|\gg\delta^{2}N^{1/2}$ such that for each $b\in\mathcal{B}$ there exists $b^{\prime}\in[N^{1/2}]$ with all of $\gcd(b,a+{b})$, $\gcd({b^{\prime}},a+{b})$, $\gcd(a+b^{\prime},a+{b})$ at most $O(\delta^{-2})$ and satisfying $\sum_{x}g_{b}(x)\overline{\tilde{h}_{b^{\prime}}(x)g_{b^{\prime}}(x)}\tilde{h}_{b}(x)\gg\delta^{2}N.$ (5.1) Expanding the definition of $\tilde{h}_{b^{\prime}}$, using the invariance of $g_{b}$ and changing variables gives $\sum_{x}\mathbb{E}_{k_{1},k_{3}\in[K]}\sum_{k_{2}}\mu_{K}(k_{2})g_{b}(x+(a+b^{\prime})k_{2}+{b^{\prime}}k_{3})\overline{h_{b^{\prime}}(x+bk_{1}+{b^{\prime}}k_{3})}\\\ \overline{g_{b^{\prime}}(x+bk_{1}+(a+b^{\prime})k_{2})}\ \tilde{h}_{b}(x+bk_{1}+(a+b^{\prime})k_{2}+{b^{\prime}}k_{3})\gg\delta^{2}N.$ Since $h_{b^{\prime}}$ is supported on $[N]$ and $b,{b^{\prime}},K\leq N^{1/2}$, there are at most $O(N)$ values of $x$ which contribute to the above sum. Applying Hölder’s inequality then gives $\sum_{x}\biggl{(}\mathbb{E}_{k_{1},k_{3}\in[K]}\sum_{k_{2}}\mu_{K}(k_{2})g_{b}(x+(a+b^{\prime})k_{2}+{b^{\prime}}k_{3})\overline{h_{b^{\prime}}(x+bk_{1}+{b^{\prime}}k_{3})}\\\ \overline{g_{b^{\prime}}(x+bk_{1}+(a+b^{\prime})k_{2})}\ \tilde{h}_{b}(x+bk_{1}+(a+b^{\prime})k_{2}+{b^{\prime}}k_{3})\biggr{)}^{8}\gg\delta^{16}N.$ The sum inside the 8th power corresponds to an integral with respect to three probability measures on $\mathbb{Z}$, with integrand amenable to Lemma A.4. Combining this with a change of variables gives $\sum_{x}\sum_{k_{1},k_{2},k_{3}}\mu_{K}(k_{1})\nu_{K}(k_{2})\mu_{K}(k_{3})\Delta_{bk_{1},(a+b^{\prime})k_{2},{b^{\prime}}k_{3}}\ \tilde{h}_{b}(x)\gg\delta^{16}N,$ where we set $\nu_{K}(k):=\sum_{k_{1}-k_{2}=k}\mu_{K}(k_{1})\mu_{K}(k_{2}).$ By Lemma 4.3, each $\tilde{h}_{b}$ is $O(K^{-1})$-Lipschitz along $(a+b)\cdot\mathbb{Z}$. Hence, if $l_{i}\in[L]$, a telescoping identity shows that $|\Delta_{h_{1}+(a+{b})l_{1},h_{2}+(a+{b})l_{2},h_{3}+(a+{b})l_{3}}\tilde{h}_{b}(x)-\Delta_{h_{1},h_{2},h_{3}}\tilde{h}_{b}(x)|\ll L/K.$ Taking $L:=c\delta^{16}K$ we obtain $\sum_{x}\sum_{k_{1},k_{2},k_{3}}\mu_{K}(k_{1})\nu_{K}(k_{2})\mu_{K}(k_{3})\mathbb{E}_{l_{1},l_{2},l_{3}\in[L]}\\\ \Delta_{bk_{1}+(a+{b})l_{1},\,(a+b^{\prime})k_{2}+(a+{b})l_{2},\,{b^{\prime}}k_{3}+(a+{b})l_{3}}\ \tilde{h}_{b}(x)\gg\delta^{16}N.$ We may replace the uniform measure on the $l_{i}$ by Fejér kernels at the cost of three applications of Cauchy–Schwarz; this gives $\sum_{x}\sum_{\begin{subarray}{c}k_{1},k_{2},k_{3}\\\ l_{1},l_{2},l_{3}\end{subarray}}\mu_{K}(k_{1})\nu_{K}(k_{2})\mu_{K}(k_{3})\mu_{L}(l_{1})\mu_{L}(l_{2})\mu_{L}(l_{3})\\\ \Delta_{bk_{1}+(a+{b})l_{1},\,(a+b^{\prime})k_{2}+(a+{b})l_{2},\,{b^{\prime}}k_{3}+(a+{b})l_{3}}\ \tilde{h}_{b}(x)\gg\delta^{128}N.$ Write $\displaystyle\lambda_{1}(h):=\sum_{bk+(a+{b})l=h}$ $\displaystyle\mu_{K}(k)\mu_{L}(l),\qquad\lambda_{2}(h):=\sum_{(a+b^{\prime})k+(a+{b})l=h}\nu_{K}(k)\mu_{L}(l),$ $\displaystyle\lambda_{3}(h):=\sum_{{b^{\prime}}k+(a+{b})l=h}\mu_{K}(k)\mu_{L}(l).$ Then $\sum_{x}\sum_{h_{1},h_{2},h_{3}}\lambda_{1}(h_{1})\lambda_{2}(h_{2})\lambda_{3}(h_{3})\\\ \Delta_{h_{1},h_{2},h_{3}}\ \tilde{h}_{b}(x)\gg\delta^{128}N.$ By Fourier inversion and extraction of a large Fourier coefficient, there exist $\alpha_{i}\in\mathbb{T}$ such that $\biggl{|}\sum_{x}\sum_{h_{1},h_{2},h_{3}}\Delta_{h_{1},h_{2},h_{3}}\ \tilde{h}_{b}(x)e(\underline{\alpha}\cdot\underline{h})\biggr{|}\prod_{i=1}^{3}\int_{\mathbb{T}}\bigl{|}\widehat{\lambda}_{i}(\beta)\bigr{|}\mathrm{d}\beta\gg\delta^{128}N.$ By our choice of $b$, $b^{\prime}$ (see the paragraph preceding (5.1)), together with Lemma 5.1, for each $i$ we have $\int_{\mathbb{T}}\bigl{|}\widehat{\lambda}_{i}(\alpha)\bigr{|}\mathrm{d}\alpha\ll\frac{\delta^{-8}}{KL}\ll\frac{\delta^{-26}}{N},$ (5.2) the latter following from the fact that $L\gg c\delta^{16}K$ and $K\geq\delta N^{1/2}$. On combining this with Lemma A.3 we obtain $\big{\|}\tilde{h}_{b}\big{\|}_{U^{3}}^{8}\gg\delta^{206}N^{4}.$ Since $\tilde{h}_{b}$ is an average of translates of $h_{b}$, we may apply the triangle inequality for the $U^{3}$-norm, together with the fact that Gowers norms are translation invariant, and conclude that $\left\|h_{b}\right\|_{U^{3}}^{8}\gg\delta^{206}N^{4}$. Summing over $b\in\mathcal{B}$ gives our final bound. ∎ Finally we synthesise Lemmas 3.3, 4.2 and 5.5. ###### Theorem 5.6 (Global $U^{5}$-control). Let $g_{0},g_{1},f:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions, each with support in $[N]$. Suppose that $\left|\Lambda_{q,N}(g_{0},g_{1},f)\right|\geq\delta\Lambda_{q,N}(1_{[N]}).$ Then $\sum_{u\in[q]}\left\|f\right\|_{U^{5}(u+q\mathbb{Z})}^{2^{5}}\gg\delta^{2^{25}}\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{5}(u+q\mathbb{Z})}^{2^{5}}.$ ###### Proof. We recall our convention (1.2) regarding $M$. We begin by applying the linearisation procedure (Lemma 3.3) to deduce that $\sum_{a,b\in(-2M,2M)}\ \biggl{|}\sum_{h}\mu_{H}(h)\sum_{x}\Delta_{q(a+b)h_{1},qbh_{2},qah_{3}}f(x)\biggr{|}\\\ \gg\delta^{32}NM^{2}.$ We note that the sum inside the absolute value is invariant under $a\mapsto-a$. Hence we may restrict to $a,b\in[0,2M]$ at the cost of changing the absolute constant. Applying Lemma 5.2 we may discard those $a,b$ for which either $\gcd(a,b)>C\delta^{-32}$ or $b<c\delta^{32}M$. Partitioning the sum over $x$ into congruence classes $u\bmod q$, the popularity principle gives: * • at least $\Omega(\delta^{32}q)$ residues $u\in[q]$; * • for each of which there is a subset of $h_{3}\in(-H,H)$ of $\mu_{H}$-measure222i.e. $\sum_{h_{3}\in\mathcal{H}}\mu_{H}(h_{3})\gg\delta^{32}$. at least $\Omega(\delta^{32})$; * • for each of which there exist $\Omega(\delta^{32}M)$ values of $a\in[2M]$; * • for each of which there are $\Omega(\delta^{32}M)$ values of $b\in[2M]$ satisfying $\gcd(a,b)\ll\delta^{-32}$ and $b\gg\delta^{32}M$; and together these satisfy $\biggl{|}\sum_{h_{1},h_{2}}\mu_{H}(h_{1},h_{2})\sum_{x}\Delta_{(a+b)h_{1},bh_{2},ah_{3}}f(qx-u)\biggr{|}\\\ \gg\delta^{32}M^{2}.$ For fixed $u,h_{3},a$ write $\tilde{f}(x):=\Delta_{ah_{3}}f(qx-u),$ so that $\tilde{f}$ has support in the interval $[(2M)^{2}]$ and $\biggl{|}\sum_{h_{1},h_{2}}\mu_{H}(h_{1},h_{2})\sum_{x}\Delta_{(a+b)h_{1},bh_{2}}\tilde{f}(x)\biggr{|}\\\ \gg\delta^{32}M^{2}.$ Set $H:=c\delta^{96}M\qquad\text{and}\qquad K:=c^{3}\delta^{160}M,$ (5.3) with $c$ sufficiently small to ensure that we may apply Lemma 4.2. This gives the existence of a 1-bounded $b$-periodic function $g_{b}$ such that on setting $\tilde{h}_{b}(x):=\sum_{k}\mu_{K}(k)\overline{\tilde{f}(x+(a+b)k)g_{b}(x+(a+b)k)}$ (5.4) we have $\sum_{x}\tilde{f}(x)g_{b}(x)\tilde{h}_{b}(x)\gg\delta^{448}M^{2}.$ Setting $\eta:=c\delta^{480}$ for some small absolute constant $c>0$, we may sum over our set of permissible $b$ to deduce that $\sum_{\begin{subarray}{c}\eta M\leq b\leq 2M\\\ \gcd(a,b)\leq\eta^{-1}\end{subarray}}\sum_{x}\tilde{f}(x)g_{b}(x)h_{b}(x)\geq\eta M^{3}.$ The hypotheses of Lemma 5.5 having been met, we conclude that $\mathbb{E}_{b\in[2M]}\big{\|}\tilde{f}g_{b}\big{\|}_{U^{3}}^{8}\gg\delta^{99,840}\left\|1_{[M^{2}]}\right\|_{U^{3}}^{8}.$ Applying Lemma 5.4 then gives $\big{\|}\tilde{f}\big{\|}_{U^{4}}^{16}\gg\delta^{2,396,160}\left\|1_{[M^{2}]}\right\|_{U^{4}}^{16}.$ Recalling that $\tilde{f}(x)=\Delta_{ah_{3}}f_{u}(x)$ where $f_{u}(x):=f(qx-u)$, we may integrate over the set of permissible $h_{3}$ and $a$, utilising positivity to extend the range of summation, and deduce that $\mathbb{E}_{a\in[2M]}\sum_{h}\mu_{H}(h_{3})\big{\|}\Delta_{ah_{3}}f_{u}\big{\|}_{U^{4}}^{16}\gg\delta^{2,396,224}\left\|1_{[M^{2}]}\right\|_{U^{4}}^{16}$ Using Lemma 5.3 and summing over the permissible range of $u$ we get that $\mathbb{E}_{u\in[q]}\left\|f_{u}\right\|_{U^{5}}^{32}\gg\delta^{28,754,720}\left\|1_{[M^{2}]}\right\|_{U^{5}}^{32},$ and the result follows. ∎ ## 6\. Degree lowering So far, we have shown that $\Lambda_{q,N}(f_{0},f_{1},f_{2})$ is controlled by $\mathbb{E}_{u\in[q]}\|f_{2}\|_{U^{5}(u+q\mathbb{Z})}^{2^{5}}$ whenever $f_{0},f_{1},$ and $f_{2}$ are $1$-bounded complex-valued functions supported on the interval $[N]$. The next step in our argument is to bound $\Lambda_{q,N}(f_{0},f_{1},f_{2})$ in terms of the $U^{5}(u+q\mathbb{Z})$-norm of the dual function $F(x):=\mathbb{E}_{y\in[M]}f_{0}(x-qy^{2})f_{1}(x+y-qy^{2}).$ (6.1) We postpone this deduction until §7. In this section we show how $U^{5}$-control of the dual implies $U^{2}$-control. Our argument combines three simple lemmas: Weyl’s inequality; what we call ‘dual–difference interchange’, which allows us to replace the difference function of the dual by the dual of the difference functions; and the fact that a function whose difference functions correlate with ‘low rank’ Fourier coefficients must have a large uniformity norm of lower degree. The following log-free variant of Weyl’s inequality can be found in [4, Lemma A.11]. ###### Lemma 6.1 (Weyl’s inequality). There exists an absolute constant $C$ such that the following holds. Let $\alpha,\beta\in\mathbb{T}$, $\delta\in(0,1)$ and let $I\subset\mathbb{Z}$ be an interval with $|I|\geq C\delta^{-6}$ and $\big{|}\mathbb{E}_{y\in I}e(\alpha y^{2}+\beta y)\big{|}\geq\delta.$ Then there exists a positive integer $q\ll\delta^{-4}$ such that $\|q\alpha\|\ll\delta^{-14}|I|^{-2}.$ This has the following consequence, which uses our convention (1.2) regarding $M$. ###### Lemma 6.2. There exist an absolute constant $C$ such that for $N\geq C(q/\delta)^{C}$ the following holds. Suppose that for $\alpha\in\mathbb{T}$ there are $1$-bounded functions $g_{0},g_{1}:\mathbb{Z}\to\mathbb{C}$ supported on the interval $[N]$ such that $\left|\sum_{x}\sum_{y\in[M]}g_{0}(qx)g_{1}(qx+y)e(\alpha(x+y^{2}))\right|\geq\delta MN/q.$ Then there exists a positive integer $q^{\prime}\ll\delta^{-4}$ such that $\|q^{\prime}q^{2}\alpha\|\ll\delta^{-14}q^{3}/N$. ###### Proof. We split the sum over $y\in[M]$ into arithmetic progressions modulo $q$ and split the sum over $x$ into intervals of length $M/q$. Hence, by the pigeon- hole principle, there exists $u\in[q]$ and an integer $m$ such that on rounding the sum over $y$ we have $\left|\sum_{x,y\in[M/q]}g_{0}(q(m+x))g_{1}(u+q(m+x+y))e\left(\alpha\left(x+(u+qy)^{2}\right)\right)\right|\\\ \gg\delta(M/q)^{2}.$ Define the functions $\displaystyle h_{0}(x):=g_{0}(q(m+x))$ $\displaystyle e(\alpha x)1_{[M/q]}(x),\qquad h_{1}(x):=g_{1}(u+q(m+x))1_{[2M/q]},$ $\displaystyle h_{2}(x):=e\left(\alpha(u+qx)^{2}\right)1_{[M/q]}(x)$ Then by orthogonality, extraction of a large Fourier coefficient and Parseval we have $\displaystyle\delta M^{2}/q^{2}\ll\left|\int_{\mathbb{T}}\hat{h}_{0}(\beta)\hat{h}_{1}(-\beta)\hat{h}_{2}(\beta)\mathrm{d}\alpha\right|\ll\big{\|}\hat{h}_{2}\big{\|}_{\infty}\big{\|}\hat{h}_{0}\big{\|}_{L^{2}}\big{\|}\hat{h}_{1}\big{\|}_{L^{2}}\ll\big{\|}\hat{h}_{2}\big{\|}_{\infty}M/q.$ It follows that there exists $\beta\in\mathbb{T}$ such that $\left|\sum_{x\in[M/q]}e\left(\alpha(u+qx)^{2}+\beta x\right)\right|\gg\delta M/q.$ Applying Weyl’s inequality, we deduce the existence of $q^{\prime}\ll\delta^{-4}$ such that $\left\|q^{\prime}q^{2}\alpha\right\|\ll\delta^{-14}(q/M)^{2}$. ∎ ###### Lemma 6.3 (Dual–difference interchange). For each $y\in[M]$, let $F_{y}:\mathbb{Z}\to\mathbb{C}$ be a 1-bounded function with support in an interval of length $N$. Set $F(x):=\mathbb{E}_{y\in[M]}F_{y}(x).$ Then for any function $\phi:\mathbb{Z}^{s}\to\mathbb{T}$ and finite set $\mathcal{H}\subset\mathbb{Z}^{s}$ we have $\left(N^{-s-1}\sum_{\underline{h}\in\mathcal{H}}\left|\sum_{x}\Delta_{\underline{h}}F(x)e\bigl{(}\phi(\underline{h})x\bigr{)}\right|\right)^{2^{s}}\ll_{s}\\\ N^{-2s-1}\sum_{\underline{h}^{0},\underline{h}^{1}\in\mathcal{H}}\left|\sum_{x}\mathbb{E}_{y\in[M]}\Delta_{\underline{h}^{0}-\underline{h}^{1}}F_{y}(x)e\bigl{(}\phi(\underline{h}^{0};\underline{h}^{1})x\bigr{)}\right|,$ where $\phi(\underline{h}^{0};\underline{h}^{1}):=\sum_{\omega\in\left\\{0,1\right\\}^{s}}(-1)^{|\omega|}\phi(\underline{h}^{\omega})\qquad\text{and}\qquad\underline{h}^{\omega}:=(h_{1}^{\omega_{1}},\dots,h_{s}^{\omega_{s}}).$ ###### Proof. We proceed by induction on $s\geq 0$, the base case being an identity. Suppose then that $s\geq 1$. For $\underline{h}\in\mathbb{Z}^{s-1}$ and $h\in\mathbb{Z}$, we note that $\Delta_{(\underline{h},h)}F(x)=\Delta_{\underline{h}}\left(\mathbb{E}_{y,y^{\prime}\in[M]}F_{y}(x)\overline{F_{y^{\prime}}(x+h)}\right)$ Hence by the induction hypothesis $\left(N^{-s-1}\sum_{h}\sum_{\begin{subarray}{c}\underline{h}\\\ (\underline{h},h)\in\mathcal{H}\end{subarray}}\left|\sum_{x}\Delta_{(\underline{h},h)}F(x)e\bigl{(}\phi(\underline{h})x\bigr{)}\right|\right)^{2^{s}}\ll_{s}\\\ \left(N^{-2s}\sum_{h}\sum_{\begin{subarray}{c}\underline{h}^{0},\underline{h}^{1}\\\ (\underline{h}^{i},h)\in\mathcal{H}\end{subarray}}\left|\sum_{x}\mathbb{E}_{y,y^{\prime}\in[M]}\Delta_{\underline{h}^{0}-\underline{h}^{1}}F_{y}(x)\overline{F_{y^{\prime}}(x+h)}e\bigl{(}\phi(\underline{h}^{0};\underline{h}^{1};h)x\bigr{)}\right|\right)^{2},$ where $\phi(\underline{h}^{0};\underline{h}^{1};h):=\sum_{\omega\in\left\\{0,1\right\\}^{s-1}}(-1)^{|\omega|}\phi(\underline{h}^{\omega},h).$ Letting $e(\psi(\underline{h}^{0};\underline{h}^{1};h))$ denote the phase of the inner absolute, we take the sum over $h$ inside and apply Cauchy–Schwarz to obtain $\left(\sum_{\underline{h}^{0},\underline{h}^{1},x}\mathbb{E}_{y,y^{\prime}\in[M]}\sum_{\begin{subarray}{c}h\\\ (\underline{h}^{i},h)\in\mathcal{H}\end{subarray}}\Delta_{\underline{h}^{0}-\underline{h}^{1}}F_{y}(x)\overline{F_{y^{\prime}}(x+h)}e\bigl{(}\phi(\underline{h}^{0};\underline{h}^{1};h)x+\psi(\underline{h}^{0};\underline{h}^{1};h)\bigr{)}\right)^{2}\\\ \leq N^{2s-1}\sum_{\underline{h}^{0},\underline{h}^{1}}\sum_{\begin{subarray}{c}h^{0},h^{1}\\\ (\underline{h}^{i},h^{j})\in\mathcal{H}\end{subarray}}\\\ \left|\sum_{x}\mathbb{E}_{y\in[M]}\Delta_{\underline{h}^{0}-\underline{h}^{1}}F_{y}(x)\overline{F_{y}(x+h^{0}-h^{1})}e\Bigl{(}\bigl{(}\phi(\underline{h}^{0};\underline{h}^{1};h^{0})-\phi(\underline{h}^{0};\underline{h}^{1};h^{1})\bigr{)}x\Bigr{)}\right|.$ The result follows. ∎ If $\phi(h_{1},\dots,h_{s-1})$ is a function of $s-1$ variables we write $\phi(h_{1},\dots,\hat{h}_{i},\dots,h_{s}):=\phi(h_{1},\dots,h_{i-1},h_{i+1},\dots,h_{s}).$ We say that $\phi(h_{1},\dots,h_{s})$ is _low rank_ if there exist functions $\phi_{i}(h_{1},\dots,h_{s-1})$ such that $\phi(h_{1},\dots,h_{s})=\sum_{i=1}^{s}\phi_{i}(h_{1},\dots,\hat{h}_{i},\dots,h_{s}).$ From the definition of the Gowers norm together with the $U^{2}$-inverse theorem (Lemma A.1), one can show that largeness of the $U^{s+2}$-norm is equivalent to the existence of $\phi:\mathbb{Z}^{s}\to\mathbb{T}$ such that $\sum_{h_{1},\dots,h_{s}}\left|\sum_{x}\Delta_{h}f(x)e(\phi(h)x)\right|\gg N^{s+1}.$ The following lemma says that if $\phi$ is low-rank, then the $U^{s+1}$-norm must also be large. ###### Lemma 6.4 (Low rank correlation implies lower degree). Let $f:\mathbb{Z}\to\mathbb{C}$ be a 1-bounded function with support in $[N]$. Then for $\phi_{1},\dots,\phi_{m}:\mathbb{Z}^{s-1}\to\mathbb{T}$ with $m\leq s$ we have $\frac{1}{N^{s+1}}\sum_{h_{1},\dots,h_{s}}\left|\sum_{x}\Delta_{h}f(x)e\left(\sum_{i=1}^{m}\phi_{i}(h_{1},\dots,\hat{h}_{i},\dots,h_{s})x\right)\right|\\\ \ll_{m}\left(\frac{\left\|f\right\|_{U^{s+1}}^{2^{s+1}}}{N^{s+2}}\right)^{2^{-m-1}}.$ (6.2) ###### Proof. We proceed by induction on $m\geq 0$, the base case corresponding to the Cauchy–Schwarz inequality. Suppose then that $m\geq 1$ and the result is true for smaller values of $m$. Letting $e(\psi(h))$ denote the phase of the inner- most sum, the left-hand side of (6.2) is equal to $\frac{1}{N^{s+1}}\sum_{h_{2},\dots,h_{s},x}\Delta_{h_{2},\dots,h_{s}}f(x)e\left(\phi_{1}(h_{2},\dots,h_{s})\right)\sum_{h_{1}}\Delta_{h_{2},\dots,h_{s}}\overline{f}(x+h_{1})\\\ e\left(\sum_{i=2}^{m}\phi_{i}(h_{1},\dots,\hat{h}_{i},\dots,h_{s})x+\psi(h_{1},\dots,h_{s})\right).$ By Cauchy–Schwarz, the square of this is at most $\frac{1}{N^{s+2}}\sum_{h_{2},\dots,h_{s}}\ \sum_{h_{1},h_{1}^{\prime}\in(-N,N)}\\\ \left|\sum_{x}\Delta_{h_{1}-h_{1}^{\prime},h_{2},\dots,h_{s}}f(x)e\left(\sum_{i=2}^{m}\left(\phi_{i}(h_{1},\dots,\hat{h}_{i},\dots,h_{s})-\phi_{i}(h_{1}^{\prime},\dots,\hat{h}_{i},\dots,h_{s})\right)x\right)\right|.$ Taking a maximum over $h_{1}^{\prime}\in(-N,N)$ and changing variables in $h_{1}$, the latter is at most an absolute constant times $\frac{1}{N^{s+1}}\sum_{h_{1},h_{2},\dots,h_{s}}\Bigg{|}\sum_{x}\Delta_{h_{1},h_{2},\dots,h_{s}}f(x)\\\ e\left(\sum_{i=2}^{m}\left(\phi_{i}(h_{1}+h_{1}^{\prime},h_{2}\dots,\hat{h}_{i},\dots,h_{s})-\phi_{i}(h_{1}^{\prime},h_{2}\dots,\hat{h}_{i},\dots,h_{s})\right)x\right)\Bigg{|}.$ This phase has lower rank than the original, hence we may apply the induction hypothesis to yield the lemma. ∎ ###### Lemma 6.5 (Degree lowering). There exists an absolute constant such that for $N\geq C(q/\delta)^{C}$ the following holds. Let $f_{0},f_{1}:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions with support in $[N]$ and define the dual $F(x):=\mathbb{E}_{y\in[M]}f_{0}(x-qy^{2})f_{1}(x+y-qy^{2}).$ If, for $s\geq 3$, we have $\sum_{u\in[q]}\left\|F\right\|_{U^{s}(u+q\cdot\mathbb{Z})}^{2^{s}}\geq\delta\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{s}(u+q\cdot\mathbb{Z})}^{2^{s}},$ then $\sum_{u\in[q]}\left\|F\right\|_{U^{s-1}(u+q\cdot\mathbb{Z})}^{2^{s-1}}\gg_{s}\delta^{4^{s+2}}\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{s-1}(u+q\cdot\mathbb{Z})}^{2^{s-1}},$ ###### Proof. Write $M:=\left\lfloor(N/q)^{1/2}\right\rfloor$. Given $u\in[q]$ let $F_{u}(x):=F(u+qx)$, a function with support in the interval $[2N/q]$. Applying the popularity principle, there exists a set of $\Omega(\delta q)$ residues $u\in[q]$ for which $\left\|F_{u}\right\|_{U^{s}}^{2^{s}}\gg\delta(N/q)^{s+1}$. Expanding the definition of the $U^{s}$-norm (1.9) we have $\sum_{h_{1},\dots,h_{s-2}}\left\|\Delta_{h_{1},\dots,h_{s-2}}F_{u}\right\|_{U^{2}}^{4}\gg\delta(N/q)^{s+1}.$ Applying the $U^{2}$-inverse theorem (Lemma A.1), there exists $\mathcal{H}\subset(-2N/q,2N/q)^{s-2}$ of size $|\mathcal{H}|\gg\delta(N/q)^{s-2}$ and a function $\phi:\mathbb{Z}^{s-2}\to\mathbb{T}$ such that for every $\underline{h}\in\mathcal{H}$ we have $\left|\sum_{x}\Delta_{\underline{h}}F_{u}(x)e\bigl{(}\phi(\underline{h})x\bigr{)}\right|\gg\delta N/q.$ (6.3) Set $T:=\left\lceil C\delta^{-1}N/q\right\rceil$, with $C$ an absolute constant taken sufficiently large to ensure that, on rounding $\phi(\underline{h})$ to the nearest fraction of the form $t/T$, the validity of (6.3) remains. Summing over $\underline{h}\in\mathcal{H}$ and applying Lemma 6.3, we deduce that $\sum_{\underline{h}^{0},\underline{h}^{1}\in\mathcal{H}}\left|\sum_{x}\mathbb{E}_{y\in[M]}\Delta_{\underline{h}^{0}-\underline{h}^{1}}f_{0}(u+qx- qy^{2})\Delta_{\underline{h}^{0}-\underline{h}^{1}}f_{1}(u+qx+y-qy^{2})\right|\\\ e\bigl{(}\phi(\underline{h}^{0};\underline{h}^{1})x\bigr{)}\gg_{s}\delta^{2^{s-1}}(N/q)^{2s-1}.$ Applying the pigeon-hole and popularity principle, there exists $\mathcal{H}^{\prime}\subset\mathcal{H}$ of size $\Omega_{s}(\delta^{2^{s-1}}(N/q)^{s-2})$ and $\underline{h}^{1}\in\mathcal{H}$ such that for every $\underline{h}^{0}\in\mathcal{H}^{\prime}$ we have $\left|\sum_{x}\sum_{y\in[M]}\Delta_{\underline{h}^{0}-\underline{h}^{1}}f_{0}(u+qx- qy^{2})\Delta_{\underline{h}^{0}-\underline{h}^{1}}f_{1}(u+qx+y-qy^{2})e\bigl{(}\phi(\underline{h}^{0},\underline{h}^{1})x\bigr{)}\right|\\\ \gg\delta^{2^{s-1}}MN/q.$ By Lemma 6.2, for each $\underline{h}^{0}\in\mathcal{H}^{\prime}$ there exists $q^{\prime}\ll\delta^{-2^{s+1}}$ such that $\left\|q^{\prime}q^{2}\phi(\underline{h}^{0},\underline{h}^{1})\right\|\ll\delta^{-2^{s}\times 7}q^{3}/N$ Notice that $\phi(\underline{h}^{0},\underline{h}^{1})$ is an element of the additive group $\left\\{t/T:t\in[T]\right\\}\subset\mathbb{T}$. Moreover, for any $Q_{i}$ we have the inclusion $\left\\{\alpha\in\mathbb{T}:\exists q^{\prime}\leq Q_{1}\text{ with }\left\|q^{\prime}q^{2}\alpha\right\|\leq Q_{2}q^{3}/N\right\\}\subset\bigcup_{\begin{subarray}{c}1\leq a\leq q\leq Q_{1}\\\ \mathrm{hcf}(a,q)=1\end{subarray}}\left[\frac{a}{q^{\prime}q^{2}}-\frac{Q_{2}}{N},\frac{a}{q^{\prime}q^{2}}+\frac{Q_{2}}{N}\right].$ By a volume packing argument, the number of $t/T$ lying in this union of intervals is at most $O\left(Q_{1}^{2}(1+\tfrac{Q_{2}T}{N})\right)$. It therefore follows from the pigeon-hole principle that there exists $\mathcal{H}^{\prime\prime}\subset\mathcal{H}^{\prime}$ of size $\Omega\left(\delta^{2^{s+3}+1-2^{s}}(N/q)^{s-2}\right)$ and $t_{0}\in[T]$ such that for any $\underline{h}^{0}\in\mathcal{H}^{\prime\prime}$ we have $\phi(\underline{h}^{0},\underline{h}^{1})=t_{0}/T$. In particular, when restricted to the set $\mathcal{H}^{\prime\prime}$, the function $\phi$ satisfies $\phi(\underline{h}^{0})=t_{0}/T-\sum_{\omega\in\left\\{0,1\right\\}^{s}\setminus\left\\{0\right\\}}(-1)^{|\omega|}\phi(\underline{h}^{\omega}).$ The right-hand side of this identity is clearly _low rank_ according to the terminology preceding Lemma 6.4. Summing over $\underline{h}\in\mathcal{H}^{\prime\prime}$ in (6.3), we deduce the existence of a low rank function $\psi:\mathbb{Z}^{s-2}\to\mathbb{T}$ such that $\sum_{\underline{h}}\left|\sum_{x}F_{u}(x)e\bigl{(}\psi(\underline{h})x\bigr{)}\right|\gg\delta^{2^{s+3}+1-2^{s}}(N/q)^{s-1}.$ Employing Lemma 6.4 then gives $\left\|F_{u}\right\|_{U^{s-1}}^{2^{s-1}}\gg\delta^{(2^{s+3}+1-2^{s})2^{s+1}}(N/q)^{s}.$ Summing over permissible $u$, then extending to the full sum over $u\in[q]$ by positivity, we obtain the bound claimed in the lemma. ∎ ## 7\. Proof of the cut norm inverse theorem In this section we complete our proof of Theorem 1.6. We first show how the dual function is controlled by the $U^{5}$-norm, and hence by the degree lowering of §6, the dual is controlled by the $U^{1}$-norm. The following can be found in the discussion following [3, Proposition 3.6]. Although the statement therein is for norms, and not seminorms, one can check that the (simple) argument remains valid in this greater generality333On occasion the relevant results in [3] appear to assume that unit balls are _bounded_ (if we take the definition of _convex body_ to be a compact convex set with non-empty interior), which may not be true for the unit ball of a seminorm. However, the boundedness assumption is not necessary in the pertinent proofs. Moreover, one could quotient by the norm zero set to obtain a genuine norm.. ###### Lemma 7.1. Let $\|\cdot\|$ be a seminorm on the space of complex-valued functions supported on $[N]$. For any such function $f$ and $\varepsilon>0$ there exists a decomposition $f=f_{str}+f_{unf}$ such that $\left\|f_{str}\right\|^{*}\leq\varepsilon^{-1}\left\|f\right\|_{2}\quad\text{and}\quad\left\|f_{unf}\right\|\leq\varepsilon\left\|f\right\|_{2}.$ ###### Lemma 7.2 ($U^{5}$-control of the dual). There exists an absolute constant $C$ such that for $N\geq Cq\delta^{-C}$ the following holds. Let $g_{0},g_{1},f:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions, each with support in $[N]$. Suppose that $\left|\Lambda_{q,N}(g_{0},g_{1},f)\right|\geq\delta\Lambda_{q,N}(1_{[N]}).$ Then, on defining the dual $G(x):=\mathbb{E}_{y\in[M]}g_{0}(x-qy^{2})g_{1}(x+y-qy^{2}),$ (7.1) we have $\sum_{u\in[q]}\left\|G\right\|_{U^{5}(u+q\cdot\mathbb{Z})}^{2^{5}}\gg\delta^{2^{26}}\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{5}(u+q\cdot\mathbb{Z})}^{2^{5}}.$ ###### Proof. Applying Lemma 7.1 to $f$ with $\left\|\cdot\right\|:=\left\|\cdot\right\|^{\sharp}_{q}$ as defined in (1.5) and $\varepsilon:=\tfrac{1}{2}\delta\Lambda_{q,N}(1_{[N]})N^{-1/2}$, we deduce that $|\Lambda_{q,N}(g_{0},g_{1},f_{str})|\geq\delta\Lambda_{q,N}(1_{[N]})-|\Lambda_{q,N}(g_{0},g_{1},f_{unf})|\\\ \geq\delta\Lambda_{q,N}(1_{[N]})-\left\|f_{unf}\right\|_{q,N}^{\sharp}\geq\tfrac{1}{2}\delta\Lambda_{q,N}(1_{[N]}).$ We note that our lower bound assumption on $N$ implies that $\Lambda_{q,N}\left(1_{[N]}\right)\gg 1$. Hence the dual inequality (1.10) gives $\delta\ll N^{-1}|\left\langle f_{str},G\right\rangle|\ll\delta^{-1}\left\|G\right\|^{\sharp}_{q}.$ Invoking Theorem 5.6 yields the result. ∎ Taken together, the work in §§3–6 gives the following. ###### Proof of Theorem 1.6. Applying Lemma 7.2, we deduce that $\sum_{u\in[q]}\left\|G\right\|_{U^{5}(u+q\cdot\mathbb{Z})}^{2^{5}}\gg\delta^{2^{26}}\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{5}(u+q\cdot\mathbb{Z})}^{2^{5}},$ where $G$ is defined as in (7.1). We now apply Lemma 6.5 three times. The first application gives $\sum_{u\in[q]}\left\|G\right\|_{U^{4}(u+q\cdot\mathbb{Z})}^{2^{4}}\gg\delta^{2^{40}}\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{4}(u+q\cdot\mathbb{Z})}^{2^{4}},$ a second replaces $U^{4}$ with $U^{3}$ at the cost of replacing $\delta^{2^{40}}$ with $\delta^{2^{52}}$. With a final application, we obtain $\sum_{u\in[q]}\left\|G\right\|_{U^{2}(u+q\cdot\mathbb{Z})}^{4}\gg\delta^{2^{62}}\sum_{u\in[q]}\left\|1_{[N]}\right\|_{U^{2}(u+q\cdot\mathbb{Z})}^{4}.$ Let $\eta:=\delta^{2^{62}}$. By the popularity principle, there are at least $\Omega(\eta q)$ values of $u\in[q]$ for which $\left\|G\right\|_{U^{2}(u+q\cdot\mathbb{Z})}^{4}\gg\eta\left\|1_{[N]}\right\|_{U^{2}(u+q\cdot\mathbb{Z})}^{4}$. The inverse theorem for the $U^{2}$-norm then gives the existence of $\phi(u)\in\mathbb{T}$ for which $\left|\sum_{x}G(u+qx)e(\phi(u)x)\right|\gg\eta^{1/2}N/q.$ (7.2) Set $T:=\left\lceil C\eta^{-1/2}N/q\right\rceil$, with $C$ an absolute constant taken sufficiently large to ensure that, on rounding $\phi(u)$ to the nearest fraction of the form $t/T$, the inequality (7.2) remains valid. By Lemma 6.2, for each $u$ satisfying (7.2), there exists a positive integer $q^{\prime}\ll\eta^{2}$ such that $\|q^{\prime}q^{2}\phi(h)\|\ll\eta^{-7}q^{3}/N$. By a volume packing argument similar to that given in the proof of Lemma 6.5, the function $\phi$ is constant on a proportion of at least $\Omega\bigl{(}\eta^{11}\bigr{)}$ of the residues $u\in[q]$ satisfying (7.2). Summing over these $u$, then extending the sum to all of $[q]$, we deduce the existence of $\alpha\in\mathbb{T}$ and $q^{\prime}\ll\eta^{-2}$ such that $\|q^{\prime}q^{2}\alpha\|\ll\eta^{-7}q^{3}/N$ and $\sum_{u\in[q]}\left|\sum_{x}G(u+qx)e(\alpha x)\right|\gg\eta^{12}N.$ (7.3) Expanding the dual function, there is a 1-bounded function $\psi(u\bmod q)$ such that the left-hand side of the above is equal to $\sum_{u\in[q]}\psi(u\bmod q)\sum_{x\equiv u(q)}\mathbb{E}_{y\in[M]}g_{0}(x-qy^{2})g_{1}(x+y-qy^{2})e(\alpha x/q)\\\ =\sum_{x}g_{0}(x)\psi(x\bmod q)e(\alpha x/q)\mathbb{E}_{y\in[M]}g_{1}(x+y)e(\alpha y^{2}).$ (7.4) Let us first suppose that $f=g_{0}$, we deal with the case $f=g_{1}$ shortly. Setting $\phi(x):=\psi(x\bmod q)e(\alpha x/q)\mathbb{E}_{y\in[M]}g_{1}(x+y)e(\alpha y^{2}),$ we have $\left\langle f,\overline{\phi}\right\rangle\gg\eta^{12}N$. Our aim is to show that $\phi$ can be approximated by a local function of the type claimed in the lemma. We begin by removing the phase from the expectation over $[M]$, at the cost of passing to shorter progressions. Let $M^{\prime}\leq M/q^{\prime}q^{2}$ be a quantity to be determined. If $y\in[M^{\prime}]$ then for any $m\in[-M,M]\cap\mathbb{Z}$ we have $\left|e(\alpha(m+q^{\prime}q^{2}y)^{2})-e(\alpha m^{2})\right|\ll\left\|\alpha\left(2mq^{\prime}q^{2}y+(q^{\prime}q^{2}y)^{2}\right)\right\|\ll q^{\prime}q^{4}\eta^{-7}M^{\prime}/M.$ (7.5) Hence, partitioning $\mathbb{Z}$ into progressions $P$ of common difference $q^{\prime}q^{2}$ and length $M^{\prime}$, there exist phases $\omega_{P}$ such that for any $x\in\mathbb{Z}$ we have $\left|\mathbb{E}_{y\in[M]}g_{1}(x+y)e(\alpha y^{2})-M^{-1}\sum_{P}\omega_{P}\sum_{y\in[M]\cap P}g_{1}(x+y)\right|\ll q^{\prime}q^{4}\eta^{-7}M^{\prime}/M.$ (7.6) Notice that there are at most $O(M/M^{\prime})$ progressions $P$ such that $P\cap[M]\neq\emptyset$ (since we are assuming $M^{\prime}\leq M/q^{\prime}q^{2}$). Next we show how the phase $e(\alpha x/q)$ is approximately periodic. Suppose that $z\in[M^{\prime\prime}]$, with $M^{\prime\prime}\leq M^{\prime}/q$ to be determined. Then for any $x\in\mathbb{Z}$ we have $\left|e\left(\alpha(x+q^{\prime}q^{3}z)/q\right)-e\left(\alpha x\right)\right|\ll\left\|\alpha q^{\prime}q^{2}\right\|M^{\prime\prime}\ll\eta^{-7}q^{3}M^{\prime\prime}/N$ and by a boundary estimate $\left|\sum_{y\in[M]\cap P}g_{1}(x+q^{\prime}q^{3}z+y)-\sum_{y\in[M]\cap P}g_{1}(x+y)\right|\ll qM^{\prime\prime}.$ It then follows from a telescoping identity that for all $x\in\mathbb{Z}$ and $z\in[M^{\prime\prime}]$ we have $\displaystyle\left|\phi(x+q^{\prime}q^{3}z)-\phi(x)\right|$ $\displaystyle\ll\frac{\eta^{-7}q^{3}M^{\prime\prime}}{N}+\frac{\eta^{-7}q^{\prime}q^{4}M^{\prime}}{M}+\frac{qM^{\prime\prime}}{M}\sum_{\begin{subarray}{c}P\\\ P\cap[M]\neq\emptyset\end{subarray}}1$ $\displaystyle\ll\frac{\eta^{-7}q^{\prime}q^{4}M^{\prime}}{M}+\frac{qM^{\prime\prime}}{M^{\prime}}.$ Taking $M^{\prime}:=c\eta^{19}M/q^{\prime}q^{4}$ and $M^{\prime\prime}:=c\eta^{12}M^{\prime}/q$ for a sufficiently small absolute constant $c>0$ we have $\left|\phi(x+q^{\prime}q^{3}z)-\phi(x)\right|\leq\eta^{12}/C\quad\text{for all }x\in\mathbb{Z}\text{ and }z\in[M^{\prime\prime}].$ (7.7) Partitioning $\mathbb{Z}$ into translates $T$ of $q^{\prime}q^{3}\cdot[M^{\prime\prime}]$ we deduce that $\sum_{T}\biggl{|}\sum_{x\in T}f(x)\biggr{|}\gg\eta^{12}N.$ Write $\chi(x)$ for the phase of the inner sum when $x\in T$. Then $\chi$ is a 1-bounded local function of modulus $q^{\prime}q^{3}$ and resolution $\Omega\left((\delta/q)^{O(1)}M\right)$ satisfying $\sum_{x}f(x)\overline{\chi(x)}\gg\delta^{2^{66}}N,$ as required. Next we give the argument for when $f=g_{1}$. Returning to (7.4) we have $\sum_{x}\left|\mathbb{E}_{y\in[M]}f(x+y)e(\alpha y^{2})\right|\gg\eta^{12}N.$ Utilising (7.5) and (7.6), we may partition $\mathbb{Z}$ into progressions $P$ of common difference $q^{\prime}q^{2}$ and length $M^{\prime}:=c\eta^{19}M/q^{\prime}q^{4}$ such that $\sum_{x}\sum_{P}\left|\mathbb{E}_{y\in[M]\cap P}f(x+y)\right|\gg\eta^{12}N.$ Since $O(M/M^{\prime})$ of the $P$ intersect $[M]$, the pigeon-hole principle gives $P^{\prime}:=P\cap[M]$ such that $\sum_{x}\left|\sum_{y\in P^{\prime}}f(x+y)\right|\gg\eta^{12}NM^{\prime}.$ In particular $|P^{\prime}|\gg\eta^{12}M^{\prime}\gg(q/\delta)^{C}M$. Partitioning $\mathbb{Z}$ into translates of $P^{\prime}$ of the form $\mathbb{Z}=\bigsqcup_{i}(a_{i}+P^{\prime}),$ the pigeon-hole principle gives $z\in P^{\prime}$ such that $\sum_{i}\left|\sum_{y\in P^{\prime}}f(a_{i}+y+z)\right|\gg\eta^{12}N.$ Writing $\chi(x)$ for the phase of the inner sum when $x\in a_{i}+P$ one sees that $\chi$ is a local function of resolution $\gg(q/\delta)^{C}M$ and modulus $q^{\prime}q^{2}$ which satisfies $\left\langle f,\chi\right\rangle\gg\eta^{12}N$. The proof is complete on noting that a local function of modulus $q^{\prime}q^{2}$ is also a local function of modulus $q^{\prime}q^{3}$. ∎ ## Appendix A Basic theory of the Gowers norms ###### Lemma A.1 (Inverse theorem for the $U^{2}$-norm). Let $f:\mathbb{Z}\to\mathbb{C}$ be a $1$-bounded function with support in $[N]$. Then there exists $\alpha\in\mathbb{T}$ such that $\|f\|_{U^{2}}^{4}\leq N\left|\sum_{x}f(x)e(\alpha x)\right|^{2}.$ ###### Proof. Using the definition of the Fourier transform (1.8), together with orthogonality of additive characters, we have $\left\|f\right\|_{U^{2}}^{4}=\int_{\mathbb{T}}\bigl{|}\hat{f}(\alpha)\bigr{|}^{4}\mathrm{d}\alpha\leq\big{\|}\hat{f}\big{\|}_{\infty}^{2}\int_{\mathbb{T}}\bigl{|}\hat{f}(\alpha)\bigr{|}^{2}\mathrm{d}\alpha\leq\big{\|}\hat{f}\big{\|}_{\infty}^{2}N.$ ∎ For each $\omega\in\\{0,1\\}^{s}$, let $f_{\omega}:\mathbb{Z}\to\mathbb{C}$ be a function with finite support. Then we define the _Gowers inner product_ by $[f_{\omega}]_{U^{s}}:=\sum_{x,h_{1},\dots,h_{s}}\prod_{\omega\in\left\\{0,1\right\\}^{s}}\mathcal{C}^{|\omega|}f_{\omega}(x+\omega\cdot h).$ Here $\mathcal{C}$ denotes the operation of complex conjugation. Notice that $[f]_{U^{s}}=\left\|f\right\|_{U^{s}}^{2^{s}}$. ###### Lemma A.2 (Gowers–Cauchy–Schwarz). For each $\omega\in\\{0,1\\}^{s}$, let $f_{\omega}:\mathbb{Z}\to\mathbb{C}$ be a function with finite support. Then we have $[f_{\omega}]_{U^{s}}\leq\prod_{\omega\in\\{0,1\\}^{s}}\|f_{\omega}\|_{U^{s}}.$ ###### Proof. See [9, Exercise 1.3.19]. ∎ ###### Lemma A.3 (Phase invariance for $s\geq 2$). Let $L\in\mathbb{R}[x,h_{1},\dots,h_{s}]$ be a linear form, with $s\geq 2$ and let $f:\mathbb{Z}\to\mathbb{C}$. Then $\biggl{|}\sum_{x,h_{1},\dots,h_{s}}\Delta_{h_{1},\dots,h_{s}}f(x)e(L(x,h_{1},\dots,h_{s}))\biggr{|}\leq\left\|f\right\|_{U^{s}}^{2^{s}}.$ ###### Proof. The linear form may be written as $L(x,h_{1},\dots,h_{s})=\alpha x+\beta_{1}(x+h_{1})+\dots+\beta_{s}(x+h_{s}),$ for some real $\alpha$ and $\beta_{i}$. Write $f_{0}(x):=f(x)e(\alpha x)$, $f_{e_{i}}(x):=f(x)e(-\beta_{i}x)$ for $i=1,\dots,s$, and for $\omega\in\left\\{0,1\right\\}^{s}\setminus\left\\{0,e_{1},\dots,e_{s}\right\\}$ set $f_{\omega}:=f$. Then by Gowers–Cauchy–Schwarz we have $\biggl{|}\sum_{x,h_{1},\dots,h_{s}}\Delta_{h_{1},\dots,h_{s}}f(x)e(L(x,h_{1},\dots,h_{s}))\biggr{|}\leq\prod_{\omega}\left\|f_{\omega}\right\|.$ It therefore suffice to prove that for a phase function $e_{\alpha}:x\mapsto e(\alpha x)$ $\left\|fe_{\alpha}\right\|_{U^{s}}=\left\|f\right\|_{U^{s}}.$ The latter follows on observing that $\Delta_{h_{1},\dots,h_{s}}(fe_{\alpha})=\left(\Delta_{h_{1},\dots,h_{s}}f\right)\left(\Delta_{h_{1},\dots,h_{s}}e_{\alpha}\right),$ and for any $x,h_{1},\dots,h_{s}$ with $s\geq 2$ we have $\Delta_{h_{1},\dots,h_{s}}e_{\alpha}(x)=1.$ ∎ ###### Lemma A.4 (Box Cauchy–Schwarz). Let $\mu_{1},\mu_{2},\mu_{3}$ be probability measures on $\mathbb{Z}$ with the discrete sigma algebra. If $F_{1},F_{2},F_{3}$ are 1-bounded function on $\mathbb{Z}^{2}$ and $F$ is a 1-bounded function on $\mathbb{Z}^{3}$ then $\left|\sum_{x\in\mathbb{Z}^{3}}F_{1}(x_{2},x_{3})F_{2}(x_{1},x_{3})F_{3}(x_{1},x_{2})F(x)\underline{\mu}(x)\right|^{8}\\\ \leq\sum_{x^{0},x^{1}\in\mathbb{Z}^{3}}\prod_{\omega\in\left\\{0,1\right\\}^{3}}\mathcal{C}^{|\omega|}F(x_{1}^{\omega_{1}},x_{2}^{\omega_{2}},x_{3}^{\omega_{3}})\mu_{1}(x_{1}^{0})\mu_{1}(x_{1}^{1})\mu_{2}(x_{2}^{0})\mu_{2}(x_{2}^{1})\mu_{3}(x_{3}^{0})\mu_{3}(x_{3}^{1}).$ ## References * BC [17] J. Bourgain and M.-C. Chang. Nonlinear Roth type theorems in finite fields. Israel J. Math., 221(2):853–867, 2017. * BL [96] V. Bergelson and A. Leibman. Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J. Amer. Math. Soc., 9(3):725–753, 1996. * Gow [10] W. T. Gowers. Decompositions, approximate structure, transference, and the Hahn-Banach theorem. Bull. Lond. Math. Soc., 42(4):573–606, 2010. * GT [08] B. Green and T. Tao. Quadratic uniformity of the Möbius function. Ann. Inst. Fourier (Grenoble), 58(6):1863–1935, 2008. * Pel [19] S. Peluse. On the polynomial szemerédi theorem in finite fields. Duke Math. J., 168(5):749–774, 04 2019. * PP [19] S. Peluse and S. Prendiville. Quantitative bounds in the non-linear Roth theorem. ArXiv e-prints, 2019. * PP [20] S. Peluse and S. Prendiville. A polylogarithmic bound in the nonlinear Roth theorem. ArXiv e-prints, 2020. * Pre [17] S. Prendiville. Quantitative bounds in the polynomial Szemerédi theorem: the homogeneous case. Discrete Anal., pages 34, Paper No. 5, 2017. * Tao [12] T. Tao. Higher order Fourier analysis, volume 142 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012. * TZ [16] T. Tao and T. Ziegler. Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors. Discrete Anal., pages 60, Paper No. 13, 2016.

2024-09-04T02:54:58.404089

2003.04122

arxiv-papers

# A polylogarithmic bound in the nonlinear Roth theorem Sarah Peluse Mathematical Institute University of Oxford UK<EMAIL_ADDRESS>and Sean Prendiville Department of Mathematics and Statistics Lancaster University UK<EMAIL_ADDRESS> ###### Abstract. We show that sets of integers lacking the configuration $x$, $x+y$, $x+y^{2}$ have at most polylogarithmic density. ###### Contents 1. 1 Introduction 2. 2 Iterating the density increment 3. 3 The cut norm inverse theorem 4. 4 A weak regularity lemma 5. 5 The density increment lemma 6. 6 Global control by major arc Fourier coefficients 7. 7 Longer progressions ## 1\. Introduction ### 1.1. Density bound In [9] the authors obtained, for the first time, an effective bound for subsets of $\left\\{1,\dots,N\right\\}$ lacking the nonlinear Roth configuration $x$, $x+y$, $x+y^{2}$. There it was established that such sets have cardinality at most $O(N/(\log\log N)^{c})$, where $c>0$ is an absolute constant. The key breakthrough of [9] was a “local $U^{1}$-control” result, from which a bound for sets lacking the nonlinear Roth configuration follows via standard methods. Here, we combine this local $U^{1}$-control result with a more sophisticated argument to remove a logarithm from the bound of [9]. ###### Theorem 1.1 (Density bound). There exists an absolute constant $c>0$ such that the following holds. Suppose that $A\subset\left\\{1,\dots,N\right\\}$ lacks configurations of the form $x,\ x+y,\ x+y^{2}\qquad(y\neq 0).$ (1.1) Then $|A|=O\left(N/(\log N)^{c}\right).$ A careful analysis shows that the exponent $c=2^{-150}$ is permissible, where 150 represents the combined number of times we utilise the Cauchy–Schwarz inequality in [9] and this paper ### 1.2. Major arc correlation The techniques which yield Theorem 1.1 also allow us to show, in a quantitatively effective manner, that the major arc Fourier coefficients of a set determine how many nonlinear Roth configurations (1.1) the set contains. ###### Theorem 1.2 (Major-arc control). Let $\delta>0$ and $f,g,h:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions with support in $\left\\{1,\dots,N\right\\}$. Suppose that $\left|\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{N}}f(x)g(x+y)h(x+y^{2})\right|\geqslant\delta N^{3/2}.$ Then either $N\ll\delta^{-O(1)}$, or there is a frequency $\alpha\in\mathbb{R}$ and a positive integer $q\ll\delta^{-O(1)}$ such that111Here $\left\|\cdot\right\|$ denotes the distance to the nearest integer, and $e(\alpha):=e^{2\pi i\alpha}$. For our conventions regarding asymptotic notation see §1.5. $\left\|q\alpha\right\|\ll\delta^{-O(1)}/N$ and $\left|\sum_{x\in\mathbb{Z}}h(x)e(\alpha x)\right|\gg\delta^{O(1)}N.$ In the nomenclature of [14], the major arc linear phases are the only obstructions to uniformity for the nonlinear Roth configuration. We emphasise that Theorem 1.2 is not used in the proof of Theorem 1.1. The major arc Fourier coefficients of a subset of $\\{1,\dots,N\\}$ essentially measure its distribution in arithmetic progressions of common difference $\ll 1$ and length $\gg N$. To illustrate this, the following definition is useful. ###### Definition 1.3 (Local function). We call a function $\phi:\mathbb{Z}\to\mathbb{C}$ a _local function of resolution $M$ and modulus $q$_ if there exists a partition of $\mathbb{Z}$ into intervals of length $M$ such that $\phi$ is constant on the intersection of every such interval with every congruence class mod $q$. ###### Corollary 1.4 (Local control of the nonlinear term). Let $\delta>0$ and $f,g,h:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions with support in $\left\\{1,\dots,N\right\\}$. Suppose that $\left|\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{N}}f(x)g(x+y)h(x+y^{2})\right|\geqslant\delta N^{3/2}.$ Then either $N\ll\delta^{-O(1)}$, or there is a 1-bounded local function $\phi$ of resolution $M\gg\delta^{O(1)}N$ and modulus $q\ll\delta^{-O(1)}$ such that $\left|\sum_{x\in\mathbb{Z}}h(x)\phi(x)\right|\gg\delta^{O(1)}N.$ One cannot hope to prove that the functions $f$ and $g$ above also correlate globally with local functions, as the following example illustrates. For any positive integers $x_{1},x_{2}\leqslant N^{1/2}$, set $f\left(x_{1}+(x_{2}-1)\left\lfloor N^{1/2}\right\rfloor\right)=\begin{cases}1&\text{ if }x_{2}\equiv 0\pmod{4},\\\ 0&\text{ if }x_{2}\equiv 1\pmod{4},\\\ -1&\text{ if }x_{2}\equiv 2\pmod{4},\\\ 0&\text{ if }x_{2}\equiv 3\pmod{4};\end{cases}$ and set $f(x)=0$ everywhere else. Taking $g:=f$ and $h:=1_{\\{1,\dots,N\\}}$, one can check that either $N\ll 1$ or $\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{N}}f(x)g(x+y)h(x+y^{2})\gg N^{3/2}.$ However, for any arithmetic progression $P\subset\\{1,\dots,N\\}$, we have $\left|\sum_{x\in P}f(x)\right|\ll N^{1/2}.$ Hence, for any 1-bounded local function $\phi$ of resolution $\geqslant\delta N$ and modulus $\leqslant\delta^{-1}$, the triangle inequality gives the discorrelation $\left|\sum_{x\in\mathbb{Z}}f(x)\phi(x)\right|\ll\delta^{-2}N^{1/2}.$ This example is a local obstruction coming from the real numbers: the nature of our counting operator means that we cannot disentangle possible correlations between the $f$ and $g$ functions on subintervals of length $N^{1/2}$. We can, however, show that these are the only other possible obstructions to uniformity. ###### Theorem 1.5 (Local control of all terms). Let $\delta>0$ and $f_{1},f_{2},f_{3}:\mathbb{Z}\to\mathbb{C}$ be 1-bounded functions with support in $\left\\{1,\dots,N\right\\}$. Suppose that $\left|\sum_{x\in\mathbb{Z}}\sum_{y\in\mathbb{N}}f_{1}(x)f_{2}(x+y)f_{3}(x+y^{2})\right|\geqslant\delta N^{3/2}.$ Then either $N\ll\delta^{-O(1)}$, or for each $i=1,2,3$ there is a 1-bounded local function $\phi_{i}$ of resolution $\gg\delta^{O(1)}N^{1/2}$ and modulus $q_{i}\ll\delta^{-O(1)}$ such that $\left|\sum_{x\in\mathbb{Z}}f_{i}(x)\phi_{i}(x)\right|\gg\delta^{O(1)}N.$ ###### Proof. This is an immediate consequence of Corollary 1.4 and Lemma 3.2. ∎ ### 1.3. Longer polynomial progressions In analogy with the first author’s generalisation [8] of [9], it is natural to ask whether the methods of this paper yield polylogarithmic bounds for sets of integers lacking longer progressions $x,\ x+P_{1}(y),\ \dots,\ x+P_{m}(y),$ (1.2) where the $P_{i}\in\mathbb{Z}[y]$ have zero constant term and $\deg P_{1}<\dots<\deg P_{m}$. As was mentioned above, the key input to this paper is the local $U^{1}$-control result [9, Theorem 7.1]. Replacing this with [8, Theorem 3.3], our argument generalises in a straightforward manner to yield polylogarithmic bounds for subsets of $\\{1,\dots,N\\}$ lacking (1.2) when $m=2$, that is, for all three-term polynomial progressions with distinct degrees and zero constant term. Obtaining polylogarithmic bounds for longer polynomial progressions requires an additional idea. We sketch a strategy in §7, which relies on obtaining an appropriate generalisation of [8, Theorem 3.3], a generalisation that would require re-running the majority of the arguments therein. ### Acknowledgements S. Peluse is supported by the NSF Mathematical Sciences Postdoctoral Research Fellowship Program under Grant No. DMS-1903038 ### 1.4. An outline of our argument Effective Szemerédi-type theorems are commonly proved via a density increment strategy, the prototypical example being the proof of Roth’s theorem [11] on three-term arithmetic progressions. This strategy begins with a set $A\subset\\{1,\dots,N\\}$ of density $\delta:=|A|/N$ that lacks the configuration in question. It then proceeds to show that there is a substructure $S\subset\\{1,\dots,N\\}$ on which $A$ has increased density $\delta+\Omega_{\delta}(1)$. One then hopes to iterate the argument with $A\cap S$ in place of $A$ and $S$ in place of $\\{1,\dots,N\\}$. One avenue to obtaining polylogarithmic bounds in a Szemerédi-type theorem is to obtain a constant proportion density increment $\delta+\Omega(\delta)$ on a substructure $S$ of polynomial size $|S|\approx N^{\Omega(1)}$. This was accomplished for three-term arithmetic progressions by Heath–Brown [7] and Szemerédi [13] (in fact, they were able to handle a smaller lower bound on $|S|$). An alternative strategy for obtaining polylogarithmic bounds is to obtain the weaker polynomial increment $\delta+\Omega(\delta^{O(1)})$, yet on a _dense_ or _global_ substructure $S$, that is, a substructure of size $|S|\geqslant\exp(-O(\delta^{-O(1)}))N$. This was accomplished by Sárközy [12] for the configuration $x,x+y^{2}$ and for three-term arithmetic progressions by Bourgain [2]. Both of these strategies are achievable for the nonlinear Roth configuration. The global structure strategy is perhaps the most natural, and may be accomplished by utilising a generalisation of Theorem 1.2. In this note we do not pursue this, and instead give details for a constant-proportion density increment, as our argument is somewhat cleaner in this form. More specifically, we show that if $A\subset\left\\{1,\dots,N\right\\}$ has density $\delta$ and lacks nontrivial configurations of the form $x,x+y,x+y^{2}$, then there exists an arithmetic progression $P$ of length $|P|\gg\delta^{O(1)}N^{1/2}$ and common difference $q\ll\delta^{-O(1)}$ such that we have the density increment $\frac{|A\cap P|}{|P|}\geqslant(1+\Omega(1))\frac{|A|}{N}.$ (1.3) As outlined in [9], the ‘almost bounded’ size of $q$ allows us to iterate this procedure. (In [9], we obtain the weaker density increment $(1+\Omega(\delta^{O(1)}))|A|/N$, which leads to the extra logarithm appearing in the bound there.) We obtain the constant-proportion increment (1.3) by combining the local $U^{1}$-control result of [9] with a strategy of Heath–Brown [7] and Szemerédi [13], which has a very robust formulation due to Green and Tao [6]. To accomplish this, we first give a structural characterisation of sets lacking the nonlinear Roth configuration (this is Lemma 3.3, whose essence is captured in the weaker Theorem 1.5). These sets resemble the level sets of the product of a function that is constant on intervals of length $N^{1/2}$ and a function that is constant on congruence classes modulo a bounded $q$. Having obtained such a structural characterisation, an energy increment procedure closely following [6] allows us to approximate an arbitrary set of integers by these level sets, up to an error that does not contribute substantially to the count of nonlinear Roth configurations. A combinatorial argument then allows us to deduce that our set must have a substantial density increment on one of these level sets, of the form $\delta+\Omega(\delta)$. As a result, our density increment procedure requires only $\log(\delta^{-1})+O(1)$ iterations, compared with the $O(\delta^{-O(1)})$ required in [9], and this yields the polylogarithmic improvement over our previous density increment iteration. The remainder of this paper is organized as follows. We derive Theorem 1.1 in §2 via a density increment iteration. Our deduction uses a density increment lemma that is established in §§3–5. We prove Theorem 1.2 and Corollary 1.4 in §6. ### 1.5. Notation #### 1.5.1. Standard conventions We use $\mathbb{N}$ to denote the positive integers. For a real number $X\geqslant 1$, write $[X]=\\{1,2,\ldots,\left\lfloor X\right\rfloor\\}$. A complex-valued function is said to be _1-bounded_ if the modulus of the function does not exceed 1. We use counting measure on $\mathbb{Z}$, so that for $f,g:\mathbb{Z}\to\mathbb{C}$, we have $\left\|f\right\|_{\ell^{p}}:=\biggl{(}\sum_{x}|f(x)|^{p}\biggr{)}^{\frac{1}{p}},\ \left\langle f,g\right\rangle:=\sum_{x}f(x)\overline{g(x)},\ \text{and}\ (f*g)(x)=\sum_{y}f(y)g(x-y).$ Any sum of the form $\sum_{x}$ is to be interpreted as a sum over $\mathbb{Z}$. The _support_ of $f$ is the set $\mathrm{supp}(f):=\left\\{x\in\mathbb{Z}:f(x)\neq 0\right\\}$. We write $\left\|f\right\|_{\infty}$ for $\sup_{x\in\mathbb{Z}}|f(x)|$. We use Haar probability measure on $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$, so that for measurable $F:\mathbb{T}\to\mathbb{C}$, we have $\left\|F\right\|_{L^{p}}:=\biggl{(}\int_{\mathbb{T}}|F(\alpha)|^{p}d\alpha\biggr{)}^{\frac{1}{p}}=\biggl{(}\int_{0}^{1}|F(\alpha)|^{p}d\alpha\biggr{)}^{\frac{1}{p}}.$ We write $\left\|\alpha\right\|_{\mathbb{T}}$ for the distance from $\alpha\in\mathbb{R}$ to the nearest integer $\min_{n\in\mathbb{Z}}|\alpha-n|.$ This remains well-defined on $\mathbb{T}$. We define the Fourier transform of $f:\mathbb{Z}\to\mathbb{C}$ by $\hat{f}(\alpha):=\sum_{x}f(x)e(\alpha x)\qquad(\alpha\in\mathbb{T}),$ (1.4) when this makes sense. Here $e(\alpha)$ stands for $e^{2\pi i\alpha}$. For a finite set $S$ and function $f:S\to\mathbb{C}$, denote the average of $f$ over $S$ by $\mathbb{E}_{s\in S}f(s):=\frac{1}{|S|}\sum_{s\in S}f(s).$ For a complex-valued function $f$ and positive-valued function $g$, write $f\ll g$ or $f=O(g)$ if there exists a constant $C$ such that $|f(x)|\leq Cg(x)$ for all $x$. We write $f=\Omega(g)$ if $f\gg g$. We subscript this notation when the implicit constant may depend on the subscripted parameters. #### 1.5.2. Local conventions Up to normalisation, all of the above are widely used in the literature. Next, we list notation specific to our paper. We have tried to minimise this in order to aid the casual reader. The quantity $(N/q)^{1/2}$ appears repeatedly, where $N$ and $q$ are integers fixed throughout the majority of our paper. We therefore adopt the convention that $M:=\left\lfloor\sqrt{N/q}\right\rfloor.$ (1.5) Assuming this, define the _counting operator_ on the functions $f_{i}:\mathbb{Z}\to\mathbb{C}$ by $\Lambda_{q,N}(f_{0},f_{1},f_{2}):=\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[M]}f_{0}(x)f_{1}(x+y)f_{2}(x+qy^{2}).$ (1.6) When $f_{0}=f_{1}=f_{2}=f$, we simply write $\Lambda_{q,N}(f)$ for $\Lambda_{q,N}(f_{0},f_{1},f_{2})$. For a real parameter $H\geqslant 1$, we use $\mu_{H}:\mathbb{Z}\to[0,1]$ to represent the following normalised Fejér kernel $\mu_{H}(h):=\frac{1}{\left\lfloor H\right\rfloor}\left(1-\frac{|h|}{\left\lfloor H\right\rfloor}\right)_{+}=\frac{(1_{[H]}*1_{-[H]})(h)}{\left\lfloor H\right\rfloor^{2}}.$ (1.7) This is a probability measure on $\mathbb{Z}$ with support in the interval $(-H,H)$. ## 2\. Iterating the density increment In this section we prove Theorem 1.1 using the following lemma, which we will devote §§3–5 to proving. ###### Lemma 2.1 (Density increment lemma). Let $q\leqslant N$ be positive integers and $\delta>0$. Suppose that $A\subset[N]$ satisfies $|A|\geqslant\delta N$ and lacks the configuration $x,\ x+y,\ x+qy^{2}\qquad(y\neq 0).$ (2.1) Then either $N\ll(q/\delta)^{O(1)}$ or there exists $q^{\prime}\leqslant\exp\left(O\left(\delta^{-O(1)}\right)\right)$ and $N^{\prime}\geqslant q^{-O(1)}\exp\left(-O\left(\delta^{-O(1)}\right)\right)N^{1/2}$ such that, for some $a\in\mathbb{Z}$, we have $|A\cap(a+qq^{\prime}\cdot[N^{\prime}])|\geqslant(1+\Omega(1))\delta N^{\prime}.$ (2.2) ###### Proof of Theorem 1.1 given Lemma 2.1. This is the same as the proof of [9, Theorem 1.1], but using the improved density increment lemma above in place of the density increment lemma of [9]. Note first that if $A$ lacks the configuration (2.1), then the set $\\{x:a+qq^{\prime}x\in A\\},$ lacks configurations of the form $x,\ x+y,\ x+q^{2}q^{\prime}y^{2}\qquad(y\neq 0).$ Let $A\subset[N]$ have size $\delta N$, and suppose that it has no non-linear Roth configurations (1.1). Setting $A_{0}:=A$, $N_{0}:=N$ and $q_{0}=1$, let us suppose we have a sequence of tuples $(A_{i},N_{i},q_{i})$ for $i=0,1,\dots,n$ that each satisfy the following: 1. (i) $A_{i}$ lacks configurations of the form $x,\ x+y,\ x+q_{0}^{2^{i}}q_{1}^{2^{i-1}}\dotsm q_{i-1}^{2}q_{i}y^{2}\qquad(y\neq 0).$ 2. (ii) $q_{i}\leqslant\exp\left(O\left(\delta^{-O(1)}\right)\right)$; 3. (iii) $A_{i}\subset[N_{i}]$ and for $i\geqslant 1$ we have $\frac{|A_{i}|}{N_{i}}\geqslant(1+c)\frac{|A_{i-1}|}{N_{i-1}},$ where $c=\Omega(1)$ is a positive absolute constant; 4. (iv) for $i\geqslant 1$ we have the lower bound $N_{i}\geqslant\frac{N_{i-1}^{1/2}}{\left(q_{0}^{2^{i-1}}\dotsm q_{i-1}\exp\left(\delta^{-O(1)}\right)\right)^{O(1)}}.$ Applying Lemma 2.1 with $q=q_{0}^{2^{i}}q_{1}^{2^{i-1}}\dotsm q_{i-1}^{2}q_{i}$, either $N_{n}\ll\left(q_{0}^{2^{n}}q_{1}^{2^{n-1}}\dotsm q_{n-1}^{2}q_{n}/\delta\right)^{O(1)},$ (2.3) or we may obtain $(A_{n+1},N_{n+1},q_{n+1})$ satisfying conditions (i)–(iv). If (2.3) holds, then our iterative process terminates at stage $n$. If the number of iterations $n$ is at least $c^{-1}$, then the density of $A_{n}$ on $[N_{n}]$ is at least $2\delta$. After an additional $\tfrac{1}{2}c^{-1}$ iterations, the density is at least $4\delta$. Hence if the number of iterations is at least $\left\lceil c^{-1}\right\rceil+\left\lceil\tfrac{1}{2}c^{-1}\right\rceil+\left\lceil\tfrac{1}{4}c^{-1}\right\rceil+\dots+\left\lceil\tfrac{1}{2^{m-1}}c^{-1}\right\rceil,$ then the density is at least $2^{m}\delta$. The density therefore exceeds one if the number of iterations exceeds $2c^{-1}+\log_{2}(\delta^{-1})$. Since this cannot happen, it follows that there exists $n\leqslant\log_{2}(\delta^{-1})+O(1)$ such that the procedure terminates at stage $n$. At the point of termination, the smallness assumption (2.3) must hold, so that $N_{n}\leqslant\exp\left(O\Bigl{(}\delta^{-O(1)}\Bigr{)}\right).$ On the other hand, iteratively applying the lower bound (iv), we have $\begin{split}N_{n}&\geqslant\frac{N_{n-1}^{1/2}}{\left(q_{0}^{2^{n-1}}\dotsm q_{n-1}\exp\left(\delta^{-O(1)}\right)\right)^{O(1)}}\\\ &\geqslant N^{1/2^{n}}\left[q_{0}^{2^{n-1}}\dotsm q_{n-1}\exp\left(\delta^{-O(1)}\right)\right]^{-O(1+\frac{1}{2}+\frac{1}{4}+\dots+2^{1-n})}\\\ &\gg\exp\left(-O\left(\delta^{-O(1)}\right)\right)N^{\Omega(\delta)},\end{split}$ where we use the upper bound (ii) on the $q_{i}$’s, together with $n\leqslant\log_{2}(\delta^{-1})+O(1)$. Taking a logarithm and comparing upper and lower bounds for $N_{n}$ gives $\log N\ll\delta^{-O(1)},$ which yields the bound claimed in Theorem 1.1. ∎ ## 3\. The cut norm inverse theorem The first step of the proof of Lemma 2.1 is to use the main technical result of [9] to prove an inverse theorem for the cut norm associated to $\Lambda_{q,N}$, which we now define. ###### Definition 3.1 (Cut norm). For positive integers $q\leqslant N$, we define the _cut norm_ of $f:\mathbb{Z}\to\mathbb{C}$ by $\left\|f\right\|_{q,N}:=\sup\\{|\Lambda_{q,N}(f,g_{1},g_{2})|,\ |\Lambda_{q,N}(g_{1},f,g_{2})|,\ |\Lambda_{q,N}(g_{1},g_{2},f)|\\},$ (3.1) where the supremum is taken over all 1-bounded functions $g_{i}:[N]\to\mathbb{C}$. We note that, in spite of our nomenclature, this is not a norm, but a seminorm. One could remedy this by summing over $y\geqslant 0$ in the counting operator (1.6). Initially, the cut norm is too restrictive for us, so we begin by working with the weaker quantity $\left\|f\right\|^{\flat}_{q,N}:=\sup\\{|\Lambda_{q,N}(f,g_{1},g_{2})|,|\Lambda_{q,N}(g_{1},f,g_{2})|:|g_{i}|\leqslant 1\text{ and }\mathrm{supp}(g_{i})\subset[N]\\},$ (3.2) which we refer to as the _partial cut norm_. The following lemma is simply a rephrasing of [9, Theorem 7.1], which is the technical heart of that paper. See Definition 1.3 for the meaning of ‘local function’. ###### Lemma 3.2 (Partial cut norm inverse theorem). Let $q\leqslant N$ be positive integers, $\delta>0$, and $f:\mathbb{Z}\to\mathbb{C}$ be a $1$-bounded function with support in $[N]$. Suppose that $\left\|f\right\|^{\flat}_{q,N}\geqslant\delta.$ Then either $N\ll(q/\delta)^{O(1)}$ or there exists a 1-bounded local function $\phi$ of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$, modulus $qq^{\prime}$ for some $q^{\prime}\ll\delta^{-O(1)}$, and such that $\sum_{x\in[N]}f(x)\phi(x)\gg\delta^{O(1)}N.$ ###### Proof. By compactness, there exist 1-bounded functions $g_{1},g_{2}:[N]\to\mathbb{C}$ such that either $|\Lambda_{q,N}(f,g_{1},g_{2})|\geqslant\delta$ or $|\Lambda_{q,N}(g_{1},f,g_{2})|\geqslant\delta.$ In the latter case, we may apply [9, Theorem 7.1] to deduce that there exist positive integers $q^{\prime}\ll\delta^{-O(1)}$ and $N^{\prime}\gg(\delta/q)^{O(1)}N^{1/2}$ such that $\sum_{x}\left|\sum_{y\in[N^{\prime}]}f(x+qq^{\prime}y)\right|\gg\delta^{O(1)}NN^{\prime}.$ In the former case, the reader may check that the argument of [9, Theorem 7.1] delivers the same conclusion222For details see the second author’s exposition [10].. To ease notation, write $Q:=qq^{\prime}$. Partitioning the integers into arithmetic progressions of length $N^{\prime}$ and common difference $Q$ gives $\delta^{O(1)}NN^{\prime}\ll\sum_{z\in[N^{\prime}]}\sum_{u\in[Q]}\sum_{x\in\mathbb{Z}}\left|\sum_{y\in[N^{\prime}]}f(Qz+QN^{\prime}x+u+Qy)\right|\\\ \leqslant N^{\prime}\max_{z}\sum_{u\in[Q]}\sum_{x\in\mathbb{Z}}\left|\sum_{y\in[N^{\prime}]}f(Qz+QN^{\prime}x+u+Qy)\right|.$ Defining $\psi_{z}(u,x)$ to be the conjugate phase of the inner sum, we deduce the existence of $z$ for which $\displaystyle\delta^{O(1)}N\ll\sum_{u\in[Q]}\sum_{x}\sum_{y\in[N^{\prime}]}f(Qz+QN^{\prime}x+u+Qy)\psi_{z}(u,x).$ The result follows on noting that every integer has a unique representation of the form $QN^{\prime}x+u+Qy$ with $u\in[Q]$, $x\in\mathbb{Z}$ and $y\in[N^{\prime}]$. Hence the map $Qz+QN^{\prime}x+u+Qy\mapsto\psi_{z}(u,x)$ is a local function of resolution $QN^{\prime}$ and modulus $Q$. ∎ Now we can prove an inverse theorem for the cut norm itself. ###### Lemma 3.3 (Full cut norm inverse theorem). Let $q\leqslant N$ be positive integers, $\delta>0$, and $f:\mathbb{Z}\to\mathbb{C}$ be a $1$-bounded function with support in $[N]$. Suppose that $\left\|f\right\|_{q,N}\geqslant\delta.$ Then either $N\ll(q/\delta)^{O(1)}$ or there exist 1-bounded local functions $\phi_{1}$ and $\phi_{2}$, of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$ and moduli $qq_{1}$ and $qq_{2}$, respectively, for some $q_{1},q_{2}\ll\delta^{-O(1)}$ such that $\left|\sum_{x\in[N]}f(x)\phi_{1}(x)\phi_{2}(x)\right|\gg\delta^{O(1)}N.$ (3.3) ###### Proof. By the definition of the cut norm (3.1) and Lemma 3.2, we may assume that there are 1-bounded functions $g,h:[N]\to\mathbb{C}$ such that $|\Lambda_{q,N}(g,h,f)|\geqslant\delta.$ (3.4) Recalling that $M:=\lfloor\sqrt{N/q}\rfloor$, define the dual function $F(x):=\mathbb{E}_{y\in[M]}h(x+y)f(x+qy^{2}).$ Re-parametrising (3.4) and applying the Cauchy–Schwarz inequality, we have that $\delta^{2}\leqslant\mathbb{E}_{x\in[N]}F(x)^{2}=\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[M]}F(x)h(x+y)f(x+qy^{2}).$ Recalling the definition of the partial cut norm (3.2), we deduce that $\left\|F\right\|_{q,N}^{\flat}\geqslant\delta^{2}.$ Applying the partial cut norm inverse theorem (Lemma 3.2), there exists a 1-bounded local function $\phi_{1}$ of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$ and modulus $qq_{1}$ for some $q_{1}\ll\delta^{-O(1)}$ such that $\left|\sum_{x\in[N]}F(x)\phi_{1}(x)\right|\gg\delta^{O(1)}N.$ Thus $|\Lambda_{q,N}(\phi_{1},h,f)|\gg\delta^{O(1)}.$ We now re-run our argument on $h$ instead of $f$, deducing the existence of a 1-bounded local function $\phi_{2}$ of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$ and modulus $qq_{2}$ for some $q_{2}\ll\delta^{-O(1)}$ such that $|\Lambda_{q,N}(\phi_{1},\phi_{2},f)|\gg\delta^{O(1)}.$ Expanding the counting operator and taking a maximum over $y\in[M]$ gives $\displaystyle\delta^{O(1)}NM$ $\displaystyle\ll\left|\sum_{y\in[M]}\sum_{x}f(x)\phi_{1}(x-qy^{2})\phi_{2}(x-qy^{2}+y)\right|$ $\displaystyle\leqslant M\left|\sum_{x}f(x)\tilde{\phi}_{1}(x)\tilde{\phi}_{2}(x)\right|,$ where both $\tilde{\phi}_{i}$ are 1-bounded local functions of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$ and moduli $qq_{i}$ for some $q_{i}\ll\delta^{-O(1)}$. ∎ ## 4\. A weak regularity lemma Much of the material is this section is standard, and closely follows the expositions in Green [4] and Green–Tao [6]. To simplify the exposition of later arguments, while the factors in [4] and [6] are $\sigma$-algebras, our factors will be the set of atoms of certain $\sigma$-algebras (which can obviously be recovered by taking the $\sigma$-algebra generated by the set of atoms). ###### Definition 4.1 (Factor). We define a _factor_ $\mathcal{B}$ of $[N]$ to be a partition of $[N]$, so that $[N]=\sqcup_{B\in\mathcal{B}}B$. We say that a factor $\mathcal{B}^{\prime}$ _refines_ $\mathcal{B}$ if every element of $\mathcal{B}$ is a union of elements of $\mathcal{B}^{\prime}$. The _join_ $\mathcal{B}_{1}\vee\dots\vee\mathcal{B}_{d}$ of factors $\mathcal{B}_{1},\dots,\mathcal{B}_{d}$ is the factor formed by taking the $d$-fold intersections of the elements of $\mathcal{B}_{1}$, …, $\mathcal{B}_{d}$, that is, $\mathcal{B}_{1}\vee\dots\vee\mathcal{B}_{d}:=\\{B_{1}\cap\dots\cap B_{d}:B_{i}\in\mathcal{B}_{i}\text{ for }i=1,\dots,d\\}.$ ###### Definition 4.2 (Measurability, projection). Given a factor $\mathcal{B}$, we say that a function $f:[N]\to\mathbb{C}$ is _$\mathcal{B}$ -measurable_ if it is constant on the elements of $\mathcal{B}$. Define the _projection_ of any function $f:[N]\to\mathbb{C}$ onto $\mathcal{B}$ by $\Pi_{\mathcal{B}}f(x)=\mathbb{E}_{y\in B_{x}}f(y),$ (4.1) where $B_{x}$ is the element of $\mathcal{B}$ that contains $x$. Notice that $\Pi_{\mathcal{B}}f$ is $\mathcal{B}$-measurable, and is just the conditional expectation of $f$ with respect to the $\sigma$-algebra generated by the elements of $\mathcal{B}$. We record some well-known properties of the projection operator $\Pi_{\mathcal{B}}$ (that is, properties of conditional expectation) in the next lemma. ###### Lemma 4.3 (Properties of the projection operator). 1. (i) The operator $\Pi_{\mathcal{B}}$ linearly projects onto the space of $\mathcal{B}$-measurable functions. 2. (ii) $\Pi_{\mathcal{B}}$ is self-adjoint with respect to the inner product $\left\langle f,g\right\rangle:=\sum_{x}f(x)\overline{g(x)}\qquad(f,g:[N]\to\mathbb{C}),$ so that $\left\langle f,\Pi_{\mathcal{B}}g\right\rangle=\left\langle\Pi_{\mathcal{B}}f,g\right\rangle$. 3. (iii) If $\mathcal{B}^{\prime}$ is a refinement of $\mathcal{B}$ then $\Pi_{\mathcal{B}^{\prime}}\Pi_{\mathcal{B}}f=\Pi_{\mathcal{B}}f.$ 4. (iv) If $\mathcal{B}^{\prime}$ refines $\mathcal{B}$ then $\Pi_{\mathcal{B}}f$ is orthogonal to $\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f$. ###### Proof. Inspecting the formula (4.1) reveals that $\Pi_{\mathcal{B}}$ is linear, that $\Pi_{\mathcal{B}}f$ is constant on elements of $\mathcal{B}$, and that if $f$ itself is constant on elements of $\mathcal{B}$, then $\Pi_{\mathcal{B}}f=f$. This establishes (i). Interchanging the order of summation gives $\begin{split}\left\langle f,\Pi_{\mathcal{B}}g\right\rangle=\sum_{B\in\mathcal{B}}|B|^{-1}\sum_{x,y\in B}f(x)\overline{g(y)}=\left\langle\Pi_{\mathcal{B}}f,g\right\rangle.\end{split}$ This proves that $\Pi_{\mathcal{B}}$ is self-adjoint. The first refinement property follows from the fact that $\Pi_{\mathcal{B}}f$ is $\mathcal{B}^{\prime}$-measurable. We utilise self-adjointness of $\Pi_{\mathcal{B}}$ and the first refinement property to conclude that $\begin{split}\left\langle\Pi_{\mathcal{B}}f,\Pi_{\mathcal{B}}f-\Pi_{\mathcal{B}^{\prime}}f\right\rangle&=\left\langle\Pi_{\mathcal{B}}f,\Pi_{\mathcal{B}}f-f\right\rangle=\left\langle f,\Pi_{\mathcal{B}}f-\Pi_{\mathcal{B}}f\right\rangle=0.\end{split}$ ∎ Now we describe the particular type of factors that will be relevant to us. ###### Definition 4.4 (Local factor). A _simple real factor_ of resolution $M$ is a factor of $[N]$ obtained by partitioning $\mathbb{R}$ into intervals all of length $M$. A _simple congruence factor_ of modulus $q$ is the factor of $[N]$ obtained by partitioning into congruence classes mod $q$. We say that $\mathcal{B}$ is a _simple local factor_ of resolution $M$ and modulus $q$ if it is the join of a simple real factor of resolution $M$ and a simple congruence factor of modulus $q$. Notice that $\mathcal{B}$ is a simple local factor if and only if it consists of the level sets of a local function (Definition 1.3) of resolution $M$ and modulus $q$. A _local factor_ of dimension $d$, resolution $M$ and modulus $q$ is the join of $d$ simple local factors $\mathcal{B}_{i}$, each of resolution $M_{i}$ and modulus $q_{i}$, where $M_{i}\geqslant M$ and $q=\mathrm{lcm}[q_{1},\dots,q_{d}]$. Local factors of large resolution and small modulus and dimension necessarily contain few sets. This fact will be useful later in the proof of Lemma 2.1. ###### Lemma 4.5 (Size of a local factor). If $\mathcal{B}$ is a local factor of dimension $d$, resolution $M$, and modulus $q$, then $|\mathcal{B}|\leqslant qd\left(\frac{N}{M}+2\right).$ ###### Proof. By the definition of a local factor, it suffices to bound the size of the join of $d$ simple real factors, and then bound the size of the join of $d$ simple congruence factors. The product of these two numbers gives us our final bound. Joining $d$ congruence simple factors with moduli $q_{1},\dots,q_{d}$ results in another congruence simple factor of modulus $q=\mathrm{lcm}[q_{1},\dots,q_{d}]$. The number of parts in such a partition is $q$. The join of $d$ simple real factors partitions $[N]$ into intervals. The upper endpoint of each of these intervals is either equal to $N$ or is equal to an endpoint of an interval in one of the original simple real factors. For a simple real factor of resolution $M$, at most $1+N/M$ upper endpoints lie in $[1,N)$. Hence the number of intervals in the join of $d$ simple real factors of resolutions $M_{1}$, …, $M_{d}$ is at most $2d+N(M_{1}^{-1}+\dots+M_{d}^{-1})$.∎ We now prove a weak regularity lemma for the cut norm via an energy increment argument. ###### Lemma 4.6 (Weak regularity). Let $q\leqslant N$ be positive integers and $\delta>0$. Either $N\ll(q/\delta)^{O(1)}$, or for any function $f:[N]\to[0,1]$ there exists a local factor $\mathcal{B}$ of dimension $d\ll\delta^{-O(1)}$, resolution $\gg(\delta/q)^{O(1)}N^{1/2}$, and modulus $qq^{\prime}$ for some $q^{\prime}\leqslant O\left(1/\delta\right)^{O(d)}$ such that $\left\|f-\Pi_{\mathcal{B}}f\right\|_{q,N}\leqslant\delta.$ (4.2) ###### Proof. We run an energy increment argument, initialising at stage $0$ with the trivial factor $\mathcal{B}_{0}:=\left\\{[N]\right\\}$. Suppose that at stage $d$ of this iteration we have a local factor $\mathcal{B}$ of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$, dimension at most $2d$, and modulus $qq^{\prime}$ for some $q^{\prime}\leqslant O(1/\delta)^{O(d)}$. In addition, suppose that we have the energy lower bound $\left\|\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}^{2}\gg d\delta^{O(1)}N.$ (4.3) With these assumptions in place, we query if the following holds $\left\|f-\Pi_{\mathcal{B}}f\right\|_{q,N}\leqslant\delta.$ (4.4) If so, then the process terminates. If not, we show how our iteration may proceed to stage $d+1$. Applying the cut norm inverse theorem (Lemma 3.3), we conclude that there exist 1-bounded local functions $\phi_{i}$ of resolution $\gg(\delta/q)^{O(1)}N^{1/2}$ and modulus $qq_{i}$ for some $q_{i}\leqslant\delta^{-O(1)}$ such that $\left|\left\langle f-\Pi_{\mathcal{B}}f,\phi_{1}\phi_{2}\right\rangle\right|=\left|\sum_{x\in[N]}(f-\Pi_{\mathcal{B}}f)(x)\phi_{1}(x)\phi_{2}(x)\right|\gg\delta^{O(1)}N.$ Let $\mathcal{B}^{\prime}$ denote the join of $\mathcal{B}$ and the simple local factors generated by $\phi_{1}$ and $\phi_{2}$, so that $\mathcal{B}^{\prime}$ is a local factor of dimension at most $2(d+1)$, resolution $\gg(\delta/q)^{O(1)}N^{1/2}$ and modulus $qq^{\prime\prime}$ for some $q^{\prime\prime}\leqslant q^{\prime}q_{1}q_{2}\leqslant O(1/\delta)^{O(d+1)}$. Since $\phi_{1}\phi_{2}$ is $\mathcal{B}^{\prime}$-measurable, we can use the properties listed in Lemma 4.3 together with the Cauchy–Schwarz inequality to deduce that $\begin{split}\left|\left\langle f-\Pi_{\mathcal{B}}f,\phi_{1}\phi_{2}\right\rangle\right|&=\left|\left\langle f-\Pi_{\mathcal{B}}f,\Pi_{\mathcal{B}^{\prime}}(\phi_{1}\phi_{2})\right\rangle\right|=\left|\left\langle\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f,\phi_{1}\phi_{2}\right\rangle\right|\\\ &\leqslant N^{1/2}\left\|\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}.\end{split}$ It follows that $\left\|\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}\gg\delta^{O(1)}N^{1/2}.$ Lemma 4.3 (iv) tells us that $\Pi_{\mathcal{B}}f$ is orthogonal to $\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f$, hence by Pythagoras’s theorem $\left\|\Pi_{\mathcal{B}^{\prime}}f\right\|_{\ell^{2}}^{2}=\left\|\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}^{2}+\left\|\Pi_{\mathcal{B}^{\prime}}f-\Pi_{\mathcal{B}}f\right\|_{\ell^{2}}^{2}.$ The energy bound (4.3) follows for $\mathcal{B}^{\prime}$, allowing us to proceed to the next stage of our iteration. Since the function $f$ is $1$-bounded, the projection $\Pi_{\mathcal{B}}f$ is also 1-bounded, hence the energy (4.3) is always bounded above by $N$. It follows that this energy increment must terminate at stage $d$ for some $d\ll\delta^{-O(1)}$, yielding the lemma.∎ ## 5\. The density increment lemma In this section we prove Lemma 2.1, modelling our argument on that given by Green and Tao [6, Corollary 5.8]. We first record, for the sake of convenience, the following immediate consequence of the triangle inequality. ###### Lemma 5.1 ($\ell^{1}$-control). Suppose that $N\geqslant q$. Then for any $f_{0},f_{1},f_{2}:[N]\to\mathbb{C}$ we have $|\Lambda_{q,N}(f_{0},f_{1},f_{2})|\leqslant N^{-1}\left\|f_{i}\right\|_{\ell^{1}}\prod_{j\neq i}\left\|f_{j}\right\|_{\infty}.$ ###### Proof. We prove the result for $i=1$, the other cases being similar. A reparametrisation gives $\displaystyle\left|\Lambda_{q,N}(f_{0},f_{1},f_{2})\right|$ $\displaystyle=\left|\mathbb{E}_{x\in[N]}f_{1}(x)\mathbb{E}_{y\in[M]}f_{0}(x-y)f_{2}(x+qy^{2}-y)\right|$ $\displaystyle\leqslant\mathbb{E}_{x\in[N]}|f_{1}(x)|\mathbb{E}_{y\in[M]}|f_{0}(x-y)||f_{2}(x+qy^{2}-y)|.$ ∎ We are now in a position to prove Lemma 2.1, and thereby complete our proof of Theorem 1.1. ###### Proof of Lemma 2.1. Let $A$ satisfy the assumptions of Lemma 2.1. Increasing $\delta$ only strengthens our conclusion, so we may assume that $|A|=\delta N$. Since $\Lambda_{q,N}(1_{A})=0$, we have that $\left|\Lambda_{q,N}(1_{A})-\Lambda_{q,N}(\delta 1_{[N]})\right|=\delta^{3}\Lambda_{q,N}(1_{[N]})\gg\delta^{3}$. Applying the weak regularity lemma (Lemma 4.6), there exists a local factor $\mathcal{B}$ of dimension $d\ll\delta^{-O(1)}$, resolution $\gg(\delta/q)^{O(1)}N^{1/2}$, and modulus $qq^{\prime}$ for some $q^{\prime}\leqslant O(1/\delta)^{O(d)}$ such that $\left\|1_{A}-\Pi_{\mathcal{B}}1_{A}\right\|_{q,N}\leqslant\tfrac{1}{6}\delta^{3}\Lambda_{q,N}({1_{[N]}}).$ Setting $f:=\Pi_{\mathcal{B}}1_{A}$, a telescoping identity thus yields $\left|\Lambda_{q,N}(f)-\Lambda_{q,N}(\delta 1_{[N]})\right|\geqslant\tfrac{1}{2}\delta^{3}\Lambda_{q,N}({1_{[N]}})\gg\delta^{3}.$ Define the $\mathcal{B}$-measurable set $S:=\left\\{x\in[N]:f(x)\geqslant(1+c)\delta\right\\},$ where $c>0$ is a sufficiently small absolute constant that will be chosen to make the following argument valid. By Lemma 5.1 and a telescoping identity, we have $\left|\Lambda_{q,N}(f)-\Lambda_{q,N}(f1_{S^{c}})\right|\leqslant 3|S|/N$, so that $\tfrac{|S|}{N}+\left|\Lambda_{q,N}(f1_{S^{c}})-\Lambda_{q,N}(\delta 1_{[N]})\right|\gg\delta^{3}.$ Yet another telescoping identity, in conjunction with Lemma 5.1, gives $\displaystyle\left|\Lambda_{q,N}(f1_{S^{c}})-\Lambda_{q,N}(\delta 1_{[N]})\right|$ $\displaystyle\ll\tfrac{\delta^{2}}{N}\left\|f1_{S^{c}}-\delta 1_{[N]}\right\|_{\ell^{1}}\leqslant\tfrac{\delta^{2}}{N}\left\|f-\delta 1_{[N]}\right\|_{\ell^{1}}+\tfrac{|S|}{N},$ so that $|S|+\delta^{2}\left\|f-\delta 1_{[N]}\right\|_{\ell^{1}}\gg\delta^{3}N.$ Since $f-\delta 1_{[N]}$ has mean zero, its $\ell^{1}$-norm is equal to twice the $\ell^{1}$-norm of its positive part. The function $\left(f-\delta 1_{[N]}\right)_{+}$ can only exceed $c\delta$ on $S$, so taking $c$ small enough gives $|S|\gg\delta^{3}N$. Letting $B$ denote the largest element of $\mathcal{B}$ for which $B\subset S$, the bound in Lemma 4.5 yields $|B|\gg q^{-O(1)}\delta^{O(d)}2^{-O(d)}N^{1/2}.$ By construction (see Definition 4.4), the set $B$ is an arithmetic progression of common difference $qq^{\prime}$ with $q^{\prime}\leqslant O(1/\delta)^{O(d)}$. Moreover, the density of $A$ on $B$ is equal to the value of $f(x)$ for any $x\in B$, and this is at least $(1+c)\delta$ by the definition of $S$. ∎ ## 6\. Global control by major arc Fourier coefficients The purpose of this section is to prove Theorem 1.2 and Corollary 1.4. We begin with an alternative version of Lemma 3.2, replacing the rigid local function found therein with something more continuous. ###### Definition 6.1 ($C$-Lipschitz). We say that $\phi:\mathbb{Z}\to\mathbb{C}$ is _$C$ -Lipschitz along $q\cdot\mathbb{Z}$_ if for any $x,y\in\mathbb{Z}$ we have $|\phi(x+qy)-\phi(x)|\leqslant C|y|.$ Recalling our definition for the Fejér kernel (1.7), we observe that a function of the form $x\mapsto\sum_{h}\mu_{H}(h)f(x+qh)$ (6.1) is Lipschitz along $q\cdot\mathbb{Z}$. ###### Lemma 6.2. Let $q,H$ be positive integers and $f:\mathbb{Z}\to\mathbb{C}$ be 1-bounded. If $\phi$ is defined as in (6.1), then $\phi$ is $O(H^{-1})$-Lipschitz along $q\cdot\mathbb{Z}$. ###### Proof. Recalling (1.7), the triangle inequality for $|\cdot|$ and $\max\\{\cdot,0\\}$ show that $|\mu_{H}(h+y)-\mu_{H}(h)|\leqslant|y|/\left\lfloor H\right\rfloor^{2}$ for all $h,y\in\mathbb{Z}$. Hence a change of variables gives $|\phi(x+qy)-\phi(x)|\leqslant\sum_{h}|\mu_{H}(h-y)-\mu_{H}(h)|\ll\frac{|y|}{H^{2}}\sum_{h\in(-H,H)\cup(y-H,y+H)}1.$ ∎ Now we prove another partial cut norm inverse theorem, this time getting correlation with functions that are Lipschitz along progressions with small common difference. ###### Lemma 6.3 (Partial cut norm inverse theorem II). Let $N$ be a positive integer, $\delta>0$, and $f,g,h:\mathbb{Z}\to\mathbb{C}$ be $1$-bounded functions with support in $[N]$. Suppose that $\left|\mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[N^{1/2}]}f(x)g(x+y)h(x+y^{2})\right|\geqslant\delta.$ Then either $N\ll\delta^{-O(1)}$, or there exists $q\ll\delta^{-O(1)}$ and a 1-bounded function $\phi$ that is $O(\delta^{-O(1)}N^{-1/2})$-Lipschitz along $q\cdot\mathbb{Z}$ such that $\sum_{x\in[N]}g(x)\phi(x)\gg\delta^{O(1)}N.$ ###### Proof. Applying [9, Theorem 7.1], we obtain positive integers $q\ll\delta^{-O(1)}$ and $N^{1/2}\geqslant M\gg\delta^{O(1)}N^{1/2}$ such that $\sum_{x}\left|\sum_{y\in[M]}g(x+qy)\right|\gg\delta^{O(1)}NM.$ By the Cauchy–Schwarz inequality and a change of variables, we have $\sum_{x}g(x)\sum_{y_{1},y_{2}\in[M]}\overline{g(x+q(y_{1}-y_{2}))}\gg\delta^{O(1)}NM^{2}.$ Setting $\phi(x):=\mathbb{E}_{y_{1},y_{2}\in[M]}\overline{g(x+q(y_{1}-y_{2}))},$ Lemma 6.2 shows this function has the required properties. ∎ Before proving Theorem 1.2, we record two standard facts. ###### Lemma 6.4. There are at most $O(N^{4})$ solutions $x\in[N]^{6}$ to the equation $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=x_{4}^{2}+x_{5}^{2}+x_{6}^{2}.$ ###### Proof. There are a number of ways to prove this. Perhaps the most robust is via the circle method, see [3]. The result can be read out of [1, Proposition 1.10]. ∎ ###### Lemma 6.5 (Weyl’s inequality). Let $P\subset\mathbb{Z}$ be an arithmetic progression with common difference $q$ and let $0<\delta\leqslant 1$. Suppose that $\left|\sum_{x\in P}e(\alpha x^{2})\right|\geqslant\delta|P|.$ Then either $|P|\ll\delta^{-O(1)}$ or there exists a positive integer $q^{\prime}\ll\delta^{-O(1)}$ such that $\|q^{\prime}q^{2}\alpha\|\ll\delta^{-O(1)}|P|^{-2}.$ ###### Proof. Let $P=x_{0}+q\cdot[N]$, so that our exponential sum becomes $\sum_{x\in P}e(\alpha x^{2})=\sum_{y\in[N]}e(\alpha q^{2}y^{2}+2\alpha qx_{0}y+\alpha x_{0}^{2}).$ Applying [5, Lemma A.11], either $N\ll\delta^{-O(1)}$ or the conclusion of our lemma follows. ∎ ###### Proof of Theorem 1.2. Write $\Lambda_{N}$ for the counting operator $\Lambda_{1,N}$ (that is, the average (1.6) with $q=1$). Let $f,g,h:[N]\to\mathbb{C}$ be 1-bounded functions satisfying $|\Lambda_{N}(f,g,h)|\geqslant\delta.$ Define the seminorm $\left\|g\right\|:=\sup\left\\{|\Lambda_{N}(g_{1},g,g_{2})|:|g_{i}|\leqslant 1\text{ and }\mathrm{supp}(g_{i})\subset[N]\right\\}.$ and the dual function $F(x):=\mathbb{E}_{y\in[N^{1/2}]}f(x-y)h(x+y^{2}-y).$ We follow the argument in the proof of Lemma 3.3 to deduce that $\left\|F\right\|\geqslant\delta^{2}.$ Hence, by Lemma 6.3, there exists $q\ll\delta^{-O(1)}$ and a 1-bounded function $\phi$ that is $O(\delta^{-O(1)}N^{-1/2})$-Lipschitz along $q\cdot\mathbb{Z}$ and satisfies $\sum_{x\in[N]}F(x)\phi(x)\gg\delta^{O(1)}N.$ Expanding the definition of the dual function, we have $\sum_{x\in[N]}\sum_{y\in[N^{1/2}]}f(x)\phi(x+y)h(x+y^{2})\gg\delta^{O(1)}N^{3/2}.$ Let us partition $\mathbb{Z}$ into arithmetic progressions $P$ each of common difference $q$ and length $M$, where $M$ will be chosen shortly. For each such arithmetic progression $P$, fix an element $y_{P}\in P$. Using the Lipschitz property of $\phi$, for any $x\in\mathbb{Z}$ and $y\in P$ we have $|\phi(x+y_{P})-\phi(x+y)|\ll\delta^{-O(1)}MN^{-1/2}.$ Hence, $\left|\sum_{P}\sum_{x\in[N]}\sum_{y\in P\cap[N^{1/2}]}f(x)[\phi(x+y)-\phi(x+y_{P})]h(x+y^{2})\right|\ll\delta^{-O(1)}MN.$ We can therefore take $M$ sufficiently small to satisfy both $M\gg\delta^{O(1)}N^{1/2}$ and $\left|\sum_{P}\sum_{x}\sum_{y\in P\cap[N^{1/2}]}f(x)\phi(x+y_{P})h(x+y^{2})\right|\gg\delta^{O(1)}N^{3/2}.$ Set $f_{P}(x):=f(x)\phi(x+y_{P})$. The number of progressions $P$ that intersect $[N^{1/2}]$ is at most $O(N^{1/2}M^{-1}+q)=O(\delta^{-O(1)})$. Therefore, the pigeon-hole principle gives a progression $P$ for which $\left|\sum_{x}\sum_{y\in P\cap[N^{1/2}]}f_{P}(x)h(x+y^{2})\right|\gg\delta^{O(1)}N^{3/2}.$ (6.2) In particular, $|P\cap[N^{1/2}]|\gg\delta^{O(1)}N^{1/2}$. Writing $S_{P}(\alpha)$ for $\sum_{y\in P\cap[N^{1/2}]}e\left(\alpha y^{2}\right)$, the orthogonality relations allow us to reformulate (6.2) as $\displaystyle\left|\int_{\mathbb{T}}\hat{f}_{P}(\alpha)\hat{h}(-\alpha)S_{P}(\alpha)d\alpha\right|\gg\delta^{O(1)}N^{3/2}.$ Let $\eta>0$ be a parameter to be determined shortly, and define the major arcs $\mathfrak{M}:=\left\\{\alpha\in\mathbb{T}:|S_{P}(\alpha)|\geqslant\eta N^{1/2}\right\\}.$ Parseval’s identity then gives $\left|\int_{\mathbb{T}\setminus\mathfrak{M}}\hat{f}_{P}(\alpha)\hat{h}(-\alpha)S_{P}(\alpha)d\alpha\right|\leqslant\eta N^{1/2}\big{\|}\hat{f}_{P}\big{\|}_{2}\big{\|}\hat{h}\big{\|}_{2}\leqslant\eta N^{3/2}.$ Hence we may take $\eta\gg\delta^{O(1)}$ and ensure that $\displaystyle\left|\int_{\mathfrak{M}}\hat{f}_{P}(\alpha)\hat{h}(-\alpha)S_{P}(\alpha)d\alpha\right|\gg\delta^{O(1)}N^{3/2}.$ By Lemma 6.4 and orthogonality, we have $\left\|S_{P}\right\|_{6}\ll N^{1/3}$. Thus, by Hölder’s inequality, we get that $\left|\int_{\mathfrak{M}}\hat{f}_{P}(\alpha)\hat{h}(-\alpha)S_{P}(\alpha)d\alpha\right|\leqslant\big{\|}\hat{f}_{P}\big{\|}_{2}\big{\|}\hat{h}\big{\|}_{2}^{2/3}\big{\|}S_{P}\big{\|}_{6}\sup_{\alpha\in\mathfrak{M}}\bigl{|}\hat{h}(-\alpha)\bigr{|}^{1/3}.$ We therefore deduce that there exists $\alpha\in\mathfrak{M}$ such that $\bigl{|}\hat{h}(-\alpha)\bigr{|}\gg\delta^{O(1)}N.$ Finally, an application of Weyl’s inequality (Lemma 6.5) shows that if $-\alpha\in\mathfrak{M}$ then $\alpha$ has the required Diophantine approximation property. ∎ ###### Proof of Corollary 1.4. Let $\alpha\in\mathbb{R}$ be the frequency and $q$ the positive integer provided by Theorem 1.2. For any integer $a$ and positive integer $M$, if $x,y\in a+q\cdot[M]$, then $\left|e(\alpha x)-e(\alpha y)\right|\leqslant 2\pi\left\|\alpha(x-y)\right\|\ll\delta^{-O(1)}MN^{-1}.$ Partitioning $\mathbb{Z}$ into arithmetic progressions of common difference $q$ and length $M$ then gives $\delta^{O(1)}N\ll\sum_{P}\Bigl{|}\sum_{x\in P}h(x)\Bigr{|}+\delta^{-O(1)}M.$ We thus take $M\gg\delta^{O(1)}N$ sufficiently small to ensure that $\delta^{O(1)}N\ll\sum_{P}\Bigl{|}\sum_{x\in P}h(x)\Bigr{|}.$ Write $\theta_{P}$ for the conjugate phase of the inner sum. Then the map $x\mapsto\sum_{P}\theta_{P}1_{P}(x)$ is a local function of resolution $\gg\delta^{O(1)}N$ and modulus $\ll\delta^{-O(1)}$, yielding the corollary. ∎ ## 7\. Longer progressions As mentioned in §1.3, the main obstacle to generalising our polylogarithmic bound to longer configurations such as (1.2) is in obtaining an appropriate generalisation of Lemma 3.3; in particular, showing that if the relevant counting operator is large, then _all_ functions must correlate with a product of a bounded number of local functions. Let us demonstrate where the argument breaks down for $m>2$. Given polynomials as in (1.2) and 1-bounded functions $f_{0},f_{1},\dots,f_{m}:[N]\to\mathbb{C}$, define the counting operator $\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},f_{1},\dots,f_{m}):=\\\ \mathbb{E}_{x\in[N]}\mathbb{E}_{y\in[N^{1/\deg P_{m}}]}f_{0}(x)f_{1}(x+P_{1}(y))\dotsm f_{m}(x+P_{m}(y)).$ Using the main technical result of [8], [8, Theorem 3.3], one can show that if $\left|\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},f_{1},\dots,f_{m})\right|\geqslant\delta,$ then both $f_{0}$ and $f_{1}$ correlate with local functions $\phi_{0}$ and $\phi_{1}$. Combining this with a dual function argument, as in our proofs of Theorem 1.2 and Lemma 3.3, one may conclude that $\left|\Lambda_{P_{1},\dots,P_{m}}^{N}(\phi_{0},\phi_{1},f_{2},\dots,f_{m})\right|\gg\delta^{O(1)},$ If $m=2$, one can then pigeon-hole in the smaller $y$ variable appearing in the counting operator (as we do in the proof of Lemma 3.3) to conclude that $f_{2}$ correlates with a product of two local functions. It is this simple pigeon-holing argument that fails when $m>2$. ### 7.1. An alternative strategy for longer progressions A more productive strategy is to follow our proof of Theorem 1.2 instead of Theorem 1.1. In proving Theorem 1.2 we replace the counting operator $\Lambda_{y,y^{2}}^{N}(f_{0},f_{1},f_{2})$ with $\Lambda_{y,y^{2}}^{N}(f_{0},\phi,f_{2})$, where $\phi$ is a local function that is constant on progressions of length $\approx N^{1/2}$ with common difference of size $\approx O(1)$. Provided that we pass to appropriate subprogressions in all of the variables appearing in our counting operator, we can exploit the properties of this local function and ‘remove’ it from our count. In effect (after passing to subprogressions of bounded common difference), we replace the count $\Lambda_{y,y^{2}}^{N}(f_{0},f_{1},f_{2})$ with one of the form $\Lambda_{Q}^{N^{\prime}}(f_{0},f_{2})$, where $Q$ is a quadratic polynomial and $N^{\prime}$ is slightly smaller than $N$. Generalising this approach, one can use [8, Theorem 3.3] to replace the counting operator $\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},f_{1},\dots,f_{m})$ with $\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},\phi,f_{2},\dots,f_{m})$, where $\phi$ is a local function. Provided that this local function has resolution $\gg N^{\deg P_{1}/\deg P_{m}}$ and common difference $q\ll 1$, we have $\phi(x+P_{1}(y))\approx\phi(x)$ for any $x\in\mathbb{Z}$ and any $y$ constrained to a subprogression of common difference $q$ and length $\approx N^{\deg P_{1}/\deg P_{m}}$. Passing to subprogressions in $x$ and $y$, one should then be able to replace the operator $\Lambda_{P_{1},\dots,P_{m}}^{N}(f_{0},\phi,f_{2},\dots,f_{m})$ by one of the form $\Lambda_{Q_{2},\dots,Q_{m}}^{N^{\prime}}(f_{0},f_{2},\dots,f_{m}).$ Applying induction on $m$ may then allow one to show that every function in the original counting operator correlates with a local function. The main impediment to carrying out this strategy is that the polynomials $Q_{2}$, …, $Q_{m}$, which arise on passing to a subprogression, may not satisfy the hypotheses required to reapply [8, Theorem 3.3]. It is likely that the polynomials are sufficiently well-behaved for the arguments of [8] to remain valid, but we leave this verification to the energetic reader. ## References * Bou [89] J. Bourgain. On $\Lambda(p)$-subsets of squares. Israel J. Math., 67(3):291–311, 1989. * Bou [99] J. Bourgain. On triples in arithmetic progression. Geom. Funct. Anal., 9(5):968–984, 1999. * Dav [05] H. Davenport. Analytic methods for Diophantine equations and Diophantine inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 2005. With a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman, Edited and prepared for publication by T. D. Browning. * Gre [07] B. Green. Montréal notes on quadratic Fourier analysis. In Additive combinatorics, volume 43 of CRM Proc. Lecture Notes, pages 69–102. Amer. Math. Soc., Providence, RI, 2007. * GT [08] B. Green and T. Tao. Quadratic uniformity of the Möbius function. Ann. Inst. Fourier (Grenoble), 58(6):1863–1935, 2008. * GT [09] B. Green and T. Tao. New bounds for Szemerédi’s theorem. II. A new bound for $r_{4}(N)$. In Analytic number theory, pages 180–204. Cambridge Univ. Press, Cambridge, 2009. * HB [87] D. R. Heath-Brown. Integer sets containing no arithmetic progressions. J. London Math. Soc. (2), 35(3):385–394, 1987. * Pel [19] S. Peluse. Bounds for sets with no polynomial progressions. ArXiv e-prints, 2019. * PP [19] S. Peluse and S. Prendiville. Quantitative bounds in the non-linear Roth theorem. ArXiv e-prints, 2019. * Pre [20] S. Prendiville. The inverse theorem for the nonlinear Roth configuration: an exposition. ArXiv e-prints, 2020. * Rot [53] K. F. Roth. On certain sets of integers. J. London Math. Soc., 28:104–109, 1953. * Sár [78] A. Sárközy. On difference sets of sequences of integers. I. Acta Math. Acad. Sci. Hungar., 31(1–2):125–149, 1978. * Sze [90] E. Szemerédi. Integer sets containing no arithmetic progressions. Acta Math. Hungar., 56(1-2):155–158, 1990. * Tao [06] T. Tao. Obstructions to uniformity and arithmetic patterns in the primes. Pure Appl. Math. Q., 2(2, Special Issue: In honor of John H. Coates. Part 2):395–433, 2006.

2024-09-04T02:54:58.416642

2003.04127

arxiv-papers

Mode-locked ultrashort pulses from an 8 µm wavelength semiconductor laser Johannes Hillbrand1,2,∗, Nikola Opačak1, Marco Piccardo2,3, Harald Schneider4, Gottfried Strasser1, Federico Capasso2, Benedikt Schwarz1,2,† 1Institute of Solid State Electronics, TU Wien, Gußhausstraße 25, 1040 Vienna, Austria 2Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, USA 3CNST – Fondazione Istituto Italiano di Tecnologia, Via Pascoli 70/3, 20133 Milano, Italy 4Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, Germany Quantum cascade lasers (QCL) have revolutionized the generation of mid- infrared light. Yet, the ultrafast carrier transport in mid-infrared QCLs has so far constituted a seemingly insurmountable obstacle for the formation of ultrashort light pulses. Here, we demonstrate that careful quantum design of the gain medium and control over the intermode beat synchronization enable transform-limited picosecond pulses from QCL frequency combs. Both an interferometric radio-frequency technique and second-order autocorrelation shed light on the pulse dynamics and confirm that mode-locked operation is achieved from threshold to rollover current. Being electrically pumped and compact, mode-locked QCLs pave the way towards monolithically integrated non- linear photonics in the molecular fingerprint region beyond 6 µm wavelength. <EMAIL_ADDRESS><EMAIL_ADDRESS> The discovery of ultrashort light pulses has led to numerous breakthroughs in science and technology, including frequency combs1, high-speed optical telecommunication2 and refractive surgery in ophthalmology3. Nowadays, optical pulses are routinely generated in mode-locked lasers operating in the visible or near-infrared range4, 5. Currently, large efforts are aimed at bringing ultrafast laser science in the mid-infrared (MIR) region to a similarly high degree of maturity6. Due to the lack of suitable gain media, methods for the generation of pulses in the molecular fingerprint region beyond 5 µm wavelength have so far relied on non-linear downconversion of near-infrared pulses 7. Established techniques such as optical parametric oscillators8 or difference frequency generation9, 10 either require sophisticated optical setups with tabletop dimensions or are restricted to mW-level of output power. Quantum cascade lasers11 (QCL) have matured to become the dominant MIR laser source. While being microchip-sized and electrically pumped, they are capable of producing Watt-level average power12, 13. Quantum engineering of the active region allows to tailor the emission wavelength throughout the entire mid- infrared region. Hence, harnessing high-performance QCL technology for the generation of MIR pulses represents a long-sought milestone in ultrafast laser science. Mode-locked QCLs could serve as monolithic pump lasers for microresonators and resonant supercontinuum generation14, paving the way towards broadband and high-brightness frequency combs. So far, the sub- picosecond carrier transport in QCL active regions has constituted a seemingly insurmountable obstacle for the formation of short light pulses15, 16, 17. To date, the only successful attempt of mode-locking in monolithic MIR QCLs was observed using a specially designed active region with strongly enhanced upper-state lifetime of the lasing transition17. However, the necessary design modifications limited mode-locked operation to cryogenic temperatures and peak power below 10 mW, thus impeding their practical use. Figure 1: Bi-functional quantum cascade lasers for mode-locking. a: Scanning electron microscope image of three adjacent laser ridges. Each laser consists of a roughly 3 mm long gain section and a shorter (320-480 µm) modulation section. b: Simulated gain and loss spectrum in a standard active region design18 depending on the applied bias. Upon decreasing the bias, the structure becomes almost transparent at the lasing wavelength $\lambda_{L}$, limiting the maximally achievable modulation depth. c: Simulated gain and loss spectrum in a bi-functional active region design12, allowing to tune the gain at 10 V continuously to absorption (shown as negative gain) at 0 V. d: Measured light-current-voltage (L-I-V) characteristics of an epi-up mounted bi-functional QCL at 15$\,{}^{\circ}$C. e: Illustration of a system of coupled oscillators. This system shows an in-phase and anti-phase synchronization state, which oscillate at different frequencies depending on the coupling. Without external stimulation, the anti-phase state is more favorable due to the damped coupling. However, both synchronization frequencies can be probed by exerting mechanical force on the platform coupling the oscillators. In the QCL, the oscillators are represented by the intermode beatings, which tend to synchronize in anti-phase due to gain damping19, 20. Both synchronization frequencies are probed by applying modulation to the laser. f: Average optical power depending on the modulation frequency and power. Two synchronization states at $f_{\mathrm{rep}}^{0}$ and 60 MHz above are observed. g: Signal of a 2-QWIP sensitive to peak power as function of modulation frequency and power. The strongly increased signal of the lobe at $f_{\mathrm{rep}}^{0}$+60 MHz indicates in-phase synchronization. In this work, we demonstrate the generation of transform limited picosecond pulses in high-performance 8 µm wavelength QCLs at room temperature both experimentally and theoretically. Mode-locking is achieved by electrically modulating the intracavity loss using a short modulation section designed for efficient radio-frequency (RF) injection (Fig. 1a). In order to achieve the large modulation depth required for stable mode-locking, close attention has to be paid to the band structure of the QCL active region. This effect is illustrated in Figs. 1a,b. As the bias applied to a standard QCL structure is decreased, it does not switch to absorption at the lasing wavelength, but becomes nearly transparent for the intracavity light due to a bias-dependent shift of the electronic levels, known as Stark effect (Fig. 1b). Hence, the modulation depth is severely limited in standard QCL designs. For this reason, we employ a bi-functional active region whose lasing wavelength and absorption wavelength at zero bias were matched to each other 21, 12 (Fig. 1c). This strategy allows to overcome the aforementioned limitations of the modulation depth caused by the Stark shift. Most importantly, the bi-functional design shows excellent overall performance, which is competitive with other state-of- the-art designs. A 3.5 mm long device mounted epitaxial-side up emits more than 130 mW average power in continuous wave at room temperature (Fig. 2d). As a first step towards pulse generation, it is essential to determine the optimal modulation frequency. For this purpose, mode-locking can be seen as synchronization of coupled oscillators20 (Fig. 1e). Each pair of neighboring cavity modes creates a beating at their difference frequency, which is equal to the cavity roundtrip frequency. These beatings can be seen as oscillators coupled by the non-linearity of the gain medium. Thanks to this coupling, the cavity modes of a free-running QCL can be locked together without modulation, thus giving rise to a self-starting frequency comb 19. Yet, this kind of frequency comb does not emit isolated pulses, but rather a quasicontinuous wave accompanied by a strong linear frequency chirp22. This corresponds to anti-phase synchronization and will be called frequency modulated (FM) comb in the following. In contrast, in-phase synchronization of the intermode beatings leads to the formation of short pulses. It is well known from coupled oscillators that the in-phase and anti-phase states synchronize at different frequencies depending on the coupling. As a consequence, while the cavity roundtrip frequency of the FM QCL comb $f_{\mathrm{rep}}^{0}$ may seem like a reasonable choice, we expect the optimal modulation frequency for generating pulses to differ from $f_{\mathrm{rep}}^{0}$. In order to investigate these two synchronization states experimentally, we start by operating the laser well above its threshold current. Subsequently, the DC bias of the modulation section is decreased to 2.8 V, where the large absorption caused by the bifunctional design (Fig. 1c) brings the QCL just slightly below lasing threshold. In these conditions, modulation at the right frequencies can provide enough additional gain to reach threshold. Fig. 1f shows the laser power depending on modulation frequency and power. At 33 dBm modulation power, the QCL reaches threshold when modulating close to $f_{\mathrm{rep}}^{0}$. Strikingly, a second modulation frequency where lasing occurs is observed almost 60 MHz higher than $f_{\mathrm{rep}}^{0}$, as predicted by the picture of synchronized oscillators. Both the range around the two synchronization frequencies, where lasing is observed, as well as the optical power grow upon increasing the modulation power. Figure 2: Mode-locked pulses from an 8 µm wavelength QCL. a: SWIFTS characterization of the QCL operated close to lasing threshold. The laser is modulated at the in-phase synchronization frequency and at 37 dBm power level. b: Reconstructed time-domain signal of the QCL, showing a train of transform limited pulses with 6.5 ps FWHM. c: Simulation of the QCL using the coherent Master equation described in supp. section 1. d: Interferometric autocorrelation (IAC) of the QCL pulses close to threshold. Red dots: envelope of the IAC reconstructed using SWIFTS. e: IAC at higher current. f: IAC at the rollover current, still displaying the 8:1 ratio. The second burst at a delay equal to the cavity roundtrip time is due to the interference of subsequently emitted pulses. Its peak value of 8 provides another proof for the coherence of the pulses because phase-decoherence would smear out the fringes of the IAC and thus decrease the peak value to smaller than 8. Inset: zoom on the interferometric fringes. Even more insight is provided by using a two-photon quantum well infrared photodetector (2-QWIP) to detect the emitted light. The signal of the 2-QWIP is proportional to the square of the intensity. This allows to identify which modulation frequency leads to in-phase and which to anti-phase synchronization (Fig. 1g). Again, two lobes appear around $f_{\mathrm{rep}}^{0}$ and 60 MHz above. Yet, the 2-QWIP signal is more than an order of magnitude larger in the lobe at higher $f_{\mathrm{mod}}$. At this frequency, the laser operates in the in-phase synchronization regime and emits intense pulses, which leads to a strongly increased 2-QWIP signal. Figure 3: Synchronization under strong modulation. a: Schematic of a 3-section QCL comprised of modulation, gain and high-speed detector sections. b: First three harmonics of the beatnote of the free-running 7 mm long QCL FM comb (red) compared to the actively mode-locked QCL (AM comb, blue). c: laser beatnote while free-running (bottom), at $f_{\mathrm{mod}}{=}f_{\mathrm{rep}}^{0}$ (middle) and at $f_{\mathrm{mod}}{=}f_{\mathrm{rep}}^{0}{+}33\,$MHz (top). While a broad pedestal is visible for $f_{\mathrm{mod}}{=}f_{\mathrm{rep}}^{0}$, the beatnote is perfectly locked for $f_{\mathrm{mod}}{=}f_{\mathrm{rep}}^{0}{+}33\,$MHz. d: RF spectrum around $f^{0}_{\mathrm{rep}}{=}$6.196 GHz as the modulation frequency is varied around $f^{0}_{\mathrm{rep}}$. The phase-noise of the RF spectrum disappears abruptly at $f_{\mathrm{mod}}{\approx}f_{\mathrm{rep}}^{0}{+}20\,$MHz, corresponding to in-phase synchronization. Here, the beatnote consists of a single narrow peak, indicating that the laser is phase-locked to the modulation. Furthermore, the sharp sidepeaks visible at $f_{\mathrm{mod}}{=}6.18\,$GHz are attributed to a periodic modulation of the QCL output, as previously observed in simulations16. In order to unequivocally prove mode-locking, we employ two independent methods to characterize the pulse dynamics at three points of operation from threshold up to the rollover current. Firstly, an interferometric RF technique called ’Shifted wave interference Fourier transform spectroscopy’ (SWIFTS)23, 24 is used to measure the phases of the QCL spectrum (details in Methods section). This information not only enables the reconstruction of the temporal waveform, but also allows to assess the phase-coherence of the pulses and whether they form a frequency comb. Secondly, we measure the interferometric autocorrelation (IAC) of the pulses using the 2-QWIP, which constitutes an additional well-established proof for mode-locking and the pulse width. Fig. 2a shows the SWIFTS characterization of the QCL operated close to threshold. In contrast to the free-running laser, the intensity spectrum consists of a single Gaussian-shaped lobe. The SWIFTS spectrum represents the part of the intensity spectrum which is beating exactly at the modulation frequency. Since the SWIFTS spectrum has the same shape as the intensity spectrum over its entire span, the QCL generates a frequency comb whose repetition frequency is given by the modulation frequency. The intermodal difference phases $\Delta\phi$, which correspond to the spectral group delay, are synchronized almost perfectly in-phase. Hence, all parts of the spectrum have the same group delay and form a pulse. Indeed, the reconstructed waveform in Fig. 2b shows the emission of 6.5 ps short pulses. The full-width-half-maximum (FWHM) of the reconstructed pulses is given by the transform limit of the spectrum in Fig. 2b, indicating that there is negligible chirp in the pulses. In these conditions, the peak power reaches almost 250 mW, which constitutes an enhancement of more than 12 compared to the average power of 20 mW. In order to model the cavity dynamics, we use a fully coherent master equation25 (supp. section 1). This single equation for the complex field replaces the entire Maxwell-Bloch system and reliably predicts the spectral shape, phase relationship and pulse width observed experimentally (Fig. 2c). Furthermore, it allows experimentally unavailable analyses, e.g. the influence of dispersion and nonlinearities (supp. Figs. 2,3,7). The IAC close to threshold (Fig. 2d) shows a prominent peak at zero path difference caused by constructive interference of the pulses after the Michelson interferometer. The ratio of this peak to the background at a delay larger than the pulse width is 8:1, which is generally regarded as the smoking gun of mode-locked pulses. Encouragingly, the measured IAC is in excellent agreement with the expected IAC, which was calculated using the pulses obtained by SWIFTS (red dots in Fig. 2d). This confirms successful mode- locking and the retrieved pulse width. The generation of pulses becomes increasingly challenging at higher gain current. Due to gain saturation, the wings of a pulse experience more gain than the peak. This effect leads to pulse broadening and can destabilize mode- locking. Fortunately, the large modulation depth provided by the bi-functional quantum design enables mode-locking over the entire lasing range from threshold to rollover. The IAC traces at 3.7 kA/cm2 and at the rollover current still show the required peak-to-background ratio of 8:1. The pulses at rollover are slightly broadened to roughly 12 ps, which is attributed partially to a slight chirp and partially to the gain saturation effect mentioned above (supp. Fig. 6). Yet, the average power is greatly increased to 62 mW, which results in over 430 mW peak power and 5 pJ pulse energy - more than an order of magnitude higher than recent reports of comparable mid- infrared semiconductor lasers emitting at shorter wavelengths 26, 27. Another fascinating aspect of bi-functional quantum design is the possibility to monolithically integrate ultrafast photodetectors. While this is particularly important in applications such as photonic integrated circuits, it also provides a tool to measure the beatnote with very large signal-to- noise ratio directly on the chip (Fig. 3a). This provides crucial information about the type of synchronization state and about its stability. Fig. 3b shows the first three harmonics of the beatnote in the free-running and the actively mode-locked regime. In the latter conditions, the beatnote amplitudes increase by 19 dB due to the much larger amplitude modulation. The zoom on the first harmonic beatnote (Fig. 3c) allows to assess the phase-coherence and stability of the frequency comb. The free-running QCL is operating in the anti-phase state showing a weak beatnote at $f_{\mathrm{rep}}^{0}$. Previous work28, 29, 30 has shown that a weak electrical modulation can be used to lock and stabilize the beatnote of the anti-phase state. However, the situation is very different when applying strong modulation at $f_{\mathrm{rep}}^{0}$. In this case, the modulation enforces an AM waveform, which is contrary to the natural anti-phase behaviour of the laser. As a result, the beatnote of this waveform is not phase-locked, as indicated by the pedestal around $f_{\mathrm{mod}}$. The situation changes completely, when the modulation frequency is tuned to the synchronization frequency of the in-phase state ($f_{\mathrm{rep}}^{0}{+}33\,$MHz). There, the strong modulation is in consensus with the natural behavior of the laser, leading to a phase-locked frequency comb with narrow beatnote. This can also be seen in Fig. 3d, which shows the laser beatnote while tuning the modulation frequency across the in- phase and anti-phase synchronization frequencies. While the frequency of the beatnote is controlled by the modulation over the entire span, a phase-locked comb is only generated around the in-phase synchronization frequency. In conclusion, our experiments provide unambiguous proof for the generation of mode-locked pulses in high-performance QCLs at room temperature - a goal which remained elusive since the invention of the QCL - and confirm stunning similarities to synchronization in coupled oscillators. These mode-locked QCLs constitute the first compact and electrically pumped source for ultrashort pulses beyond 5 µm wavelength, demonstrating that they are a highly promising technology for ultrafast laser science in the long-wave infrared region. The availability of such a source paves the way towards a semiconductor-based platform for non-linear photonics, potentially enabling broadband mid-infrared frequency combs and supercontinuum generation. ## Methods QCLs optimized for RF modulation: The QCLs are processed as buried heterostructures. The width of the QCL ridges is 12 µm and the facets are left as cleaved. The area of the top contact of the modulation section is minimized to decrease its parasitic capacitance, which increases the RF injection efficiency. Ground contacts for the modulation are provided by etching through the Fe-doped InP layer next to the laser ridges. The modulation signal is provided by a HP8341B synthesized sweeper, amplified up to roughly 5 W and injected via coplanar tips. The insertion loss at 12 GHz is 14 dB including cables and bias-tee. SWIFTS and IAC: The light emitted by the QCL is shone through a Bruker Vertex 70v FTIR spectrometer. In order to perform SWIFTS, the light is then detected by a home-built fast QWIP at the exit of the FTIR. The optical beating obtained from the QWIP is subsequently amplified and mixed down to roughly 10 MHz using a local oscillator. A Zurich Instruments HF2LI lock-in amplifier and the trigger of the FTIR are used to record the SWIFTS and intensity interferograms in rapid scan mode. The IAC is obtained by detecting the pulses at the exit of the FTIR using the 2-QWIP and recording its photocurrent depending on the path delay of the FTIR. ## References * 1 Udem, T., Holzwarth, R. & Hänsch, T. W. Optical frequency metrology. _Nature_ 416, 233–237 (2002). * 2 Hasegawa, A. & Tappert, F. 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Theory of Frequency-Modulated Combs in Lasers with Spatial Hole Burning, Dispersion, and Kerr Nonlinearity. _Physical Review Letters_ 123 (2019). * 26 Feng, T., Shterengas, L., Hosoda, T., Belyanin, A. & Kipshidze, G. Passive Mode-Locking of 3.25 $\upmu$m GaSb-Based Cascade Diode Lasers. _ACS Photonics_ 5, 4978–4985 (2018). * 27 Hillbrand, J. _et al._ Picosecond pulses from a mid-infrared interband cascade laser. _Optica_ 6, 1334 (2019). * 28 Hillbrand, J., Andrews, A. M., Detz, H., Strasser, G. & Schwarz, B. Coherent injection locking of quantum cascade laser frequency combs. _Nature Photonics_ 13, 101–104 (2018). * 29 St-Jean, M. R. _et al._ Injection locking of mid-infrared quantum cascade laser at 14 GHz, by direct microwave modulation. _Laser & Photonics Reviews_ 8, 443–449 (2014). * 30 Forrer, A. _et al._ Photon-Driven Broadband Emission and Frequency Comb RF Injection Locking in THz Quantum Cascade Lasers. _ACS Photonics_ (2020). ## Acknowledgements This work was supported by the Austrian Science Fund (FWF) in the framework of ”Building Solids for Function” (Project W1243), the projects ”NanoPlas” (P28914-N27) and ”NextLite” (F4909-N23). ## Author contributions J.H. processed the QCLs and carried out the experiments. B.S. and J.H. built up the SWIFTS and IAC setups. N.O. and B.S. developed the simulation tool. M.P. carried out the temporal reconstruction using the IAC data. H.S. provided the 2-QWIP. J.H. wrote the manuscript with editorial input from N.O., B.S., G.S and F.C. All authors analysed the results and commented on the paper.

2024-09-04T02:54:58.433052

2003.04133

arxiv-papers

# Frequency and duration of low-wind-power events in Germany Nils Ohlendorf<EMAIL_ADDRESS>Wolf-Peter Schill<EMAIL_ADDRESS>Mercator Research Institute on Global Commons and Climate Change (MCC), EUREF Campus 19, Torgauer Straße 12-15, 10829 Berlin, Germany German Institute for Economic Research (DIW Berlin), Mohrenstrasse 58, 10117 Berlin, Germany Energy Transition Hub, Climate & Energy College, The University of Melbourne ###### Abstract In the transition to a renewable energy system, the occurrence of low-wind- power events receives increasing attention. We analyze the frequency and duration of such events for onshore wind power in Germany, based on 40 years of reanalysis data and open software. We find that low-wind-power events are less frequent in winter than in summer, but the maximum duration is distributed more evenly between months. While short events are frequent, very long events are much rarer. Every year, a period of around five consecutive days with an average wind capacity factor below 10% occurs, and every ten years a respective period of nearly eight days. These durations decrease if only winter months are considered. The longest event in the data lasts nearly ten days. We conclude that public concerns about low-wind-power events in winter may be overrated, but recommend that modeling studies consider multiple weather years to properly account for such events. ###### keywords: Wind power; Low-wind-power events; Reanalysis data; ††journal: arXiv ## 1 Introduction The Paris Agreement calls for an extensive decarbonization of the global economy. A major strategy for achieving this goal is a massive expansion of variable renewable energy sources, in particular solar photovoltaics (PV) and wind power [de Coninck et al., 2018]. While power generation from solar PV largely follows diurnal and seasonal cycles with annually repeating patterns, wind power is subject to more irregular inter-annual as well as intra-annual variations which are relevant from a security of supply perspective. In countries with growing shares of wind power, the occurrence of low-wind-power (LWP) events thus receives increasing attention. This is particularly true in Germany. In the context of its energy transition, Germany is one of the global front-runners in wind power deployment. In 2018, a total capacity of 52.5 GW of onshore wind power was installed in Germany, generating 90.5 TWh of electricity. This corresponds to 15% of German gross electricity consumption [BMWi, 2019]. Given the government’s targets to expand the share of renewables in electricity consumption to 65% by 2030 and at least 80% by 2050 [Bundesregierung, 2019], the dependence of the German energy system on wind power is set to increase strongly in the future. Concerns about LWP events have been discussed in German media [Wetzel, 2017, 2019] and in the German parliament [Deutscher Bundestag, 2019a], and LWP events are also mentioned in the government’s energy transition reporting [Deutscher Bundestag, 2019b]. In this context, the term Dunkelflaute is increasingly used. It refers to a persistent situation with very low power generation from wind and solar PV, which would be especially challenging in the German winter season where PV availability is low and electric load has its peak. Yet no clear definition of this concept has been provided so far [Wissenschaftliche Dienste, 2019], and quantitative evidence on the frequency and duration of such events is missing. In Table $15$ of Deutscher Bundestag [2019b], an independent expert commission generally assumes a no-wind-no-solar period of two weeks. Yet research on LWP events is sparse so far. In this paper, we contribute to filling this gap, focusing on onshore wind power in Germany. We provide an in- depth analysis of the frequency, duration, and magnitude of LWP events, making use of reanalysis data for 40 full years (1980 to 2019) and power curves of recently installed wind turbines. In doing so, we propose two definitions of LWP events and investigate three different thresholds of capacity factors (2%, 5% and 10%). We also compare the spatial distributions of the most persistent LWP event and the mean electricity generation. Parts of our analysis explicitly focus on winter months: these are particularly relevant, as power generation from solar PV is relatively low during this season, while the German peak load also occurs in winter. In order to allow for the highest degree of transparency and reproducibility, we provide the source code of our analysis under a permissive open-source license [Ohlendorf, 2020]. There are only few dedicated analyses on the frequency and duration of LWP events. Early contributions address reliability aspects of spatially dispersed wind power in California [Kahn, 1979] or in the midwestern United States [Archer and Jacobson, 2007]. Analyses explicitly focusing on LWP events only recently emerged. Yet these differ from our work, amongst other factors, with respect to geographical and temporal coverage, data sources used, and methodologies applied. In particular, previous low-wind analyses mostly draw on local measurement data and either evaluate wind speeds [Leahy and McKeogh, 2013, Patlakas et al., 2017] or wind power [Handschy et al., 2017, Kruyt et al., 2017]. Leahy and McKeogh [2013] and Patlakas et al. [2017] investigate low-wind events for Ireland and the North Sea area, respectively. Both studies firstly evaluate low-wind events that are constantly below a given wind speed threshold, and secondly determine annual minimum moving average wind speeds for given durations, using extreme value distributions. Kruyt et al. [2017] and Handschy et al. [2017] go one step further and calculate respective power generation from wind speeds for Switzerland and the United States, using a power curve. While the findings of these studies are necessarily idiosyncratic to the specific geographical applications, some common findings emerge. First, low-wind events are less frequent and less persistent if more, and spatially more dispersed, measurement stations are used. Second, there are generally less events in winter than in summer. The measurement-based analyses face challenges related to their data sources. In general, studies that draw on measured wind speeds are spatially biased, have low measurement densities, and extrapolation from measurement height to hub height is challenging because of distorting effects of terrain, elevations or buildings [Sharp et al., 2015]. Measurement data may further be subject to inconsistencies caused by changing equipment and measurement errors. Extreme event analyses further require consistent measurements over large time periods to sufficiently capture climatic variations. These issues can be addressed by using long-term meteorological reanalysis data. Such data is increasingly applied for onshore wind energy modelling. Several studies focus on data accuracy and on validating models of wind power generation [Decker et al., 2012, Sharp et al., 2015, Olauson and Bergkvist, 2015, Rose and Apt, 2015, Staffell and Pfenninger, 2016, González-Aparicio et al., 2017, Germer and Kleidon, 2019]. Other analyses deal with variability aspects of wind power, but do not focus on extreme low-wind events. For example, Grams et al. [2017] explain longer-term fluctuations in European wind power generation with different types of weather regimes, based on MERRA-2 data. With similar approaches, Collins et al. [2018] investigate inter-annual variations of European wind and solar power, and Santos-Alamillos et al. [2017] explore optimal allocations of renewable generation capacity in a European super grid. For the contingent U.S. states, Shaner et al. [2018] investigate the reliability of future power systems dominated by wind and/or solar PV, and Kumler et al. [2019] explore inter-annual renewable variability for Texas. Yet none of these studies explicitly focuses on the frequency and duration of extreme low-wind-power events. A notable reanalysis study that does focus on extreme wind events is conducted by Cannon et al. [2015] for Great Britain. Using 33 years of MERRA as well as ERA-Interim data, the authors conclude that the frequency and duration of low- wind-power events can be approximated by a Poisson-like process. Weber et al. [2019] also use ERA-Interim data for a superstatistical analysis of extreme wind power events at nine specific European sites, including one German onshore location. They find that the distribution of low-wind events has a heavy tail, as low-wind events may result from a combination of different weather and circulation patterns.111Weber et al. [2019] base their analysis on wind speeds, not wind power generation, with a cut-off threshold of $4$ m/s. In another analysis based on ERA-Interim reanalysis data and other sources, Raynaud et al. [2018] define and investigate the occurrence of renewable “energy droughts”, which are measured relative to average daily generation. They find that wind power droughts are both relatively frequent and relatively short in most European countries, compared to hydro power droughts. We contribute to this emerging literature with a dedicated open-source, reanalysis-based study that investigates LWP events in Germany in detail. To the best of our knowledge, we are the first to use MERRA-2 data in this context, i.e., spatially and temporally consistent reanalysis data covering 40 years at 50 m above surface. Compared to Cannon et al. [2015], we also make use of not only one, but three recent power curves to represent different types of wind turbines that are characteristic for different locations defined by mean wind speeds. Complementary to Raynaud et al. [2018], we further present an alternative approach to defining and evaluating LWPs by looking either at hours that are constantly below a threshold, or at hours with a mean below a threshold. ## 2 Methods and data ### 2.1 General approach Based on wind speeds and power curves, we derive an hourly aggregated time series of capacity factors for wind power in Germany. First, we take wind speeds at 50 m above surface from the MERRA-2 reanalysis dataset, which covers 40 years from 1980 to 2019, and extrapolate to hub heights.222See Section A for further information on the use of reanalysis data for energy modelling. Second, capacity factors of each MERRA-2 grid cell are calculated based on power curves of recently installed wind turbines. Third, we spatially aggregate these capacity factors using a weighting scheme that considers the current spatial distribution of onshore wind power capacity in Germany. Finally, we investigate the resulting time series of hourly aggregated capacity factors by applying a narrower and a wider definition of LWP events. ### 2.2 Wind speeds derived from reanalysis data We use the MERRA-2 dataset provided by NASA [Gelaro et al., 2017]. Data is available starting from the year 1980. In contrast to several other global reanalysis datasets which have time resolutions of 3 to 6 hours and provide wind speeds at 10 m above surface, MERRA-2 includes hourly wind speed data at 50 m, which allows better modelling of wind power generation. Figure 1: MERRA-2 grid points (blue) and grid cells that intersect with Germany. The MERRA-2 grid consists of 576 longitudinal and 361 latitudinal horizontal grid points, i.e., a resolution of $0.625^{\circ}$ x $0.5^{\circ}$ which for Germany roughly corresponds to 50 x 50 km [Bosilovich et al., 2016]. Figure 1 shows the grid points in blue and all grid cells extrapolated from these points that intersect with Germany. For each grid cell, MERRA-2 provides hourly northward and eastward wind speed data at 50 m above surface. Our dataset further includes surface roughness data for the year 2019. ### 2.3 Aggregated wind power derived from wind speeds using power curves We calculate the magnitude of the horizontal wind speed ($U$) for each MERRA-2 grid point based on northward ($u$) and eastward components ($v$) at 50 m (Equation 1). $U=\sqrt{(u^{2}+v^{2})}$ (1) In line with Kruyt et al. [2017], we use the logarithmic power law to extrapolate wind speeds to hub-height ($h$) with $U_{hub}$ as the wind speed at hub height and $z_{0}$ as the surface roughness data for every grid point and each hour of the year 2019 (Equation 2). $U_{hub}=w\frac{\ln\frac{h}{z_{0}}}{\ln\frac{50}{z_{0}}}$ (2) Figure 2: Wind speed zones in Germany. Dark blue implies high mean wind speeds, blue medium wind speeds, and light blue low mean wind speeds. We define three types of wind zones, based on mean local wind speeds over 40 years for each MERRA-2 grid cell (Figure 2), and assign typical hub heights for wind turbines. For high-wind-speed sites, we assign a hub height of 100 m, for medium-wind-speed sites of 125 m, and for low-wind-speed sites of 139 m [Wallasch et al., 2015]. These values reflect the mean hub heights of recently installed wind power plants in respective German wind speed zones. We calculate hourly capacity factors for each grid cell by applying power curves characteristic for the three wind zones. The power curves are based on manufacturer data of currently available wind turbines for low-, medium- and high-wind sites, respectively. Both the low- and high-wind site power curves represent an average of four wind turbines of similar diameters and capacities. We consider turbines from six manufacturers (see B), among them four large companies which cover 87% of the capacity installed in Germany in 2015 [Lüers, 2016]. Manufacturers generally provide discrete capacity factors ($CF_{disc}$) for wind speed intervals of 1 m/s. For both the low- and high-wind curves, we first calculate discrete mean capacity factors for each wind speed and then calculate continuous capacity factors using a generalized logistic function (Equation 3). $CF_{cont}=A+\frac{C}{(1+Te^{-B(U_{hub}-M)})^{1/T}}$ (3) Here, $CF_{cont}$ is the continuous capacity factor and $A$, $B$, $C$, $M$ and $T$ are fitted coefficients based on minimising the squared deviations between $CF_{disc}$ and $CF_{cont}$. For both the low- and the high-wind power curve, cut-in wind speeds of around 3 m/s emerge, and the resulting capacity factors are capped at 0% and 100%. The medium-wind power curve represents the average of the low- and high-wind curves (Figure 3). Figure 3: Power curves of three types of wind turbines Aggregated hourly capacity factor time series for overall Germany are derived by weighting all grid cells with the current distribution of installed wind power generation capacity. The latter is extracted from Open Power System Data [Open Power System Data, 2017, Wiese et al., 2019] and open-source GIS data. The red points in Figure 4 indicate the installed wind capacity of locally aggregated wind power plant sites in Germany and the blue squares show the corresponding relative capacity distribution of the MERRA-2 grid cells. Grid cells only partly intersecting with the German land area receive lower weights according to the overlapping area. We implicitly assume that the transmission infrastructure allows geographical balancing of wind power in Germany, which is currently largely the case.333This assumptions is particularly valid for low-wind periods. During high-wind, high-load periods, the German transmission grid can be constrained in North-South direction. Figure 4: Distribution of currently installed wind power capacity in Germany. Darker colors indicate a larger share of total or relative installed capacity. ### 2.4 Definition of low-wind-power events We propose two different measures of low-wind-power periods, a narrower and a wider one (Figure 5). We further consider three alternative capacity factor thresholds of 2%, 5%, and 10%. As for the narrower definition, we consider LWP events to be consecutive hours in which the aggregated capacity factors are Constantly Below the Threshold (CBT). This concept bears some resemblance to the “runs analysis” by Leahy and McKeogh [2013] or the “duration given intensity” method by Patlakas et al. [2017]. Starting in the first hour, we list annual LWP events for durations starting from five consecutive hours and report the number of hours constantly below the given capacity factor threshold. We then increase the duration in hourly steps and repeat until there are no further events listed. To provide a wider definition, we consider LWP events to consist of consecutive hours in which the moving average of capacity factors is under the same threshold, i.e., Mean Below the Threshold (MBT). Again, we list all LWP periods until we reach the threshold value, ensuring that LWP periods do not overlap. By definition, the MBT method results in more low-wind-power events for a given duration and also results in longer events for each threshold, compared to CBT. Figure 5: Illustration of the two LWP event definitions The average annual amount of LWP events per duration over all 40 years equals the expected value of events per year. Further, the reciprocal value of the annual average provides the return period, that is the expected temporal distance between two similar reoccurring events. Periods overlapping annually or monthly are assigned to the year or month in which more than 50% of the hours are located444 Accounting for annually overlapping periods requires December data from the previous year, and January data from the subsequent year. For the two boundary years 1980 and 2019, we substitute the missing data for December 1979 (January 2020) with data from December 1980 (January 2019). . ## 3 Results ### 3.1 Seasonal distribution and frequency of low-wind-power events Figure 6 shows that LWP events are generally most frequent in summer (here defined as June-August) and least frequent in winter (December-February). The results for spring (March-May) and autumn (September-November) are mostly close to the annual average. Accordingly, respective findings made for other European countries [Leahy and McKeogh, 2013, Cannon et al., 2015, Kruyt et al., 2017] are also valid for Germany. Figure 6: Average seasonal duration (horizontal axis) and frequency (vertical axis) of LWP events in Germany The frequency of events for a given duration is about 1.5-3 times higher for the wider MBT definition compared to the narrower CBT concept. For both metrics, the frequency of LWP events increases substantially with the capacity factor threshold value. For example, a 10-hour event below a capacity factor of 2% occurs on average around 0.2 times per winter for CBT and slightly less than once per winter for MBT. For a 10% capacity factor threshold, there are on average around eight such winter events for CBT and 13 for MBT. In general, we find that short LWP events with a duration of up to around half a day are relatively frequent and may occur several times per year, especially under the wider MBT definition. Longer LWP events, in contrast, are much less frequent. To provide a complementary perspective, we calculate the return periods for different durations of LWP events (Table 1). The return periods are the reciprocal of the average (annual or seasonal) frequency of LWP events for different durations, considering both definitions and all three thresholds (cf. Figure 6). For example, an LWP event with an average frequency of 0.2 for a given duration leads to a return period of 5 years for this specific duration. The longer a given duration, the lower its average frequency and the longer its return period. For a return period of ten years, we find a duration of 17 hours (2% capacity factor threshold), 41 hours (5%) and 77 hours (10%) under the narrower CBT definition, and a duration of 34 hours (2%), 79 hours (5%) and 188 hours (10%) under the wider MBT concept. In other words, every ten years the German energy system has to deal with a period of nearly eight days of average wind power generation (MBT) below 10% of the installed capacity. Table 1: Duration in hours for LWP events in winter or in any season for different return periods | Constantly below threshold (CBT) | Mean below threshold (MBT) ---|---|--- | Winter | Any season | Winter | Any season Return period | 2% | 5% | 10% | 2% | 5% | 10% | 2% | 5% | 10% | 2% | 5% | 10% 1 year | 5 | 15 | 29 | 11 | 23 | 45 | 8 | 30 | 63 | 18 | 58 | 122 2 years | 7 | 21 | 40 | 13 | 32 | 57 | 12 | 45 | 92 | 21 | 69 | 144 3 years | 8 | 23 | 44 | 14 | 33 | 60 | 14 | 52 | 101 | 23 | 71 | 161 4 years | 9 | 30 | 48 | 14 | 33 | 63 | 16 | 62 | 112 | 27 | 72 | 173 5 years | 10 | 32 | 57 | 15 | 35 | 65 | 22 | 68 | 113 | 28 | 75 | 178 6 years | 10 | 32 | 57 | 15 | 35 | 67 | 25 | 69 | 114 | 29 | 76 | 182 7 years | 12 | 33 | 60 | 15 | 36 | 67 | 27 | 70 | 114 | 31 | 76 | 186 8 years | 14 | 33 | 63 | 17 | 37 | 69 | 28 | 72 | 117 | 33 | 79 | 186 9 years | 14 | 33 | 63 | 17 | 37 | 69 | 28 | 72 | 117 | 33 | 79 | 186 10 years | 14 | 33 | 64 | 17 | 41 | 77 | 28 | 72 | 126 | 34 | 79 | 188 15 years | 17 | 36 | 67 | 18 | 41 | 77 | 31 | 76 | 129 | 38 | 82 | 189 20 years | 19 | 41 | 77 | 19 | 49 | 81 | 34 | 79 | 131 | 45 | 89 | 221 25 years | 19 | 41 | 77 | 19 | 49 | 81 | 34 | 79 | 131 | 45 | 89 | 221 30 years | 19 | 41 | 77 | 19 | 49 | 81 | 34 | 79 | 131 | 45 | 89 | 221 To better interpret these return periods, we provide an example for the German onshore wind power capacity of 52.5 GW installed in 2018. For this wind turbine fleet, average power generation is expected to not exceed around five GW, i.e., 10% of capacity, during a period of around five consecutive days every year (122 hours, MBT for ’Any Season’ in 1). Every ten years, this period increases to nearly eight days, and every twenty years to more than nine full days. Looking only at LWP events in winter, these durations decrease to less than three days every winter, less than five days every tenth winter, and around five and a half days every twentieth winter. The remaining load has to be covered by other generators, energy storage or demand-side measures. However, wind power still contributes some generation capacity above the 10% threshold during some of these hours, as indicated by much lower CBT return periods. ### 3.2 Magnitude of the most extreme low-wind-power events The most extreme LWP events over the entire 40 years analyzed can be interpreted as worst cases from an energy system planning perspective. In an annual perspective, the most extreme events occurred in 1985 for all capacity factor thresholds (Figure 7). Under the narrower CBT definition, there are nearly four consecutive days with wind power generation constantly below 10% in 1985, and still around two consecutive days with generation constantly below 5%. Under the wider MBT definition, the duration of this most extreme event increases to nearly ten days (10%) or around four days (5%). Figure 7: Most extreme LWP events per year. The vertical axis shows the duration of the longest event per year for the three capacity factor thresholds. While this 1985 event is the most extreme one under both CBT and MBT, the ranking of the second most extreme yearly events differs between the LWP definitions. For example, the second-longest event occurred in 1984 under the CBT definition. Yet under MBT, the duration of the most extreme event in 1984 is only average. In general, the definition of LWP events and the chosen thresholds have a substantial impact on quantitative results. Under MBT, the most extreme annual events are generally around twice as high compared to CBT. We further find very large inter-annual variations. Considering the 10% threshold, the longest event for the MBT definition lasted for almost 10 days in 1985, but in 2005 the longest duration was only three days for the same threshold. The relative difference between the longest events for each year increases with the threshold. These large variations of the most extreme annual LWP events complement the findings made by Collins et al. [2018], who determine large inter-annual variations of average renewable availability. We next look at the most extreme LWP event in a monthly perspective, irrespective of the year in which these occur (Figure 8). The most extreme events for the 10% threshold occur in March for both definitions. This is the 1985 event discussed above, with durations of nearly four (CBT) or nearly ten consecutive days (MBT). Figure 8: Most extreme LWP events per month. The vertical axis shows the duration of the longest event of all respective months for the three capacity factor thresholds. Considering all thresholds and both LWP definitions, there is no clear trend of the most extreme monthly LWP events. That is, substantial extreme events may occur throughout the year, and also in winter months. This contrasts the previous finding that the frequency of LWP events is generally much higher in summer than in winter, as shown in Section 3.1. Under CBT, the most extreme events in each of the winter months are even longer than those in summer months for the 10% capacity threshold. This finding is, however, not confirmed under the MBT definition. ### 3.3 Spatial distribution of wind power during most extreme LWP event To also explore the spatial dimension of LWP events, we compare the distribution of capacity factors during the most extreme LWP of 1985 to the distribution of annual mean capacity factors in the same year (Figure 9). Figure 9: Spatial distribution of wind power. Left: Average wind power during most extreme LWP event (10% capacity factor, MBT) in dataset in March 1985 (Scale: From 0% to 20% of mean capacity factors). Right: Mean wind power in the entire year 1985 (Scale: From 5% to 50% of mean capacity factors). The spatial pattern of annual mean capacity factors (Figure 9, right panel) largely resembles that of average wind speeds in Germany (Figure 2). Mean capacity factors are generally higher in Northern than in Southern Germany. They are highest close to the Northern and the Baltic Sea, and lowest in the southern Alpine region. The spatial pattern of mean capacity factors during the most extreme LWP event (Figure 9, left panel) substantially deviates from the distribution of the means. In particular, capacity factors of the north-eastern region and parts of the northern region are relatively low. The respective spatial distributions of capacity factors for other thresholds under both the CBT and MBT definitions of the same event also show substantial deviations from annual means. Accordingly, the spatial distribution of capacity factors during extreme LWP events does not necessarily correspond to the annual mean pattern. This indicates that low-wind events can be very pronounced even in regions with very good average wind resources. ## 4 Conclusions We analyze the seasonal distribution, frequency and magnitude of onshore low- wind-power events in Germany, as well as spatial aspects of the most extreme events, based on MERRA-2 reanalysis data and open software. We propose and evaluate two definitions of low-wind-power events for three capacity factor thresholds. We synthesize three key results from the analysis. First, LWP events are generally most frequent in summer and least frequent in winter. Nonetheless, substantial events occur in all months of the year, and also in winter. The most persistent LWP event in the dataset occurred in March. Second, while short events with a duration of up to around half a day are relatively frequent, very long events are much rarer.555Weber et al. [2019] argue that low-wind event statistics do not follow a simple exponential distribution, but have “heavy tails”, i.e. the probability decreases rather slowly with increasing duration. Every year, the German energy system has to deal with a period of around five consecutive days during which average wind power generation is below 10% of the installed capacity. Every ten years, a respective period of nearly eight days is to be expected. Looking only at winter months, the durations of these expected events decrease to less than three days every winter and less than five days every tenth winter. The most persistent low-wind event in the entire dataset has a duration of nearly ten consecutive days of average wind power generation below a 10% capacity factor. Third, the spatial pattern of LWP events may be very different from the one of average wind power resources. During the most persistent LWP event, we find average generation to be particularly low in several regions which have some of the best wind resources. We conclude that energy modeling studies that only consider one historic weather year are likely to substantially underestimate the occurrence of low- wind-power events and related system implications. In particular, analyses with an energy system planning perspective should take less frequent LWP events into account, e.g., the discussed events with a return period of ten years, or even the most extreme event identified here. This is particularly important when the complementary role of other variable and dispatchable generators, energy storage, or demand-side measures in highly-renewable energy systems is to be explored.666This is demonstrated, for example, by Schill and Zerrahn [2018] in an analysis of storage requirements for renewable energy integration in a sensitivity analysis with one artificial no-wind week. Further, analyses dealing with the pros and cons of either more decentralized or more centralized renewable energy systems should consider the spatial dimension of LWP events. Although not in the focus of our analysis, our results indicate that LWP events are more pronounced for smaller geographic areas. From an energy policy perspective, our findings on LWP events occurring in winter may be most relevant. Our analysis indicates that concerns about frequent and persistent LWP events in German winters appear to be overrated, considering that the longest event with an average capacity factor below 10% and a ten-year return period in winter has a duration of less than five days. We further recommend that policy makers or regulators develop a proper definition of the Dunkelflaute term, which currently appears to be used in a rather qualitative way. Our two definitions of LWP events proposed here may be useful in this context. While our analysis deliberately focuses on LWP events of onshore wind power in Germany, we see an avenue for future research that would ideally combine the analysis of low production periods of onshore and offshore wind power as well as solar PV with time series of load, while expanding the geographic focus beyond Germany. The open-source provision of the tool used for the present analysis may be a useful starting point for such research. ## Acknowledgment This analysis is a result of the research project P2X, funded by the German Federal Ministry of Education and Research (FKZ 03SFK2B1). Wolf-Peter Schill carried out parts of the work during a research stay at the University of Melbourne. Nils Ohlendorf mainly worked on this project while employed at DIW Berlin, and partly also after being employed at MCC. We thank the participants of the DIW Sustainability Cluster Seminar in April 2017, Strommarkttreffen Berlin November 2017, IAEE International Conference Groningen 2018 and Enerday Dresden 2018 for valuable comments on earlier drafts. ## Data availability statement The data that support the findings of this study have been created with software that is openly available under an MIT license at https://doi.org/10.5281/zenodo.3694373. ## References * Archer and Jacobson [2007] Archer, C.L., Jacobson, M.Z., 2007\. Supplying baseload power and reducing transmission requirements by interconnecting wind farms. Journal of Applied Meteorology and Climatology 46, 1701–1717. doi:10.1175/2007JAMC1538.1. * BMWi [2019] BMWi, 2019. Zeitreihen zur Entwicklung der erneuerbaren Energien in Deutschland. Bundesministerium für Wirtschaft und Energie (Federal Ministry for Economic Affairs and Energy). 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The underlying global circulation models extrapolate measurement station data on wind speeds, temperature, moisture and surface pressure as well as data from satellites and precipitation measurements [Decker et al., 2012]. Several publicly available second-generation global reanalysis datasets have been released since the early 2000s. We use MERRA-2, which builds on and improves the previous MERRA dataset, using advanced models and data sources [Molod et al., 2015]. Decker et al. [2012] evaluate the accuracy of several reanalysis datasets (MERRA, NCEP, ERA-40, ERA-Interim, CFSR and GLDAS) using flux tower measurements in the Northern Hemisphere. Almost all products overestimate the monthly and 6-hourly wind speeds and their variability. MERRA and ERA-Interim show the lowest values root-mean-square error and bias for diurnal cycles. Sharp et al. [2015] review other data validation studies of different reanalysis datasets. Three studies derive Pearson’s correlation coefficients for MERRA between 0.75 and 0.89 based on measurement stations in Sweden, Portugal, Norway and Denmark [Liléo and Petrik, 2011, Liléo et al., 2013, Carvalho et al., 2014]. Staffell and Pfenninger [2016] propose country- specific wind speed bias correction factors for MERRA and MERRA-2 to increase the correlation with national capacity factors. Without such correction, average capacity factors for Germany based on raw MERRA or MERRA-2 wind speeds would be overestimated. Staffell and Green [2014] make a similar point for the UK. In contrast, Cannon et al. [2015] do not use correction factors in a UK application. Even if MERRA wind speeds turn out to be not particularly valid for single measurement points, spatial aggregation of mean wind speed over all stations results in a correlation coefficient of 0.94. This indicates a high validity of MERRA data for large-scale wind patterns. Following Cannon et al. [2015], we also refrain from introducing correction factors and instead make use of the error-smoothing effect of spatial aggregation. In doing so, we also avoid model artefacts, particularly as the usefulness of correction factors has only been demonstrated for average wind speeds, but not for extreme values. ## Appendix B Wind power turbines The low- and high-wind power curves used in our analysis are based on data of eight wind power turbines by six manufacturers, namely Nordex, Senvion, Enercon, Vestas, Gamesa and Vensys. Specifically, we use the following high- wind power turbines: * 1. Nordex N90-2.5MW * 2. Vestas V90-2.0MW * 3. Gamesa G97-2MW * 4. Vensys 100-2.5MW Analogously, we use following low-wind power turbines: * 1. Nordex N131-3.3MW * 2. Senvion 3.2M122 * 3. Enercon E126 EP4/4.2MW * 4. Vestas V126-3.3MW ## Appendix C Discussion of limitations We briefly discuss some limitations of our analysis and how these may qualitatively impact results. First, there are general limitations of using reanalysis data which have been discussed in the literature, for example spatial biases or issues with upscaling to hub heights [Sharp et al., 2015, Olauson and Bergkvist, 2015, Rose and Apt, 2015, Staffell and Pfenninger, 2016]. It is, however, not clear if there are specific distortions with respect to extreme low-wind events derived from reanalysis data. A limitation specific to the MERRA-2 dataset is the relatively coarse 50x50 km grid cell size, which insufficiently represent local impacts on wind speeds. Regional reanalysis data with more refined geographical resolutions may resolve this issue, e.g. COSMO-REA2 with 2x2 km, or COSMO-REA6 with 6x6 km [Hans Ertel Zentrum, 2019], yet these are only available for shorter periods of time. The global coverage of MERRA-2 further allows repeating our open-source analysis for other countries and world regions. Second, we use power curves of currently available wind turbines and assume hub-heights of recently constructed plants. We may thus overestimate wind power generation compared to the currently existing fleet of wind turbines in Germany, which includes many older and smaller turbines, and in turn underestimate the magnitude of current LWP events. Conversely, we may underestimate power generation of future turbines, and accordingly overestimate the magnitude of future low-wind-power events, assuming that turbine efficiency and hub height increases further, with corresponding upward shifts in the power curves. Once LWP events become more relevant for the overall energy system, this may also trigger specific technology improvements toward lower cut-in speeds and a steeper slope of the power curve on the very left-hand side. Quantifying the potentially mitigating effects of such developments on LWP periods is left for future research. Third, we use the current spatial capacity distribution of German wind power plants for deriving an aggregated capacity factor time series. We implicitly assume that this distribution also persists in the future. In reality, a relative increase of wind power deployment at sites with lower wind resources may occur, for example in southern Germany. From the results presented in Section 3.1, we infer that a more even spatial dispersion of wind turbines could slightly mitigate LWP events. Next, climate change has an impact on wind speeds. Future time series of wind power capacity factors will thus differ from the historic ones investigated here. Tobin et al. [2016] find that wind power variability in Europe may generally increase, but Schlott et al. [2018] conclude that this has no substantial effect on optimal deployment of onshore wind power in highly renewable future scenarios. Moemken et al. [2018] find that climate change will increase the occurrence of low wind speeds. Finally, the focus of this analysis is a detailed but selective investigation of onshore LWP events in Germany. This geographic focus helps to keep the analysis tractable and avoids making implicit assumptions on continental electricity transmission infrastructure. It is also relevant from an energy policy perspective, which often includes national energy security considerations. Yet expanding the geographic scope of the analysis would allow raising complementary insights on larger-scale spatial patterns of extreme LWP events. Focusing on onshore wind power, and not including other renewable energy sources such as offshore wind power and solar PV, allows for parsimonious model assumptions, and findings remain valid for any level of installed capacity. Analyses that would combine periods of low production from various renewable energy sources, and also explore their correlation with electric load, appear to be a promising field for future research. The work of Raynaud et al. [2018], albeit with lower temporal and spatial detail compared to our analysis, can be considered as a first step in this direction.

2024-09-04T02:54:58.461125

2003.04239

arxiv-papers

# A shadow of algebraic topology and variational method - Prandtl Batchelor problem Debajyoti Choudhuri111Corresponding author<EMAIL_ADDRESS>ORCID ID: 0000-0001-8744-9350 Department of Mathematics, National Institute of Technology Rourkela, India ###### Abstract In this paper we study the existence of nontrivial weak solution to a Prandtl- Batchelor type free boundary value elliptic problem involving a $p$-Laplacian operator and a power nonlinearity. Topics from algebraic topology will be used to establish the existence of a solution to the approximating problem, whereas, the variational technique will be used to fix the claim of existence of a solution to the main problem. In the process, a couple of classical results were also improved to suit the purpose of establishing the existence of a nontrivial solution. Keywords: Dirichlet free boundary value problem, Sobolev space, Morse relation, cohomology group. AMS Classification: 35J35, 35J60. ## 1 Introduction We will investigate the existence of solution to the following free boundary value problem. $\displaystyle\begin{split}-\Delta_{p}u&=\lambda\chi_{\\{u>1\\}}(u-1)_{+}^{q-1},~{}\text{in}~{}\Omega\setminus H(u),\\\ |\nabla u^{+}|^{p}-|\nabla u^{-}|^{p}&=\frac{p}{p-1},~{}\text{in}~{}H(u)\\\ u&=0,\text{on}~{}\partial\Omega.\end{split}$ (1.1) Here, $\lambda>0$ is a parameter, $(u-1)_{+}=\max\\{u-1,0\\}$ and $H(u)=\partial\\{u>1\\}.$ Also $\nabla u^{\pm}$ are the limits of $\nabla u$ from the sets $\\{u>1\\}$ and $\\{u\leq 1\\}^{\circ}$ respectively. The domain $\Omega\subset\mathbb{R}^{N}(N\geq 2)$ is bounded with a sufficiently smooth boundary $\partial\Omega$. The relation between the exponents are assumed in the order $1<p\leq q-1$, with $q<p^{*}=\dfrac{Np}{N-p}$. The solution(s) satisfy the free boundary condition in the following sense: for all $\vec{\phi}\in C_{0}^{1}(\mathbb{R}^{N})$ such that $u\neq 1$ a.e. on the support of $\vec{\phi}$, $\displaystyle\underset{\epsilon^{+}\rightarrow 0}{\lim}\int_{u=1+\epsilon^{+}}\left(\frac{p}{p-1}-|\nabla u|^{p}\right)\vec{\phi}\cdot\hat{n}dS-\underset{\epsilon^{-}\rightarrow 0}{\lim}\int_{u=1-\epsilon^{-}}|\nabla u|^{p}\vec{\phi}\cdot\hat{n}dS$ $\displaystyle=0,$ (1.2) where $\hat{n}$ is the outward drawn normal to $\\{1-\epsilon^{-}<u<1+\epsilon^{+}\\}$. Note that the sets $\\{u=1\pm\epsilon^{\pm}\\}$ are smooth hypersurfaces for almost all $\epsilon^{\pm}>0$ by the Sard’s theorem. The limit above in (1.2) is taken by running such $\epsilon^{\pm}>0$ towards zero. A rich literature survey has been done in the book due to Perera et al. [10] where the author has discussed problems of several variety involving the $p$-Laplacian operators which could be studied using the Morse theory. The motivation for the current work has been drawn from the work due to Perera [13]. The treatment used to address the existence of atleast one (or two) solution(s) to the approximating problem may be classical (section $3$, Theorems 3.3 and 3.5) but the result concerning the reguarity of the free boundary is very new and the question of existence of solution to the problem (1.1) has not been answered till now (section $4$, Lemma 4.1), to the best of my knowledge. Two more results due to Alt-Caffarelli [1] (section $4$, Lemma 4.2) and Caffarelli et al. [6] (Appendix, Lemma 4.3) were improved to the best possible extent to suit the purpose of the problem in this paper. ### 1.1 A physical motivation Consider the problem $\displaystyle\begin{split}-\Delta u&=\lambda\chi_{\\{u>1\\}}(x),~{}\text{in}~{}\Omega\setminus H(u),\\\ |\nabla u^{+}|^{2}-|\nabla u^{-}|^{2}&=2,~{}\text{in}~{}H(u)\\\ u&=0,\text{on}~{}\partial\Omega.\end{split}$ (1.3) This is the well known Prandtl-Batchelor free boundary value problem, where the phase $\\{u>1\\}$ is a representation of the vortex patch bounded by the vortex line $u=1$ in a steady fluid flow for $N=2$ (refer Batchelor [2, 3]). Thus the current problem is a more generalized version of (1.3). For a more physical application to this problem we direct the reader’s attention to the work due to Caflisch [4], Elcrat and Miller [7]. Another instance of occurrence of such a phenomena is in the non-equilibrium system of melting of ice. In a given block of ice, the heat equation can be solved with a given set of appropriate initial/boundary conditions in order to determine the temperature. However, if there exists a region of ice in which the temperature is greater than the melting point of ice, this subdomain will be filled with water. The boundary thus formed due to the ice-water interface is controlled by the solution of the heat equation. Thus encountering a free boundary in the nature is not unnatural. The problem in this paper is a large enough generalization to this physical phenomena which besides being a new addition to the literature can also serve as a note to bridge the problems in elliptic PDEs with algebraic topology. ## 2 Preliminaries We begin by giving the relevant definitions and results besides defining the function space which will be used very frequently in the article. Let $X$ be a topological space and $A\subset X$ be a topological subspace. Roughly, a homology group is an algebraic group constructed from a topological object or a space. Following is the fundamental tool that will be used to work with, namely the homology theory [12]. ###### Definition 2.1. A homology group on a family of pairs of spaces $(X,A)$ consists of: 1. 1. A sequence $\\{H_{k}(X,A)\\}_{k\in\mathbb{N}_{0}}$ of abelian groups is known as homology group for the pair $(X,A)$ (note that for the pair $(X,\phi)$, we write $H_{k}(X),k\in\mathbb{N}_{0}$). Here $\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. 2. 2. To every map of pairs $\varphi:(X,A)\rightarrow(Y,B)$ is associated a homomorphism $\varphi^{*}:H_{k}(X,A)\rightarrow H_{k}(Y,B)$ for all $k\in\mathbb{N}_{0}$. 3. 3. To every $k\in\mathbb{N}_{0}$ and every pair $(X,A)$ is associated a homomorphism $\partial:H_{k}(X,A)\rightarrow H_{k-1}(A)$ for all $k\in\mathbb{N}_{0}$. These items satisfy the following axioms. ($A_{1}$) If $\varphi=id_{X}$, then $\varphi_{*}=id|_{H_{k}(X,A)}$. ($A_{2}$) If $\varphi:(X,A)\rightarrow(Y,B)$ and $\psi:(Y,B)\rightarrow(Z,C)$ are maps of pairs, then $(\psi\circ\varphi)_{*}=\psi_{*}\circ\varphi_{*}$. ($A_{3}$) If $\varphi:(X,A)\rightarrow(Y,B)$ is a map of pairs, then $\partial\circ\varphi_{*}=(\varphi|_{A})_{*}\circ\partial$. ($A_{4}$) If $i:A\rightarrow X$ and $j:(X,\phi)\rightarrow(X,A)$ are inclusion maps, then the following sequence is exact $...\xrightarrow[]{\partial}H_{k}(A)\xrightarrow[]{i_{*}}H_{k}(X)\xrightarrow[]{j_{*}}H_{k}(X,A)\xrightarrow[]{\partial}H_{k-1}(A)\rightarrow...$ Recall that a chain $...\xrightarrow[]{\partial_{K+1}}C_{k}(X)\xrightarrow[]{\partial_{k}}C_{K-1}(X)\xrightarrow[]{\partial_{k-1}}C_{k-2}(X)\xrightarrow[]{\partial_{k-2}}...$ is said to be exact if $im(\partial_{k+1})=ker(\partial_{k})$ for each $k\in\mathbb{N}_{0}$. ($A_{5}$) If $\varphi,\psi:(X,A)\rightarrow(Y,B)$ are homotopic maps of pairs, then $\varphi_{*}=\psi_{*}$. ($A_{6}$) (Excision): If $U\subseteq X$ is an open set with $\bar{U}\subseteq\text{int}(A)$ and $i:(X\setminus U,A\setminus U)\rightarrow(X,A)$ is the inclusion map, then $i_{*}:H_{k}(X\setminus U,A\setminus U)\rightarrow H_{k}(X,A)$ is an isomorphism. ($A_{7}$) If $X=\\{*\\}$, then $H_{k}({*})=0$ for all $k\in\mathbb{N}$. ###### Definition 2.2. A continuous map $F:X\times[0,1]\to X$ is a deformation retraction of a space $X$ onto a subspace $A$ if, for every $x\in X$ and $a\in A$, $F(x,0)=x$, $F(x,1)\in A$, and $F(a,1)=a$. A crucial notion in analysis is the idea of compactness and the Palais-Smale condition is a special type of compactness which is given as follows. ###### Definition 2.3. (S. Kesavan [11]) Let $V$ be a Banach space and $J:V\rightarrow\mathbb{R}$ a $C^{1}$ functional. It is said to satisfy the Palais-Smale condition (PS) if the following holds: whenever $(u_{n})$ is a sequence in $V$ such that $(J(u_{n}))$ is bounded and $J^{\prime}(u_{n})\rightarrow 0$ in $V^{\prime}$ (the dual space of $V$), then $(u_{n})$ has a strongly convergent subsequence. The following is a deformation lemma which will be quintessential in computing the homology groups. ###### Lemma 2.4. (S. Kesavan [11]) Let $J:V\rightarrow\mathbb{R}$ be a $C^{1}$ functional satisfying the Palais-Smale condition. Let $c,a$ be real numbers. Define $K_{J,c}=\\{v\in X:J(u)=c,J^{\prime}(v)=0\\}$, $K^{a}=\\{v\in X:J(v)\leq a\\}$ (likewise we define $K_{a}=\\{v\in X:J(v)\geq a\\}$). Let $K_{J,c}=\O$. Then there exists $\epsilon^{\prime}>0$ and a continuous homotopy $\eta:[0,1]\times V\rightarrow V$ such that $\forall~{}0<\epsilon\leq\epsilon^{\prime}$ 1. 1. $\eta(0,v)=v$ for all $v\in X$. 2. 2. $\eta(t,v)=v$ for all $t\in[0,1]$, $v\neq J^{-1}([c-\epsilon,c+\epsilon])$. 3. 3. $\eta(1,K^{c+\epsilon})\subset K^{c-\epsilon}$. ###### Definition 2.5. Morse index of a functional $J:V\rightarrow\mathbb{R}$ is defined to be the maximum subspace of $V$ such that $J^{\prime\prime}$, the second Fréchet derivative, is negative definite on it. ### 2.1 Space description We begin by defining the standard Lebesgue space $L^{p}(\Omega)$ for $1\leq p<\infty$ as $L^{p}(\Omega)=\left\\{u:\Omega\rightarrow\mathbb{R}:u\;{\text{is measurable and}}\int_{\Omega}|u|^{p}dx<\infty\right\\}$ endowed with the norm $\|u\|_{p}=\left(\int_{\Omega}|\nabla u|^{p}dx\right)^{\frac{1}{p}}$. We will define the Sobolev space as $W^{1,p}(\Omega)=\\{u\in L^{p}(\Omega):\nabla u\in(L^{p}(\Omega)^{N}\\}$ with the norm $\|u\|_{1,p}^{p}=\|u\|_{p}+\|\nabla u\|_{p}$. We further define $W_{0}^{1,p}(\Omega)=\\{u\in W^{1,p}(\Omega):u=0~{}\text{on}~{}\partial\Omega\\}.$ The associated norm is $\|u\|^{p}=\|\nabla u\|_{p}$. With these norms, $L^{p}(\Omega)$, $W^{1,p}(\Omega)$ and $W_{0}^{1,p}(\Omega)$ are separable, reflexive Banach spaces([11]). We now state the Hölder’s inequality and embedding results in the following propositions. ###### Proposition 2.6. For any $u\in L^{p}(\Omega)$ and $v\in L^{p^{\prime}}(\Omega)$, where $L^{p^{\prime}}(\Omega)$ is the conjugate space of $L^{p}(\Omega)$ such that $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, $\big{|}\int_{\Omega}uv\;dx\big{|}\leq\|u\|_{p}\|v\|_{p^{\prime}}$ ###### Proposition 2.7. If $p<N$, then $W^{1,p}(\Omega)\hookrightarrow L^{r}(\Omega)$ is continuous for $r\in[p,p^{*}]$ and compact for $r\in[p,p^{*})$. If $p=N$, then $W^{1,p}(\Omega)\hookrightarrow L^{r}(\Omega)$ is continuous and compact for $r\in[p,\infty)$. Further, if $p>N$, then $W^{1,p}(\Omega)\hookrightarrow C^{1-\left[\frac{N}{p}\right]}(\bar{\Omega})$. ## 3 The way to tackle the problem using Morse theory We at first define an energy functional associated to the problem in (1.1) which is as follows. $\displaystyle\begin{split}I(u)&=\int_{\Omega}\frac{|\nabla u|^{p}}{p}dx+\int_{\Omega}\chi_{\\{u>1\\}}(x)dx-\lambda\int_{\Omega}\frac{(u-1)_{+}^{q}}{q}dx.\end{split}$ This functional is not even differentiable and hence poses serious issues as far as the application of variational theorems are concerned. Thus we approximate $I$ using the following functionals that varies with respect to a parameter $\alpha>0$. This method is adapted from the work of Jerison-Perera [9]. We define a smooth function $g:\mathbb{R}\rightarrow[0,2]$ as follows: $g(t)=\begin{cases}0,&\text{if}~{}t\leq 0\\\ \text{a positive function},&\text{if}~{}0<t<1\\\ 0,&\text{if}~{}t\geq 0\end{cases}$ and $\int_{0}^{1}g(t)dt=1$. We further let $G(t)=\int_{0}^{t}g(t)dt$. Clearly, $G$ is smooth and nondecreasing function such that $G(t)=\begin{cases}0,&\text{if}~{}t\leq 0\\\ \text{a positive function}<1,&\text{if}~{}0<t<1\\\ 1,&\text{if}~{}t\geq 0.\end{cases}$ We thus define $\displaystyle\begin{split}I_{\alpha}(u)&=\int_{\Omega}\frac{|\nabla u|^{p}}{p}dx+\int_{\Omega}G\left(\frac{u-1}{\alpha}\right)dx-\lambda\int_{\Omega}\frac{(u-1)_{+}^{q}}{q}dx.\end{split}$ This functional $I_{\alpha}$, is of at least $C^{2}$ class and hence $\displaystyle\langle I_{\alpha}^{\prime\prime}(u)v,w\rangle=$ $\displaystyle\int_{\Omega}[|\nabla u|^{p-2}\nabla v\cdot\nabla w+(p-2)|\nabla u|^{p-4}(\nabla u\cdot\nabla v)(\nabla u\cdot\nabla w)]dx$ $\displaystyle+\int_{\Omega}\frac{1}{\alpha^{2}}g^{\prime}\left(\frac{u-1}{\alpha}\right)vwdx-\lambda\int_{\Omega}(u-1)_{+}^{q-2}vwdx.$ Following is an important result in Morse theory which explains the effect of the associated Homology groups on the set $K_{J,(-\infty,a]}=\\{x\in V:f(x)\leq a\\}$. ###### Theorem 3.1. Let $J\in C^{2}(V)$ satisfy the Palais-Smale condition and let ‘$a$’ be a regular value of $J$. Then if, $H_{*}(V,J^{a})\neq 0$, implies that $K_{J,(-\infty,a]}\neq\emptyset$. ###### Remark 3.2. Before we apply the Morse lemma we recall that for a Morse function the following holds 1. 1. $H_{*}(J^{c},f^{c}\setminus\text{Crit}(J,c))=\oplus_{j}H_{*}(J^{c}\cap N_{j},J^{c}\cap N_{j}\setminus\\{x_{j}\\}),$ where $\text{Crit}(J,c)=\\{x\in V:J(x)=c,J^{\prime}(x)=0\\}$, $N_{j}$ is a neighbourhood of $x_{j}$. 2. 2. $H_{k}(J^{c}\cap N,J^{c}\cap N\setminus\\{x\\})=\begin{cases}\mathbb{R},&k=m(x)\\\ 0,&\text{otherwise}\end{cases}$ where $m(x)$ is a Morse index of $x$, a critical point of $J$. 3. 3. Further $H_{k}(J^{a},J^{b})=\oplus_{\\{i:m(x_{i})=k\\}}\mathbb{R}=\mathbb{R}^{m_{k}(a,b)}$ where $m_{k}(a,b)=n(\\{i:m(x_{i})=k,x_{i}\in K_{J,(a,b)}\\})$. Here $n(S)$ denotes the number of elements present in the set $S$. 4. 4. Morse relation $\sum_{u\in K_{J,[a,b]}}\sum_{k\geq 0}\text{dim}(C_{k}(J,u))t^{k}=\sum_{k\geq 0}\text{dim}(H_{k}(J^{a},J^{b}))t^{k}+(1+t)\mathcal{Q}_{t}$ for all $t\in\mathbb{R}$. Here $Q_{t}$ is a nonnegative polynomial in $\mathbb{N}_{0}[t]$. ###### Theorem 3.3. The functional $I_{\alpha}$ has at least one nontrivial critical point when $0<\lambda\leq\lambda_{1}$, $\lambda_{1}$ being the first eigen value of $(-\Delta)_{p}$. ###### Proof. We observe that $I_{\alpha}(tu)\rightarrow-\infty$ as $t\rightarrow\infty$. A key observation here is that there exists $R$ sufficiently small such that $I_{\alpha}(u)\geq\alpha>0$ whenever $\|u\|=R$. We choose $\epsilon>0$ such that $c=\epsilon$ is a regular value of $I_{\alpha}$. Thus, $I_{\alpha}^{\epsilon}$ is not path connected since it has at least two path connected components namely in the form of a neighbourhood of $0$ and a set $\\{u:\|u\|\geq R\\}$ for $R$ suffciently large. From the theory of homology groups we get that $\text{dim}(H_{0}(I_{\alpha}^{\epsilon}))\geq 2$, ‘dim’ denoting the dimension of the Homology group. From the Definition 2.1, let us consider the following exact sequence $...\rightarrow H_{1}(W_{0}^{1,p(x)}(\Omega),I_{\alpha}^{\epsilon})\xrightarrow[]{\partial_{1}}H_{0}(I_{\alpha}^{\epsilon},\emptyset)\xrightarrow[]{i_{0}}H_{0}(W_{0}^{1,p(x)}(\Omega),\emptyset)\rightarrow...$ Obviously $\text{dim}(H_{0}(W_{0}^{1,p(x)}(\Omega),\emptyset))=1$ and $\text{dim}(H_{0}(I_{\alpha}^{\epsilon}))\geq 2$. Due to the exactness of the sequence we conclude that $\text{dim}H_{1}(W_{0}^{1,p(x)}(\Omega),I_{\alpha}^{\epsilon})\geq 1$. Thus by the Remark we have $K_{I_{\alpha},(-\infty,\epsilon]}\neq\emptyset$. Suppose that the only critical point to (1.1) is $u=0$ at which the energy of the functional $I_{\alpha}$ is also $0$. Thus from the discussion above and the Remark (3.2)-(4) we have from the Morse relation we have the following identity over $\mathbb{R}$ $1=t+\mathcal{P}(t)+(1+t)\mathcal{Q}_{t},$ $g$ being a power series in $t$, $\mathcal{Q}_{t}\geq 0$. This is a contradiction. Thus there exists at least one $u\neq 0$ which is a critical point to $I_{\alpha}$ whenever $\lambda\leq\lambda_{1}$. ∎ ###### Definition 3.4 (Krasnoselskii genus). Let $V$ be a Banach space and $S\subset V$. A set $S$ is said to be symmetric if $u\in S$ implies $-u\in S$. Let $S$ be a close, symmetric subset of $V$ such that $0\notin S$. We define a genus $\gamma(S)$ of $S$ by the smallest integer $k$ such that there exists an odd continuous mapping from $S$ to $\mathbb{R}^{k}\setminus\\{0\\}$. We define $\gamma(S)=\infty$, if no such $k$ exists. To each closed and symmetric subsets $M$ of $W_{0}^{1,p}(\Omega)$ with the Krasnoselskii genus $\gamma(M)\geq k$, define $\lambda_{k}=\inf_{M\in\mathfrak{F}_{k}}\sup_{u\in M}I_{\alpha}(u).$ Here $\mathfrak{F}_{k}=\\{M\subset W_{0}^{1,p}(\Omega),~{}\text{closed and symmetric}:\gamma(M)\geq k\\}$. A natural question at this point will be to ask if the same conclusion can be drawn when $\lambda_{k}<\lambda\leq\lambda_{k+1}$. We will define $\lambda_{0}=0$. The next theorem answers this question. ###### Theorem 3.5. The problem in (1.1) has at least one nontrivial solution when $\lambda_{k}<\lambda\leq\lambda_{k+1}$, $\lambda_{k}$ being as defined above. ###### Proof. We at first show that $H_{k}(W_{0}^{1,p}(\Omega),I_{\alpha}^{-a})=0$ for all $k\geq 0$. Pick a $u\in\\{v:\|v\|=1\\}=\partial B^{\infty}$, where $B^{\infty}=\\{v:\|v\|\leq 1\\}$. Then $I_{\alpha}(tu)=\int_{\Omega}\frac{|\nabla(tu)|^{p}}{p}dx+\int_{\Omega}G\left(\frac{tu-1}{\alpha}\right)dx-\lambda\int_{\Omega}\frac{(tu-1)_{+}^{q}}{q}dx<-a<0$ for all $t\geq t_{0}$. It can be easily seen that for a fixed $u$, we have $I^{\prime}(tu)>0$. Further, for any $t\geq t_{0}$ we have $I_{\alpha}(tu)<-a<0$. Thus, there exists $t(u)$ such that $I_{\alpha}^{\prime}(tu)=0$ by the continuity of $I_{\alpha}^{\prime}$. We can thus say that there exists a $C^{1}$-function $T:W_{0}^{1,p}(\Omega)\setminus\\{0\\}\rightarrow\mathbb{R}^{+}$. We now define a standard deformation retract $\eta$ of $W_{0}^{1,p}(\Omega)\setminus B_{R^{\prime}}(0)$ into $I_{\alpha}^{-a}$ as follows (refer Definition 2.2). $\eta(s,u)=\begin{cases}(1-s)u+sT\left(\frac{u}{\|u\|}\right)\frac{u}{\|u\|},&\|u\|\geq R^{\prime},I_{\alpha}(u)\geq-a\\\ u,&I_{\alpha}(u)\leq-a.\end{cases}$ It is not difficult to see that $\eta$ is a $C^{1}$ function over $[0,1]\times W_{0}^{1,p}(\Omega)\setminus B_{R^{\prime}}(0)$. On using the map $\delta(s,u)=\dfrac{u}{\|u\|}$, for $u\in W_{0}^{1,p}(\Omega)\setminus B_{R^{\prime}}(0)$ we claim that $H_{k}(W_{0}^{1,p}(\Omega),W_{0}^{1,p}(\Omega)\setminus B_{r}(0))=H_{k}(B^{\infty},S^{\infty})$ for all $k\geq 0$. This is because, $H_{k}(B^{\infty},S^{\infty})\cong H_{k}(*,0)$. From elementary computation of homology groups with two $0$-dimensional simplices it is easy to see that $H_{k}(*,0)=\\{0\\}$ for each $k\geq 0$. A result in [10] says that $C_{m}(I,u)=\begin{cases}\mathbb{R},&\text{if}~{}m(u)=m\\\ 0,&\text{otherwise}\end{cases}$ Therefore, from the Morse relation in the Remark (3.2)-4 and the result above, we have for $b>0$ $\displaystyle\sum_{u\in K_{I,[-a,\infty)}}\sum_{k\geq 0}\text{dim}(C_{k}(I,u))t^{k}$ $\displaystyle=t^{m(u)}+p(t)$ (3.1) where $m(u)$ is the Morse index of $u$ and $\mathcal{P}(t)$ contains the rest of the powers of $t$ corresponding to the other critical points, if any. The Morse index is finite because of the following reason. From the argument which helped in establishing a ‘maxima’, say $u_{0}$, using the mountain pass geometry around $0$, we had to assume $\lambda<C^{-q}\frac{q}{p}\|u\|^{p-q}$. Owing to $u_{0}$ being a maxima, we have $I_{\alpha}^{\prime\prime}(u_{0})<0$ which necessarily requires $\lambda>C^{-q}\frac{p-1}{q-1}\|u\|^{p-q}$. Thus we have $C^{-q}\frac{p-1}{q-1}\|u\|^{p-q}<\lambda<C^{-q}\frac{q}{p}\|u\|^{p-q}.$ This implies that $\lambda_{i}<\lambda<\lambda_{j}$ for some $i,j\in\mathbb{N}_{0}$. On further using the the Morse relation we obtain $\displaystyle t^{m(u)}+\mathcal{P}(t)$ $\displaystyle=(1+t)\mathcal{Q}_{t}.$ (3.2) This is because the $H_{k}$s are all trivial groups. Hence, $Q_{t}$ either contains $t^{m(u)}$ or $t^{m(u)-1}$ or both. Thus there exists at least one nontrivial $u\in K_{I_{\alpha},[-a,\infty)}$ with $m(u)\leq n+1$. ∎ ###### Remark 3.6. If $0<\lambda\leq\lambda_{k+1}$, then there exists at least $k$ solutions to the equation (1.1). ## 4 Existence of solution to the main problem (1.1) and smoothness of the boundary $\partial\\{u>1\\}$ ###### Lemma 4.1. Let $\alpha_{j}\rightarrow 0$ ($\alpha_{j}>0$) as $j\rightarrow\infty$ and $u_{j}$ be a critical point of $I_{\alpha_{j}}$. If $(u_{j})$ is bounded in $W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$, then there exists $u$, a Lipschitz continuous function, on $\bar{\Omega}$ such that $u\in W_{0}^{1,p}(\Omega)\cap C^{2}(\bar{\Omega}\setminus H(u))$ and a subsequence (still denoted by $(u_{j})$) such that 1. (i) $u_{j}\rightarrow u$ uniformly over $\bar{\Omega}$, 2. (ii) $u_{j}\rightarrow u$ locally in $C^{1}(\bar{\Omega}\setminus\\{u=1\\})$, 3. (iii) $u_{j}\rightarrow u$ strongly in $W_{0}^{1,p}(\Omega)$, 4. (iv) $I(u)\leq\lim\inf I_{\alpha_{j}}(u_{j})\leq\lim\sup I_{\alpha_{j}}(u_{j})\leq I(u)+|\\{u=1\\}|$, i.e. $u$ is a nontrivial function if $\lim\inf I_{\alpha_{j}}(u_{j})<0$ or $\lim\sup I_{\alpha_{j}}(u_{j})>0$. Furthemore, $u$ satisfies $-\Delta_{p}u=\lambda\chi_{\\{u>1\\}}(x)(u-1)_{+}^{q-1}$ classically in $\Omega\setminus H(u)$, the free boundary condition is satisfies in the generalized sense and vanishes continuously on $\partial\Omega$. In the case of $u$ being nontrivial, then $u>0$ in $\Omega$, the set $\\{u<1\\}$ is connected and the set $\\{u>1\\}$ is nonempty. An important result that will be used to pass the limit in the proof of the Lemma 4.1 is the following theorem which is in line to the theorem due to Caffarelli et al. in [6, Theorem $5.1$]. ###### Lemma 4.2. Let $u$ be a Lipschitz continuous function on the unit ball $B_{1}(0)\subset\mathbb{R}^{N}$ satisfying the distributional inequalities $\pm\Delta_{p}u\leq A\left(\dfrac{1}{\alpha}\chi_{\\{|u-1|<\alpha\\}}(x)+1\right)$ for constants $A>0$ and $0<\alpha\leq 1$. Then there exists a constant $C>0$ depending on $N,A$ and $\int_{{B_{1}}(0)}u^{p}dx$, but not on $\alpha$, such that $\underset{x\in B_{\frac{1}{2}}(0)}{\text{esssup}}\\{|\nabla u(x)|\\}\leq C.$ ###### Proof. Given that $u$ is a Lipschitz continuous function on the unit ball $B_{1}(0)\subset\mathbb{R}^{N}$, so $u$ is also bounded in the unit ball say by a constant $M_{0}$. Not just that, $u$ is also differentiable a.e. in $B_{1}(0)$. We will prove the result stated in the lemma for $u_{+}$, as the proof for $u_{-}$ will follow suit. Denote $v(x)=\frac{15}{\alpha}u_{-}(\alpha x/15)$ and $v_{1}=v+\underset{B_{1/4}}{\max}\\{v^{-}\\}.$ Therefore, $0\leq v_{1}\leq M_{1}$. Let us choose a test function $\eta\in C_{0}^{\infty}(B_{1/4})$ which is such that $0\leq\eta\leq 1$ in $B_{3/4}$ and $\eta=1$ in $B_{1/2}$. Thus $\displaystyle\begin{split}\int_{\Omega}\eta^{p}|\nabla v_{1}|^{p}=&-\int_{\Omega}(pv_{1}\eta^{p-1}|\nabla v_{1}|^{p-2}(\nabla v_{1}\cdot\nabla\eta)+\eta^{p}v_{1}\Delta_{p}v_{1}dx)dx\\\ \leq&\frac{1}{p}\int_{\Omega}\eta^{p}|\nabla v_{1}|^{p}dx+p\int_{\Omega}v_{1}^{p}|\nabla\eta|^{p}dx\\\ &+AM_{1}\int_{\Omega}\eta^{p}\left(\frac{1}{\alpha}\chi_{\\{|u-1|<\alpha\\}}(x)+1\right)dx\\\ \leq&\frac{1}{p}\int_{\Omega}\eta^{p}|\nabla v_{1}|^{p}dx+pM_{1}^{p}\int_{\Omega}|\nabla\eta|^{p}dx\\\ &+AM_{1}\int_{\Omega}\eta^{p}\left(\frac{1}{\alpha}\chi_{\\{|u-1|<\alpha\\}}(x)+1\right)dx.\end{split}$ (4.1) It is now established that $\displaystyle\frac{p-1}{p}\int_{B_{1/2}}|\nabla v_{1}|^{p}dx$ $\displaystyle\leq M_{2}.$ (4.2) However, $u$ being Lipschitz continuous, the gradient $\nabla u$ is bounded a.e. in $B_{1}(0)$ and hence in $B_{1/2}(0)$. Thus $\underset{B_{1/2}(0)}{\text{esssup}}\\{|\nabla u|\\}\leq C$, for some $C>0$. ∎ ###### Proof of Lemma 4.1. Let $0<\alpha_{j}<1$. Consider the problem sequence $(P_{j})$ $\displaystyle\begin{split}-\Delta_{p}u_{j}&=-\frac{1}{\alpha_{j}}g\left(\frac{(u_{j}-1)_{+}}{\alpha_{j}}\right)+\lambda(u-1)_{+}^{q-1}~{}\text{in}~{}\Omega\\\ u_{j}&>0~{}\text{in}~{}\Omega\\\ u_{j}&=0~{}\text{on}~{}\partial\Omega.\end{split}$ (4.3) The nature of the problem being a sublinear one allows us to conclude by an iterative technique that the sequence $(u_{j})$ is bounded in $L^{\infty}(\Omega)$. Therefore, there exists $C_{0}$ such that $0\leq g\left(\frac{(u_{j}-1)_{+}}{\alpha_{j}}\right)(u-1)_{+}^{q-1}\leq C_{0}$. Let $\varphi_{0}$ be a solution of $\displaystyle\begin{split}-\Delta_{p}\varphi_{0}&=\lambda C_{0}~{}\text{in}~{}\Omega\\\ \varphi_{0}&=0~{}\text{on}~{}\partial\Omega.\end{split}$ (4.4) Now since $g\geq 0$, we have that $-\Delta_{p}u_{j}\leq\lambda C_{0}=-\Delta\varphi_{0}$ in $\Omega$. Therefore by the maximum principle, $\displaystyle 0\leq u_{j}(x)\leq\varphi_{0}(x)~{}\forall x\in\Omega.$ (4.5) Since $\\{u_{j}\geq 1\\}\subset\\{\varphi_{0}\geq 1\\}$, hence $\varphi_{0}$ gives a uniform lower bound, say $d_{0}$, on the distance from the set $\\{u_{j}\geq 1\\}$ to $\partial\Omega$. Thus $(u_{j})$ is bounded with respect to the $C^{2,a}$ norm. Therefore, it has a convergent subsequence in the $C^{2}$-norm in a $\dfrac{d_{0}}{2}$ neighbourhood of the boundary $\partial\Omega$. Obviously $0\leq g\leq 2\chi_{(-1,1)}$ and hence $\displaystyle\begin{split}\pm\Delta u_{j}&=\pm\frac{1}{\alpha_{j}}g\left(\frac{(u_{j}-1)_{+}}{\alpha_{j}}\right)\mp\lambda(u_{j}-1)_{+}^{q-1}\\\ &\leq\frac{2}{\alpha_{j}}\chi_{\\{|u_{j}-1|<\alpha_{j}\\}}(x)+\lambda C_{0}.\end{split}$ (4.6) Since, $(u_{j})$ is bounded in $L^{2}(\Omega)$ and by Lemma 4.2 it follows that there exists $A>0$ such that $\displaystyle\underset{x\in B_{\frac{r}{2}}(x_{0})}{\text{esssup}}\\{|\nabla u_{j}(x)|\\}$ $\displaystyle\leq\frac{A}{r}$ (4.7) for a suitable $r>0$ such that $B_{r}(0)\subset\Omega$. However, since $(u_{j})$ is a sequence of Lipschitz continuous functions that are also $C^{1}$, therefore $\displaystyle\underset{x\in B_{\frac{r}{2}}(x_{0})}{\sup}\\{|\nabla u_{j}(x)|\\}$ $\displaystyle\leq\frac{A}{r}.$ (4.8) Thus $(u_{j})$ is uniformly Lipschitz continuous on the compact subsets of $\Omega$ such that its distance from the boundary $\partial\Omega$ is at least $\frac{d_{0}}{2}$ units. Thus by the Ascoli-Arzela theorem applied to $(u_{j})$ we have a subsequence, still named the same, such that it converges uniformly to a Lipschitz continuous function $u$ in $\Omega$ with zero boundary values and with strong convergence in $C^{2}$ on a $\frac{d_{0}}{2}$-neighbourhood of $\partial\Omega$. By the Eberlein-Šmulian theorem we conclude that $u_{j}\rightharpoonup u$ in $W_{0}^{1,p}(\Omega)$. We now prove that $u$ satisfies $\displaystyle-\Delta_{p}u$ $\displaystyle=\alpha\chi_{\\{u>1\\}}(x)(u-1)_{+}^{q-1}$ (4.9) in the set $\\{u\neq 1\\}$. Let $\varphi\in C_{0}^{\infty}(\\{u>1\\})$ and therefore $u\geq 1+2\delta$ on the support of $\varphi$ for some $\delta>0$. On using the convergence of $u_{j}$ to $u$ uniformly on $\Omega$ we have $|u_{j}-u|<\delta$ for any sufficiently large $j,\delta_{j}<\delta$. So $u_{j}\geq 1+\delta_{j}$ on the support of $\varphi$. On testing (4.9) with $\varphi$ yields $\displaystyle\int_{\Omega}|\nabla u_{j}|^{p-2}\nabla u_{j}\cdot\nabla\varphi dx$ $\displaystyle=\lambda\int_{\Omega}(u_{j}-1)_{+}^{q-1}\varphi dx.$ (4.10) On passing the limit $j\rightarrow\infty$ to (4.9), we get $\displaystyle\int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi dx$ $\displaystyle=\lambda\int_{\Omega}(u_{j}-1)_{+}^{q-1}\varphi dx.$ (4.11) To arrive at (4.11) we have used the weak convergence of $u_{j}$ to $u$ in $W_{0}^{1,p}(\Omega)$ and the uniform convergence of the same in $\Omega$. Hence $u$ is a weak solution of $-\Delta_{p}u=\lambda(u-1)_{+}^{q-1}$ in $\\{u>1\\}$. Since $u$ is a Lipschitz continuous function, hence by the Schauder estimates we conclude that it is also a classical solution of $-\Delta_{p}u=\lambda(u-1)_{+}^{q-1}$ in $\\{u>1\\}$. Similarly on choosing $\varphi\in C_{0}^{\infty}(\\{u<1\\})$ one can find a $\delta>0$ such that $u\leq 1-2\delta$. Therefore, $u_{j}<1-\delta$. Let us now analyze the nature of $u$ in the set $\\{u\leq 1\\}^{\circ}$. On testing (4.9) with any nonnegative function and passing the limit $j\rightarrow\infty$ and using the fact that $g\geq 0$, $G\leq 1$ we can show that $u$ satisfies $\displaystyle-\Delta_{p}u$ $\displaystyle\leq\lambda(u-1)_{+}^{q-1}~{}\text{in}~{}\Omega$ (4.12) in the distributional sense. Furthermore, $\mu=\Delta_{p}u$ is a positive Radon measure supported on $\Omega\cap\partial\\{u<1\\}$ (refer Lemma 4.3 in Appendix). From (4.12), the positivity of the Radon measure $\mu$ and the usage of Section $9.4$ in Gilbarg-Trudinger [8] we conclude that $u\in W_{\text{loc}}^{2,p}(\\{u\leq 1\\}^{\circ})$, $1<p<\infty$. Thus $\mu$ is supported on $\Omega\cap\partial\\{u<1\\}\cap\partial\\{u>1\\}$ and $u$ satisfies $-\Delta_{p}u=0$ in the set $\\{u\leq 1\\}^{\circ}$. In order to prove $(ii)$, we will show that $u_{j}\rightarrow u$ locally in $C^{1}(\Omega\setminus\\{u=1\\})$. Note that we have already proved that $u_{j}\rightarrow u$ in the $C^{2}$ norm in a neighbourhood of $\partial\Omega$ of $\bar{\Omega}$. Suppose $M\subset\subset\\{u>1\\}$. In this set $M$ we have $u\geq 1+2\delta$ for some $\delta>0$. Thus for sufficiently large $j$, with $\delta_{j}<\delta$, we have $|u_{j}-u|<\delta$ in $\Omega$ and hence $u_{j}\geq 1+\delta_{j}$ in $M$. From (4.3) we have $-\Delta_{p}u_{j}=\lambda(u_{j}-1)_{+}^{q-1}~{}\text{in}~{}M.$ Clearly, $(u_{j}-1)_{+}^{q-1}\rightarrow(u-1)_{+}^{q-1}$ in $L^{p}(\Omega)$ for $1<p<\infty$ and $u_{j}\rightarrow u$ uniformly in $\Omega$. This analysis says something more stronger - since $(-\Delta_{p})u_{j}=\lambda(u_{j}-1)_{+}^{q-1}$ in $M$, we have that $u_{j}\rightarrow u$ in $W^{2,p}(M)$. By the embedding $W^{2,p}(M)\hookrightarrow C^{1}(M)$ for $p>2$, we have $u_{j}\rightarrow u$ in $C^{1}(M)$. This shows that $u_{j}\rightarrow u$ in $C^{1}(\\{u>1\\})$. Working on similar lines we can also show that $u_{j}\rightarrow u$ in $C^{1}(\\{u<1\\})$. We will now prove $(iii)$. Since $u_{j}\rightharpoonup u$ in $W_{0}^{1,p}(\Omega)$, we have that by the weak lower semicontinuity of the norm $\|\cdot\|$ that $\|u\|\leq\lim\inf\|u_{j}\|.$ It is sufficient to prove that $\lim\sup\|u_{j}\|\leq\|u\|$. To achieve this, we multiply (4.3) with $(u_{j}-1)$ and then integrate by parts. We will also use the fact that $tg\left(\frac{t}{\delta_{j}}\right)\geq 0$ for any $t\in\mathbb{R}$. This gives, $\displaystyle\begin{split}\int_{\Omega}|\nabla u_{j}|^{p}dx&\leq\lambda\int_{\Omega}f(u_{j}-1)_{+}^{q}dx-\int_{\partial\Omega}\frac{\partial u_{j}}{\partial\hat{n}}dS\\\ &\rightarrow\lambda\int_{\Omega}(u-1)_{+}^{q}dx-\int_{\partial\Omega}\frac{\partial u}{\partial\hat{n}}dS\end{split}$ (4.13) as $j\rightarrow\infty$. Here $\hat{n}$ is the outward drawn normal to $\partial\Omega$. ∎ We choose $\vec{\varphi}\in C_{0}^{1}(\Omega,\mathbb{R}^{N})$ such that $u\neq 1$ a.e. on the support of $\vec{\varphi}$. On multiplying $\nabla u_{n}\cdot\vec{\varphi}$ to the weak formulation of (4.3) and integrating over the set $\\{1-\epsilon^{-}<u_{n}<1+\epsilon^{+}\\}$ gives $\displaystyle\begin{split}\int_{\\{1-\epsilon^{-}<u_{n}<1+\epsilon^{+}\\}}\left[-\Delta_{p}u_{n}+\frac{1}{\alpha_{n}}g\left(\frac{u_{n}-1}{\alpha_{n}}\right)\right]\nabla u_{n}\cdot\vec{\varphi}dx\\\ =\int_{\\{1-\epsilon^{-}<u_{n}<1+\epsilon^{+}\\}}(u_{n}-1)_{+}^{q-1}\nabla u_{n}\cdot\vec{\varphi}dx.\end{split}$ (4.14) The term on the left hand side of (4.14) can be expressed as follows. $\displaystyle\nabla\cdot\left(\frac{1}{p}|\nabla u_{n}|^{p}\vec{\varphi}-(\nabla u_{n}\cdot\vec{\varphi})|\nabla u_{n}|^{p-2}\nabla u_{n}\right)+(\nabla\vec{\varphi}\cdot\nabla u_{n})\cdot\nabla u_{n}|\nabla u_{n}|^{p-2}$ $\displaystyle-\frac{1}{p}|\nabla u_{n}|^{p}\nabla\cdot\vec{\varphi}$ $\displaystyle+\nabla G\left(\frac{u_{n}-1}{\alpha_{n}}\right)\cdot\vec{\varphi}.$ (4.15) Using (4) and on integrating by parts we obtain $\displaystyle\begin{split}\int_{\\{u_{n}=1+\epsilon^{+}\\}\cup\\{u_{n}=1-\epsilon^{-}\\}}\left[\frac{1}{p}|\nabla u_{n}|^{p}\vec{\varphi}-(\nabla u_{n}\cdot\vec{\varphi})|\nabla u_{n}|^{p-2}\nabla u_{n}+G\left(\frac{u_{n}-1}{\alpha_{j}}\right)\hat{\varphi}\right]\cdot\hat{n}dS\\\ =\int_{\\{1-\epsilon^{-}<u_{n}<1+\epsilon^{+}\\}}\left(\frac{1}{p}|\nabla u_{n}|^{p}\nabla\cdot\vec{\varphi}-(\nabla\vec{\varphi}\cdot\nabla u_{n})|\nabla u_{n}|^{p-2}\nabla u_{n}\right)dx\\\ +\int_{\\{1-\epsilon^{-}<u_{n}<1+\epsilon^{+}\\}}\left[G\left(\frac{u_{n}-1}{\alpha_{n}}\right)\nabla\cdot\vec{\varphi}+\lambda(u_{n}-1)_{+}^{q-1}(\nabla u_{n}\cdot\vec{\varphi})\right]dx.\end{split}$ (4.16) The integral on the left of equation (4.16) converges to $\displaystyle\int_{\\{u_{n}=1+\epsilon^{+}\\}\cup\\{u_{n}=1-\epsilon^{-}\\}}\left(\frac{1}{p}|\nabla u|^{p}\vec{\varphi}-(\nabla u_{n}\cdot\vec{\varphi})|\nabla u_{n}|^{p-2}\nabla u_{n}\right)\cdot\hat{n}dS+\int_{\\{u_{n}=1+\epsilon^{+}\\}}\vec{\varphi}\cdot\hat{n}dS$ (4.17) $\displaystyle=\int_{\\{u_{n}=1+\epsilon^{+}\\}}\left[1-\left(\frac{p-1}{p}\right)|\nabla u_{n}|^{p}\right]\vec{\varphi}\cdot\hat{n}dS-\int_{\\{u_{n}=1-\epsilon^{-}\\}}\left(\frac{p-1}{p}\right)|\nabla u_{n}|^{p}\vec{\varphi}\cdot\hat{n}dS.$ (4.18) Thus the equation (4.17) under the limit $\epsilon\rightarrow 0$ becomes $\displaystyle 0=\underset{\epsilon\rightarrow 0}{\lim}\int_{\\{u=1+\epsilon^{+}\\}}\left[\left(\frac{p}{p-1}\right)-|\nabla u|^{p}\right]\vec{\varphi}\cdot\hat{n}dS-\underset{\epsilon\rightarrow 0}{\lim}\int_{\\{u=1-\epsilon^{-}\\}}|\nabla u|^{p}\vec{\varphi}\cdot\hat{n}dS$ (4.19) This is because $\hat{n}=\pm\dfrac{\nabla u}{|\nabla u|}$ on the set $\\{u=1+\epsilon^{+}\\}\cup\\{u=1-\epsilon^{-}\\}$. This proves that $u$ satisfies the free boundary condition. The solution cannot be trivial as it satisfies the free boundary condition. Thus a solution to (1.1) exists that obeys the free boundary condition besides the Dirichlet boundary condition. ## Appendix ###### Lemma 4.3. $u$ is in $W_{\text{loc}}^{1,p}(\Omega)$ and the Radon measure $\mu=\Delta_{p}u$ is nonnegative and supported on $\Omega\cap\\{u<1\\}$. ###### Proof. We follow the proof due to Alt-Caffarelli [1]. Choose $\delta>0$ and a test function $\varphi^{p}\chi_{\\{u<1-\delta\\}}$ where $\varphi\in C_{0}^{\infty}(\Omega)$. Therefore, $\displaystyle\begin{split}0&=\int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla(\varphi^{p}\min\\{u-1+\delta,0\\})dx\\\ &=\int_{\Omega\cap\\{u<1-\delta\\}}|\nabla u|^{p-2}\nabla u\cdot\nabla(\varphi^{p}\min\\{u-1+\delta,0\\})dx\\\ &=\int_{\Omega\cap\\{u<1-\delta\\}}|\nabla u|^{p}\varphi^{p}dx+p\int_{\Omega\cap\\{u<1-\delta\\}}\varphi^{p-1}(u-1+\delta)|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi dx,\end{split}$ (4.20) and so by Caccioppoli like estimate we have $\displaystyle\begin{split}\int_{\Omega\cap\\{u<1-\delta\\}}|\nabla u|^{p}\varphi^{p}dx&=-p\int_{\Omega\cap\\{u<1-\delta\\}}\varphi^{p-1}(u-1+\delta)|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi dx\\\ &\leq c\int_{\Omega}u^{p}|\nabla\varphi|^{p}dx.\end{split}$ (4.21) Since $\int_{\Omega}|u|^{p}dx<\infty$, therefore on passing the limit $\delta\rightarrow 0$ we conclude that $u\in W_{\text{loc}}^{1,p}(\Omega)$. Furthermore, for a nonnegative $\zeta\in C_{0}^{\infty}(\Omega)$ we have $\displaystyle\begin{split}-\int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla\zeta dx=&\left(\int_{\Omega\cap\\{0<u<1-2\delta\\}}+\int_{\Omega\cap\\{1-2\delta<u<1-\epsilon\\}}+\int_{\Omega\cap\\{1-\delta<u<1\\}}\right.\\\ &\left.+\int_{\Omega\cap\\{u>1\\}}\right)\\\ &\left[|\nabla u|^{p-2}\nabla u\cdot\nabla\left(\zeta\max\left\\{\min\left\\{2-\frac{1-u}{\delta},1\right\\},0\right\\}\right)\right]dx\\\ \geq&\int_{\Omega\cap\\{1-2\delta<u<1-\delta\\}}\left[|\nabla u|^{p-2}\nabla u\cdot\left(2-\frac{1-u}{\delta}\right)\nabla\zeta+\frac{\zeta}{\delta}|\nabla u|^{p}\right]dx\geq 0.\end{split}$ (4.22) On passing the limit $\delta\rightarrow 0$ we obtain $\Delta_{p}(u-1)_{-}\geq 0$ in the distributional sense and hence there exists a Radon measure $\mu$ (say) such that $\mu=\Delta(u-1)_{-}\geq 0$. ∎ ## Acknowledgement The author thanks the community of the free boundary value problems for injecting a new lease of life to the study of elliptic PDEs and the CSIR, India (25(0292)/18/EMR-II) for the financial support. ## References * [1] Alt, H.W., Caffarelli, L.A., Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325, 105-144, 1981. * [2] Batchelor, G.K., On steady state laminar flow with closed streamlines at large Reynolds number, J. Fluid mech., 1, 177-190, 1956. * [3] Batchelor, G.K., A proposal concerning laminar wakes behind bluff bodies at large Reynolds number, J. Fluid mech., 1, 388-398, 1956. * [4] Caflisch, R.E., Mathematical analysis of vortex dynamics. In: Mathematical Aspects of Vortex Dynamics (Leesburg VA, 1988), pp 1-24, SIAM, Philadelphia, PA, 1989. * [5] Caffarelli, L.A., Peral, I., On $W^{1,p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51, 1-21, 1998. * [6] Caffarelli, L.A., Jerison, D., Kenig, C.E., Some new monotonicity theorems with applications to free boundary problems, Ann. Math. (2), 155(2), 369-404, 2002. * [7] Elcrat, A.R., Miller, K.G., Variational formulas on Lipschitz domains, Trans. Am. Math. Soc., 347(7), 2669-2678, 1995. * [8] Gilbarg, D., Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg, 2001. * [9] Jerison, D., Perera, K., A multiplicity result for the Prandtl-Batchelor free boundary problem (Preprint arXiv:2003.05921). * [10] Kanishka Perera, Ravi P. Agarwal and Donald O’Regan, Morse Theoretic Aspects of $p$-Laplacian Type Operators, Mathematical surveys and Monographs, Amer. Math. Soc., 161, 2010. * [11] Kesavan, S., Topics in functional analysis and applications, New Age International (P) Ltd., 2003. * [12] Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš, Nonlinear Analysis - Theory and Methods, Springer, 2019. * [13] Perera, K., On a class of elliptic free boundary problems with multiple solutions, Nonlinear Differ. Equ. Appl., 28, Art: 36, 2021.

2024-09-04T02:54:58.480708

2003.04294

arxiv-papers

# Optimizing Streaming Parallelism on Heterogeneous Many-Core Architectures: A Machine Learning Based Approach Peng Zhang, Jianbin Fang, Canqun Yang, Chun Huang, Tao Tang, Zheng Wang Peng Zhang, Jianbin Fang, Canqun Yang, Chun Huang, and Tao Tang are with National University of Defense Technology, China. E-mail: {zhangpeng13a, j.fang, canqun<EMAIL_ADDRESS>Zheng Wang is with University of Leeds, United Kingdom. E-mail<EMAIL_ADDRESS> ###### Abstract As many-core accelerators keep integrating more processing units, it becomes increasingly more difficult for a parallel application to make effective use of all available resources. An effective way for improving hardware utilization is to exploit spatial and temporal sharing of the heterogeneous processing units by multiplexing computation and communication tasks – a strategy known as heterogeneous streaming. Achieving effective heterogeneous streaming requires carefully partitioning hardware among tasks, and matching the granularity of task parallelism to the resource partition. However, finding the right resource partitioning and task granularity is extremely challenging, because there is a large number of possible solutions and the optimal solution varies across programs and datasets. This article presents an automatic approach to quickly derive a good solution for hardware resource partition and task granularity for task-based parallel applications on heterogeneous many-core architectures. Our approach employs a performance model to estimate the resulting performance of the target application under a given resource partition and task granularity configuration. The model is used as a utility to quickly search for a good configuration at runtime. Instead of hand-crafting an analytical model that requires expert insights into low-level hardware details, we employ machine learning techniques to automatically learn it. We achieve this by first learning a predictive model offline using training programs. The learnt model can then be used to predict the performance of any unseen program at runtime. We apply our approach to 39 representative parallel applications and evaluate it on two representative heterogeneous many-core platforms: a CPU-XeonPhi platform and a CPU-GPU platform. Compared to the single-stream version, our approach achieves, on average, a 1.6x and 1.1x speedup on the XeonPhi and the GPU platform, respectively. These results translate to over 93% of the performance delivered by a theoretically perfect predictor. ###### Index Terms: Heterogeneous computing; Parallelism; Performance Tuning; Machine learning ## 1 Introduction Heterogeneous many-cores, as representative by GPGPUs and Intel’s XeonPhi, are widely used for accelerating parallel applications [1, 2, 3]. As users demand higher performance, many-core accelerators have become more powerful by providing more and more processing units. While the abundant computing resources offer the potential for higher performance, it becomes harder for a parallel application to utilize all the available computing resources [4, 5]. As a result, many parallel applications fail to fully unlock the performance potential of a many-core accelerator. One way for improving heterogeneous many-core utilization is to exploit spatial and temporal sharing of processing resources. This strategy is also known as heterogeneous streaming [6]. The idea is to exploit the computation and communication independency of task parallelism to improve hardware utilization. It works by partitioning the processor cores to allow independent communication and computation tasks (i.e. streams) to run concurrently on different hardware resources, which effectively overlaps the concurrent kernel execution with data movements. Representative heterogeneous streaming implementations include CUDA Streams [7], OpenCL Command Queues [8], and Intel heterogeneous streams library (hStreams) [9, 6]. These implementations allow a parallel program to spawn more than one stream (or pipeline) so that the data movement stage of one pipeline overlaps the kernel execution stage of another. Prior work on heterogeneous streaming mainly targets GPUs [10, 11, 12]. Compared to GPU implementations, OS-enabled coprocessors, like the Intel XeonPhi, provides some unique features that are currently unavailable on the GPU. For example, besides specifying the number of streams, developers can explicitly map streams to different groups of cores on XeonPhi to control the number of cores of each hardware partition. This parameter is not exposed to programmers on GPUs, making previous work on GPU-based parallel streaming optimizations infeasible to fully exploit Xeon-Phi-like many-core accelerators (see also Section 6.3). On the other hand, ample evidence is showing that choosing the right stream configuration, i.e., the number of processor core partitions and the number of concurrent tasks of a multi-stream application, values, has a significant impact the application’s performance on many-core architectures [13, 14, 15]. However, attempting to find the optimal values through exhaustive profiling would be ineffective, because the range of the possible values for the two parameters is huge. What we need is a technique that automatically determines the optimal stream configuration for any streamed application in a fast manner. This article presents a novel approach to determine the right number of processor core partitions and tasks for heterogeneous streams, targeting heterogeneous many-core architectures. Our key insight is to use a performance model to quickly search for the optimal stream configuration. The performance model estimates the resulting performance of the target streamed application when it runs under a given stream configuration. If the prediction can be performed quickly with low overhead, we can then quickly explore a large configuration space. Instead of hand-crafting the performance model that requires human modification whenever the architecture evolves (i.e., when the number and types of cores change), we employ machine learning techniques to automatically construct a predictive model. Our predictor is first trained _off-line_. Then, using code and dynamic runtime features of the program, the model predicts performance for a _new_ , _unseen_ program under a given stream configuration. Our prior work [16] develops a machine learning based classifier to predict the optimal stream configuration. However, this approach can only choose from a limited set of configurations seen during the training phase. Unlike a classification-based approach, the approach presented in the article allows us to explore a larger number of stream configurations (including those that are not seen during the training phase) with negligible runtime overhead. This advantage significantly improves the generalization ability of the proposed approach (Section 3). Due to the newness of heterogeneous streaming execution model, there are very few multi-stream benchmarks available. To evaluate our approach on a wide range of applications, we have developed a compiler-based tool to automatically translate standard OpenMP benchmarks into their streamed variants for the backends of XeonPhi and GPU architectures (Section 4). With the help of this code generator, we can apply our approach to 39 parallel benchmarks. We argue that this tool can help generate more streamed code and thus is an added value to the community. We evaluate our approach on two representative heterogeneous many-core platforms: a 57-core Intel XeonPhi and an NVIDIA 1080Ti GPU platforms. We achieve, on average, a 1.6x and 1.1x speedup over the single-stream execution on the XeonPhi and the GPU platforms, respectively. This translates to over 93% of the best available performance. The core contribution of this paper is a novel machine-learning-guided approach for automatically determining the optimal stream configuration on heterogeneous many-cores. We show that our approach delivers good performance across benchmarks and heterogeneous many-core platforms. While we do not seek to advance the machine learning algorithm itself, our work shows how machine learning can be used to address the challenging problem of tuning fine-grained streaming parallelism on heterogeneous many-core architectures. In this work, we demonstrate the usefulness of our approach on XeonPhi and an NVIDIA GPU, but our approach is equally applicable on other heterogeneous platforms like AMD GPUs. ## 2 Background and Overview In this section, we first give a brief introduction of heterogeneous streaming; we then define the scope of this work, before motivating the need of our scheme and providing an overview of our approach. ### 2.1 Heterogeneous Streaming The idea of heterogeneous streaming is to exploit spatial and temporal sharing of computing resources to utilize the hardware resources to improve application performance. Spatial Sharing. Modern many-core accelerators offer a large number of processing units. Since many applications cannot fully utilize all the cores at a time, we can partition the computing units into multiple groups to concurrently execute multiple tasks. In this way, the computing resource is spatially shared across concurrently-running application tasks. The key to spatial sharing is to determine the right number of partitions, because over- provisioning of processing units would waste computing resources but under- provisioning would lead to slowed down performance. Temporal Sharing. Code written for heterogeneous computing devices typically consists of several stages, such as host device communication and computation. Using temporal sharing, one can overlap some of these stages to exploit pipeline parallelism to improve performance by overlapping the host-device communication and kernel execution. ⬇ 1//setting the partition-size and task granularity hStreams_app_init(partition_size,streams_p_part); 3 //stream queue id 5stream_id = 0; for(…){ 7 //enquque host-device transfer to current stream hStreams_app_xfer_memory(,,, stream_id, HSTR_SRC_TO_SINK,…); 9 … //enqueue computation to the current stream 11 hStreams_EnqueueCompute(stream_id, ”kernel1”, …); … 13 //move to the next stream stream_id = (stream_id++) % MAX_STR; 15} //transfer data back to host 17hStreams_app_xfer_memory(,,, HSTR_SINK_TO_SRC,…); Figure 1: Heterogeneous streaming using hStreams as an example. (a) binomial (b) prefixsum Figure 2: Heatmaps show the resultant speedup (over single-stream) of binomial and prefixsum under different stream configurations. The #partitions and #tasks have a significant impact on the resultant performance, and the sweet spots are sparse and vary across programs. ### 2.2 Problem Scope Our work aims to improve the performance of a data parallel application by exploiting spatial and temporal sharing of heterogeneous streams. We do so by determining at runtime how many partitions should be used to group the cores (_#partitions_) and how many data parallel tasks (_#tasks_) should be used to run the application. Our current implementation is applicable to XeonPhi and GPUs by using different runtime back-ends (hStream for XeonPhi, and CUDA or OpenCL for GPUs). Code Example. Figure 1 gives a simplified code example written with Intel’s hStreams APIs that can run on the XeonPhi many-core. At line 2 we initialize the stream execution by setting the number of partitions and tasks/streams per partition. This initialization process essentially creates multiple processor domains and determines how many logical streams can run on a partition. In the _for_ loop (lines 7-14) we enqueue the communication and computation tasks to a number of streams identified by the stream_id variable. In this way, communication and computation of different streams can be overlapped during execution (temporal sharing); and streams on different processor domains (or partitions) can run concurrently (spatial sharing). Our predictive model determines the #partitions and the #tasks before invoking the hStreams initialization routine, hStreams_app_init(). Figure 3: Color table showing the speedups of best-performing configurations across inputs for dct. Each cell shows the performance for one of the 16 best- performing configurations, $Cn$, on a given input, $Dn$. The best configuration varies across inputs and a good configuration on one input can give poor performance on another dataset. ### 2.3 Motivating Examples Consider Figure 2 which shows the resultant performance improvement given by multi-stream parallelism over the single-stream version of the code for two applications on a 57-core Intel XeonPhi system. We use two streamed programs from prior work [13]: binomial computes the price evolution over a given period and prefixSum calculates the prefix sum for a sequence of numbers. It is observed from this example that not all multi-stream configurations give improved performance. As can be seen from the diagrams, the search space of multi-stream configurations is huge but good configurations are sparse. The performance varies significantly over stream configurations (#partitions, #tasks). The optimal #tasks for binomial ranges from 1 to 30, and the best #partitions is between 1 and 40. In contrast to binomial, prefixsum benefits from fine-grained parallelism when using a larger #tasks (220 to 224) and #partitions (60 to 80). However, the stream configurations that are effective for prefixsum give no speedup over the single-stream version for binomial. Now consider Figure 3 that shows the speedups of dct under 16 multi-stream configurations over the single-stream version, where each configuration is found to give the best-performance for one of the 16 inputs. In the color table, each cell shows the performance of a stream configuration ($C1,...,C16$) on a specific input dataset ($D1,...,D16$); and the values along the diagonal line represent the best-available performance (found through profiling) for an input. As can be seen from the figure, the best stream configuration can vary across inputs for the same benchmark. For example, while $C4$ gives a speedup of 1.33x over the baseline for dataset $D4$, it delivers a poor performance for dataset $D14$ by doubling the execution time over the single-stream version. This diagram also suggests that no single configuration can give improved performance for all inputs. Lesson Learned. These two examples show that choosing the stream configuration has a great impact on performance and the best configuration must be determined on a per-program and per-dataset basis. Later, we will show that this observation is not unique to XeonPhi but also holds for GPUs. Attempting to find the optimal configuration through means of an exhaustive search would be ineffective, and the overhead involved would be far bigger than the potential benefits. Online search algorithms, while can speed up the search process, the overhead can still outweigh the benefit. For example, when applying simulated annealing to binomial, the best-found configuration only reaches 84% of the best-available performance after 310,728 iterations111In Section 6.1, we show that our approach achieves 93% of the best-available performance for binomial on XeonPhi.. Classical hand-written heuristics are not ideal either, as they are not only complex to develop, but are likely to fail due to the variety of programs and the ever-changing hardware architecture. An alternate approach, and the one we chose to use, is to use machine learning to automatically construct a performance model to estimate the benefit of any candidate configuration, providing minimal runtime overhead for searching for a good configuration, and having little development cost when targeting new architectures. ### 2.4 Overview of Our Approach Our library-based approach, depicted in Figure 4, is completely automated. To determine the best streaming configuration, our approach follows a number of steps described as follows. We use a set of information or _features_ to capture the characteristics of the program. We develop a LLVM [17] compiler pass to extract static code features at compile time, and a low-overhead profiling pass to collect runtime information at execution time (i.e., during the first few loop iterations). Because profiling also contributes to the final program output, no computation cycle is wasted. At runtime, we search for a good configuration through an offline trained performance model to estimate the resulting performances for all candidate configurations. The performance model takes in the feature values, a given configuration of resource partition and task granularity and estimates the potential speedup for the given configuration over the single-stream version. The overhead of runtime feature collection and search is a few milliseconds, which is included in all our experimental results. Since our training process can be performed automatically, we can easily target our performance model for different architectures. Figure 4: Our machine learning based performance model (trained _offline_) predicts the speedup based on the extracted feature values of the code and a given stream configuration. We use the predictions to quickly rank candidate configurations at runtime to choose the one with the best predicted performance. ## 3 Performance Modeling At the core of our approach is a machine learned performance model built upon the Multi-layer Perceptron (MLP) artificial neural network (ANN). Our prototype is implemented using the Python scikit-learn machine learning package [18]. It is to note that our prior work [16] uses a Support Vector Machine (SVM) based classifier. However, such an approach can only make predictions on a limited set of configurations seen at the training time. Unlike a classification-based approach, the new approach presented in this article is a _regression-based_ model which can make predictions on any stream configuration. This new approach thus has a better generalization ability for various heterogeneous architectures. We have also evaluated a number of alternative modeling techniques, including MLP, SVM, and decision trees. We chose MLP because it gives the best performance and has modest training overhead (see Section 6.6.1). Our performance model takes as input the feature values and a given configuration (e.g., #partitions and #tasks for XeonPhi and #tasks for GPUs). It predicts the speedup for the given configuration. Building and using such a model follows a 3-step process for supervised learning: (i) generate training data (ii) train a performance model (iii) use the performance model, described as follows. Figure 5: The training process of our performance model. ### 3.1 Training the Performance Model Our method for model training is shown in Figure 5. To learn a regression model, we first need to profile the execution time (in order to calculate the speedup over the single-stream version) of all candidate configurations for each training program, and extract the feature values from the program. We then use the feature values, configuration settings and speedups to train a model. #### 3.1.1 Generating Training Data To generate training data, we apply _cross-validation_ to 39 benchmarks, i.e., by excluding the testing benchmarks from the training dataset (see also Section 5.3.1). We execute each training program and benchmark a number of times until the gap of the upper and lower confidence bounds is smaller than 5% under a 95% confidence interval setting. We then calculate the average speedup for a given stream configuration over the single-stream version. We exhaustively execute each training program across a wide range of stream configurations, and record the performance of each. Next, we calculate the speedup for each configuration, program and dataset. Finally, we extract the values of our selected set of features from each program and dataset. We stress that the trained model can be applied to stream configurations that are not seen in the training phase. #### 3.1.2 Profiling Configurations During the training phase, we exhaustively execute each training program across a set of streamed configurations. On XeonPhi, we profile each training program using the _#partitions_ ranging from 1 to 224 (the maximum number of physical threads on XeonPhi) and the _#tasks_ ranging from 1 to 256 222We chose these values because configuration settings beyond these values give a poor performance during our initial evaluation.. On GPUs, we cannot configure the number of partitions currently, we set the _#partitions_ to the same as _#tasks_ to be consistent with XenPhi. On this platform, we also set the _#tasks_ to be range between $2^{0}$ and $2^{10}$, which is big enough to include the optimal values according to our experiments. Note that these parameter ranges can be configured by the user. #### 3.1.3 Building The Model Each evaluated configuration is appended to the feature value vector of a training program to form a model input. The model inputs and the corresponding speedups (i.e., ground truths) for all training programs are passed to a learning algorithm. The algorithm finds a correlation between the input vector and the desired prediction. The output of our learning algorithm is an MLP model where the weights of the model are determined from the training data. Model parameter tuning is performed on the training dataset for each targeting hardware architecture, using cross-validation (see also Section 6.6.3). In our case, the overall training process for all the 39 training programs (which is dominated by training data generation) takes less than a week on a single machine. Since training is performed only once “at the factory”, this is a _one-off_ cost. ### 3.2 Features Our performance models are based exclusively on code and dynamic features of the target programs. Code features are extracted from the program source code, and dynamic features are collected using hardware performance counters during the initial profiling run of the target application. We restrict us in using hardware performance counters that are commonly available on modern processors such as the data cache misses to ensure that our approach can be applied to a wide range of many-core architectures. We considered 38 candidate raw features in this work. Some features were chosen from our intuition based on factors that can affect the performance such as dts (host-device data transfer size) and #xfer_mem, while other features were chosen based on previous work [19, 20]. #### 3.2.1 Feature Selection To build an accurate model through supervised learning, the training sample size typically needs to be at least one order of magnitude greater than the number of features. In this work, we start from 311 training samples and 38 raw features, so we would like to reduce the number of features in use. Our process for feature selection is fully automatic, described as follows. We first combine several raw features to form a set of combined normalized features, which are able to carry more information than the individual parts. For example, instead of reporting raw branch hit and miss counts, we use the branch miss rate. Next, we removed raw features that carried similar information which is already captured by chosen features. To find which features are closely correlated, we constructed a correlation coefficient matrix using the Pearson correlation coefficient [21]. The closer a coefficient between two features is to +/-1, the stronger the correlation between the two input features. We removed any feature which had a correlation coefficient (taking the absolute value) greater than 0.7. Similar features include the number of executed instructions and the number of E-stage cycles that were successfully completed. Our feature selection process reduces the number of features to 10 for XeonPhi (see Table I) and 10 for the NVIDIA Titan 1080Ti GPU (see Table II), where some features are shared. Since our approach for feature selection is automatic, the approach can be applied to other sets of candidate features. It is to note that feature selection is also performed using cross-validation (see also Section 5.2). Table I: Chosen features for XeonPhi performance model Feature | Description ---|--- loop nest | at which level the outermost parallelizable loop lies on loop count | # of the parallel loop iterations #xfer_mem | # of host-device transfer API calls dts | total host-device transfer size redundant transfer size | host-device transfer size among overlapping tasks max blocks | the maximum number of tasks of the application min task unit | the minimum task granularity for a partition # instructions | the total number of instructions of the kernel branch miss | branch miss rate L1 DCR | L1 Data cache miss rate Table II: Chosen features for GPU programs Feature | Description ---|--- Access type 1 | # array access, whose fastest varying index is an affine function of the block id Access type 2 | #array accesses, whose second or higher dimensional index is an affine function of the block id #xfer_mem | # of host-device transfer API calls host to device transfer size | total host to device transfer size device to host transfer size | total device to host transfer size redundant transfer size | host-device transfer size among overlapping tasks max blocks | the maximum number of tasks # instructions | the total number of instructions of the kernel divergent branches | # divergent branches L2 read miss rate | L2 cache read miss rate #### 3.2.2 Feature Standardization Supervised learning typically requires the feature values to lie in a certain range. Therefore, we scaled the value for each of our features between the range of 0 and 1. We record the maximum and minimum value of each feature found at the training phase, and use these values to scale features extracted from a new application after deployment. We truncate a value during deployment if the value is outside the minimum/maximum value range seen during training. It is to note that we also use the same approach to normalize the model predictions (speedups) to the range of 0 and 1. In this work, we choose Z-score to standardize the training data, and the details of quantifying the impact of feature engineering methods can be found in Section 6.6.2. (a) XeonPhi (b) NVIDIA GPU Figure 6: Feature importance on (a) XeonPhi and (b) NVIDIA GPU. #### 3.2.3 Feature Importance To understand the usefulness333In Section 6.6.4, we give a further breakdown of the impact of individual feature to the model performance on a per benchmark basis. of each feature, we apply a factor analysis technique called Varimax rotation [22] to the feature space transformed by the principal component analysis (PCA). This technique quantifies the contribution of each feature to the overall variance in each of the PCA dimensions. Intuitively, the more variances a feature brings to the space, the more useful information the feature carries. As an example, Figure 6 shows the top features chosen for XeonPhi and NVIDIA GPU architectures. For the XeonPhi platform, features that capture the parallelism degree (e.g. max blocks), host-device communication (e.g. redundant transfer size), and computation (e.g. #instructions) are found to be important. Other features such as L1 DCR and loop nest are useful, but are less important compared to others. On the NVIDIA GPU platform, we note that the parallelism degree is important, and the other features are equally useful (Figure 6b). This figure shows that prediction can accurately draw upon a subset of aggregated feature values. ### 3.3 Runtime Deployment Once we have built and trained our performance model as described above, we can use it as a cost function to search for the best stream configuration for any _new_ , _unseen_ program. Feature values are extracted from the single- stream version of the code. Static code features (such as loop count) are extracted from the program source at compile time. Dynamic features (such as branch miss) are extracted by profiling the program without partitioning for a few loop iterations (which typically translate to several microseconds). After feature collection, we feed the feature values to the search engine to rank all candidate configurations using the performance model. The top-ranked stream configuration is then used for the target program. In Section 4.4, we provide further details on how the performance model can be integrated with the host code generation process. #### 3.3.1 Adapt to Changing Program Phases Our current implementation chooses a configuration for each kernel and does not change the configuration throughout the kernel execution. Therefore, it can adapt to different behaviors across kernels because predictions are performed on a per-kernel basis. We found that this strategy is sufficient for many data-parallel kernels targeted in this work. Our approach can be extended to adapt phase or program behavior changes within a kernel. One way of doing this is to first partition the input data into groups and then perform configuration selection before launching the kernel that performs on an input data group. To reduce the prediction and configuration overhead, we can sample periodically to see if the performance counter readings are significantly different from the ones used for the current prediction to trigger re-configuration. Dynamic re-configuration of a running kernel will require extending the underlying runtime (e.g., hStreams or CUDA) to adjust thread mapping and having hardware support to stop and resume the execution contexts. We leave this as future work. ## 4 OpenMP to Streamed Code Generator Figure 7: Work flow for translating OpenMP programs to streamed programs using our automatic code generator. Currently, there are very few publicly available benchmarks for utilizing the streaming capability of heterogeneous many-core architectures, in particular, XeonPhi. To evaluate our approach on a diverse set of benchmarks, we have developed a compiler-based code generator, autostreamer, to automatically translate OpenMP programs onto streamed code depending on the target architecture. Our code generator is open sourced444Available at: https://github.com/wisdom-moon/autostreamer.. Our implementation currently supports converting OpenMP code to hStreams, CUDA and OpenCL programs. While we do not claim novelty on this as several works on source-to-source translation from OpenMP to CUDA[23, 24, 25, 26] or OpenCL[20, 27] exist, we believe the tool could serve as a useful utility for translating OpenMP programs to exploit multi-stream performance on heterogeneous many-core architectures. ### 4.1 Code Generator Overview Figure 7 depicts our source to source code generator for translating OpenMP code to streamed programs. We use LLVM’s Clang front-end to convert OpenMP code into the abstract syntax tree (AST). We then traverse the AST to obtain the information to generate candidate streamed kernels and host-device management code. The generated kernel and host code make use of exiting programming models for kernel launching and communication management. We use hStreams for XeonPhi and CUDA or OpenCL for GPUs. Our current implementation supports the translation of OpenMP parallel loops, i.e., loops annotated with omp for or omp for reduction constructs. For each parallel loop, we outline the loop body and translate it into an individual kernel function. We then replace the original loop body with a function call (running on the host CPU) to launch the generated kernel. We also generate management code for streaming context initialization, data partitioning, data movements between the host and the accelerator, etc. Our code generator relies on the native host/device compiler to optimize the generated code. We have also compared our automatically generated code against the manually translated code used in our prior work [16] and found that there is little difference in performance for the set of OpenMP benchmarks used in this work. ### 4.2 Preprocessing As an example, Figure 8 illustrates how an OpenMP parallel loop can be translated into hStreams code for XeonPhi. Note that a similar code generation process is implemented for GPUs, using CUDA for NVIDIA GPU architectures and OpenCL for other GPU platforms. For each OpenMP parallel loop, we extract information of loop iterations from the loop head. In this work, partitioning is achieved by splitting the loop iteration space. Furthermore, we collect all the variables needed by the hStreams kernel. Because hStreams requires kernel parameters to be passed as the uint64_t (lines 1-2 of Figure 8b), the kernel parameters will be cast into this type. The kernel parameters need to be packed into an array (line 21 in Figure 8c). Then the hStreams library will unpack kernel parameters from the array and pass the parameters to kernel function. During the preprocessing stage, we also extract the static code feature values of each target parallel loop. The code feature values will be encoded into the source code during host code generation. It is to note that our approach can be easily applied to existing hStreams programs – by first gathering feature values from an hStreams kernel, and then storing the extracted information in an auxiliary file or source code through a compiler front-end pass. ⬇ 1// An OpenMP C code for vector addition float * hostOutput = (float *) malloc(inputLength*sizeof(float)); 3… #pragma omp parallel for 5for(int i=0; i<inputLength; i++) { 7 hostOutput[i] = hostInput1[i] + hostInput2[i]; } 9… (a) OpenMP code. ⬇ 1COINATIVELIBEXPORT void kernel (uint64_t arg0, uint64_t arg1, … uint64_t arg5) 3{ int _start = (int) arg0; 5 … float *hostInput2 = (float *) arg5; 7 #pragma omp parallel for 9 for(int i= _start; i< _end; i++) hostOutput[i] = hostInput1[i] + hostInput2[i]; 11} (b) hStreams kernel code. ⬇ 1//output buffer float * hostOutput = (float *) malloc(inputLength*sizeof(float)); 3 //Feature update and prediction 5Stream config; 7conf_search(&config, &kernel_1_features, kernel_1_profile_runs); int partitions = config.partitions; 9int tasks = config.tasks; 11//hStreams Initialization hStreams_app_init(partitions, 1); . 13… hStreams_app_create_buf((float *)hostInput1, …); 15… 17//Work partition int sub_blocks = inputLength / tasks; 19int remain_index = inputLength % tasks; 21//Initialize kernel arguments uint64_t args[6]; args[2] = (uint64_t) inputLength; 23… for (int idx = 0; idx < tasks; idx++) { 25 args[0] = (uint64_t) _start; _end = _start + sub_blocks; 27 if (idx < remain_index) 29 _end ++; 31 args[1] = (uint64_t) _end; hStreams_app_xfer_memory(&hostInput1[_start], &hostInput1[_start], (_end- _start)*sizeof(float), idx % partitions, HSTR_SRC_TO_SINK, NULL); 33 hStreams_app_xfer_memory(&hostInput2[_start], …); 35 //Kernel launch hStreams_EnqueueCompute(idx % partitions, ”kernel_1”, 3, 3, args, …); 37 //Read back results 39 hStreams_app_xfer_memory(&hostOutput[_start], …); _start = _end; 41} … 43//hStreams cleanup code hStreams_app_fini(); (c) hStreams host code. Figure 8: A running example of translating (a) an OpenMP parallel loop to (b) hStreams kernel and (c) host management code. ### 4.3 Kernel Code Generation Generating a streamed kernel function is straightforward as much of the OpenMP code can be re-used. Figure 8b gives an example of the automatically generated kernel for the OpenMP loop given in Figure 8a for hStreams kernels. For the example given in Figure 8, an hStreams kernel starts with a pre- processor macro COINATIVELIBEXPORT (lines 1-2 in Figure 8b). The number and the type of the kernel parameters are loop-specific and are automatically determined by our code generator. Within the generated kernel, all the function parameters are cast from uint64_t into an appropriate type before they are used. Note that the OpenMP parallel for pragmas are kept in the generated kernel code per hStreams requirement (line 8 in Figure 8b). With our code generator, the original outer-most loop iteration space will be partitioned among parallel streams. The amount of work given to a specific stream is determined by the _start and _end variables, which define which part of the loop iteration space a stream instance will work on. A similar kernel code generation approach is implemented for GPUs using CUDA or OpenCL. ### 4.4 Host Code Generation To generate host code, we replace the original OpenMP parallel loop with a function call to invoke the generated kernel (e.g., hStreams_EnqueueCompute in Figure 8c)) together with additional code to initialize the host context and to manage data transfer. #### 4.4.1 Feature Value Collection Static code features, extracted by our code generator, will be encoded as a feature vector of real values. The feature vector will be passed to our configuration search engine to find the optimal stream configuration at runtime. Dynamic feature values are automatically collected by running the generated streamed kernel for 5 iterations under the single-stream configuration. As some loop bounds are dependent on the input, we might be unable to determine certain feature values at compile time. These features are represented as static symbolic pre-computation of loop bound variables, which will be updated using runtime values at runtime. #### 4.4.2 Setting Stream Configurations To partition tasks among streams, we break the loop iterations into a number of chunks of an equal size of subtask. We then group the hardware processor cores into partitions, where each partition contains a fixed set of streams. Processor partitioning and streams creation are achieved by calling the hStreams_app_init (line 12 in Figure 8c) function for XeonPhi (and cudaStreamCreate and clCreateCommandQueue for CUDA and OpenCL programs respectively) by passing the stream configuration given by our search engine. To overlap host device communications, we further split the input/output data arrays to multiple data blocks (lines 32-39 in Figure 8c) where each task operates on one block at a time while another data block is transferring between the host and the accelerator. The number of data blocks is determined by the stream configuration chosen at program runtime. The amount of work per task and the size of transferred data can be determined with kernel parameters. For example, in _for-loop_ at line 24 of Figure 8c, we calculate them with the starting position (_start) and the block size (sub_block). Thereafter, we schedule tasks and transfer the corresponding data blocks onto streams in a round-robin fashion. #### 4.4.3 Runtime Prediction When a streamed (e.g., hStreams or CUDA) kernel is invoked, the configuration selection engine library will choose a stream configuration (line 7 in Figure 8c) for the kernel. It uses the performance model to rank the candidate stream configurations and returns the optimal configuration (_#partitions_ and _#tasks_ for the example shown in Figure 8). The returned values are then used to initialize the streamed context (lines 8-9 of Figure 8c). The overhead of prediction is negligible (a few milliseconds) and is included in the results. #### 4.4.4 Supporting OpenMP Constructs OpenMP variables may have additional type information specified by directives, including default, share, private, firstprivate, lastprivate, copyin and threadprivate. Our generator uses these directives to map data onto the accelerator memory space. Each variable with the share or default directive will be translated into a global variable shared by all parallel threads. Variables declared as private and threadprivate are translated such that there is a private copy for each streamed kernel; no memory transfer between the host and the accelerator is needed. For each variable specified as copyin or first private, we create a private copy for each streamed kernel but initialize each copy using explicit memory transfers before its first use. Similarly, we create a private copy of a last private variable and the original variable is updated by a stream that executes the last iteration. Our implementation also supports a number of synchronization and thread constructs. Structured blocks identified with master, and single directives are executed by one thread on the host multi-core. barrier is implemented by splitting up the parallel loop into smaller tasks to create synchronization points among multiple streams. critical is implemented by using a mutex lock to restrict the execution of the associated structured blocks to a single thread at a time. The atomic and flush directives are already supported by hStreams, CUDA or OpenCL. #### 4.4.5 Host-Accelerator Communication Optimization For each buffer that is used by both the host and the accelerator, we manage two copies: one on the host memory and the other on the accelerator memory. Our runtime records the status of each variable and checks whether the copy on a device memory space is valid or not. No memory transfer is needed as long as the copy in the target memory space is valid. We currently use a conservative approach: if an element of an buffer has been updated, the entire buffer needs to be synchronized before it can be used by threads running on a different device. We also avoid unnecessary device to host data transfer by tracking the data dependence between the kernel and the host program. For example, when there are data-dependencies between two kernels but the host does not access this data in between the two kernels, we directly pass the memory address of the buffer to the later kernel (without moving the data back to the host). ## 5 Experimental Setup ### 5.1 Hardware, Systems Software and Benchmarks Table III: Our evaluation platforms | CPU-XeonPhi | CPU-GPU ---|---|--- CPU | 8-core Xeon CPU @ 2.6 GHz | Core i7-8700K CPU @ 3.7 GHz Accelerator | Intel Xeon 31SP Phi | NVIDIA GeForce GTX 1080 Ti GPU Platforms. We evaluate our approach on two heterogeneous many-core platforms: one is a CPU-XeonPhi platform and the other is a CPU-GPU platform. Table III gives details of our hardware platforms. Systems software. On the CPU-XeonPhi platform, the host CPU and the accelerator are connected through PCIe. The host runs Redhat Linux v7.0 (with kernel v3.10). The coprocessor runs a customized uOS (v2.6.38.8). We use Intel’s MPSS (v3.6) to communicate between the host and the coprocessor. We use the Intel hStreams library (v3.6) and Intel ICC (v16.0.3) for compilation (with -O3 as the compiler option). The CPU-GPU platform runs Ubuntu 16.04 (with kernel v4.15). We use CUDA v10.0 and gcc v5.3 as the host compiler with option “-O3”. Benchmarks. We use our code generator to translate 37 OpenMP applications from commonly used benchmark suites into hStreams and CUDA programs. We have excluded benchmarks where the data transfer cannot be overlapped with the kernel execution, which do not benefit from streamed parallelization. Table IV gives the full list of these benchmarks. Among them, convolutionFFT2d and convolutionSeparable have algorithm-dependent parameters, which are regarded as different benchmarks in the experiments. This setting gives us a total of 39 programs. We run the majority of the programs using over 25 different datasets, except for some applications where we used around 10 datasets because the algorithmic constraints prevent us from using a larger number of inputs. Table IV: Streamed benchmarks used in our experiments. Suite | Name | Acronym | Name | Acronym ---|---|---|---|--- | convol.Separable | convsepr1(8) | dotProduct | dotprod | convolutionFFT2d | fftx1y1(4y3) | fwt | fwt | MonteCarlo | montecarlo | matVecMul | mvmult | scalarProd | scalarprod | transpose | transpose NVIDIA SDK | vectorAdd | vecadd | | AMD SDK | binomial | binomial | BlackScholes | blackscholes dct | dct | prefixSum | prefix | bfs | bfs | histo | histo | lbm | lbm | mri-q | mri-q | mri-gridding | mri-gridding | sad | sad Parboil | sgemm | sgemm | spmv | spmv POLY BENCH | 2mm | 2mm | 3mm | 3mm adi | adi | correlation | correlation covariance | covariance | deriche | deriche gemm | gemm | gemver | gemver gesummv | gesummv | heat-3d | heat-3d jacobi-1d | jacobi-1d | jacobi-2d | jacobi-2d mvt | mvt | syr2k | syr2k syrk | syrk | | ### 5.2 Competitive Approaches We compare our regression-based approach against our preliminary work that employs an SVM-based classifier to predict the optimal stream configuration [16]. We denote our prior approach as SVM-classifier. We also compare our approach against two recent models for predicting the optimal stream configuration on GPUs. As it is currently not possible to configure the number of processor partitions on GPUs, the relevant GPU models can only predict the number of tasks. _Liu et al._ In [12], Liu _et al._ use linear regression models to search for the optimal number of tasks for GPU programs [12]. The approach employs several analytic models, described as follows. For a task with an input data size of $m$, the transferring time between the CPU and the accelerator, $T_{t}$, is determined as $T_{t}=\alpha\cdot m+\beta$, and the computation time, $T_{c}$, is calculated as: $T_{c}=\eta\cdot m+\gamma$ where the model coefficients, $\alpha$, $\beta$, $\eta$ and $\gamma$, are determined through empirical experiments. For a given kernel with $N$ input data elements running using $n$ streams, this approach partitions the computation into $n$ tasks, where the data size for each task, $m$, is equal to $N$/$n$. For the programs which kernel dominated, the total execution time, $T_{total}$, can be determined by: $T_{total}=T_{t}+nT_{c}=\alpha\cdot m+\frac{N\gamma}{m}+N\eta+\beta$ For the programs which data transfer dominated: $T_{total}=\alpha\cdot N+2\frac{N}{m}\beta$ By calculating the partial differential and second-order partial differential of $T_{total}$ with respect to $m$, we can obtain the optimal task-granularity as $m=\sqrt{\frac{N\gamma}{\alpha}}$. Then we can calculate the number of tasks ($n$). Note that $m=N/2$ is the optimal parameter for programs which data transfer dominated, i.e., the optimal number of tasks is 2. Another problem of this model is that it does not consider scenarios where communications in different direction (i.e., host to device and device to host) can overlap with each other. Note that we set the #partitions to be the same as $n$ for XeonPhi. _Werkhoven et al._ The work presented by Werkhoven _et al._ models the performance of data transfers between the CPU and the GPU [10]. They use the LogGP model to estimate the host-device data transfer time. Specifically, the model estimates the data transfer time using five parameters: the communication latency ($L$), overhead ($o$), the gap ($g$), the number of processors ($P$), and the PCIe bandwidth ($G$). Let $B_{hd}$ denotes the amount of data transferred from the host to the device and $B_{dh}$ denotes vice versa, and $T_{kernel}$ donates the kernel execution time. For the dominant transfer scenario, the optimal number of tasks(i.e., _#tasks_), $N_{s}$, can be estimated by solving the following equations: $B_{dh}*G_{dh}+g*(N_{s}-1)=\begin{cases}\frac{T_{kernel}}{N_{s}}+\frac{B_{dh}}{N_{s}}*G_{dh},&\text{if}B_{dh}>B_{hd}\\\ \frac{B_{hd}}{N_{s}}*G_{hd}+\frac{T_{kernel}}{N_{s}},&\text{otherwise}\end{cases}$ This model does not consider the dominant kernel scenario, as it assumes the kernel execution time will increase as the number of streams increases and can not model the kernel execution time. Here, we use the same equation to calculate the optimal number of tasks. For this model, we also set the #partitions to be equal to the optimal $N_{s}$ value on XeonPhi. ### 5.3 Evaluation Methodology #### 5.3.1 Model Evaluation We use cross-validation to evaluate our machine learning models. To test the portability of our approach, we apply _leave-one-out_ cross-validation, described as follows. We exclude the target program for predictions from the training program set, and learn a model using the _remaining_ programs. We then apply the learned model to the testing program. We repeat this process until each benchmark is tested once. This is a standard evaluation methodology, providing an estimate of the generalization ability of a machine learned model in predicting _unseen_ data. Note that we exclude both convolutionFFT2d and convolutionSeparable from the training set when one of the two is evaluated, and we make sure all approaches are trained on the same benchmarks for fair comparisons. #### 5.3.2 Performance Report We run each program under a stream configuration multiple times and report the _geometric mean_ of the runtime. Compared to the arithmetic mean, the geometric mean is often considered as a more suitable metric for reporting program performance, as it can better minimize the impact of outliers [28]. To determine how many runs are needed, we calculated the confidence range using a 95% confidence interval and make sure that the difference between the upper and lower confidence bounds is smaller than 5%. ## 6 Experimental Results In this section, we first present the overall performance of our approach on both platforms. We then compare our approach to that uses fixed stream configurations, two prior analytical models and our previous work. We futher discuss the benefit sources of the streaming parallelism and the working mechanism of our approach. At last, we demonstrate the tunning process of our model. ### 6.1 Overall Performance (a) XeonPhi (b) NVIDIA GPU Figure 9: Overall performance of our approach over a single-stream version on XeonPhi (a) and NVIDIA GPU (b). Our approach achieves, on average, 93.7% and 97.9% of the oracle performance on XeonPhi and NVIDIA GPU, respectively. The min-max bars show the range of performance achieved across different inputs. In this experiment, we exhaustively profiled each application with all possible stream configurations and report the best-found performance as the _Oracle_ performance. The Oracle gives an indication of how close our approach is to a _theoretically perfect_ solution. The baseline used to calculate the speedup is running the application using a single-stream without processor core or task partitioning. The overall result is shown in Figure 9. The min-max bar on the diagram shows the range of speedups per application across all evaluated inputs. Overall, our approach achieves an average speedup of 1.57$\times$ and 1.1$\times$ over the single-stream configuration on XeonPhi and the GPU respectively. This translates to 93.7% and 97.9% of the Oracle performance on XeonPhi and the GPU respectively. On XeonPhi, the performance improvement of our approach comes from two factors. First, by predicting the right processor partition size, our approach allows effective overlapping of the host-device communication and computation. Second, by matching task parallelism to the number of available processor cores, our approach can reduce the overhead of thread management, compared to the single-stream execution. When the host-device communication time dominates the streaming process, the performance improvement mainly comes from computation-communication overlapping and the speedup from streaming is consistently less than 2$\times$. When the kernel execution time dominates the stream process, the application can benefit from the overhead reduction of thread management. In this case, the speedup can be as large as 5$\times$. We provide a further discussion on this later in Section 6.5.1. On the GPU, we can exploit bidirectional data transfer between the host and the device by using pined memory which is not supported by hStreams. The support of bidirectional data transfer allows us to obtain further performance gains by overlapping host-device data transfer and computation. The theoretically up-bound speedup on the GPU platform is 3$\times$, when data transfer is perfectly overlapped with computation. The representative sample is fftx4y3 with the larges dataset, the data transfer time in the two directions is the same, and the kernel execution time is 1.5 times of the data transfer time. The oracle speedup is 2.3$\times$, and our approach achieves a speedup of 2.2 $\times$. On the other hand, because the current GPU implementation does not support processor core partition, the kernel execution time benefits less from using multiple streams. Programs which the kernel execution time dominated have no speedup using multiple streams, such as bfs, MonteCarlo. ### 6.2 Comparison to Fixed Stream Configurations Our approach predicts from a wide range of stream configurations, which configuration is likely to give the best performance for a given program and dataset. A natural question to ask is that: is there a fixed stream configuration that gives reasonable good performance across benchmarks and datasets? To answer this question, we compare our predictive modeling based approach to two specific configurations on each of our evaluation platforms. Our justification for why we selecting the fixed configurations are described as follows. On XeonPhi, our initial results in Section 2 indicate that using the stream configuration of $(4,16)$, i.e. partitioning the cores to 4 groups and running 4 tasks on each partition (16 tasks in total), gives good performance. The statistics obtained from the training data suggest that the configuration of $(17,85)$ give the best average performance across training samples. On the GPU, several programs support a maximum of 4 tasks. Thus we select the two configurations $(2,2)$ and $(4,4)$. The results are shown in Figure 10. (a) XeonPhi (b) NVIDIA GPU Figure 10: Comparing the performance with two fixed configurations on XeonPhi (a) and NVIDIA GPU (b): config. $(4,16)$ of 4 partitions and 4 tasks per partition, config. $(17,85)$ of 17 partitions and 5 tasks per partition, config. $(2,2)$ of 2 partitions and 1 tasks per partition, and config. $(4,4)$ of 4 partitions and 1 tasks per partition. (a) XeonPhi (b) NVIDIA GPU Figure 11: Violin plot showing the distribution of speedups per scheme across benchmarks and datasets on XeonPhi (a) and GPU (b). The shape of the violin corresponds to the speedup distribution to the oracle performance. The thick black line shows where 50% of the data lies. #### 6.2.1 XeonPhi On XeonPhi, we observe improved performance for several benchmarks such as mri-gridding, transpose, sad, under both configurations, but slower performance for dotprod, vecadd, blackscholes, lbm, and mir-q (Figure 10a). For prefix, configuration $(17,85)$ delivers improved performance while configuration $(4,16)$ leads to slowdown performance. Overall, none of the two fixed configurations give an improved performance on average. On average, our approach outperforms the two fixed configurations by a factor of 1.4, and delivers consistently improved performance across benchmarks and datasets. The violin plot in Figure 11a shows how far is each of the three schemes to the Oracle performance across benchmarks and datasets. Our approach not only delivers the closest performance to the Oracle, but also has the largest number of samples whose performance is next to the Oracle. By contrast, the performance given by the fixed configurations for many samples is farther from the Oracle performance. #### 6.2.2 GPU On the GPU, in most cases, the performance of configuration $(2,2)$ is moderate, not great, but not much worse than single-version, leading to an average speedup 1.03$\times$ (Figure 10b). By contrast, although configuration $(4,4)$ performs poorly on two programs, it delivers a slightly larger averaged speedup of 1.04$\times$. By choosing the stream configuration on a per-program basis, our approach outperforms the two fixed configurations, achieving an averaged speedup 1.10$\times$. On only four programs, our approach delivers slightly worse performance with a small margin. The violin plot in Figure 11b also confirms the strengths of our approach by presenting the distribution of performance improvement. The results on the diagram are normalized to the Oracle (best-available) performance. For most of the programs, the two fixed configurations deliver 80% to 100% to the Oracle performance. However, configuration $(4,4)$ can lead to rather poor performance (less than 40% to the best available performance) on some programs. Compared to the fixed configurations, the performance distribution given by our approach is centralized on a range between 90% to 100%, where most programs are within this percentile range. Furthermore, compared to the fixed configurations, our approach has a fewer number of performance outliers, which have less serious performance slowdown. Therefore, our approach delivers consistently better performance compared with the fixed configurations. #### 6.2.3 Summary This experiment confirms that a fixed configuration fails to deliver improved performance across applications and datasets, and selecting a right stream configuration on a per program, per dataset basis is thus required. ### 6.3 Comparison to Analytical Models (a) XeonPhi (b) NVIDIA GPU Figure 12: Comparing against _Liu et al._ and _Werkhoven et al._ on XeonPhi (a) and NVIDIA GPU (b). In this experiment, we compare our approach to the two recent analytical models described in Section 5.2. The results are shown in Figures 12 and 13. On XeonPhi, both competitive models prefer using $2$ tasks across benchmarks and datasets. This is because that many programs are kernel dominated, the analytical models simply assume that task partition has no effect on kernel’s performance, and do not consider the thread management overhead. On the GPU, the model proposed by _Liu et al._ tends to use $2$ tasks across benchmarks and datasets. This is due to the fact that most programs are data transfer dominated and this model ignores the overlap of the bidirectional data transfers between the host and the device. XeonPhi. Figure 12a demonstrates that our approach gives better performance for nearly all programs on XeonPhi. For the remaining handful programs, all three approaches deliver comparable performance. Compared to the results Figure 10, we can find the performance of the analytical models is similar to fixed stream configurations. This is because the performance of the seven programs, such as binomial, changes dramatically with different stream configurations (see also Figure 2). The performance of the remaining programs is not sensitive to the variation of stream configurations. From Figure 13a, we can further see that _Liu et al._ and _Werkhoven et al._ deliver a speedup within a range on 20% to 80%, while the performance of our approach is centralized on a range between 80% to 100%. Thus, our approach delivers consistently better performance compared with the alternative models. GPU. Figure 12b shows that our approach delivers better performance for around 75% of the programs on the GPU. Since _Werkhoven et al._ and _Liu et al._ are manually tuned for the GPUs, they give better performance on some benchmarks over our approach. However, our approach has the advantages of being automatically learned from training data, with little expert involvement. The performance of our approach can be further improved by using more training examples to better cover the program space. Figure 13b shows that _Liu et al._ and _Werkhoven et al._ delivers a speedup within a range of 5% to 80%, and 70% to 100%, respectively. By contrast, the performance of our approach is centralized within a range between 90% to 100% for more programs. Therefore, overall, our approach delivers better average performance compared with the alternative models. (a) XeonPhi (b) NVIDIA GPU Figure 13: Violin plots showing the distribution of speedups across benchmarks and datasets on XeonPhi (a) and GPU (b). ### 6.4 Comparison to Classification-based Approach (a) XeonPhi (b) NVIDIA GPU Figure 14: Comparing against a classification based approach on XeonPhi (a) and NVIDIA GPU (b). Our prior work uses a SVM classifier to predict the configurations [16]. Compared with it, the regression-based model presented in this article has several advantages. A classification model predicts which of a set of predefined labels the input belongs to. Using this strategy, we will need to label each unique stream configuration. This will lead to a total of 175 labels for 311 profiling samples on the XeonPhi, and 11 labels on the GPU. On the XeonPhi, the ratio of samples to labels is too small to build an accurate model. As a result, we have to merge labels in our prior work [16] at the cost of losing accuracy. Classification is a constraint optimization problem where the model has to know all the possible configurations during training. Our new regression-based approach avoids this pitfall by directly modeling the impact of the stream configuration; it thereby can be used on any stream configuration as the configuration is the model’s input. Figure 14a presents results obtained on the XeonPhi. Our regression-based approach outperforms the SVM-classifier in 21 of the 39 programs and achieves over 5% performance improvement for 13 programs. It is to note that the overhead for ranking stream configurations is included in the experimental results. Overall, our regression-based approach improves the SVM-classifier by, on average, 3% (up to 46%). Unlike XeonPhi, we were able to obtain sufficient training samples per label (because the optimization space is smaller) on the GPU to build a more accurate classification model. As can be seen from Figure 14b, the average speedup of SVM-classifier and the regression-based approach is comparable. Compared to a classifier, our regression-based approach has the advantage of being able to be applied to configurations that were not seen during the training phase. Therefore, our approach has a better generalization ability. ### 6.5 Further Analysis of Performance Results We now take a closer look into the performance results, using XeonPhi as a case study. Figure 15: Reduction of kernel computation time over a single-stream execution on XeonPhi. The performance improvement comes from the reduction of the threading overhead. A stream configuration is annotated as (_#partitions_ , _#tasks_). #### 6.5.1 High Speedup Cases On XeonPhi, bidirectional data transfer between the host and the accelerator cannot be overlapped, i.e., we can only issue data transfer from the host to the device or vice versa at once but not simultaneously. As a result, the theoretical up-bound speedup for overlapping computation and communication is 2$\times$, when the computation is perfectly overlapped with the data transfer time. It is interesting to observe that several benchmarks achieve a speedup of over 2$\times$ on XeonPhi (see Figure 9a). After having a closer investigation, we notice that such performance is attributed to the reduction in the kernel execution time in additional to the overlapping of communication and computation. To quantify the benefit of kernel time reduction, we measure the kernel execution time with and without multiple streams and calculate the speedup between them. Note that we _exclude the host-device communication time in this case_ to isolate the contributing factors. The kernel time improvement for transpose, binomial, and fftx1y1 is shown in Figure 15. As can be seen from the diagram, choosing a good stream configuration can lead to more than 4x reduction on the kernel execution time. This is because these benchmarks are implemented by parallelizing the inner loop within a nested loop. During runtime, the parallel threads working on the inner loop will be created, synchronized, or destroyed for each outer loop iteration. Such threading overhead could be significant when the outer loop iterates a large number of times. With multiple streams, we divide the whole outer loop iteration space into multiple smaller iterations. This allows multiple groups of threads to be managed simultaneously, leading to a significant decrease in threading overhead and faster kernel execution time. On the other hand, using too many streams and partitions will lead to a performance decrease. This is because stream management also comes at a cost, which increases as the number of partitions grows. Nonetheless, for applications where the kernel computation dominates the program execution time, by reducing the kernel time can lead to additional improvement, yielding more than 2x speedups. (a) XeonPhi (b) NVIDIA GPU Figure 16: Violin plot showing the distribution of speedups per benchmark across datasets on XeonPhi (a) and NVIDIA GPU (b). The shape of the violin corresponds to the speedup distribution. The thick black line shows where 50% of the data lies. #### 6.5.2 Speedup Distribution Figure 16 gives the speedup per benchmark across datasets on XeonPhi and the GPU. The shape of the violin plot corresponds to the speedup distribution. On XeonPhi, we see that the speedups of montecarlo and prefix distribute fairly uniformly while the data distribution of fftx1y1 and fftx4y3 is multimodal (i.e. it has two peaks). Further, the input datasets have little impact on the behavior of fwt and lbm, so the speedups remain constant across datasets. On the GPU, the speedups of dotprod, vecadd, blackscholes and mri-q distribute fairly uniformly while the data distribution of convsepr1, convsepr8, fftx1y1, fftx4y3 and dct is unimodal (i.e. it has one peak). Furthermore, the input datasets have a very slight impact on the performance behaviors of montecarlo, scalarprod, transpose and binomial. Thus, their speedups remain constant across datasets. To conclude, the streaming speedups of some applications are sensitive to their input datasets whereas the others are not. And the distribution of speedups on the GPU is more concentrated than XeonPhi. This is because the current GPU implementation does not support processor core partition, the kernel execution time benefits less from multiple streams than XeonPhi. Figure 17: The relation between computation-communication ratio and the speedup. The computation-communication ratio is normalized using the natural logarithm function. Thus, the kernel computation time equals the host-device communication time when $ratio=0$. In general, a higher computation- communication ratio leads to a better speedup. #### 6.5.3 Correlation Analysis Figure 17 shows the relation between the computation-communication ratio and the achieved speedup when using heterogeneous streams across all benchmarks and datasets on XeonPhi. We see that the computation-communication ratio varies over the benchmarks and the speedup changes accordingly, but in general, a higher computation-to-communication ratio leads to a greater speedup. As explained in Section 6.5.1, in addition to overlapping computation and communication, our approach can also reduce the kernel computation time by choosing the right stream configuration. Therefore, benchmarks with a high computation-communication ratio also benefit from a reduction in the kernel computation time. To quantify the relation between the computation-communication ratio and the speedup, we calculate the Pearson correlation coefficient of the two variables. The calculation gives a correlation coefficient of 0.7, indicating that the two variables (the computation-communication ratio and the speedup) have a strong linear correlation. By carefully selecting the stream configuration, our approach tries to maximize the overlap between communication and computation, which thus leads to favourable performance. #### 6.5.4 Impact of Streaming Parallelism Figure 18: Breakdown of program execution time ($T$), host-device data transfer time ($T_{m}$), kernel execution time ($T_{k}$), hStreams context initialization overhead ($T_{c}$) and communication-computation overlapping time ($T_{o}$) for single and best-performing multi-stream configurations. Our earlier experiments show that by carefully exploiting streaming parallelism, we can significantly improve application performance. We now take a closer look at three representative benchmarks, fftx1y1, fwt and gesummv, to get a better understanding of streaming performance on XeonPhi. These benchmarks represent different degrees of benefits obtained from streamed parallelism (with a speedup of 2$\times$, 1.5$\times$ and 1$\times$, respectively). We use the following analytical model to breakdown the execution time of a multi-stream program: $T=T_{m}+T_{k}+T_{c}-T_{o}$ (1) where $T_{m}$ is host-device data transfer time, $T_{k}$ is kernel execution time, $T_{c}$ is the overhead for initializing the context, and $T_{o}$ is overlapping time between data transfer and kernel execution. We measure $T$, $T_{m}$, $T_{k}$, and $T_{c}$, and use the measurements to calculate $T_{o}$. Figure 18 gives the breakdown for the five components in Equation 1. For each testing program, we compare the single-stream configuration against the best- performing multi-stream configuration. The host-device data transfer time, $T_{m}$, is nearly constant among a single and a multiple stream configuration, but multi-streaming can reduce the kernel execution time, $T_{k}$, by exploiting the spatial sharing of processing resources among computation tasks. The overhead of initializing the hStreams context, $T_{c}$, depends on the kernel execution time. For fftx1y1 and fwt, whose kernels run for a sufficiently long time, this one-off runtime overhead is negligible. However, for gesummv, this overhead cannot be ignored due to the relatively short kernel running time. The contribution for overlapping host-device communications with kernel execution, $T_{o}$, varies across programs. For fftx1y1 and fwt, it accounts for around 50% of $T_{m}$, suggesting that by exploiting temporal sharing to overlap communication with kernel execution can amortize the host-device communication overhead. For gesummv, $T_{o}$ is small due to little alignment between data transfer and kernel execution. As such, there is little benefit for exploiting temporal sharing for this program. This experiment gives a more detailed analysis for the benefits of exploiting multiple streams. The results reinforce our claim that the benefit for streaming parallelism depends on the computation kernel and hence an adaptive scheme for choosing the optimal stream configuration is needed. Our work aims to offer such a capability. ### 6.6 Analysis of Predictive Modeling Techniques In this section, we analyse the working mechanism of our predictive model, using XeonPhi as an evaluation platform. #### 6.6.1 Comparison to Alternative Modeling Techniques We compare our MLP-based model against four widely used regression methods: the DCT (Decision Tree), the RF (Random Forest), the XGB (eXtreme Gradient Boosting) and SVM (Support Vector Machine) as well as four classification models: SVM, DCT, MLP and KNN (K-Nearest Neighbors). We use the Radial basis function kernel for the SVM models. For each technique, we follow the same training methodology and use the same features and training examples to build a model. For classification models, we apply the label merging process described in our prior work [16] to improve the prediction accuracy. Table V compares the training overhead, average prediction time and achieved average speedup for each model. We note that training a regression-based SVM model has the largest overhead. Although training a DCT has less overhead over our MLP-based regression model, MLP gives better prediction performance. The RF and XGB models are based on DCT, but they do not yield a better performance. Compared to regression models, a classification model takes less time to train and make predictions. However, classification models give worse performance over regression models as they require more training data to cover the optimization space. Overall, we choose to use a regression-based approach and employ MLP because it gives the best overall prediction performance and has modest training overhead. Table V: Comparison to alternative modeling techniques Technique | Training time | Avg. pred. time | Avg. speedup ---|---|---|--- SVM (regression) | 100 hours | 2280 ms | 1.56 DCT (regression) | 65.57 seconds | 0.74 ms | 1.51 RF (regression) | 317.89 seconds | 11.94 ms | 1.51 XGB (regression) | 28.46 seconds | 0.74 ms | 1.49 MLP (regression, ours) | 245.8 seconds | 0.76 ms | 1.57 SVM (classifier) | 1.28 seconds | 0.10 ms | 1.53 DCT (classifier) | 0.79 seconds | 0.05 ms | 1.38 MLP(classifier) | 46.45 seconds | 0.05 ms | 1.41 KNN (classifier) | 0.22 seconds | 0.23 ms | 1.43 #### 6.6.2 Feature Engineering Feature engineering has a significant impact on the performance of a machine learning model (Section 3.2). Here we quantify the impact of feature engineering methods. In this work, we consider three standard feature engineering approaches including standardization, normalization and dimension reduction. Standardization converts all features value to be in a common range, e.g., between 0 and 1. The idea is to prevent the feature value range to dominate the importance of that feature. In this work we apply a commonly used standardization method called _Z-score_ [29] to standardize the raw feature values and the speedups (i.e., prediction targets) in the training data. We found that feature standardization improves the achieved speedup by 3% on average, and speedup standardization improves the achieved speedup by 5% on average. Normalization scales the feature values to make them follow the normal distribution. We have tested a range of normalization methods including the square root, the reciprocal of square root and the natural logarithm transformation. However, we found that normalization does not improve our model prediction accuracy. Dimension reduction reduces the number of features, which is often useful when the number of training examples is not proportional to the number of feature dimensions. In this work, we apply factor analysis (FA) [30] and principal component analysis (PCA) [31] to the raw features. Applying PCA and using 9 PCA components gives the best overall result, by improving the average speedup by 17%. PCA outperforms FA which gives an average 3% improvement on the achieved speedup. #### 6.6.3 MLP Parameters Tuning We now discuss the impact of the MLP parameter choices. There are four configurable parameters for an MLP model: the activation function, the number of hidden layers, the number of neurons, and the learning algorithm (i.e., the solver). For activation functions, we consider identity, logistic, tanh and relu. For hidden layers and neurons, we vary the number of hidden layers from 1 to 5 and the number of neurons per layer from 3 to 100. For the solver, we consider three commonly used weight optimizers: lbfgs, sgd and and adam. We use scikit-learn implementations for the activation function and the solver. Our experimental results suggest that the best-performing activation function and solver are tanh and adam respectively, and using three hidden layers with 9 neurons per layers gives the best overall results on our training data. Overall, tuning MLP model parameter improves the average speedup by 5% over the default parameter setting. Figure 19: A Hinton diagram showing the impact of each feature used by the performance model to the resultant application performance. The larger the box, the more likely a feature has a greater impact on the performance of the respective benchmark. #### 6.6.4 Impact of Individual Feature In this experiment, we consider the impact of a specific feature to the resultant performance. Figure 19 presents a Hinton diagram illustrating how important a feature contribution to the performance model prediction accuracy (which in turns affects the resulting application performance). The larger the box, the more significant a feature for a given program’s performance. Here, the x-axis denotes the programs, and the y-axis denotes the features used by our performance model. The impact of a feature is quantified by measuring how much speedup improvement can be obtained if that feature is used by the performance model. Note that this is a post-hoc analysis and, in general, we cannot know in advance the importance of a feature on _unseen_ programs. Figure 19 shows that all the features are important for the set of benchmarks targeted in the work, but the importance of features varies across programs. This diagram illustrates how hard it is to develop an analytical model to capture the diverse behaviors and characteristics of streaming programs. ## 7 Related Work Our work builds upon the following past foundation, while qualitatively differing from each. Task Scheduling. There is considerable work on distributing work across heterogeneous processors to improve application performance [32, 33, 34]. Prior work in the area typically assumes that the processor configuration is fixed and relies on the operating system to schedule parallel tasks across processing units. Recent studies show that by partitioning the processing units into groups it is possible to significantly improve the application performance by overlapping the host-device communication and computation on coprocessors like Intel XeonPhi [14, 6]. However, existing approaches rely on static tuning to find the processor partition and the best number of streams to run within a partition. As a result, previous approaches cannot adapt to the change of program inputs. As a departure from prior work, we develop an automatic approach to dynamically adjust the processor partition and task- granularity during runtime, considering the characteristics of applications and input datasets; our approach thereby can adapt to the change of program inputs. Domain-specific Optimizations There is considerable work on domain-specific optimization on Intel XeonPhi. Cheng _et al._ [35] and Jha _et al._ [36] show that in-memory database applications suffer from under-utilization of processor resources and hence a fine-grained tuning approach is required. Mrphi is a framework for optimizing MapReduce workload on the XeonPhi [37]. It employs a set of techniques to improve the resource utilization to obtain higher application performance. Other works look at performance optimization for numerical solvers [38], sparse matrix vector multiplication [39, 40], and dynamic stochastic economic models [39]. Ferrão _et al._ [41] and Memeti _et al._ [42] develop a stream processing framework for XeonPhi to increase the programming productivity. The runtime can automatically distribute workloads across CPUs and accelerating devices. These approaches improve the processor utilization by adjusting the algorithmic design, which are complementary to our work on tuning multi-streaming parallelism for data parallel applications. Multiple Streams Modeling. Gomez-Luna _et al._ [11] develop a set of models to estimate the asynchronous data transfer overhead on different GPU architectures. The models can be used to estimate the optimal number of streams to use on a given GPU platform. Werkhoven _et al._ [10] present an analytical model to determine when to apply an overlapping method on GPUs. Liu _et al._ [12] also develop an analytical based approach to determine the optimal number of streams to use on GPUs. However, none of these approaches considers the processor partition. As we have shown in Section 6.3, ignoring the processor partitioning parameter can lead to poor performance on Intel XeonPhi. Furthermore, these hand-crafted models have the drawback of being not portable across architectures as the model is tightly coupled to a specific GPU architecture. Our work advances prior work by employing machine learning to automatically learn the optimal processor partition and the number of streams/tasks to use. Since our models are automatically learned from empirical observations, one can easily re-learn a model for a new architecture. Predictive Modeling. Recent studies have shown that machine learning based predictive modeling is effective in code optimization [43, 44], performance predicting [45, 46], parallelism mapping [47, 48, 20, 49, 50], and task scheduling [51, 52, 53, 54, 55, 56]. Its great advantage is its ability to adapt to the ever-changing platforms as it has no prior assumption about their behavior. The work presented by Wen _et al._ [57] employs SVMs to develop a binary classifier to predict that if a given OpenCL kernel can achieve a large speedup or not. Our work differs from [57] in that it targets a different architecture and programming model, and it predicts from a larger number of configurations instead of making a binary prediction. Our prior work developed an SVM based classifier to predict the optimal stream configuration for Intel XeonPhi [16]. However, it requires having sufficient training data samples to cover all possible stream configurations. Our approach improves the prior work by directly modeling the impact of the stream configuration. As a result, our approach can make predictions for any stream configuration (even those are not seen in the training data). Autotuning Parallel Programs. Our approach is closely related to autotuning that searches for the best-performing optimization configuration [58, 59]. This technique is demonstrated to be effective for choosing algorithmic choices [60], tuning GPU code [61, 62, 63], optimizing structured parallel programs [64, 65, 66] and non-uniform memory access (NUMA) architectures [67], and more recently for deep neural networks [68]. Many of the prior works in this area employ an evolutionary-based approach by applying and profiling candidate optimization options to choose a good option to use. One of the key changes of autotuning is how to avoid the profiling overhead which could be prohibitively expensive. We do so by using a performance model to quickly evaluate the profitability of a candidate optimization option. We show that our approach has low runtime overhead, which thus permits us to apply it at runtime to best match the optimization strategy to the program input. Furthermore, our work is the first for tuning heterogeneous streaming parallelism on heterogeneous many-cores (XeonPhis and GPUs). Automatic Generation of Parallel Programs. The OpenMPC compiler [69] translates OpenMP to CUDA programs. Wang _et al._ [24, 20, 70] translates OpenMP to OpenCL programs and use machine learning to select the most suitable device from the host CPU and the GPU to run the code. Rawat _et al._ presents an automatic approach to generate GPU code from a domain-specific language (DSL) for stencil programs [71]. All of the above approaches target GPUs, and do not utilize the multi-streaming strategy. ## 8 Conclusion This article has presented an automatic approach to exploit streaming parallelism on heterogeneous many-cores. Central to our approach is a machine learning-based model that predicts the resulting performance when running the target application under a given streamed configuration. The performance predictor is then used as a cost function to quickly rank candidate configurations at runtime, to determine which stream configuration should be used on a per-program per-dataset basis. We have evaluated our approach on an Intel XeonPhi and an NVIDIA GTX 1080 Ti GPU, with 39 representative benchmarks. Experimental results show that our approach delivers an average speedup of 1.6x and 1.1x on XeonPhi and the GPU, respectively. 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2024-09-04T02:54:58.500113

2003.04352

arxiv-papers

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) ​​​ CERN-EP-2020-020 LHCb-PAPER-2019-043 9 March 2020 Search for the lepton flavour violating decay ${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$ using $B_{s2}^{*0}$ decays LHCb collaboration†††Authors are listed at the end of this paper. A search is presented for the lepton flavour violating decay ${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$ using a sample of proton–proton collisions at centre-of-mass energies of 7, 8, and $13\text{\,}\mathrm{Te\kern-1.00006ptV}$, collected with the LHCb detector and corresponding to a total integrated luminosity of 9$\text{\,fb}^{-1}$. The $\tau$ leptons are selected inclusively, primarily via decays with a single charged particle. The four-momentum of the $\tau$ lepton is determined by using ${B}^{+}$ mesons from ${B_{s2}^{*0}}\rightarrow{{B}^{+}}{{K}^{-}}$ decays. No significant excess is observed, and an upper limit is set on the branching fraction ${\mathcal{B}}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}})<$3.9\text{\times}{10}^{-5}$\text{ at 90\% confidence level}.$ The obtained limit is comparable to the world-best limit. Submitted to JHEP © 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence. ## 1 Introduction A number of experimental hints of lepton flavour universality violation in the semileptonic transitions $b\\!\rightarrow s\ell^{+}\ell^{-}$ [1, 2, 3] and $b\\!\rightarrow c\ell^{-}{\overline{\nu}}_{\ell}$ [4, 5, 6, 7, 8, 9] have recently been found.111The inclusion of charge-conjugate processes is implied throughout. In general, physics beyond the Standard Model that generates lepton flavour non-universality is likely to also produce direct lepton flavour violation [10]. Theoretical models seeking to simultaneously explain all these anomalies, for example with a vector leptoquark, often lead to relatively large branching fractions for the decays ${B}\\!\rightarrow{K}\mu^{\pm}\tau^{\mp}$ [11, 12, 13, 14, 15, 16]. The branching fractions for the two $\mu\tau$ charge combinations are not in general the same, as they depend on the details of the physics mechanism producing the decay. In this paper, we present a search for the decay ${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$. From an experimental point of view, this combination is preferred over ${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{+}}{\tau^{-}}$ as it has a lower background from semileptonic ${B}\\!\rightarrow{\kern 1.79993pt\overline{\kern-1.79993ptD}}X{\mu^{+}}{{\nu}_{\mu}}$ decays, because Cabibbo-favoured decays of the charm meson are likely to lead to kaons of the same charge as the muon. An upper limit on the branching fraction for the signal decay has been previously set by the BaBar collaboration [17] ${\mathcal{B}}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}})<$2.8\text{\times}{10}^{-5}$$ at 90% confidence level (CL). We reconstruct the full four-momentum of the $\tau$ lepton using ${B}^{+}$ mesons from the decay ${B_{s2}^{*0}}\\!\rightarrow{{B}^{+}}{{K}^{-}}$, which amounts to about 1% of ${B}^{+}$ production. By reconstructing the decay vertex of the ${B}^{+}$ meson from the ${{K}^{+}}{\mu^{-}}$ pair and the momentum of the ${K}^{-}$ meson, it is possible to determine the momentum of the ${B}^{+}$ meson up to a quadratic ambiguity by imposing mass constraints on the $B_{s2}^{*0}$ and ${B}^{+}$ mesons [18]. This technique was first used to study relative branching fractions in ${{B}^{+}}\\!\rightarrow{{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}X{\mu^{+}}\nu$ decays [19]. We then search for a peak in the missing-mass squared distribution corresponding to the $\tau$ mass squared, $m_{\tau}^{2}$. Even signal ${B}^{+}$ mesons not coming from a $B_{s2}^{*0}$ decay show a peak at $m_{\tau}^{2}$. We account for the contribution of these non-$B_{s2}^{*0}$ candidates in the analysis. The $\tau$ leptons are selected inclusively, as we only require one additional charged track near the ${{K}^{+}}{\mu^{-}}$ pair to help discriminate against background. To normalise the branching fraction, we use the decay ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$, with ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$. The normalisation channel is also used to quantify the contributions from $B_{s2}^{*0}$ decays, as well as non-$B_{s2}^{*0}$ candidates with nearby kaons. In addition to providing the missing-mass discriminating variable, this method allows us to study the control sample composed of same-sign ${{B}^{+}}{{K}^{+}}$ decays, which does not include any $B_{s2}^{*0}$ component. We use this sample to optimise the signal selection, and motivate our description of the background missing-mass shape. ## 2 Detector, data samples, and simulation The LHCb detector [20, 21] is a single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the study of particles containing $b$ or $c$ quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the $pp$ interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three stations of silicon-strip detectors and straw drift tubes placed downstream of the magnet. The tracking system provides a measurement of the momentum, $p$, of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV.222Natural units with $c=1$ are used throughout. The minimum distance of a track to a primary $pp$ interaction vertex (PV), the impact parameter, is measured with a resolution of $(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$ is the component of the momentum transverse to the beam, in GeV. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers. The online event selection is performed by a trigger, which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. At the hardware trigger stage, events are required to have a muon with high $p_{\mathrm{T}}$ or a hadron, photon or electron with high transverse energy deposited in the calorimeters. The software trigger requires a two-, three- or four-track secondary vertex with a significant displacement from any primary vertex. We use data samples collected from 2011 to 2018, at centre-of-mass energies of 7, 8, and $13\text{\,}\mathrm{Te\kern-1.00006ptV}$, corresponding to an integrated luminosity of 9$\text{\,fb}^{-1}$. We model signal and normalisation decays using simulation. In the simulation, $pp$ collisions are generated using Pythia [22, *Sjostrand:2007gs] with a specific LHCb configuration [24]. Decays of hadronic particles are described by EvtGen [25]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [26, *Agostinelli:2002hh] as described in Ref. [28]. For the signal, we consider both a phase space model and variations of the decay kinematics with effective operators for the $b\\!\rightarrow s{\mu^{+}}{\tau^{-}}$ interaction and their corresponding Wilson coefficients using the distributions from Ref. [29] and the form factors from Ref. [30]. The branching fraction limit is determined for various hypotheses: for the phase-space decay, for a decay via the vector or axial-vector operators $\mathcal{O}_{9}^{(^{\prime})}$ or $\mathcal{O}_{10}^{(^{\prime})}$, and for a decay using the scalar or pseudoscalar operators $\mathcal{O}^{(^{\prime})}_{S}$ or $\mathcal{O}^{(^{\prime})}_{P}$ [29]. ## 3 Selection and missing mass calculation The selection of ${B}^{+}$ candidates begins with a ${{K}^{+}}{\mu^{-}}$ pair with an invariant mass $m_{{{K}^{+}}{\mu^{-}}}>$1800\text{\,}\mathrm{Me\kern-1.00006ptV}$$ to reduce background from semileptonic charm decays. The ${K}^{+}$ and $\mu^{-}$ candidates are formed from high-quality tracks consistent with kaon and muon hypotheses and inconsistent with being produced at any PV in the event. The ${{K}^{+}}{\mu^{-}}$ vertex must be of high quality and well separated from any PV. To better separate signal candidates with $\tau$ leptons from background, we require an additional track, labelled $t^{+}$, with charge opposite to that of the muon. By adding this third track, we also fully reconstruct the normalisation mode ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$, with ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$. Many background candidates are expected to come from $B$-meson decays of the form ${B}\\!\rightarrow{\kern 1.79993pt\overline{\kern-1.79993ptD}}\quantity(\\!\rightarrow{{K}^{+}}X{\mu^{-}}){{K}^{+}}Y$, where $X$ and $Y$ refer to any number of additional particles. In these cases the kaon originating from the $\kern 1.79993pt\overline{\kern-1.79993ptD}$ meson is assigned as the additional track. Since only approximately 2% of $\tau$ decays contain a charged kaon, we apply particle identification requirements so that the track is unlikely to be a charged kaon. Events in which a candidate ${\tau^{+}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\overline{\nu}}_{\tau}}$ decay is found are not used in this search to avoid overlap with ongoing searches at LHCb exclusively using this decay channel. In addition, events in which we find multiple candidates are not used in this analysis. These requirements do remove signal with multi-prong $\tau$ decays, with an overall loss of less than 3%. Multiple candidate events are more likely to come from background, however. We split the data samples into signal and normalisation regions based on the invariant mass of the ${{K}^{+}}{\mu^{-}}t^{+}$ triple, using the muon hypothesis for the third track. Candidates with $m_{K\mu\mu}<$4800\text{\,}\mathrm{Me\kern-1.00006ptV}$$ fall into the signal region, while candidates with $5180<m_{K\mu\mu}<$5380\text{\,}\mathrm{Me\kern-1.00006ptV}$$ and $\absolutevalue{m_{\mu\mu}-m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi}}}}<$40\text{\,}\mathrm{Me\kern-1.00006ptV}$$ fall into the normalisation region. The ${B}^{+}$ candidate direction is estimated using the PV and ${{K}^{+}}{\mu^{-}}$ vertex positions. We next consider prompt tracks, _i.e._ those that are consistent with being produced at that PV. Those tracks identified as kaons, with a charge opposite to that of the kaon in the ${{K}^{+}}{\mu^{-}}$ pair and a small perpendicular momentum relative to the ${B}^{+}$ candidate direction, are combined with the ${B}^{+}$ candidates to form $B_{s2}^{*0}$ candidates. We refer to this sample as the opposite-sign kaon (OS$K$) sample. Additionally, we select a control sample, referred to as same-sign kaon (SS$K$) sample, by adding prompt kaons of the same sign as the kaon in the ${{K}^{+}}{\mu^{-}}$ pair. From Ref. [19], the two $B$-meson energy solutions are $\displaystyle E_{B}$ $\displaystyle=\frac{\Delta^{2}}{2E_{K}}\frac{1}{1-\quantity(p_{K}/E_{K})^{2}\cos^{2}\theta}\quantity[1\pm\sqrt{d}],\text{ where}$ (1) $\displaystyle d$ $\displaystyle=\frac{p_{K}^{2}}{E_{K}^{2}}\cos^{2}\theta-\frac{4m_{B}^{2}p_{K}^{2}\cos^{2}\theta}{\Delta^{4}}\quantity(1-\frac{p_{K}^{2}}{E_{K}^{2}}\cos^{2}\theta),$ (2) $\displaystyle\Delta^{2}$ $\displaystyle=m_{BK}^{2}-m_{B}^{2}-m_{K}^{2},$ (3) where $m_{BK}=m_{{B_{s2}^{*0}}}$ is the assumed ${{B}^{+}}{{K}^{-}}$ mass, $p_{K}$ and $E_{K}$ are the reconstructed prompt kaon momentum and energy, and $\theta$ is the laboratory frame angle between the prompt kaon and $B$-meson directions. The missing four-momentum of the $\tau$ lepton, $P_{\text{miss}}$, is then reconstructed as $P_{B}-P_{{{K}^{+}}{\mu^{-}}}$, where $P_{B}$ and $P_{{{K}^{+}}{\mu^{-}}}$ are the four-momenta of the $B$ meson and ${{K}^{+}}{\mu^{-}}$ pair. The missing mass squared is calculated using the lowest energy, real solution for which the resulting missing energy is greater than the reconstructed energy of the third track under a pion mass hypothesis. With this choice, we correctly reconstruct the energy of signal decays in simulation in more than 75% of cases. About 9% of all signal decays have no such solution and are lost. Both signal and normalisation candidates, as well as the SS$K$ control-sample candidates, are required to pass this procedure. Candidates in the signal region are additionally required to have the residual missing mass squared, defined as the four-momentum difference of the $B$ meson and ${{K}^{+}}{\mu^{-}}t^{+}$ triple, $\quantity(P_{B}-P_{{{K}^{+}}{\mu^{-}}}-P_{t})^{2}$, greater than $-0.5\text{\,}{\mathrm{Ge\kern-1.00006ptV}}^{2}$. This requirement removes background and only poorly reconstructed signal candidates which do not peak at the $\tau$ mass squared. The minimum mass difference, defined in Ref. [19] as $\Delta m_{\mathrm{min}}=\sqrt{m_{B}^{2}+m_{K}^{2}+2m_{B}\sqrt{p_{K}^{2}\sin^{2}\theta+m_{K}^{2}}}-m_{B}-m_{K},$ (4) is required to be greater than $30\text{\,}\mathrm{Me\kern-1.00006ptV}$. This removes contributions from $B_{s1}^{0}$ and ${B_{s2}^{*0}}\\!\rightarrow{B}^{*+}{{K}^{-}}$ decays, as well as background in which a kaon from the $B$ decay is wrongly associated to the primary vertex. Missing-mass distributions for the signal simulation and the full data sample after the above selection are shown in Fig. 1. All signal decays, whether they come from a $B_{s2}^{*0}$ meson or not, peak at the known $m_{\tau}^{2}$, however the non-$B_{s2}^{*0}$ candidates have a much wider peak than the $B_{s2}^{*0}$ ones. The data distributions are shown for both the OS$K$ and SS$K$ samples. They have similar shapes with a broad hump centred near $5\text{\,}{\mathrm{Ge\kern-1.00006ptV}}^{2}$. We note that the OS$K$ sample has a higher yield than the SS$K$; this excess has been observed in both fully and partially reconstructed decays [31, 19]. Figure 1: Missing mass squared, $m_{\mathrm{miss}}^{2}$, distributions for (left) simulated signal ${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$ decays and (right) all selected candidates in data before applying the signal optimisation described in Sect. 5. ## 4 Normalisation We determine the yield of the normalisation decay, as well as the relative efficiency of the signal modes with respect to the normalisation mode, separately for each data-taking year. For the normalisation mode, we determine the inclusive yield of ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ decays, whether or not they originate from a $B_{s2}^{*0}$ meson, by a binned maximum- likelihood fit to the ${{K}^{+}}{\mu^{-}}t^{+}$ mass distribution, where we assign the muon mass hypothesis to the third track. The signal is described with a Gaussian distribution, and the background with a linear model. We determine the fraction of the normalisation candidates coming from $B_{s2}^{*0}$ decays using a ${{K}^{+}}{\mu^{-}}t^{+}$ mass fit for the combined-years data sample using the same model as the separated-years samples, along with a binned maximum-likelihood fit to the measured mass- difference distribution $m_{{{B}^{+}}{{K}^{-}}}-m_{{{B}^{+}}}-m_{{{K}^{-}}}$ around the $B_{s2}^{*0}$ peak. For the latter fit, we describe the signal peak with a Gaussian core that transitions to an exponential tail on each side, and we model the background with a third-degree polynomial. The results of these fits are shown in Fig. 2. The total data sample contains $4240\pm 70$ ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ decays; the fraction originating from $B_{s2}^{*0}$ decays is $f_{B_{s2}^{*0}}=$25.4\pm 1.8\text{\,}\mathrm{\char 37\relax}$$, where the uncertainty combines the statistical and systematic uncertainties from the choice of fit function. The year-to-year variation is not found to be statistically significant, so we use the value obtained from the combined dataset for all years. Figure 2: Distributions of normalization candidates in (left) mass, $m_{{{K}^{+}}{\mu^{-}}{\mu^{+}}}$, and (right) the mass difference, $m_{{{B}^{+}}{{K}^{-}}}-m_{{{B}^{+}}}-m_{{{K}^{-}}}$. The result of each fit is shown as a solid line, with the background component as a dashed line. The relative efficiency of the signal and normalisation modes is determined using simulation with corrections from data. For $B_{s2}^{*0}$ decays the relative efficiencies in different years average around 30%, with an absolute year-to-year variation of less than 3%. Different signal decay models change the relative efficiency by approximately 10%, with the decays via scalar and pseudoscalar operators having a lower overall efficiency. Signal events in which the ${B}^{+}$ meson does not originate from a $B_{s2}^{*0}$ decay have a lower selection efficiency, primarily because fewer of these candidates pass the residual missing-mass requirement and fall into the missing-mass fit range. Using simulation, we derive an additional efficiency factor for this signal component of $r_{\text{non-{$B_{s2}^{*0}$}}}=0.849\pm 0.007$. ## 5 Multivariate signal selection We further improve the signal selection using a Boosted Decision Tree (BDT) classification with the Adaboost algorithm [32]. The BDT inputs are primarily chosen to distinguish additional tracks coming from signal $\tau$ lepton decays from various sources of background. Some examples are semileptonic $b$-hadron decays to charm where the charm hadron produces a kaon with charge opposite that of the muon, or $b$-hadron decays where the muon is produced in the semileptonic decay of a child charm hadron. The background training sample is taken from the SS$K$ sample in the $m_{\mathrm{miss}}^{2}$ region around $m_{\tau}^{2}$. This focuses the training on the sources of background which fall near the signal peak. We describe the signal with simulation samples that include only $B_{s2}^{*0}$ decays; the effect of the BDT on non-$B_{s2}^{*0}$ signal simulation is then estimated separately. The training makes use of different topological reconstructions of the ${{K}^{+}}{\mu^{-}}t^{+}$ triple: in addition to the signal selection, we also first combine either the kaon and the track or the muon and the track into a pair before adding the third particle. The pair masses and the flight distance of the pair in each topology help to distinguish the signal from background, for instance when the pair comes from a charm hadron decay. We also include the flight distance of the $\tau$, which we reconstruct as the distance along the $\tau$ trajectory found in the missing-mass calculation from the ${{K}^{+}}{\mu^{-}}$ vertex to the point of closest approach of the third track. The result of a separate isolation discriminant is included to reduce background with additional charged tracks; this discriminant is trained to distinguish additional tracks belonging to the same $b$-hadron decay from other tracks in the event based on kinematic and topological variables. We perform the rest of the analysis in four bins of the signal optimisation BDT output, keeping about 70% of all simulated $B_{s2}^{*0}$ signal candidates and about 40% of non-$B_{s2}^{*0}$ signal candidates. The bins are chosen by optimising the expected upper limit using a number of background events derived from the OS$K$ and SS$K$ $m_{\mathrm{miss}}^{2}$ sidebands. ## 6 Background studies The background in this analysis is composed of a large number of different partially reconstructed $b$-hadron decays. None of them, however, produce a narrow peak in $m_{\mathrm{miss}}^{2}$. Only ${B}^{+}$ mesons produced from $B_{s2}^{*0}$ decays have a resolution comparable to the signal. Furthermore, if there is more than one missing particle then the true missing-mass distribution will be much wider than the expected signal peak. Charm hadrons have masses close to the $\tau$ mass, however there is no Standard Model decay ${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{{D}^{+}}$. Because of their low branching fraction, we are not sensitive to decays such as ${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{D}^{+}}$, where the pion is misidentified as a muon. We expect that the missing-mass distribution, summed over many different background components, is smooth, and we model it as a polynomial. These assumptions are tested using simulation and data. We produce fast simulation samples with RapidSim [33] of a number of potential exclusive background sources from ${B}^{+}$, ${B}^{0}$, ${B}^{0}_{s}$, and ${\mathchar 28931\relax}^{0}_{b}$ hadrons; the true missing-mass distributions for these decays are smeared to estimate their shapes in data. No sign of any sharply peaking component is found. In data we consider a number of different control samples, namely all possible $K\mu t$ charge combinations in both OS$K$ and SS$K$ samples, excluding the signal selection of ${{K}^{+}}{\mu^{-}}t^{+}$ in the OS$K$ sample. There is no sign of any narrow peak in any of the distributions, even after applying a tight requirement on the BDT output. Maximum-likelihood fits to the SS$K$ sample using polynomials of different degrees in the restricted $m_{\mathrm{miss}}^{2}$ range from $16\text{\,}{\mathrm{Ge\kern-1.00006ptV}}^{2}$ are used to study the background shape in more detail. The optimal number of free polynomial parameters in the most signal-like BDT output bin, based on the best-fit value of $-2\log\mathcal{L}$, penalised by one for each additional parameter, is four. We further study the effect of background modelling by performing a large number of pseudoexperiments, both background-only and with injected signal at branching fractions of $1\text{\times}{10}^{-5}$ and $2\text{\times}{10}^{-5}$. In these studies, we first fit a background model of some polynomial degree to one of the control samples. From this background model we generate many pseudodatasets that we fit with a model of a different degree. Based on these studies, we take into account the systematic uncertainty due to the background modelling by reporting the weakest limit using background descriptions of third, fourth, or fifth degree polynomials, all of which well describe the background shapes in the pseudoexperiments. ## 7 Fit description We search for the ${{K}^{+}}{\mu^{-}}{\tau^{+}}$ missing-mass peak with an unbinned maximum-ikelihood fit simultaneously in four bins of BDT output in the OS$K$ ${{K}^{+}}{\mu^{-}}t^{+}$ signal channel. The fit is performed in the missing-mass range $1<m_{\mathrm{miss}}^{2}<$6\text{\,}{\mathrm{Ge\kern-1.00006ptV}}^{2}$$. The parameter of interest is the branching fraction ${\mathcal{B}}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}})$. We describe the $m_{\mathrm{miss}}^{2}$ shape for the signal component with a generalized hyperbolic distribution with shape parameters obtained from simulation. Two signal shapes are used: one for $B_{s2}^{*0}$ decays, and one for the wider non-$B_{s2}^{*0}$ contribution. We determine the shapes separately in each bin of BDT response. The signal decay model does not significantly affect the signal missing-mass shape. The background is described by polynomial functions which vary independently in each BDT output bin. We base the normalization of the signal components on the yields of the ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ decays determined in data year-by-year. We combine this together with the relative efficiencies, $\varepsilon_{\text{rel}}$; the known ${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ with ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ combined branching fraction, abbreviated as ${\mathcal{B}}\quantity({{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}})$; and the parameter of interest to derive a total number of ${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$ signal decays. This total is divided between $B_{s2}^{*0}$ and non-$B_{s2}^{*0}$ decays based on the observed fraction in the normalization channel, and then distributed across the four BDT bins. This gives yields in each BDT bin $j$ of $\displaystyle N_{j}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}|{B_{s2}^{*0}})={}$ $\displaystyle\varepsilon_{{B_{s2}^{*0}},j}\frac{{\mathcal{B}}\quantity({{K}^{+}}{\mu^{-}}{\tau^{+}})}{{\mathcal{B}}\quantity({{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}})}f_{{B_{s2}^{*0}}}\times{}$ $\displaystyle\sum_{i\in\text{years}}\varepsilon_{\text{rel},i}N_{i}\quantity({{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}),$ (5) $\displaystyle N_{j}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}|\text{non-{$B_{s2}^{*0}$}})={}$ $\displaystyle\varepsilon_{\text{non-{$B_{s2}^{*0}$}},j}\frac{{\mathcal{B}}\quantity({{K}^{+}}{\mu^{-}}{\tau^{+}})}{{\mathcal{B}}\quantity({{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}})}\quantity(1-f_{{B_{s2}^{*0}}})\times{}$ $\displaystyle\sum_{i\in\text{years}}\varepsilon_{\text{rel},i}r_{\text{non-{$B_{s2}^{*0}$}}}N_{i}\quantity({{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}),$ (6) where $\varepsilon_{{B_{s2}^{*0}},j}$ and $\varepsilon_{\text{non-{$B_{s2}^{*0}$}},j}$ are the separate efficiencies for each signal component to be found in BDT bin $j$. The main parameters of the fit are thus the ${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$ branching fraction, four parameters for the background normalisation in each BDT bin, and up to five parameters describing the polynomial background shapes in each BDT bin. The largest systematic uncertainty comes from the choice of background model. The fifth degree background description obtains the weakest limit among the tested background models. We include the effects of other systematic uncertainties using Gaussian-constrained nuisance parameters. These nuisance parameters modify the normalisation yield, the relative efficiency of the signal and normalisation channels, the signal yield in each BDT bin, and the signal shapes. The largest effects come from the modelling of the kinematics of $B_{s2}^{*0}$ decays in simulation, which results in 5% changes in the relative efficiency and in the signal fractions in each bin of BDT response. The relative statistical uncertainty of the $B_{s2}^{*0}$ fraction taken from the normalisation channel is also approximately 5%. Altogether, the total effect of these systematic uncertainties on the final limit is small, at the $10^{-6}$ level. ## 8 Results and conclusion The result at the best fit point is shown in Fig. 3. The obtained value for the signal branching fraction from the maximum-likelihood fit is ${\quantity(1.9\pm 1.5)\times 10^{-5}}$. No significant excess is observed, and we set upper limits on the branching fraction using the CLs method [34]. We perform a scan in the signal branching fraction, obtaining the signal and background $p$-values from the distributions of a one-sided profile- likelihood-ratio test statistic obtained with pseudoexperiments in which we vary the constraints on the systematic uncertainties. The scan used to determine the observed limits, compared to the expected one, is shown in Fig. 4. The expected upper limit at 90% CL is $2.3\text{\times}{10}^{-5}$. The observed 90% and 95% CL limits, assuming a phase space signal decay model, are: $\displaystyle{\mathcal{B}}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}})$ $\displaystyle<$3.9\text{\times}{10}^{-5}$\text{ at 90\% CL},$ $\displaystyle<$4.5\text{\times}{10}^{-5}$\text{ at 95\% CL}.$ An identical limit is obtained when the decay is generated from the effective operators $\mathcal{O}_{9}^{(^{\prime})}$ or $\mathcal{O}_{10}^{(^{\prime})}$. If instead it is produced from $\mathcal{O}^{(^{\prime})}_{S}$ or $\mathcal{O}^{(^{\prime})}_{P}$, the obtained limit is ${\mathcal{B}}\quantity({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}})<$4.4\text{\times}{10}^{-5}$$ at 90% CL and ${}<$5.0\text{\times}{10}^{-5}$$ at 95% CL. Figure 3: Fits to the missing-mass-squared distribution OS$K$ signal sample in each bin of BDT output included in the final fit. The best fit is overlaid. BDT bin 1 is the most background-like. The fit is performed using a fifth degree polynomial description of the background. Figure 4: Scan of the $p$-value in the signal branching fraction used to determine the CLs upper limits, compared to the expected one. The horizontal red line shows a $p$-value of 0.1, used to define the 90% CL upper limit. This is the first result from the LHCb experiment for the lepton-flavour violating decay ${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{-}}{\tau^{+}}$. By studying ${B}^{+}$ mesons from $B_{s2}^{*0}$ decays, we are able to make the first analysis at LHCb of a $B$ hadron decay using inclusive $\tau$ decays. This provides complementary information to searches for lepton-flavour violation at LHCb with three-prong $\tau$ decays, for example $B_{(s)}^{0}\\!\rightarrow\tau^{\pm}\mu^{\mp}$ decays [35]. We observe no significant signal, and set an upper limit slightly above that obtained by the BaBar collaboration [17]. ## Acknowledgements We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL- GRID (Poland) and OSC (USA). We are indebted to the communities behind the multiple open-source software packages on which we depend. Individual groups or members have received support from AvH Foundation (Germany); EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU, and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society and the Leverhulme Trust (United Kingdom). ## References * [1] LHCb collaboration, R. 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Zucchelli19,e. 1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil 2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 3Center for High Energy Physics, Tsinghua University, Beijing, China 4School of Physics State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing, China 5University of Chinese Academy of Sciences, Beijing, China 6Institute Of High Energy Physics (IHEP), Beijing, China 7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, China 8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy, France 9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France 10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France 11Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France , Orsay, France 12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3, Paris, France 13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany 14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany 15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany 16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 17School of Physics, University College Dublin, Dublin, Ireland 18INFN Sezione di Bari, Bari, Italy 19INFN Sezione di Bologna, Bologna, Italy 20INFN Sezione di Ferrara, Ferrara, Italy 21INFN Sezione di Firenze, Firenze, Italy 22INFN Laboratori Nazionali di Frascati, Frascati, Italy 23INFN Sezione di Genova, Genova, Italy 24INFN Sezione di Milano-Bicocca, Milano, Italy 25INFN Sezione di Milano, Milano, Italy 26INFN Sezione di Cagliari, Monserrato, Italy 27INFN Sezione di Padova, Padova, Italy 28INFN Sezione di Pisa, Pisa, Italy 29INFN Sezione di Roma Tor Vergata, Roma, Italy 30INFN Sezione di Roma La Sapienza, Roma, Italy 31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands 32Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, Netherlands 33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Kraków, Poland 34AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science, Kraków, Poland 35National Center for Nuclear Research (NCBJ), Warsaw, Poland 36Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania 37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI), Gatchina, Russia 38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute (ITEP NRC KI), Moscow, Russia, Moscow, Russia 39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia 40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS), Moscow, Russia 41Yandex School of Data Analysis, Moscow, Russia 42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia 43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI), Protvino, Russia, Protvino, Russia 44ICCUB, Universitat de Barcelona, Barcelona, Spain 45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela, Santiago de Compostela, Spain 46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia - CSIC, Valencia, Spain 47European Organization for Nuclear Research (CERN), Geneva, Switzerland 48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland 49Physik-Institut, Universität Zürich, Zürich, Switzerland 50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine 51Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine 52University of Birmingham, Birmingham, United Kingdom 53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom 54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom 55Department of Physics, University of Warwick, Coventry, United Kingdom 56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom 57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 58School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom 59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom 60Imperial College London, London, United Kingdom 61Department of Physics and Astronomy, University of Manchester, Manchester, United Kingdom 62Department of Physics, University of Oxford, Oxford, United Kingdom 63Massachusetts Institute of Technology, Cambridge, MA, United States 64University of Cincinnati, Cincinnati, OH, United States 65University of Maryland, College Park, MD, United States 66Los Alamos National Laboratory (LANL), Los Alamos, United States 67Syracuse University, Syracuse, NY, United States 68Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria, associated to 2 69School of Physics and Astronomy, Monash University, Melbourne, Australia, associated to 55 70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2 71Guangdong Provencial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou, China, associated to 3 72School of Physics and Technology, Wuhan University, Wuhan, China, associated to 3 73Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia, associated to 12 74Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16 75Van Swinderen Institute, University of Groningen, Groningen, Netherlands, associated to 31 76National Research Centre Kurchatov Institute, Moscow, Russia, associated to 38 77National University of Science and Technology “MISIS”, Moscow, Russia, associated to 38 78National Research University Higher School of Economics, Moscow, Russia, associated to 41 79National Research Tomsk Polytechnic University, Tomsk, Russia, associated to 38 80University of Michigan, Ann Arbor, United States, associated to 67 aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil bLaboratoire Leprince-Ringuet, Palaiseau, France cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia dUniversità di Bari, Bari, Italy eUniversità di Bologna, Bologna, Italy fUniversità di Cagliari, Cagliari, Italy gUniversità di Ferrara, Ferrara, Italy hUniversità di Genova, Genova, Italy iUniversità di Milano Bicocca, Milano, Italy jUniversità di Roma Tor Vergata, Roma, Italy kUniversità di Roma La Sapienza, Roma, Italy lAGH - University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Kraków, Poland mDS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain nHanoi University of Science, Hanoi, Vietnam oUniversità di Padova, Padova, Italy pUniversità di Pisa, Pisa, Italy qUniversità degli Studi di Milano, Milano, Italy rUniversità di Urbino, Urbino, Italy sUniversità della Basilicata, Potenza, Italy tScuola Normale Superiore, Pisa, Italy uUniversità di Modena e Reggio Emilia, Modena, Italy vUniversità di Siena, Siena, Italy wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines xNovosibirsk State University, Novosibirsk, Russia yINFN Sezione di Trieste, Trieste, Italy zSchool of Physics and Information Technology, Shaanxi Normal University (SNNU), Xi’an, China aaPhysics and Micro Electronic College, Hunan University, Changsha City, China abUniversidad Nacional Autonoma de Honduras, Tegucigalpa, Honduras

2024-09-04T02:54:58.521393

2003.04360

arxiv-papers

# GenNet: Reading Comprehension with Multiple Choice Questions using Generation and Selection model Vaishali Ingale Department of Information Technology Army Institute of Technology, Pune <EMAIL_ADDRESS> Pushpender Singh Department of Information Technology Army Institute of Technology, Pune <EMAIL_ADDRESS> ###### Abstract Multiple-choice machine reading comprehension is difficult task as its required machines to select the correct option from a set of candidate or possible options using the given passage and question.Reading Comprehension with Multiple Choice Questions task, required a human (or machine) to read a given passage, question pair and select the best one option from n given options. There are two different ways to select the correct answer from the given passage. Either by selecting the best match answer to by eliminating the worst match answer. Here we proposed GenNet model, a neural network-based model. In this model first we will generate the answer of the question from the passage and then will matched the generated answer with given answer, the best matched option will be our answer. For answer generation we used S-net (Tan et al.,, 2017) model trained on SQuAD and To evaluate our model we used Large-scale RACE (ReAding Comprehension Dataset From Examinations) (Lai et al.,, 2017). ## 1 Introduction Reading comprehension is one of the fundamental skills for human, which one learn systematically since the elementary school. Reading comprehension give human the ability of reading texts, understanding their meanings,and with the help of given context answering questions. When machines are required to comprehend texts, they first need to understand the unstructured text and do reasoning based on given text (Chen et al.,, 2016)(Wang et al., 2018b, ).Answering questions based a passage requires an individual unique skill set. It requires ability to perform basic mathematical operations and logical ability (e.g. to answer questions like how many times Amit visited sweet shop?), look-up ability, ability to deduce, ability to gather information contained in multiple sentences and passages. This diverse and unique skill set makes question answering a challenging task.There are several variants of this task, For example, if we have a given passage and a question, the answer could either (i) be generated from the passage (ii) match some span in the passage (iii) or could be one of the n number of given candidate answers. The last variant is mostly used in various high school, quiz , middle school, and different competitive examinations. This variant of Reading Comprehension generally referred as Reading Comprehension with Multiple Choice Questions (RC-MCQ).In the given figure 1 We have a passage and a question and 4 candidate answers. Task here defined is to find the most suitable answer from the passage for given question. While answering such Multiple Choice Questions (MCQs) figure 1, humans typically use a combination of option elimination and option selection or sometimes they find answer from the passage i.e they generate the answer of the question from passage and match the generated answer with given options and they choose more close candidate as correct answer. Here we proposed model which mimic the answer generation and then matching human process.First the span where possible answer in the passage is computed. we first compute a question-aware representation of the passage (which essentially tries to retain portions of the passage which are only relevant to the question). Then we use answer generation using state-of-art S-Net model (Tan et al.,, 2017)which extract and generate answer figure 2. After we have answer generated from the passage now we weight every given candidate option and select the best matched option. That best matched option was our answer figure 3. Figure 1: An example multiple-choice reading comprehension question. Figure 2: Overview of S-Net.(Tan et al.,, 2017) Figure 3: Overview of option matching and selection. ## 2 Related Work Datasets played an important role in machine reading comprehension, there were different type of datasets designed to solve different variant of machine reading comprehension. SQuAD dataset(Rajpurkar et al.,, 2016) was designed to answer simple question answer reading comprehension that aims to answer a question with exact text spans in a passage. Later MS-MACRO dataset(Nguyen et al.,, 2016) was designed for multi-passage reading comprehension. CNN/ Dailymail (Chen et al.,, 2016) and Who did what dataset(Onishi et al.,, 2016) designed for cloze variant reading comprehension. MCtest(Richardson et al.,, 2013) and RACE dataset(Lai et al.,, 2017) are released for Multiple choice question variant reading comprehension. Similar work in reading comprehension where Multiple choice variant of Comprehension considered includes Hierarchical Attention Flow model(Zhu et al.,, 2018), in this model the candidate options leverage to model the interaction between question options and passage.This was a option selection model which select the correct option from the given candidate options. Other work relatable to this paper was eliminating options model(Parikh et al.,, 2019) which eliminate the wrong answer from the candidate answer.Multi matching network(Tang et al.,, 2019) models interaction relationship between passage, questions and candidate answer. It take different paradigm of matching into account. Option comparison Network (Ran et al.,, 2019) compares between options at word level and identify correlation to help buildup logic and reasoning. Co-matching model (Wang et al., 2018a, ) is used to match between answer and question and passage pair. It match for the relationship between question and answer with the passage. Bidirectional co-matching based model (Zhang et al.,, 2019) matched passage and question, answer bidirectionally. The Convolutional Spatial Attention (CSA) model (Chen et al.,, 2019) form the enriched representaion by fully extract the mutual information among the passage, question, and the candidates. To generate answer several models are there like QANet (Yu et al.,, 2018) which combined local Convolution with Global Self-Attention and its encoder consist exclusively of convolution and self-attention.Bidirectional Attention Flow model (Seo et al.,, 2016) use to focus on large span of passage. BIDAF network is a multi stage hierarchical process and use bidirection attention flow to obtain a query-aware context representation. But the reason to use S-Net model as answer generation model because S-Net not only find the answer from the passage but it can also synthesise passage when required. Some questions are tricky and there answer lies in different span of passage. In such situation S-Net is useful as it remember the past context for longer time as it have GRU as basic component. ## 3 Proposed model There are two tasks needs to be performed in this model. First is Answer extraction and Answer Synthesis/Generation and then option selection. Answer extraction and Generation will be done using state-of-art S-NET model(Tan et al.,, 2017). S-Net first pull out evidence snippets by matching the question and passage respectively, and then generates the answer by filtering the question, passage, and evidence snippets. consider a passage $P=[p_{1},p_{2},p_{3},...p_{p}]$ of word length P, Question $Q=[Q_{1},Q_{2},Q_{3},...Q_{q}]$ of word length Q, and n options $Z_{n}=[z_{1},z_{2},z_{3},...z_{k}]$ where n > 1 and word length k. We first convert the words to their word-level embedding and character-level embedding using GLOVE(Pennington et al.,, 2014).The encoding and embedding layers take in a series of tokens and represent it as a series of vectors. The character- level embeddings are cause by taking the final hidden states of a bi- directional GRU applied to embedding of characters in the token. They then use a bi-directional Gated Recurrent Unit to give rise to new depiction $u_{1}^{p},u_{2}^{p},u_{3}^{p},...u_{p}^{p}$ for questions as well as $u_{1}^{q},u_{2}^{q},u_{3}^{q},...u_{q}^{q}$ for passages too and $u_{1}^{z},u_{2}^{z},u_{3}^{z},...u_{z}^{z}$ for options as well. The embedding matrix is boot only once and not trained in the entire learning process. As shown in Figure 4 S-NET uses the series-to-series model to incorporate the answer with the extracted evidences as features. They first produce the depiction It first produce the depiction $h_{p}^{t}$ and $h_{q}^{t}$ of all words in the question and passage respectively. When giving out the answer depiction, it merge the basic word embedding $e_{p}^{t}$ with some added features $f_{s}^{t}$ and $f_{e}^{t}$ to indicate the end and start place of the evidence snippet given out by evidence extraction model. $f_{s}^{t}$=1 and $f_{e}^{t}$=1 mean the position t is the start and end of the evidence span, respectively. $h_{t}^{p}=BiGRU(h_{t-1}^{p},[e_{t}^{p},f_{t}^{s},f_{t}^{e}])$ (1) $h_{t}^{q}=BiGRU(h_{t-1}^{q},e_{t}^{q})$ (2) On top of the encoder, S-Net uses GRU with attention as the decoder to produce the answer. At each decoding time step t , the GRU reads the previous word embedding $w_{t-1}$ and previous context vector $c_{t-1}$ and finally produced answer. Figure 4: Answer Synthesis/Generation Model(Tan et al.,, 2017) The produced answer will be stored in Answer vector. $A_{n}=[a_{1},a_{2},a_{3},...a_{a}]$ where a is length of the answer.Figure 3 shows the overview of selection module. The selection module will take the refined answer representation $a_{t}$ and computes its bi-linear similarity with each option representation. $score(i)=a_{t}W_{att}z_{t_{i}}$ (3) where i is the number of option, $a_{t}$ is generated answer vector, $z_{t_{i}}$ is option vector and $W_{att}$ is a matrix which needs to be learned. We select the option which gives the highest score as computed above. We train the model using the cross entropy loss by normalizing the above scores (using softmax) first to obtain a probability distribution. ## 4 Experimental Setup Here we discussed about the dataset used to evaluate our model, Training procedure, result comparison and future work. ### 4.1 Dataset We evaluate our model on RACE dataset(Lai et al.,, 2017) Race is a large-scale reading comprehension dataset with more than 28,000 passages and nearly 100,000 questions. The dataset is collected from English examinations in China, which are designed for middle school and high school students. Each passage is a JSON file. The JSON file contains fields (i) article: A string, which is the passage (ii) questions: A string list. Each string is a query. There are two types of questions. First one is an interrogative sentence. Another one has a placeholder, which is represented by _. (iii)options: A list of the options list. Each options list contains 4 strings, which are the candidate option. (iv) answers: A list contains the golden label of each query.(v) id: Each passage has a unique id in this dataset. RACE has wide variety of questions like Summarization, Inference, Deduction and Context matching. Figure 5: Statistic information about Reasoning type in RACE dataset ### 4.2 Training Procedures and Hyper-parameter We integrate two different model into once. First we train our model on S-Net. To train model on S-Net we process dataset differently. We only consider passage and question and correct option to train model on S-Net. Later we pass the result on to next stage on our model where we train model using generated answer and all candidate options. To train the model, we used stochastic gradient descent with ADAM optimizer.(Kingma and Ba,, 2014) We initialize learning rate with 0.005. Gradients are clipped in L2-norm to no larger than 10\. To update model parameter per step,we used A mini-batch of 32 samples. We have created a vocabulary using top 65k words from passage and questions and if a new out-of-vocabulary(OOV) word encountered we add a special token UNK. We use the same vocabulary for the passage, question, and options vector embedding. We tune all our models based on the accuracy achieved on the validation set. We use 300 dimensional Glove embedding (Pennington et al.,, 2014) for word embedding and word and character encoding.We experiment with both fine-tuning and not fine-tuning these word embedding. We train all our models for upto 80 epochs as we do not see any benefit of training beyond 80 epochs as result were starting recurrence.The hidden state size of all GRU network is 128. We apply dropout (Srivastava et al.,, 2014)to word embeddings and BiGRU’s outputs with a drop rate of 0.45. ### 4.3 Results and Future Work Model | RACE-Mid | RACE-High | RACE ---|---|---|--- Random* | 24.6 | 25.0 | 24.9 Sliding Window* | 37.3 | 30.4 | 32.2 GA Reader (100D)* | 43.7 | 44.2 | 44.1 Stanford AR (100D)* | 44.2 | 43.0 | 43.3 Sliding Window* | 37.3 | 30.4 | 32.2 GenNet | 79.6 | 75.4 | 77.3 Table 1: Accuracy on test set of RACE-M, RACE-H and RACE. * indicates the results from (Lai et al.,, 2017) which are trained with 100D pre-trained Glove word embeddings The Human Ceiling Performance reported by CMU on RACE dataset is 94.2. Our model gives accuracy of 79.6 % on RACE-M 75.4 % on RACE-H and 77.3% on RACE FULL which outperform several other model. Since in this model first answer are generated and then option is selected such model can be used to solve such multiple choice question whose answer option is not present or MCQ with "none of the above" or "No answer" type multiple choice questions. ## 5 Conclusion In this paper, we present the GenNet model for multiple-choice reading comprehension. Specifically, the model uses a combination of Generation and selection to arrive at the correct option. This is achieved by first generating the answer for the questions from the passage and then matching generated answer with the options.At last, the proposed model achieves overall sate-of-the-art accuracy on RACE and significantly outperforms neural network baselines on RACE-M, RACE-H and RACE FULL.As future work, we would like to work towards unanswerable questions or questions where no option matched. ## References * Chen et al., (2016) Chen, D., Bolton, J., and Manning, C. D. (2016). A thorough examination of the cnn/daily mail reading comprehension task. arXiv preprint arXiv:1606.02858. * Chen et al., (2019) Chen, Z., Cui, Y., Ma, W., Wang, S., and Hu, G. (2019). Convolutional spatial attention model for reading comprehension with multiple-choice questions. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 6276–6283. * Kingma and Ba, (2014) Kingma, D. P. and Ba, J. (2014). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980. * Lai et al., (2017) Lai, G., Xie, Q., Liu, H., Yang, Y., and Hovy, E. (2017). Race: Large-scale reading comprehension dataset from examinations. arXiv preprint arXiv:1704.04683. * Nguyen et al., (2016) Nguyen, T., Rosenberg, M., Song, X., Gao, J., Tiwary, S., Majumder, R., and Deng, L. (2016). Ms marco: a human-generated machine reading comprehension dataset. * Onishi et al., (2016) Onishi, T., Wang, H., Bansal, M., Gimpel, K., and McAllester, D. (2016). Who did what: A large-scale person-centered cloze dataset. arXiv preprint arXiv:1608.05457. * Parikh et al., (2019) Parikh, S., Sai, A. B., Nema, P., and Khapra, M. M. (2019). Eliminet: A model for eliminating options for reading comprehension with multiple choice questions. arXiv preprint arXiv:1904.02651. * Pennington et al., (2014) Pennington, J., Socher, R., and Manning, C. D. (2014). Glove: Global vectors for word representation. In Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP), pages 1532–1543. * Rajpurkar et al., (2016) Rajpurkar, P., Zhang, J., Lopyrev, K., and Liang, P. (2016). Squad: 100,000+ questions for machine comprehension of text. arXiv preprint arXiv:1606.05250. * Ran et al., (2019) Ran, Q., Li, P., Hu, W., and Zhou, J. (2019). Option comparison network for multiple-choice reading comprehension. arXiv preprint arXiv:1903.03033. * Richardson et al., (2013) Richardson, M., Burges, C. J., and Renshaw, E. (2013). Mctest: A challenge dataset for the open-domain machine comprehension of text. In Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing, pages 193–203. * Seo et al., (2016) Seo, M., Kembhavi, A., Farhadi, A., and Hajishirzi, H. (2016). Bidirectional attention flow for machine comprehension. arXiv preprint arXiv:1611.01603. * Srivastava et al., (2014) Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I., and Salakhutdinov, R. (2014). Dropout: a simple way to prevent neural networks from overfitting. The journal of machine learning research, 15(1):1929–1958. * Tan et al., (2017) Tan, C., Wei, F., Yang, N., Du, B., Lv, W., and Zhou, M. (2017). S-net: From answer extraction to answer generation for machine reading comprehension. arXiv preprint arXiv:1706.04815. * Tang et al., (2019) Tang, M., Cai, J., and Zhuo, H. H. (2019). Multi-matching network for multiple choice reading comprehension. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 7088–7095. * (16) Wang, S., Yu, M., Chang, S., and Jiang, J. (2018a). A co-matching model for multi-choice reading comprehension. arXiv preprint arXiv:1806.04068. * (17) Wang, Y., Li, R., Zhang, H., Tan, H., and Chai, Q. (2018b). Using sentence-level neural network models for multiple-choice reading comprehension tasks. Wireless Communications and Mobile Computing, 2018. * Yu et al., (2018) Yu, A. W., Dohan, D., Luong, M.-T., Zhao, R., Chen, K., Norouzi, M., and Le, Q. V. (2018). Qanet: Combining local convolution with global self-attention for reading comprehension. arXiv preprint arXiv:1804.09541. * Zhang et al., (2019) Zhang, S., Zhao, H., Wu, Y., Zhang, Z., Zhou, X., and Zhou, X. (2019). Dual co-matching network for multi-choice reading comprehension. arXiv preprint arXiv:1901.09381. * Zhu et al., (2018) Zhu, H., Wei, F., Qin, B., and Liu, T. (2018). 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2024-09-04T02:54:58.531454

2003.04376

arxiv-papers

# Not all disadvantages are equal: Racial/ethnic minority students have largest disadvantage of all demographic groups in both STEM and non-STEM GPA Kyle M. Whitcomb Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15260 Chandralekha Singh Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15260 ###### Abstract An analysis of institutional data to understand the outcome of the many obstacles faced by students from historically disadvantaged backgrounds is important in order to work towards promoting equity and inclusion for all students. We use 10 years of institutional data at a large public research university to investigate the grades earned (both overall and in STEM courses only) by students categorized on four demographic characteristics: gender, race/ethnicity, low-income status, and first-generation college student status. We find that on average across all years of study and for all clusters of majors, underrepresented minority students experience a larger penalty to their mean overall and STEM GPA than even the most disadvantaged non-URM students. Moreover, the underrepresented minority students with additional disadvantages due to socioeconomic status or parental education level were even further penalized in their average GPA. Furthermore, we also find that while women in all demographic groups had a higher average overall GPA, these gender differences are almost completely non-existent in STEM GPA except among the most privileged students. These findings suggest that there is need to provide support to bridge the gaps that emanate from historical disadvantages to certain groups. ## I Introduction and Theoretical Framework The importance of evidence-based approaches to improving student learning and ensuring that all students have the opportunity to excel regardless of their background is becoming increasingly recognized by Science, Technology, Engineering, and Mathematics (STEM) departments across the US Johnson (2012); Johnson _et al._ (2017); Maltese and Tai (2011); Borrego _et al._ (2008); Borrego and Bernhard (2011); Borrego and Henderson (2014); Henderson and Dancy (2008); Dancy and Henderson (2010); Henderson _et al._ (2012). With advances in digital technology in the past few decades, institutions have been keeping increasingly large digital databases of student records. We have now reached the point where there is sufficient data available for robust statistical analyses using data analytics that can provide valuable information useful for transforming learning for all students Baker and Inventado (2014); Papamitsiou and Economides (2014). This has lead to many recent studies utilizing many years of institutional data to perform analyses that were previously limited by statistical power Lord _et al._ (2009, 2015); Ohland and Long (2016); Matz _et al._ (2017); Witherspoon and Schunn (2019). Therefore, here we focus on harnessing institutional data to investigate the obstacles faced by students with various disadvantages who must overcome obstacles in their pursuit of higher education. The theoretical framework for this study has two main foundations: critical theory and intersectionality. Critical theories of race, gender, etc. identify historical sources of inequity within society, that is, societal norms that perpetuate obstacles to the success of certain groups of disadvantaged people Crenshaw _et al._ (1995); Kellner (2003); Yosso (2005); Gutiérrez (2009); Taylor _et al._ (2009); Tolbert _et al._ (2018); Schenkel and Calabrese Barton (2020). Critical theory tells us that the dominant group in a society perpetuates these norms, which are born out of their interests, and pushes back against support systems that seek to subvert these norms Crenshaw _et al._ (1995); Kellner (2003); Yosso (2005). These highly problematic societal norms are founded in the historical oppression of various groups of people, and manifest today in many ways including economic disadvantages, stereotypes about who can succeed in certain career paths, and racist and/or sexist barriers to opportunity, including educational advancement. While these norms are, by definition, specific to a particular culture or even country, they are nonetheless pervasive and oppressive and demand attention to rectify these historical wrongs. Much important work has been done on building critical race and/or gender theories of STEM education Johnson (2012); Johnson _et al._ (2017); Solorzano _et al._ (2000); Lewis _et al._ (2009); Bang and Medin (2010); Estrada _et al._ (2018); Ong _et al._ (2018); Tolbert _et al._ (2018); Green _et al._ (2019); Mutegi _et al._ (2019); Sheth (2019); Schenkel and Calabrese Barton (2020). In one study, Bancroft (2018) lays out a “critical capital theory,” using varying forms of capital (economic, social, and cultural) to examine persistence through graduation in STEM doctoral programs and to contextualize the mechanisms behind racial inequities in STEM education Bancroft (2018). The idea that race, gender, or another demographic characteristic alone cannot fully explain the intricacies of the obstacles that students face is rooted in the framework of intersectionality Crenshaw (1990); Cho _et al._ (2013); Mitchell _et al._ (2014); Charleston _et al._ (2014); Morton and Parsons (2018). In particular, the combination of different aspects of an individual’s social identity (e.g., gender, race, first-generation college status, and socioeconomic status) leads to unique levels of disadvantages that cannot be explained by simply adding together the effects of the individual components of identity Crenshaw (1990). For example, according to the framework of intersectionality, in many STEM disciplines where the societal norm expects that students are white men, the experience of a black woman is not a simple sum of the experiences of white women and black men Charleston _et al._ (2014); Morton and Parsons (2018). With an eye toward this intersectional approach to critical theory, we seek to understand the relationship between four different aspects of student identity that can lead to obstacles in STEM education: race/ethnicity, gender, low- income status, and first-generation college student status. The students disadvantaged by low-income or first-generation status are likely to experience a lack of resources relative to their more privileged peers Lam _et al._ (2005); Dika and D’Amico (2016); Katrevich and Aruguete (2017). Women and underrepresented minority students are susceptible to additional stress and anxiety from stereotype threat (i.e., the fear of confirming stereotypes pertaining to their identity) which is not experienced by their majority group peers Lewis _et al._ (2009); Johnson (2012); Green _et al._ (2019); Mutegi _et al._ (2019); Sheth (2019); Astin (1993); Cross (1993); Felder _et al._ (1995, 1998); Bianchini _et al._ (2002); Britner and Pajares (2006); Bianchini (2013); Basile and Lopez (2015); Cheryan _et al._ (2017); Hilts _et al._ (2018). In summary, the different mechanisms by which students belonging to each demographic characteristic can be disadvantaged are as follows. * • Race/Ethnicity: Students belonging to underrepresented minority (URM) groups may experience stereotype threat that causes anxiety and robs the students of their cognitive resources, particularly during high-stakes testing. * • Gender: There are pervasive societal biases against women succeeding in many STEM disciplines which can result in stereotype threat. * • Low-Income Status: Low-Income (LI) students are more likely to need to work to support themselves, reducing their time and energy available to devote to their studies, in addition to anxiety due to the financial burden of attending college. These burdens are in addition to other factors that low-income students may be more likely to face, such as lower quality preparation for college. * • First-Generation Status: First-Generation (FG) students may lack the resources of encouragement, advice, and support that are available more readily to students with degree-holding parents. This lack of resources can make FG students more susceptible to the stress of the unknown in college. All of these mechanisms can produce an inequitable learning environment wherein students belonging to any of these groups are forced to work against obstacles that their peers do not have. The framework of intersectionality asserts that for students that belong to more than one of these groups, complex interactions between these different obstacles can result in compounded disadvantages that are not a simple sum of the individual effects Crenshaw (1990); Cho _et al._ (2013); Mitchell _et al._ (2014); Charleston _et al._ (2014); Morton and Parsons (2018). In order to measure the long-term effects of these systemic disadvantages, we will investigate the academic achievement of students belonging to these various demographic groups over the course of their studies at one large public research university using 10 years of institutional data. By grouping students according to their demographic background, we will be able to investigate how different combinations of obstacles affect student grade point averages. ## II Research Questions Our research questions regarding the intersectional relationships between demographic characteristics and academic achievement are as follows. 1. RQ1. Are there differences in the overall or STEM grades earned by students belonging to different demographic groups (i.e., underrepresented minority, low-income status, and first-generation college student status)? 2. RQ2. Do any patterns observed in RQ1 differ for men and women? 3. RQ3. Do grades earned in STEM courses alone exhibit similar demographic patterns as grades earned in all courses? 4. RQ4. What are the trends over time in the mean GPA of these different demographic groups among different clusters of majors (i.e., computer science, engineering, mathematics, and physical science majors, other STEM majors, and non-STEM majors)? ## III Methodology ### III.1 Sample Using the Carnegie classification system, the university at which this study was conducted is a public, high-research doctoral university, with balanced arts and sciences and professional schools, and a large, primarily residential undergraduate population that is full-time and reasonably selective with low transfer-in from other institutions Indiana University Center for Postsecondary Research (2018). The university provided for analysis the de-identified institutional data records of students with Institutional Review Board approval. In this study, we examined these records for $N=24,567$ undergraduate students enrolled in three colleges within the university: the colleges of Arts and Sciences, Computing and Information, and Engineering. This sample of students includes all of those from ten cohorts who met several selection criteria, namely that the student had first enrolled at the university in a Fall semester from Fall 2005 to Fall 2014, inclusive, and the institutional data on the student was not missing or unspecified for any of the following measures: gender, race/ethnicity, parental education level, and family income. This sample of students is $50\%$ female and had the following race/ethnicities: 79% White, 9% Asian, 7% Black, 3% Hispanic, and 2% other or multiracial. Further, this sample is $16\%$ first-generation college students and $21\%$ “low-income” students (to be defined in the following section). We acknowledge that gender is not a binary construct, however in self- reporting their gender to the university students were given the options of “male” or “female” and so those are the two self-reported genders that we are able to analyze. There were $39$ students who had met all other selection criteria but who had not indicated any gender on the survey, these students were removed from the sample and are not included in the reported sample size or any analyses. ### III.2 Measures #### III.2.1 Demographic Characteristics Four primary measures are the demographic characteristics mentioned in the previous section, namely gender, race/ethnicity, parental education level, and family income. All of these were converted into binary categories intended to distinguish between the most and least privileged students on each measure. * • Gender. Gender was reported as a binary category to begin with (either “male” or “female”), therefore no further steps were required. * • First-generation. Students for whom both parents had a highest completed level of education of high school or lower were grouped together as “first- generation” (FG) college students and correspondingly students for whom at least one parent had earned a college degree were labeled non-FG. * • Low-income. Students whose reported family Adjusted Gross Income (AGI) was at or below 200% of the federal U.S. poverty line were categorized as “low- income” (LI), and those above 200% of the poverty line as non-LI Cauthen and Fass (2007); Jiang _et al._ . * • Underrepresented minority. All students who identified as any race or ethnicity other than White or Asian were grouped together as “underrepresented minority” (URM) students, including multiracial students who selected White and/or Asian in addition to another demographic option. Students who only identified as White and/or Asian students were categorized as non-URM students. #### III.2.2 Academic Performance Measures of student academic performance were also included in the provided data. High school GPA was provided by the university on a weighted scale from 0-5 that includes adjustments to the standard 0-4 scale for Advanced Placement and International Baccalaureate courses. The data also include the grade points earned by students in each course taken at the university. Grade points are on a 0-4 scale with $\text{A}=4$, $\text{B}=3$, $\text{C}=2$, $\text{D}=1$, $\text{F}=0$, where the suffixes “$+$” and “$-$” add or subtract, respectively, $0.25$ grade points (e.g. $\text{B}-=2.75$), with the exception of $\text{A}+$ which is reported as the maximum 4 grade points. The courses were categorized as either STEM or non-STEM courses, with STEM courses being those courses taken from any of the following departments: biological sciences, chemistry, computer science, economics, any engineering department, geology and environmental science, mathematics, neuroscience, physics and astronomy, and statistics. We note that for the purposes of this paper, “STEM” does not include the social sciences other than economics, which has been included due to its mathematics-intensive content. #### III.2.3 Year of Study Finally, the year in which the students took each course was calculated from the students’ starting term and the term in which the course was taken. Since the sample only includes students who started in fall semesters, each “year” contains courses taken in the fall and subsequent spring semesters, with courses taken over the summer omitted from this analysis. For example, if a student first enrolled in Fall 2007, then their “first year” occurred during Fall 2007 and Spring 2008, their “second year” during Fall 2008 and Spring 2009, and so on in that fashion. If a student is missing both a fall and spring semester during a given year but subsequently returns to the university, the numbering of those post-hiatus years is reduced accordingly. If instead a student is only missing one semester during a given year, no corrections are made to the year numbering. In this study we consider up through the students’ sixth year of study or the end of their enrollment at the studied institution, whichever comes first. ### III.3 Analysis The primary method by which we grouped students in this analysis was by their set of binary demographic categories. This grouping was performed in two different ways. First, use of all four binary categories (gender, FG, LI, URM) resulted in sixteen mutually exclusive groups (e.g., “female, FG+URM” or “male, LI”). Second, use of all categories except gender resulted in eight mutually exclusive categories. We calculated each student’s yearly (i.e., not cumulative) grade point average (GPA) across courses taken in each year of study from the first to sixth years. In addition, we calculated the student’s yearly STEM GPA, that is, the GPA in STEM courses alone. Then, using the aforementioned grouping schemes, we computed the mean GPA in each demographic group as well as the standard error of the mean separately for each year of study Freedman _et al._ (2007). Further, in the case of grouping by gender, we computed the effect size of the gender differences within each demographic group using Cohen’s $d$, which is typically interpreted using minimum cutoff values for “small” ($d=0.20$), “medium” ($d=0.50$), and “large” ($d=0.80$) effect sizes Cohen (1988); Neter _et al._ (2004); Montgomery _et al._ (2012). All analyses were conducted using R R Core Team (2019), making use of the package tidyverse Wickham (2017) for data manipulation and plotting. ## IV Results ### IV.1 GPA Trends by Demographic Group: “Dinosaur Plots” In order to answer RQ1, we plotted in Fig. 1 the mean GPA earned by students in each demographic group, including gender as a grouping characteristic. We start with overall GPA, rather than STEM GPA alone, in order to provide context for the results in STEM GPA and identify trends that may or may not be present when viewing STEM grades alone. Groups are ordered from left to right first by the ascending number of selected characteristics and then alphabetically. Mean GPA is plotted separately (i.e., not cumulatively) for each year of study from the first to sixth year. Setting aside the gender differences for a moment, we note that the general GPA trends by demographic group in Fig. 1 follow a shape resembling the neck, back, and tail of a sauropod, and so accordingly we refer to the plots in Fig. 1 as “dinosaur plots.” This shape is clearest in the plots for the first through fourth years, as the sample size drops significantly in the fifth year as the majority of students graduate. Looking more closely at Fig. 1, particularly the first four years, we see that the “neck” is consistently comprised of the group of students with the most privileges, namely those students that are non-FG, non-LI, and non-URM. Following this, the “back” is relatively flat across the next four groups, namely students that are FG only, LI only, URM only, or FG and LI. Notably, the URM group of students typically have the lowest mean GPA within this set of demographic groups. Finally, the “tail” consists of the final three groups, FG+URM, LI+URM, and FG+LI+URM. The mean GPA in this set of groups tends to decrease from left to right in the plots. Notably, the four groups that contain URM students are consistently in the lowest four or five mean GPAs. Figure 1: Average GPA of each demographic group. Students are binned into separate demographic groups based on their status as first-generation (FG), low-income (LI), and/or underrepresented minority (URM) students. The men and women in each demographic group are plotted separately. The mean GPA in all courses taken by students in each demographic group is plotted along with the standard error on the mean, with a separate plot for each of the (a) first, (b) second, (c) third, (d) fourth, (e) fifth, and (f) sixth years. The sample size is reported by each point, and Cohen’s $d$ Cohen (1988) measuring the effect size of the gender difference in each group is reported. ### IV.2 Intersectionality with Gender We now turn our attention to the differences between men and women in Fig. 1 in order to answer RQ2. We note in particular that across all demographic groups women’s mean GPA is roughly 0.2 grade points higher than men’s. The effect sizes (Cohen’s $d$) of this difference range from small to medium Cohen (1988). This difference in mean GPA earned is substantial enough to indicate a change in letter grade, given that the grading system at the studied university uses increments of 0.25 grade points for letter grades containing “$+$” or “$-$.” Further, this trend holds in the fifth year (Fig. 1e) and sixth year (Fig. 1f), with some exceptions in demographic groups with particularly low sample sizes after the fourth year. ### IV.3 STEM GPA Trends In order to answer RQ3, Figure 2 plots students’ mean STEM GPA in a similar manner to Fig. 1. We note that the general “dinosaur” pattern discussed in Fig. 1 also holds at least for the first and second years (Figs. 2a and 2b, respectively). In the third year and beyond, the general features of the trend continue to hold, with the most privileged students having the highest mean GPA, followed by those with one disadvantage as well as the first-generation and low-income group, followed by the remaining groups of URM students with one or more additional disadvantages. However, in these later years, the finer details of the plots noted before fall away in favor of a sharper mean GPA decrease for URM students with at least one additional disadvantage in the third year (Fig. 2c) and a more gradual decrease across all groups in the fourth year (Fig. 2d) and fifth year (Fig. 2e). When restricting the GPA calculations to STEM courses, the sample size becomes too small in the sixth year (Fig. 2f) to draw meaningful conclusions. Figure 2: Average STEM GPA of each demographic group. Students are binned into separate demographic groups based on their status as first-generation (FG), low-income (LI), and/or underrepresented minority (URM) students. The men and women in each demographic group are plotted separately. The mean GPA in all courses taken by students in each demographic group is plotted along with the standard error on the mean, with a separate plot for each of the (a) first, (b) second, (c) third, and (d) fourth, (e) fifth, and (f) sixth years. The sample size is reported by each point, and Cohen’s $d$ Cohen (1988) measuring the effect size of the gender difference in each group is reported. We further observe a trend of students earning higher grades on average in later years, although the rise from the first to the fourth year is somewhat lower in STEM GPA than in overall GPA. Notably, while in overall GPA this trend seemed to be somewhat universal across demographic groups, in Fig. 2 we see a quicker rise in mean STEM GPA over time for the more privileged students than the less privileged students, particularly comparing the leftmost and rightmost groups. Regarding gender differences, Fig. 2 shows smaller gender differences in STEM GPA than those observed in overall GPA in Fig. 1. While in overall GPA women earned roughly 0.2 grade points more than men on average, in STEM GPA that difference is much less consistent and typically ranges from 0 to 0.1 grade points. For many demographic groups we see no significant differences between men and women’s mean STEM GPA. We do see that there is still a consistent STEM GPA gender difference, albeit smaller than in Fig. 1, among the group of the most privileged students (i.e., those with “None” of the disadvantages). There is also a STEM GPA gender difference among first-generation low-income but non-URM students, however this difference is less consistent and in fact briefly vanishes in the third year. ### IV.4 GPA Trends By Major Over Time In order to better understand the trends over time in both overall and STEM GPA and answer RQ4, we plotted the mean GPA by year in Fig. 3 and mean STEM GPA by year in Fig. 4. In these plots, we have not separated men and women and instead focus on the other demographic characteristics while further grouping students into three different groups of majors in order to understand if these trends differ for students in different areas of study. Further, since the sample size becomes quite small in years five and six for many of the demographic groups of interest, we plot only the mean GPA over the first four years. In Figs. 3a and 4a, we plot the mean overall and STEM GPA, respectively, of all students. In the other subfigures, we plot the mean GPA earned by students majoring in different clusters of majors. In particular, we plot the mean GPA of engineering (including computer science), mathematics, and physical science (i.e., chemistry and physics) majors in Figs. 3b and 4b, the remaining STEM majors in Figs. 3c and 4c, and non-STEM majors in Figs. 3d and 4d. Figure 3: Students are binned into separate demographic groups as in Fig. 1, but not separated by gender. The mean GPA in all courses of each group is plotted over time from year one to four, along with the standard error of the mean. The plots show this for four subpopulations: (a) all students; (b) chemistry, computer science, engineering, mathematics, and physics students; (c) biology, economics, geology, neuroscience, and statistics students; and (d) non-STEM students including psychology. Figure 4: Students are binned into separate demographic groups as in Fig. 2, but not separated by gender. The mean GPA in STEM courses of each group is plotted over time from year one to four along with the standard error of the mean. The plots show this for four subpopulations: (a) all students; (b) chemistry, computer science, engineering, mathematics, and physics students; (c) biology economics, geology, neuroscience, and statistics students; and (d) non-STEM students including psychology. These plots make clearer some of the trends noted earlier, especially the rise in mean GPA over time from the first to the fourth year. However, we can now see that this is not universally true since the first-generation URM students have a drop in mean GPA in the second year for physical science majors (Fig. 3b), and in the third year for other STEM majors (Fig. 3c). This trend is even more noticeable in STEM GPA (Fig. 4), where the mean STEM GPA of the group of first-generation URM students drops in the third year for every subpopulation by major. ## V Discussion To start, we consider how much the current system disadvantages students who are first-generation, low-income, or underrepresented minority but not a combination of the two. Discussing these groups first is helpful in setting the stage for a more complex discussion of the intersectionality of these various demographic characteristics. We find in Figs. 1 and 2 that not all of these disadvantages are equal. In particular, non-URM students who have one disadvantage, namely the first-generation (but not low-income) and low-income (but not first-generation) students, still earn slightly higher grades than even the URM students who are not low-income or first-generation. Notably, this trend (the “back” of the dinosaur plots) is similar in both overall grades (Fig. 1) and in STEM grades alone (Fig. 2). The size of this mean grade difference varies from year to year, but in STEM grades it can reach as high as about 0.25 grade points, which at the studied institution is the difference between, for example, a B and B$+$ or B$-$ grade. The group with the grades most similar to these non-first-generation, non-low- income URM students are the first-generation, low-income non-URM students, who earn both overall (Fig. 1) and STEM (Fig. 2) grades similar to or very slightly higher than the URM students. One explanation could be that the lack of resources available due to being first-generation or low-income is not as severe an obstacle as the stereotype threat experienced by URM students. Turning then to the “tail” in the dinosaur plots, we find that consistently the most disadvantaged students in both overall grades (Fig. 1) and STEM grades (Fig. 2) are the URM students with at least one additional obstacle. In this case, it appears that the intersection of being low-income and URM is the most disadvantageous combination, with no notable difference in either Fig. 1 or Fig. 2 among these students whether or not they are also first-generation. Meanwhile, the first-generation URM students who are not low-income sometimes have a slightly higher mean GPA than the low-income URM students (Fig. 1). Another avenue to investigate intersectionality is how gender interacts with the other demographic groups. Interestingly, in overall GPA (Fig. 1), gender appears to have about the same effect across all demographic groups. That is, there does not appear to be an intersectional effect of gender identity with other identities as measured by overall GPA. However, Fig. 2 shows that this is a context-dependent effect, with the gender gap substantially and unevenly reduced across all groups in mean STEM GPA. For most demographic groups in Fig. 2, the higher overall GPA earned by women in Fig. 1 has vanished completely in STEM GPA. This is consistent with stereotype threat being the mechanism of disadvantage for women, where stereotypes surrounding STEM disciplines unfairly cause stress and anxiety for women Astin (1993); Cross (1993); Felder _et al._ (1995, 1998); Britner and Pajares (2006); Basile and Lopez (2015); Cheryan _et al._ (2017); Hilts _et al._ (2018). Notably, while the gender gap is reduced nearly to zero for most groups in Fig. 2, there does remain a small consistent gender gap favoring women in the most privileged group of students. In other groups the gender gap in Fig. 2 is inconsistent across years. One explanation could be that the wealth of resources available to them may help to alleviate the stereotype threat. Taking a more temporal view of these GPA trends, Fig. 3 (overall GPA) and Fig. 4 (STEM GPA) have grouped men and women together in order to focus on the other demographic characteristics more closely. In these plots, the most noteworthy trend is again that, with the sole exception of the first year in Fig. 3b, the four groups with the lowest mean GPA (Fig. 3) and STEM GPA (Fig. 4) across the first four years are always the four groups containing URM students. Notably, this trend is true regardless of which group of majors we investigate. The consistency of this result is particularly striking, showing that the most otherwise disadvantaged non-URM students have fewer obstacles to success than even the most privileged URM students among all students. Focusing further on the STEM GPA of STEM majors in Figs. 4b and 4c, we see that while non-URM students consistently rise in mean GPA over time, the same is not true for all URM students. In particular, the first-generation URM students who major in chemistry, computer science, engineering, mathematics, or physics (Fig. 4b) experience a steady decline in mean STEM GPA from year one to two and year two to three. While the standard error of those means is quite large due to a relatively small sample size, that lack of representation for these students could itself be what is hindering their coursework by causing a stereotype threat. Based upon the frameworks of critical theory and intersectionality, the main implication of these findings is that many students who come from less privileged backgrounds are not being adequately supported in college in order to catch up with the privileged students Crenshaw _et al._ (1995); Kellner (2003); Yosso (2005); Gutiérrez (2009); Taylor _et al._ (2009); Tolbert _et al._ (2018); Schenkel and Calabrese Barton (2020); Johnson (2012); Johnson _et al._ (2017); Crenshaw (1990); Cho _et al._ (2013); Mitchell _et al._ (2014); Charleston _et al._ (2014); Morton and Parsons (2018). The disadvantages of these less privileged students manifest as lower mean overall and STEM GPA for those demographic groups. In order to promote equity and inclusion, it is crucial that these students are provided appropriate mentoring, guidance, scaffolding, and support in college so that these obstacles can be cleared for students who have been put at a disadvantage relative to their peers through no fault of their own Birt _et al._ (2019). We note that these demographic groups with more disadvantages are likely to consist of students who had K-12 education from schools with fewer resources and less well-prepared teachers than those of the more privileged students, with high school being an especially important time for disadvantages related to STEM learning increasing Bianchini _et al._ (2003); Maltese and Tai (2011); Means _et al._ (2017); Bottia _et al._ (2018); Daley (2019); Dou _et al._ (2019). 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2024-09-04T02:54:58.547629

2003.04389

arxiv-papers

# Discovering Representations for Black-box Optimization Adam Gaier Inria, CNRS, Université de LorraineBonn-Rhein-Sieg University of Applied Sciences<EMAIL_ADDRESS>, Alexander Asteroth Bonn-Rhein-Sieg University of Applied SciencesSankt AugustinGermany53757 <EMAIL_ADDRESS>and Jean-Baptiste Mouret Inria, CNRS,Université de LorraineNancyFrance54000<EMAIL_ADDRESS> (2020; 2020) ###### Abstract. The encoding of solutions in black-box optimization is a delicate, handcrafted balance between expressiveness and domain knowledge — between exploring a wide variety of solutions, and ensuring that those solutions are useful. Our main insight is that this process can be automated by generating a dataset of high- performing solutions with a quality diversity algorithm (here, MAP-Elites), then learning a representation with a generative model (here, a Variational Autoencoder) from that dataset. Our second insight is that this representation can be used to scale quality diversity optimization to higher dimensions — but only if we carefully mix solutions generated with the learned representation and those generated with traditional variation operators. We demonstrate these capabilities by learning an low-dimensional encoding for the inverse kinematics of a thousand joint planar arm. The results show that learned representations make it possible to solve high-dimensional problems with orders of magnitude fewer evaluations than the standard MAP-Elites, and that, once solved, the produced encoding can be used for rapid optimization of novel, but similar, tasks. The presented techniques not only scale up quality diversity algorithms to high dimensions, but show that black-box optimization encodings can be automatically learned, rather than hand designed. ††copyright: rightsretained††doi: 10.1145/nnnnnnn.nnnnnnn††isbn: 978-x-xxxx- xxxx-x/YY/MM††conference: the Genetic and Evolutionary Computation Conference 2020; July 8–12, 2020; Cancún, Mexico††journalyear: 2020††price: 15.00††journalyear: 2020††copyright: rightsretained††conference: Genetic and Evolutionary Computation Conference; July 8–12, 2020; Cancún, Mexico††booktitle: Genetic and Evolutionary Computation Conference (GECCO ’20), July 8–12, 2020, Cancún, Mexico††doi: 10.1145/3377930.3390221††isbn: 978-1-4503-7128-5/20/07 Figure 1. Data-Driven Encoding MAP-Elites (DDE- Elites) searches the space of representations to search for solutions. A data- driven encoding (DDE) is learned by training a VAE on the MAP-Elites archive. High fitness solutions, which increase the bias of the DDE toward performance, are found using the DDE. Novel solutions, which increase the range of solutions which can be expressed, are found using mutation operators. UCB1, a bandit algorithm, balances the mix of these explorative and exploitative operators. ## 1\. Introduction The method of encoding solutions is one of the most critical design decisions in optimization, as the representation defines the way an algorithm can move in the search space (Rothlauf, 2006). Work on representations tends to focus on encoding priors or innate biases: aerodynamic designs evolved with splines to encourage smooth forms (Olhofer et al., 2001), Compositional Pattern Producing Networks (CPPNs) with biases for symmetry and repetition in images and neural network weight patterns (Stanley, 2007; Stanley et al., 2009), modularity induced in evolved neural networks (Mouret and Doncieux, 2008; Durr et al., 2010; Doncieux and Meyer, 2004), or neural network structures which encode strong enough biases to perform without training (Gaier and Ha, 2019). The best representations balance a bias for high performing solutions, so they can easily be discovered, and the ability to express a diversity of potential solutions, so the search space can be widely explored. At the one extreme, a representation which only encodes the global optimum is easy to search but useless for finding any other solution. At the other, a representation which can encode anything presents a difficult and dauntingly vast search space. Given a large set of example solutions, representations could be learned from data instead of being hand-tailored by trial-and-error: a learned representation would replicate the same biases toward performance and the same range of expressivity as the source data set. For instance, given a dataset of face images, a Variational Autoencoder (VAE) (Kingma and Welling, 2014) or a Generative Adversarial Network (GAN) (Goodfellow et al., 2014) can learn a low-dimensional latent space, or encoding, that makes it possible to explore the space of face images. In essence, the decoder which maps the latent space to the phenotypic space learns the “recipe” of faces. Importantly, the existence of such a low-dimensional latent space is possible because _the dataset is a very small part of the set of all possible images_. However, using a dataset of preselected high-performing solutions “traps” the search within the distribution of solutions that are already known: a VAE trained on white faces will never generate a black face. This limits the usefulness of such data-driven representations for discovering _novel_ solutions to hard problems. In this paper, we propose the use of the MAP-Elites algorithm (Mouret and Clune, 2015) to automatically generate a dataset for representations using only a performance function and a diversity space. Quality diversity (QD) algorithms (Cully and Demiris, 2018; Pugh et al., 2016) like MAP-Elites are a good fit for representation discovery: creating archives of diverse high- performing solutions is precisely their purpose. Using the MAP-Elites archive as a source of example solutions, we can capture the genetic distribution of the highest performing solutions, or elites, by training a VAE and obtaining a latent representation. As the VAE is only trained on elites, this learned representation, or Data-Driven Encoding (DDE), has a strong bias towards solutions with high fitness; and because the elites have varying phenotypes, the DDE is able to express a range of solutions. Though the elites vary along a phenotypic continuum, they commonly have many genotypic similarities (Vassiliades and Mouret, 2018), making it more likely to find a well- structured latent space. Nonetheless, MAP-Elites will struggle to find high-performing solutions without an adequate representation. Fortunately, the archive is produced by MAP-Elites in an iterative, any-time fashion, so there is no “end state” to wait for before a DDE can be trained — a DDE can be trained during optimization. The DDE can then be used to enhance optimization. By improving the quality of the archive the DDE improves the quality of its own source data, establishing a virtuous cycle of archive and encoding improvement. A DDE based on an archive will encounter the same difficulty as any learned encoding: the DDE can only represent solutions that are already in the dataset. How then, can we discover new solutions? Fundamentally, to search for an encoding we need to both _exploit the best known representation_ , that is, create better solutions according to the current best “recipes”, and also _explore new representations_ — solutions which do not follow any “recipe”. In this paper, we address this challenge by mixing solutions generated with the DDE with solutions obtained using standard evolutionary operators. Our algorithm applies classic operators, such as Gaussian mutation, to create candidates which could not be captured by the current DDE. At the same time we leverage the DDE to generalize common patterns across the map and create new solutions that are likely to be high-performing. To avoid introducing new hyper-parameters, we tune this exploration/exploitation trade-off optimally using a multi-armed bandit algorithm (Garivier and Moulines, 2011). This new algorithm, DDE-Elites, reframes optimization as a search for representations (Figure 1). Integrating MAP-Elites with a VAE makes it possible to apply quality diversity to high-dimensional search spaces, and to find effective representations for future uses. We envision application to domains that have straightforward but expansive low-level representations, for instance: joints positions at 20Hz for a walking robot ($12\times 100=1200$ joint positions for a 5-second gait of a robot with $12$ degrees of freedom), 3D shapes in which each voxel is encoded individually (1000-dimensional for a $10\times 10\times 10$ grid), images encoded in the pixel-space, etc. Ideally, the generated DDE will capture the main regularities of the domain. In robot locomotion, this could correspond to periodic functions, since we already know that a $36$-dimensional controller based on periodic functions can produce the numerous joint commands required every second to effectively drive a 12-joint walking robot in many different ways (Cully et al., 2015). In many domains the space of possible solutions can be vast, while the inherent dimensionality of interesting solutions is still compact. By purposefully seeking out a space of solutions, rather than the solutions themselves, we can solve high-dimensional problems in a lower dimensional space. ## 2\. Background ### 2.1. Optimization of Representations In his 30 year perspective on adaptation in evolutionary algorithms, Kenneth De Jong identified representation adaptation as ”perhaps the most difficult and least understood area of EA design.” (De Jong, 2007) Despite the difficulty of creating adaptive encodings, the potential rewards have lured researchers for decades. Directly evolving genotypes to increase in complexity has a tradition going back to the eighties (Goldberg et al., 1989; Altenberg, 1994). The strategy of optimizing a solution at low complexity and then adding degrees of freedom has proved effective on problems from optimal control (Gaier and Asteroth, 2014), to aerodynamic design (Olhofer et al., 2001), to neural networks (Stanley and Miikkulainen, 2002). Evolving the genome’s structure is particularly important when the structure itself is the solution, such as in genetic programming (Koza, [n. d.]) or neural architecture search (Elsken et al., 2019; Miikkulainen et al., 2019; Gaier and Ha, 2019). Recent approaches toward representation evolution have focused on genotype- phenotype mappings (Bongard and Pfeifer, 2003). Neural networks, which map between inputs and outputs, are a natural choice for such ‘meta- representations’. These mappings can evolve with the genome (Scott and Bassett, 2015; Simões et al., 2014), or fix the genome and evolve only the mapping (Stanley et al., 2009; Stanley, 2007). Supervised methods have been previously applied to learn encodings. These approaches require a set of example solutions for training. Where large, well- curated data sets are available this strategy has proven effective at creating representations well suited to optimization (Volz et al., 2018; Bontrager et al., 2018b; Bontrager et al., 2018a), but where a corpus of solutions does not exist it must be created. In (Scott and De Jong, 2018; Moreno et al., 2018) these solutions were collected by saving the champion solutions found after repeatedly running an optimizer on the problem, with the hope that the learned representation would then be effective in similar classes of problems. ### 2.2. MAP-Elites MAP-Elites (Mouret and Clune, 2015) is a QD algorithm which uses a niching approach to produce high-performing solutions which span a continuum of user- defined phenotypic dimensions. These phenotypic dimensions, or behavior descriptors, describe the way the problem is solved, and are often orthogonal to performance. MAP-Elites has been used in such diverse cases as optimizing the distance traveled by a walking robot using different legs (Cully et al., 2015), the drag of aerodynamic designs with varied volumes and curvatures (Gaier et al., 2017), and the win rate of decks composed of different cards in deck-building games (Fontaine et al., 2019). MAP-Elites is a steady-state evolutionary algorithm which maintains a population in a discretized grid or ‘archive’. This grid divides the continuous space of possible behaviors into bins, or ‘niches’ with each bin holding a single individual, or ‘elite’. These elites act as parents, and are mutated to form new individuals. These child individuals are evaluated and assigned a niche based on their behavior. If the niche is empty the child is placed inside; if the niche is already occupied, the individual with higher fitness is stored in the niche and the other discarded. By repeating this process, increasingly optimal solutions which cover the range of phenotype space are found. The MAP-Elites algorithm is summarized in Algorithm 1. Algorithm 1 MAP-Elites 1:function MAP-Elites($fitness()$, $variation()$, $\mathcal{X}_{initial}$) 2: $\mathcal{X}\leftarrow\emptyset$, $\mathcal{F}\leftarrow\emptyset$ $\triangleright$ Map of genomes $\mathcal{X}$, and fitnesses $\mathcal{F}$ 3: $\mathcal{X}\leftarrow\mathcal{X}_{initial}$ $\triangleright$ Place initial solutions in map 4: $\mathcal{F}\leftarrow fitness(\mathcal{X}_{initial})$ 5: for iter = $1\to I$ do 6: $\mathbf{x^{\prime}}\leftarrow variation(\mathcal{X})$ $\triangleright$ Create new solution from elites 7: $\mathbf{p^{\prime}},\mathbf{b^{\prime}}\leftarrow fitness(\mathbf{x^{\prime}})$ $\triangleright$ Get performance and behavior 8: if $\mathcal{F}(\mathbf{b^{\prime}})=\emptyset$ or $\mathcal{F}(\mathbf{b^{\prime}})<\mathbf{f^{\prime}}$ then $\triangleright$ Replace if better 9: $\mathcal{F}(\mathbf{b^{\prime}})\leftarrow\mathbf{f^{\prime}}$ 10: $\mathcal{X}(\mathbf{b^{\prime}})\leftarrow\mathbf{x^{\prime}}$ 11: end if 12: end for 13: return $(\mathcal{X}$, $\mathcal{F})$ $\triangleright$ Return illuminated map 14:end function Though phenotypically diverse the elites are often genotypically similar, existing in an “elite hypervolume”, a high performing region of genotype space (Vassiliades and Mouret, 2018). Just as in nature, where species as diverse as fruit flies and humans share nearly 60 percent of their genome (Adams et al., 2000), the “recipe” for high performance is often composed of many of the same ingredients. This insight was leveraged in (Vassiliades and Mouret, 2018) to create a new variation operator which considers the correlation among elites. Genes which vary little across the elites, and so are likely common factors that produce high performance, are also subject to the smallest amount of perturbation — lowering the chance their children stray from the elite hypervolume. Biasing mutation in this way ensures that exploration is focused on factors which induce phenotypic variation without drifting into regions of poor performance. ### 2.3. Variational Autoencoders Autoencoders (AEs) (Hinton and Salakhutdinov, 2006) are neural networks designed to perform dimensionality reduction. AEs are composed of two components: an encoder, which maps the input to a lower dimensional latent space; and a decoder, which maps the latent space back to the original space. The decoder is trained to reconstruct the input through this lower dimensional latent “bottleneck”. The encoder component can be viewed as a generalization of Principal Component Analysis (Wold et al., 1987), with the latent space approximating principal components. Though the AE is able to represent the data at a lower dimensionality, and reproduce it with minimal loss, it can still be a poor representation for optimization. An important quality of representations is ‘locality’, that a small change in the genotype induces a small change in the phenotype (Rothlauf, 2006). When AEs are trained only to minimize reconstruction error they may overfit the distribution of the training data and create an irregular latent space. The low-locality of such latent spaces limits their usefulnesses in optimization: nearby points in latent space may decode to very different solutions, meaning even a small mutation could have a large effect. Variational autoencoders (VAEs) (Kingma and Welling, 2014) are AEs whose training is regularized to ensure a high-locality latent space. The architecture is broadly the same: an encoder and decoder mediated by a bottleneck, but rather than encoding the input as a single point it is encoded as a normal distribution in the latent space. When training the model a point from this input distribution is sampled, decoded, and the reconstruction error computed. By encoding the input as a normal distribution we induce the distributions produced by the encoder to be closer to normal. VAEs are trained by minimizing two terms: (1) the reconstruction error, and (2) the Kullback- Liebler (KL) divergence (Kullback and Leibler, 1951) of the latent space to a unit Gaussian distribution, giving the loss function: (1) $loss=\|x-\hat{x}\|^{2}+KL\left[N\left(\mu_{x},\sigma\right),N(0,1)\right]$ Inducing solutions to be encoded in the form of a normal distribution structures the latent space in a continuous and overlapping way, creating a local encoding better suited to optimization. ## 3\. DDE-Elites Figure 2. DDE-Elites Algorithm (1) A VAE is trained on the archive, and used to create a ‘reconstructive crossover’ operator which creates new solutions by averaging the parameters of an individual with its own reconstruction; (2) the mix of exploitative and explorative variation operators predicted to have the most success is chosen by the multi-armed bandit algorithm UCB1 and used to create new solutions; (3) the new solutions are added to the archive and the success rate of the applied variation operator is updated. Every representation biases optimization in some way, improving optimization by limiting the range of solutions that can be expressed to those which are valid or high-performing (Rothlauf, 2006). But finding a balance between expressivity and bias is an arduous task requiring considerable domain expertise. Our method, DDE-Elites, automates the process of representation design and learns new encodings in tandem with search — allowing optimization and representation learning to improve each other in a self-reinforcing cycle. DDE-Elites learns an encoding from examples of high performing solutions. To create these examples we use MAP-Elites, which produces a variety of high performing solutions rather than converging to a single optima. The variety produced by MAP-Elites is critical — the expressivity of any learned encoding is limited by the variety of examples. That MAP-Elites not only produces a variety of solutions, but allows us to define the nature of that variety, makes it particularly powerful for crafting useful representations. By defining the type of variety we want to explore we are defining the biases and expressivity we encode in our representation. DDE-Elites is a variant of the MAP-Elites algorithm. The core component of competition within a niched archive is maintained, but novel methods of producing child solutions are introduced. Child solutions are created using an encoding learned from the archive. This encoding is refined as the archive improves, which in turn improves the optimization process. DDE-Elites optimizes an archive of varied solutions by reframing optimization as a search for the best representation, rather than the best solution. The DDE-Elites algorithm proceeds as follows (see Figure 2 and Algorithm 2): (1) a DDE and reconstructive crossover operator is created by training a VAE on the archive; (2) the probability of using each variation operator is determined by the UCB1 bandit algorithm; (3) MAP-Elites is run with the chosen variation operator probabilities. The success rate of the variation operators to create solutions is used to update the bandit and the improved archive is used to create a new DDE and reconstructive crossover operator. #### Data Driven Encoding The MAP-Elites archive is a record of the highest-performing solutions yet found in each bin. When the archive is updated the VAE is trained to reconstruct the individuals in the archive. Reconstruction is a mapping from one phenotype to another, mediated through latent space; and the mapping from latent space to phenotype space analogous to a genotype-phenotype mapping, which we refer to as a Data-Driven Encoding (DDE). Features common in high performing solutions will be the most successfully compressed and reconstructed — and features widely shared by high performing solutions are likely to lead to high performance. Critically, by training the encoding only on high-performing solutions we bias the space of solutions the DDE can express to those with high performance. #### Reconstructive Crossover By limiting the range of solutions which can be expressed by a representation, we are able to bias the solutions found during search. When a solution is reconstructed with the VAE it is mapped onto the restricted space of solutions expressible by the DDE — a space characterized by high performance. Reconstructing individuals with the VAE can create new solutions with higher fitness than the originals, but cannot create novel solutions. Solutions created by the DDE are based on those already in the archive, so cannot reach solutions which lie outside of the encoded distribution. At early stages of optimization when there are few example solutions, using only reconstruction to create new solutions would doom our encoding to a small region of expression. Rather than completely replacing individuals with their reconstructions we instead shift them closer to forms expressible by the DDE with a new variation operator, reconstructive crossover. Child solutions are created by performing crossover with two parents: a parent chosen from the archive and its reconstruction. Crossover takes the form of an element-wise mean of the parameter vectors. (2) $\mathbf{x}_{i}^{(t+1)}=\frac{1}{2}*(\mathbf{x}_{i}^{(t)}+VAE.Decode(VAE.Encode(\mathbf{x}_{i}^{(t)})))$ The reconstructive crossover operator slows the loss of diversity by only moving an individual toward the distribution of solutions encoded by the DDE, not directly into it. By only shifting solutions rather than replacing them, we allow exploration outside of the distribution to continue. Even when there is little gain in fitness, solutions that are the result of reconstructive crossover have a lower inherent dimensionality, on the account of having parents pass through the compressive bottleneck of the VAE. In this way the reconstructive crossover operator not only spreads globally advantageous genes throughout the archive, but also pulls the archive towards more easily compressed solutions. #### Line Mutation Reconstructive crossover enables effective optimization within the range of solutions that the DDE can express, but explorative operators are required to widen the pool of example solutions and improve the DDE. So when creating new solutions we choose to either produce them through reconstructive crossover, or through random mutation. In addition to isometric Gaussian mutation commonly used in MAP-Elites, we apply the line mutation operator proposed in (Vassiliades and Mouret, 2018). Line mutation imposes a directional component on the Gaussian perturbations. During mutation the parent genome is compared to a random genome from the archive. The variance of mutation in each dimension is then scaled by the difference in each gene: (3) $\mathbf{x}_{i}^{(t+1)}=\mathbf{x}_{i}^{(t)}+\sigma_{1}\mathcal{N}(0,\mathbf{I})+\sigma_{2}\left(\mathbf{x}_{j}^{(t)}-\mathbf{x}_{i}^{(t)}\right)\mathcal{N}(0,1)$ where $\sigma_{1}$ and $\sigma_{2}$ are hyperparameters which define the relative strength of the isometric and directional mutations. Intuitively, when two genes have similar values the spread of mutation will be small, when the values are very different the spread will be large. In many cases certain parameter values will be correlated to high fitness, regardless of the individual’s place in behavior space. The line operator is a simple way of exploiting this similarity, but in contrast to reconstructive crossover does not limit expressivity – allowing it to be used as a method of exploring new solutions. Though both the reconstructive crossover and line mutation operators take advantage of the similarities between high performing individuals, their differing approaches allow them to be effectively combined as explorative and exploitative operators. #### Parameter Control DDE-Elites explores the space of representations with the exploitative operator of reconstructive crossover, which finds high performing solutions similar to those already encoded by the DDE, and explorative operators of mutation, which expand the space of solutions beyond the range of the DDE. The optimal ratio to use these operators is not only domain dependent, but dependent on the stage of the algorithm. When the archive is nearly empty, it makes little sense to base a representation on a few randomly initialized solutions; once the behavior space has been explored, it is beneficial to continue optimization through the lens of the DDE; and when the archive is full of solutions produced by the DDE it is more useful to expand the range of possible solutions with mutation. These stages are neither predictable nor clear cut, complicating the decision of when to use each operator. Faced with a trade-off between exploration and exploitation we frame the choice of operators as a multi-armed bandit problem (Auer et al., 2002). Multi-armed bandits imagine sets of actions as levers on a slot machine, each with their own probability of reward. The goal of a bandit algorithm is to balance exploration, trying new actions, and exploitation, repeating actions that yield good rewards. Bandit approaches are straightforward to implement and have been previously used successfully to select genetic operators (DaCosta et al., 2008). We define a set of possible actions as usage ratios between reconstructive crossover, line mutation, and isometric mutation. The ratio of $[\frac{1}{4},\frac{3}{4},0]$, for example, would have solutions created by reconstructive crossover with a probability of $\frac{1}{4}$, line mutation with a probability of $\frac{3}{4}$, and never with isometric mutation. Each action is used to create a batch of child solutions and a reward is assigned in proportion to the number of children who earned a place in the archive. At each generation a new action is chosen, and the reward earned for that action recorded. Actions are chosen based on UCB1 (Auer et al., 2002), a simple and effective bandit algorithm which minimizes regret. Actions with the greatest potential reward are chosen, calculated as: (4) $Q(a)+\sqrt{(2\log t)/(N_{t}(a))}$ where $Q(a)$ is the reward for an action $a$, $t$ is the total number of actions that have been performed, and $N_{t}(a)$ the number of times that action has been performed. UCB1 is an optimistic algorithm which rewards uncertainty — given two actions with the same mean reward, the action which has been tried fewer times will be chosen. Our archive is in constant flux, and so the true reward of each mix of operators changes from generation to generation. To handle the non-stationary nature of the problem we use a sliding window (Garivier and Moulines, 2011), basing our predictions only on the most recent generations. Algorithm 2 DDE-Elites 1:function DDE-Elites($fitness()$ $\mathcal{X}_{initial}$) 2: $\mathcal{X}\leftarrow\mathcal{X}_{initial}$ 3: $\mathcal{V}$: Possible Variation Operator Probabilities (vector) 4: (e.g., [0,0.5,0.5], [0.8,0.0,0.2], [1.0,0.0,0.0] for [xover,line,iso]) 5: successes $\leftarrow zeros(len(\mathcal{V}))$ $\triangleright$ # successes for each option 6: selection $\leftarrow zeros(len(\mathcal{V}))$ $\triangleright$ # selections for each option 7: for iter = $1\to I$ do 8: — Train VAE on Current Archive — 9: VAE.Train ($\mathcal{X}$) 10: — Choose Variation Based on UCB1 — 11: $i\leftarrow\arg\max\left(\frac{\text{successes}[s]}{\text{ selected }[s]}+\sqrt{\frac{2\ln(\text{sum}(\text{successes}))}{\text{selected}[s]}}\right)$ 12: — Run MAP-Elites Using Chosen Variation — 13: $variation()\leftarrow\mathcal{V}[i]$ 14: $\mathcal{X^{\prime}}\leftarrow$MAP- Elites$(fitness(),variation(),\mathcal{X}$) 15: — Track Performance of Chosen Variation — 16: $selection[i]\leftarrow selection[i]+1$ 17: $successes[i]\leftarrow successes[i]+nImproved(\mathcal{X^{\prime}},\mathcal{X})$ 18: end for 19: DDE $\leftarrow$ VAE.Decode() 20: return $\mathcal{~{}X}$, DDE 21:end function 1:function Isometric Mutation($\mathcal{X}$) 2: $\mathbf{x~{}}\leftarrow random\\_selection(\mathcal{X})$ 3: return $\mathbf{x}+\sigma\mathcal{N}(0,\mathbf{I})$ 4:end function 1:function Line Mutation($\mathcal{X}$) 2: $\mathbf{x~{},y~{}}\leftarrow random\\_selection(\mathcal{X})$ 3: return $\mathbf{x}+\sigma_{1}\mathcal{N}(0,\mathbf{I})+\sigma_{2}(\mathbf{x}-\mathbf{y})\mathcal{N}(0,1)$ 4:end function 1:function Reconstructive Crossover($\mathcal{X}$) 2: $\mathbf{x~{}}\leftarrow random\\_selection(\mathcal{X})$ 3: $\mathbf{y~{}}\leftarrow VAE.Decode(VAE.Encode(\mathbf{x}))$ $\triangleright$ VAE Reconstruction 4: return $(\mathbf{x}+\mathbf{y})/2$ 5:end function Figure 3. Archive Illumination Archive illumination performance of MAP-Elites with different variation operators: standard isometric mutation (MAP-Elites), line mutation (ME-Line), reconstructive crossover (DDE-XOver) and DDE-Elites, which uses the UCB1 bandit algorithm to choose between the three at every generation. We measure fitness as the mean fitness of all solutions in the archive; coverage as the fraction of behavior space bins which contain solutions. Results over 20 replicates with lines indicating medians and quartile bounds shaded. The median of DDE-Elites, our approach, is additionally noted with black dots. All final results are significantly different ($p<0.01$ Mann-Whitney U) in fitness and coverage. Progress is shown in evaluations (0 to 1 million); a batch size of 100 evaluations per generation was used, so this scale corresponds to generations from 0 to 10,000. ## 4\. Experiments #### Planar Arm Inverse Kinematics 111see Figure 5 for a visualization of this domain We demonstrate the effectiveness of DDEs and DDE-Elites on in the inverse kinematics (IK) problem of a 2D robot arm, a common QD benchmark problem (Cully and Demiris, 2018; Vassiliades and Mouret, 2018). Given target coordinates a configuration of joint angles should be found to place the end effector at the target. To solve this task, a discretized behavior space is defined over the x,y plane and MAP-Elites finds a configuration of joint angles which places the end effector in each bin. The location of the end effector is derived for an arm with $n$ joints with angles $y$ with using the forward kinematics equation: $\mathbf{b}(\mathbf{y})=\left[\begin{array}[]{c}{l_{1}\cos(y_{1})+l_{2}\cos(y_{1}+y_{2})+\cdots+l_{n}\cos(y_{1}+\cdots+y_{n})}\\\ {l_{1}\sin(y_{1})+l_{2}\sin(y_{1}+y_{2})+\cdots+l_{n}\sin(y_{1}+\cdots+y_{n})}\end{array}\right]$ There are many solutions to this IK problem, but solutions with lower joint variance are preferred to allow for smoother transitions between configurations. We define fitness as the negative joint variance: $-\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\mu)^{2}$ where $(\mu=\sum_{i=1}^{n}y_{i})$. To summarize: the phenotype is the angle of each joint, the behavior is the x,y coordinates of the end effector, and the fitness the negative variance of the joint angles. The difficulty of the problem can be easily scaled up by increasing the number of joints in the arm: we solve this task with 20, 200, and 1000 joints. When a DDE is used 10 latent dimensions are used for the 20D arm, and 32 dimensions for the 200 and 1000D arms. The same archive structure is used for all domains. A unit circle is divided into 1950 bins, with each bin defined by the Voronoi cell (Vassiliades et al., 2017) with centers placed in a ring formation222See supplementary material for a visualization of this structure. Figure 4. Archive Recreation with Data-Driven Encoding Performance of MAP-Elites algorithm when run with direct or data-driven encoding. When using the direct encoding, MAP-Elites was given one order of magnitude more evaluations (note logarithmic scale of evaluations). Fitness is measured as the mean fitness of all solutions in the archive, coverage as the fraction of behavior space bins which contain solutions. Results over 50 replicates with dotted lines indicating medians and quartile bounds shaded. ### 4.1. Archive Illumination We first demonstrate the ability of DDE-Elites to scale up illumination to high-dimensional problems. The performance of DDE-Elites is compared to three algorithmic variants: the canonical MAP-Elites algorithm using isometric mutation (MAP-Elites); MAP-Elites using line, or directional, mutation (ME- Line); and MAP-Elites using the reconstructive crossover (DDE-XOver). Our proposed approach DDE-Elites uses all operators at a ratio determined by the UCB1 bandit algorithm. These treatments are summarized in Table 1. | | Isometric --- Mutation | Line --- Mutation | Reconstructive --- Crossover MAP-Elites | X | | ME-Line | | X | DDE-XOver | | | X DDE-Elites | X | X | X Table 1. Algorithm variants. DDE-Elites is our approach. These variants are compared based on the quality of the archive at each generation (Figure 3). Archives are judged based on two metrics: (1) coverage, the number of bins filled, and (2) performance, the mean fitness of solutions.333Sixty-four core machines were used to evaluate 100 individuals in parallel, requiring $\sim$0.2s, $\sim$0.8s, $\sim$1.6s, for the arm at 20d, 200d, and 1000D arm respectively. In every case the VAE required $\sim$2.4s to train on a single CPU core. In the 20-dimensional case ME-Line quickly fills the map with high performing solutions. In only a one hundred thousand evaluations ME-Line creates an archive unmatched by MAP-Elites even after one million evaluations. When only the reconstructive crossover operator is used, despite promising early progress, a chronic lack of exploration results in archives which are worse than the standard MAP-Elites. DDE-Elites, with access to all operators, explores as quickly as ME-Line and creates archives of similar quality. When the dimensionality of the arm is scaled up to 200D, we see the convergence rate of ME-Line slow down considerably. While still reaching high levels of performance it does so only after one million evaluations, a tenth of the evaluations required in in the 20D case — suggesting that the effectiveness of ME-Line scales linearly with the dimensionality of the problem. In contrast DDE-Elites is barely affected by a ten-fold increase in parameters — exploration is only slightly slowed, and high-performing solutions are found from the very earliest iterations. The effects of scaling can be observed even more clearly in the 1000D case: ME-Line illuminates the archive only very slowly, while the performance of DDE-Elites is marked by the same burst of exploration and consistently high fitness solutions that characterized its performance in lower dimensions. The line mutation operator is clearly able to leverage the similarities in high performing solutions across the archive — in every case performing far better than the isometric mutation operator. The mechanism for doing this, adjusting the range of parameter mutations, does not appear to scale well enough to handle very high dimensional problems. The reconstructive crossover operator is able to rapidly find high-performing solutions even in high- dimensional spaces, but is poor at exploring. Search with reconstructive crossover is confined to the distribution of genes that already exist in the archive, if used exclusively that distribution of genes is limited to the initial population. By combining these operators — expanding the range of genes in the archive with mutation, and spreading high performing genes with reconstructive crossover — DDE-Elites is able to create high-performing archives even in high-dimensional problems. Figure 5. Optimization with Direct and Data-Driven Encodings CMA-ES is given a set budget to find a solution with a target behavior, and searches with either a direct encoding or a DDE. Left: Example solutions for target matching with the direct and data driven encodings. End effectors in yellow, targets in red. Top: Optimization over time of median distance (dotted line) to the 18 targets over 50 replicates (quartiles shaded). Bottom: The final distance to the targets, and a characteristic of the solution. These characteristics were not optimized by CMA-ES, but optimized during the creation of the DDE, biasing the solutions produced. ### 4.2. Archive Recreation DDE-Elites is as much a method of optimizing representations as solutions. By learning a representation from the archive, we create an encoding that is biased towards high performance and has a range of expression matching the defined behavior space. In these experiments, our DDE encodes smooth joint configurations which place an arm’s end effector anywhere in its reach. To demonstrate that DDE-Elites does more than guide search, but learns a representation, we search the space again, using the found DDE in place of the direct encoding. We run the standard MAP-Elites algorithm, with isometric mutation only, using a learned DDE444The decoder network of the VAE found in the highest coverage replicate of DDE-Elites. acting as our genome. In the 20D arm this DDE has 10 parameters, in the 200D and 1000D arms the DDE has 32 parameters. No previous solutions are maintained, only the trained DDE. For reference we compare to the MAP-Elites algorithm using the direct encoding. An order of magnitude fewer evaluations were budgeted when using the DDE. In every case the DDE far outperforms the direct encoding, reaching the same levels of fitness and coverage with several orders of magnitude fewer evaluations (Figure 4). The DDE can express the same range of solutions as were found in the original archive, and finds them rapidly. Archives were recreated after only 10,000 evaluations — a rate of about 5 evaluations per bin.55510,000 individuals/1950 bins $\approx$ 5 evaluations/bin discovered. The found solutions are also high performing. Such improvement cannot be explained away by the decrease in dimensionality of the search. In both low and high dimensional cases the bias toward high performance is also apparent: the mean fitness curve is nearly flat at the optima, indicating that when new solutions are added to the map they are already near optimal. The contrast with the direct encoding is stark, with the direct encoding considerable effort is taken to search for good solutions, the DDE finds little else. DDE-Elites not only produces solutions, but learns domain-specific representation. ### 4.3. Optimization with Learned Encodings Beyond its place in the DDE-Elites optimization loop, the produced DDE is a powerful representation with high expressivity and built in biases. Though created by MAP-Elites, the DDE is not tied to it. Once discovered, a DDE can be used as a representation for any black box optimization algorithm. We illustrate this generality by using again solving the arm inverse kinematics problem with the black-box optimizer CMA-ES (Hansen and Ostermeier, 2001). A set of target positions for the end effector is defined (Figure 5, left), and CMA-ES used to find a joint configuration which reaches each target. In one case optimization is performed using the DDE; in the other the direct encoding is used. When optimizing with the DDE, CMA-ES quickly finds solutions to the target hitting problems with a precision never matched with the direct encoding (Figure 5, top). Moreover, a bias for how the problem is solved is built into the representation (Figure 5, bottom). As the DDE was trained only on solutions with low joint variance, this same property is found in the solutions found by CMA-ES with the DDE — even without searching for them. With the DDE CMA-ES not only finds solutions to the IK problem, the built-in priors of the DDE ensures we find kind of solutions we want. ## 5\. Discussion Learning representations by combining quality diversity (here, MAP-Elites) and generative models (here, a VAE) opens promising research avenues for domains in which optimizations of the same cost function are launched continuously. This is, for example, the case of Model Predictive Control (Mayne et al., 2000), in which the sequence of actions for the next seconds is optimized at every time-step of the control loop, or the case of shape optimization in interactive design tools (Hoyer et al., 2019; Bendsøe and Sigmund, 1995), in which each modification by the user requires a novel optimization. In preliminary experiments, we searched for an encoding to describe action sequences for a walking robot. The results show that using MAP-Elites to generate a diversity of sequences, then using a VAE to learn a representation leads to an encoding that can accelerate future optimizations by several orders of magnitude. Nevertheless, using the representation during optimization, as described in this paper, did not accelerate the quality diversity optimization as much as in the high-dimensional arm used here. One hypothesis is that the regularities in action sequences are harder to recognize than in the arm experiments, especially at the beginning of the process. For instance, it might help to use an auto-encoder that is especially designed for sequences (Vaswani et al., 2017; Co-Reyes et al., 2018). For other tasks, appropriate generative models could be explored, for example convolutional models for tasks with spatial correlations (Salimans et al., 2015). In addition, though the latent spaces created by VAEs are easier to navigate than those created by normal autoencoders, even better models offer the opportunities for further improvements. Much work has been done to create VAEs which have even better organized latent spaces (Higgins et al., 2017; Burgess et al., 2018; Chen et al., 2018; Kim and Mnih, 2018), ideally with each dimension responsible for a single phenotypic feature such as the lighting or color of an image. A second research avenue is to improve the bandit algorithm used to balance between operators. In theory, it should ensure that adding new operators can only aid optimization, since useless or detrimental operators would rarely be selected. However, we observed that it is not always effective: in some cases, using only the line mutation outperformed DDE-Elites, whereas DDE-Elites could revert to using only line mutation with a perfect bandit. Our hypothesis is that this is a sign that “successes” — child solutions which discover new bins or improve on existing solutions — is not the perfect measure of utility for a QD algorithm. In the case of our experiments, it may be that reconstructive crossover consistently improves solutions, but may only do so slightly. According to the “success” metric, a tiny improvement is worth the same as a large one. To best utilize the bandit, other methods of judging performance in QD algorithms should be explored. Beyond performance advantages, for both the current and future optimizations, these “disentangled” representations offer even more interesting opportunities. Reducing the dimensionality of the search space into meaningful components would allow rapid model-based optimization of single solutions (Shahriari et al., 2015), or entire archives (Gaier et al., 2018). Engineers could interactively explore and understand such encodings, laying bare the underlying properties responsible for performance and variation — and so from encodings receive, rather than provide, insight and domain knowledge. ## Acknowledgements This work received funding from the European Research Council (ERC) under the EU Horizon 2020 research and innovation programme (grant agreement number 637972, project ”ResiBots”) and the German Federal Ministry of Education and Research (BMBF) under the Forschung an Fachhochschulen mit Unternehmen programme (grant agreement number 03FH012PX5, project ”Aeromat”). ## Source Code The source code used to produce the results in this paper is available at https://github.com/resibots/2020_gaier_gecco ## References * (1) * Adams et al. (2000) Mark D Adams, Susan E Celniker, Robert A Holt, Cheryl A Evans, Jeannine D Gocayne, Peter G Amanatides, Steven E Scherer, Peter W Li, Roger A Hoskins, Richard F Galle, et al. 2000\. 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Hyperparameters of DDE Experiments Hyperparameter | Value ---|--- Isometric Mutation Strength | 0.003 Line Mutation Strength | 0.1 Batch Size | 100 Bandit Options, | | [0.00:0.00:1.00], [0.25:0.00:0.75], --- [0.50:0.00:0.50], [0.75:0.00:0.25], [1.00:0.00:0.00], [0.00:0.25:0.75], [0.00:0.50:0.50], [0.00:0.75:0.25], [0.00:1.00:0.00] Bandit Window Length | 1000 Generations per VAE Training | 1 Epochs per VAE Training | 5 Mutation Strength when Searching DDE | 0.15 Latent Vector Length [Arm20] | 10 Latent Vector Length [Arm200] | 32 Latent Vector Length [Arm1000] | 32

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arxiv-papers

Int. J. Business Process Integration and Management Ngo, V.M., Le-Khac, N.A. and Kechadi M.T. Int. J. Business Process Integration and Management 10 1 2020 Vuong M. Ngo E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS> Nhien-An Le-Khac E-mail<EMAIL_ADDRESS> M-Tahar Kechadi E-mail<EMAIL_ADDRESS> Ho Chi Minh City Open University, HCMC, Vietnam University College Dublin, Belfield, Dublin 4, Ireland # Data Warehouse and Decision Support on Integrated Crop Big Data ###### Abstract In recent years, precision agriculture is becoming very popular. The introduction of modern information and communication technologies for collecting and processing Agricultural data revolutionise the agriculture practises. This has started a while ago (early 20th century) and it is driven by the low cost of collecting data about everything; from information on fields such as seed, soil, fertiliser, pest, to weather data, drones and satellites images. Specially, the agricultural data mining today is considered as Big Data application in terms of volume, variety, velocity and veracity. Hence it leads to challenges in processing vast amounts of complex and diverse information to extract useful knowledge for the farmer, agronomist, and other businesses. It is a key foundation to establishing a crop intelligence platform, which will enable efficient resource management and high quality agronomy decision making and recommendations. In this paper, we designed and implemented a continental level agricultural data warehouse (ADW). ADW is characterised by its (1) flexible schema; (2) data integration from real agricultural multi datasets; (3) data science and business intelligent support; (4) high performance; (5) high storage; (6) security; (7) governance and monitoring; (8) consistency, availability and partition tolerant; (9) cloud compatibility. We also evaluate the performance of ADW and present some complex queries to extract and return necessary knowledge about crop management. Data warehouse, decision support, crop Big Data, smart agriculture. to this paper should be made as follows: Ngo, V.M., Le-Khac, N.A. and Kechadi, M.T. (2020) ‘Data Warehouse and Decision Support on Integrated Crop Big Data’, Int. J. Business Process Integration and Management, Vol. 10, No. 1, pp. 17–28. Vuong M. Ngo received the B.E, M.E and PhD degrees in computer science at HCMC University of Technology in 2004, 2007 and 2013 respectively. He is currently a Senior Researcher at UCD and HCMC Open University. His research interests include information retrieval, sentiment analysis, data mining, graph matching and data Nhien-An Le-Khac is currently a Lecturer at the School of Computer Science, UCD and a Programme Director of MSc programme in forensic computing and cybercrime investigation. He obtained his PhD in computer science in 2006 at the Institut National Polytechnique Grenoble, France. His research interest spans the area of cybersecurity and digital forensics, data mining/distributed data mining for security, grid and high performance computing. M-Tahar Kechadi was awarded PhD and Master degrees in computer science from University of Lille 1, France. He joined the UCD School of Computer Science in 1999. He is currently Professor of Computer Science at UCD. His research interests span the areas of data mining, data analytics, distributed data mining, heterogeneous distributed systems, grid and cloud Computing, cybersecurity, and digital forensics. He is a Principal Investigator at Insight Centre for Data Analytics and CONSUS project. He is a member of IEEE and ACM. ## 1 Introduction Annual world cereal productions were $2,608$ million tons and $2,595$ million tons in $2017$ and $2018$, respectively (USDA report, 2018; FAO-CSDB report, 2018). However, there were also around $124$ million people in $51$ countries faced food crisis and food insecurity (FAO-FSIN report, 2018). According to United Nations (UN document, 2017), we need an increase $60\%$ of cereal production to meet $9.8$ billion people needs by $2050$. To satisfy the huge increase demand for food, crop yields must be significantly increased using modern farming approaches, such as smart farming also called precision agriculture. As highlighted in the European Commission report (EC report, 2016), precision agriculture is vitally important for the future and can make a significant contribution to food security and safety. The precision agriculture’s current mission is to use the decision-support system (DSS) based on Big Data approaches to provide precise information for more control of waste and farming efficiency, such as soil nutrient (Rogovska and et al., 2019), early warning (Rembold and et al., 2019), forecasting (Bendre and et al., 2015), irrigation systems (Huang and et al., 2013), evapotranspiration prediction (Paredes and et al., 2014), soil and herbicide, insecticide optimisation (Ngo and Kechadi, 2020), awareness (Lokers and et al., 2016), supply chain (Protopop and Shanoyan, 2016) and financial services (Ruan and et al., 2019). Normally, the DSSs implement a knowledge discovery process also called data mining process, which consists of data collection and data modelling, data warehousing, data analysis (using machine learning or statistical techniques), and knowledge deployment (Dicks and et al., 2014). Hence, designing and implementing an efficient agricultural data warehouse (ADW) is one of the key steps of this process, as it defines a uniform data representation through its schema model and stores the derived datasets so that they can be analysed to extract useful knowledge. However, currently, this step was not given much attention. Therefore, there are very few reports in the literature that focus on the design of efficient ADWs with the view to enable Agricultural Big Data analytics and mining. The design of large scale ADWs is very challenging. Because, the agricultural data is spatial, temporal, complex, heterogeneous, non-standardised, high dimensional, collected from multi-sources, and very large. In particular, it has all the features of Big Data; volume, variety, velocity and veracity. Moreover, the precision agriculture system can be used by different kinds of users at the same time, for instance by farmers, policymakers, agronomists, and so on. Every type of user needs to analyse different information, sets thus requiring specific analytics. Unlike in any other domains; health-care, financial data, etc, the data and its warehousing in precision agriculture are unique. This is because, there are very complex relationships between the agricultural data dimensions. The data sources are very diversified and varying levels of quality. Precision agriculture (PA) warehousing has many decision-making processes and each needs different levels of data access and different needs of analysis. Finally, there are many stakeholders involved in the data ownership and exploitation. So, the data has significant number of uncertainties. For examples, the quality of data collected by farmers depends directly on their knowledge, routines and frequency of information recording, and support tools, etc. All these issues make the PA data unique when it becomes to its storage, access, and analysis. These issues may exist in other domains, but not at the same scale and as in agriculture practices. In this research, we firstly analyse real-world agricultural Big Data to build the effective constellation schema. From this schema, some simple questions can be easily answered directly from the modelled data. These questions include: (1) For a given field, what kind of crops are suitable to grow? (2) Which companies can purchase a specific crop with the highest price in the past season? (3) List the history of soil texture and applied fertilisers for a given field; (4) List costs of production for wheat and barley in the last 5 years, and so on. Secondly, the proposed ADW has enough main features and characteristics of Big Data Warehouse (BDW). These are (1) high storage capacity, high performance and cloud computing compatibility; (2) flexible schema and integrated storage structure; (3) data ingestion, monitoring, and security to deal with the data veracity. Besides, an experimental evaluation is conducted to study the performance of ADW storage. The rest of this paper is organised as follows: in the next Section, we reviewed the related work about decision support systems and data warehouses in agriculture. In Sections 3, 4 and 5, we presented big data aspects of PA, our ADW architecture and its modules. In Sections 6, 7, 8 and 9, the quality criteria, implementation, performance analysis and decision-making applications of the proposed ADW are presented respectively. Section 10 gives some concluding remarks and future research directions. Finally, a concrete example about the ADW and its operational average run-times are shown in the appendix. ## 2 Related Work In precision agriculture, DSSs are designed to support different stakeholders such as farmers, advisers and policymakers to optimise resources, farms’ management and improve business practices (Gutierreza and et al., 2019). For instance, DSSs were built to 1) manage microbial pollution risks in dairy farming (Oliver and et al., 2017); 2) analyse nitrogen fertilisation from satellite images (Lundstrom and Lindblom, 2018); 3) control pest and disease under uncertainty in climate conditions (Devitt and et al., 2017); 4) manage drip irrigation and its schedule (Friedman and et al., 2016); 5) predict and adopt climate risks (Han and et al., 2017). However, the datasets that were used in the mentioned studies are small. Besides, they focused on using visualisation techniques to assist end-users understand and interpret their data. Recently, many papers have been published on how to exploit intelligent algorithms on sensor data to improve agricultural economics Pantazi (2016), Park and et al. (2016), Hafezalkotob and et al. (2018), Udiasa and et al. (2018) and Rupnik and et al. (2019). In Pantazi (2016), the authors predicted crop yield by using self-organising-maps; namely supervised Kohonen networks, counter-propagation artificial networks and XY-fusion. In Park and et al. (2016), one predicted drought conditions by using three rule-based machine learning; namely random forest, boosted regression trees, and Cubist. To select the best olive harvesting machine, the authors in Hafezalkotob and et al. (2018) applied the target-based techniques on the main criteria, which are cost, vibration, efficiency, suitability, damage, automation, work capacity, ergonomics, and safety. To provide optimal management of nutrients and water, the paper Udiasa and et al. (2018) exploited the multi-objective genetic algorithm to implement an E-Water system. This system enhanced food crop production at river basin level. Finally, in Rupnik and et al. (2019) the authors predicted pest population dynamics by using time series clustering and structural change detection which detected groups of different pest species. However, the proposed solutions are not scalable enough to handle agricultural Big Data; they present weaknesses in one of the following aspects: data integration, data schema, storage capacity, security and performance. From a Big Data point of view, the papers Kamilaris and et al. (2018) and Schnase and et al. (2017) have proposed “smart agricultural frameworks”. In Kamilaris and et al. (2018), the authors used Hive to store and analyse sensor data about land, water and biodiversity which can help increase food production with less environmental impact. In Schnase and et al. (2017), the authors moved toward a notion of climate analytics-as-a-service, by building a high-performance analytics and scalable data management platform, which is based on modern cloud infrastructures, such as Amazon web services, Hadoop, and Cloudera. However, the two papers did not discuss how to build and implement a DW for a precision agriculture. The proposed approach, inspired from Schulze and et al. (2007), Schuetz and et al. (2018), Nilakanta and et al. (2008) and Ngo and et al. (2018), introduces ways of building agricultural data warehouse (ADW). In Schulze and et al. (2007), the authors extended entity-relationship concept to model operational and analytical data; called multi-dimensional entity-relationship model. They also introduced new representation elements and showed how can be extended to an analytical schema. In Schuetz and et al. (2018), a relational database and an RDF triple store were proposed to model the overall datasets. The data is loaded into the DW in RDF format, and cached in the RDF triple store before being transformed into relational format. The actual data used for analysis was contained in the relational database. However, as the schemas used in Schulze and et al. (2007) and Schuetz and et al. (2018) were based on entity- relationship models, they cannot deal with high-performance, which is the key feature of a data warehouse. In Nilakanta and et al. (2008), a star schema model was used. All data marts created by the star schemas are connected via some common dimension tables. However, a star schema is not enough to present complex agricultural information and it is difficult to create new data marts for data analytics. The number of dimensions of the DW proposed in Nilakanta and et al. (2008) is very small; only 3-dimensions – Species, Location, and Time. Moreover, the DW concerns livestock farming. Overcoming disadvantages of the star schema, the authors of Ngo and et al. (2018) and Ngo and Kechadi (2020) proposed a constellation schema for an agricultural DW architecture in order to satisfy the quality criteria. However, they did not describe how to design and implement their DW. ## 3 Crop Big Data ### 3.1 Crop Datasets The datasets were primarily obtained from an agronomy company, which extracted it from them operational data storage systems, research results, and field trials. Especially, we were given real-world agricultural datasets on iFarms, Business-to-Business (B2B) sites, technology centres and demonstration farms. Theses datasets were collected from several European countries and they are presented in Figures 1 and 2 (Origin report, 2018). These datasets describe more than $112$ distribution points, $73$ demonstration farms, $32$ formulation and processing facilities, $12.7$ million hectares of direct farm customer footprint and $60,000$ trial units. Figure 1: Data from UK and Ireland. Figure 2: Data in Continental Europe. There is a total of 29 datasets. On average, each dataset contains $18$ tables and is about $1.4$ GB in size. Each dataset focuses on a few information that impact the crop. For instance, the weather dataset includes information on location of weather stations, temperature, rainfall and wind speed over time. Meanwhile, soil component information in farm sites, such as mineral, organic matter, air, water and micro-organisms, were stored in the soil dataset. The fertiliser dataset contains information about field area and geographic position, crop name, crop yield, season, fertiliser name and quantity. ### 3.2 Big Data Challenges Raw and semi-processed agricultural datasets are usually collected through various sources: Internet of Thing (IoT) devices, sensors, satellites, weather stations, robots, farm equipment, farmers and agronomists, etc. Besides, agricultural datasets are very large, complex, unstructured, heterogeneous, non-standardised, and inconsistent. Hence, it has all the features of Big Data. 1. 1. Volume: The amount of agricultural data is increasing rapidly and is intensively produced by endogenous and exogenous sources. The endogenous data is collected from operational systems, experimental results, sensors, weather stations, satellites, and farming equipment. The systems and devices in the agricultural ecosystem can be connected through IoT. The exogenous data concerns the external sources, such as government agencies, retail agronomists, and seed companies. They can help with information about local pest and disease outbreak tracking, crop monitoring, food security, products, prices, and knowledge. 2. 2. Variety: Agricultural data has many different forms and formats, structured and unstructured data, video, imagery, chart, metrics, geo-spatial, multi- media, model, equation, text, etc. 3. 3. Velocity: The collected data increases at very high rate, as sensing and mobile devices are becoming more efficient and cheaper. The datasets must be cleaned, aggregated and harmonised in real-time. 4. 4. Veracity: The tendency of agronomic data is uncertain, inconsistent, ambiguous and error prone because the data is gathered from heterogeneous sources, sensors and manual processes. ### 3.3 ADW Schema Figure 3: A part of ADW schema for Precision Agriculture The DW uses schema to logically describe the entire datasets. A schema is a collection of objects, including tables, views, indexes, and synonyms which consist of some fact and dimension tables (Oracle document, 2017). The DW schema can be designed based on the model of source data and the user requirements. There are three kind of models, namely star, snowflake and fact constellation. With the its various uses, the ADW schema needs to have more than one fact table and should be flexible. So, the constellation schema, also known galaxy schema should be used to design the ADW schema. Figure 4: Field and Crop dimension tables Figure 5: Soil and Pest dimension tables We developed a constellation schema for ADW and it is partially described in Figure 3. It includes few fact tables and many dimension tables. FieldFact fact table contains data about agricultural operations on fields. Order and Sale fact tables contain data about farmers’ trading operations. The key dimension tables are connected to their fact table. There are some dimension tables connected to more than one fact table, such as Crop and Farmer. Besides, CropState, Inspection, Site, and Weather Reading dimension tables are not connected to any fact table. CropState and Inspection tables are used to support Crop table. While, Site and Weather Reading tables support Field and WeatherStation tables. FieldFact fact table saves the most important facts about teh field; yield, water volume, fertiliser quantity, nutrient quantity, spray quantity and pest number. While, in Order and Sale tables, the important facts needed by farm management are quantity and price. Table 1: Descriptions of other dimension tables No. | Dim. tables | Particular attributes ---|---|--- 1 | Business | BusinessID, Name, Address, Phone, Mobile, Email 2 | CropState | CropStateID, CropID, StageScale, Height, MajorStage, MinStage, MaxStage, Diameter, MinHeight, MaxHeight, CropCoveragePercent 3 | Farmer | FarmerID, Name, Address, Phone, Mobile, Email 4 | Fertiliser | FertiliserID, Name, Unit, Status, Description, GroupName 5 | Inspection | InspectionID, CropID, Description, ProblemType, Severity, ProblemNotes, AreaValue, AreaUnit, Order, Date, Notes, GrowthStage 6 | Nutrient | NutrientID, NutrientName, Date, Quantity 7 | Operation Time | OperationTimeID, StartDate, EndDate, Season 8 | Plan | PlanID, PName, RegisNo, ProductName, ProductRate, Date, WaterVolume 9 | Product | ProductID, ProductName, GroupName 10 | Site | SiteID, FarmerID, SiteName, Reference, Country, Address, GPS, CreatedBy 11 | Spray | SprayID, SprayProductName, ProductRate, Area,Date, WaterVol, ConfDuration, ConfWindSPeed, ConfDirection, ConfHumidity, ConfTemp, ActivityType 12 | Supplier | SupplierID, Name, ContactName, Address, Phone, Mobile, Email 13 | Task | TaskID, Desc, Status, TaskDate, TaskInterval, CompDate, AppCode 14 | Trans Time | TransTimeID, OrderDate, DeliverDate, ReceivedDate, Season 15 | Treatment | TreatmentID, TreatmentName, FormType, LotCode, Rate, ApplCode, LevlNo, Type, Description, ApplDesc, TreatmentComment 16 | Weather Reading | WeatherReadingID, WeatherStationID, ReadingDate, ReadingTime, AirTemperature, Rainfall, SPLite, RelativeHumidity, WindSpeed, WindDirection, SoilTemperature, LeafWetness 17 | Weather Station | WeatherStationID, StationName, Latitude, Longitude, Region The dimension tables contain details on each instance of an object involved in a crop yield or farm management. Figure 4 describes attributes of Field and Crop dimension tables. Field table contains information about name, area, co- ordinates (being longitude and latitude of the centre point of the field), geometric (being a collection of points to show the shape of the field) and site identify the site that the field it belongs to. While, Crop table contains information about name, estimated yield of the crop (estYield), BBCH Growth Stage Index (BbchScale), harvest equipment and its weight. These provide useful information for crop harvesting. Figure 5 describes attributes of Soil and Pest dimension tables. Soil table contains information about PH value (a measure of the acidity and alkalinity), minerals (nitrogen, phosphorus, potassium, magnesium and calcium), its texture (texture label and percentage of Silt, Clay and Sand), cation exchange capacity (CEC) and organic matter. Besides, information about recommended nutrient and testing dates ware also included in this table. In Pest table contains name, type, density, coverage and detected dates of pests. For the remaining dimension tables, their main attributes are described in Table 1. ## 4 ADW Architecture A DW is a federated repository for all the data that an enterprise can collect through multiple heterogeneous data sources; internal or external. The authors in Golfarelli and Rizzi (2009) and Inmon (2005) defined DW as a collection of methods, techniques, and tools used to conduct data analyses, make decisions and improve information resources. DW is defined around key subjects and involves data cleaning, data integration and data consolidations. Besides, it must show its evolution over time and is not volatile. The general architecture of a typical DW system includes four separate and distinct modules; Raw Data, Extraction Transformation Loading (ETL), Integrated Information and Data Mining (Kimball and Ross, 2013), which is illustrated in Figure 6. In that, Raw Data (source data) module is originally stored in various storage systems (e.g. SQL, sheets, flat files, …). The raw data often requires cleansing, correcting noise and outliers, dealing with missing values. Then it needs to be integrated and consolidated before loading it into a DW storage through ETL module. Figure 6: Agricultural Data Warehouse Architecture. The Integrated Information module is a logically centralised repository, which includes the DW storage, data marts, data cubes and OLAP engine. The DW storage is organised, stored and accessed using a suitable schema defined by the metadata. It can be either directly accessed or used to create data marts, which is usually oriented to a particular business function or an enterprise department. A data mart partially replicates DW storage’s contents and is a subset of DW storage. Besides, the data is extracted in a form of data cube before it is analysed in the data mining module. A data cube is a data structure that allows advanced analysis of data according to multiple dimensions that define a given problem. The data cubes are manipulated by the OLAP engine. The DW storage, data mart and data cube are considered as metadata, which can be applied to the data used to define other data. Finally, Data Mining module contains a set of techniques, such as machine learning, heuristic, and statistical methods for data analysis and knowledge extraction at multiple level of abstraction. ## 5 ETL and OLAP The ETL module contains Extraction, Transformation, and Loading tools that can merge heterogeneous schemata, extract, cleanse, validate, filter, transform and prepare the data to be loaded into a DW. The extraction operation allows to read, retrieve raw data from multiple and different types of data sources systems and store it in a temporary staging. During this operation, the data goes through multiple checks – detect and correct corrupted and/or inaccurate records, such as duplicate data, missing data, inconsistent values and wrong values. The transformation operation structures, converts or enriches the extracted data and presents it in a specific DW format. The loading operation writes the transformed data into the DW storage. The ETL implementation is complex, and consuming significant amount of time and resources. Most DW projects usually use existing ETL tools, which are classified into two groups. The first is a commercial and well-known group and includes tools such as Oracle Data Integrator, SAP Data Integrator and IBM InfoSphere DataStage. The second group is famous for it open source tools, such as Talend, Pentaho and Apatar. OLAP is a category of software technology that provides the insight and understanding of data in multiple dimensions through fast, consistent, interactive access, management and analysis of the data. By using roll-up (consolidation), drill-down, slice-dice and pivot (rotation) operations, OLAP performs multidimensional analysis in a wide variety of possible views of information that provides complex calculations, trend analysis and sophisticated data modelling quickly. The OLAP systems are divided into three categories: 1) Relational OLAP (ROLAP), which uses relational or extended- relational database management system to store and manage the data warehouse; 2) Multidimensional OLAP (MOLAP), which uses array-based multidimensional storage engines for multidimensional views of data, rather than in a relational database. It often requires pre-processing to create data cubes. 3) Hybrid OLAP (HOLAP), which is a combination of both ROLAP and MOLAP. It uses both relational and multidimensional techniques to inherit the higher scalability of ROLAP and the faster computation of MOLAP. In the context of agricultural Big Data, HOLAP is more suitable than both ROLAP and MOLAP because: 1) ROLAP has quite slow performance and does not meet all the users’ needs, especially when performing complex calculations; 2) MOLAP is not capable of handling detailed data and requires all calculations to be performed during the data cube construction; 3) HOLAP inherits advantages of both ROLAP and MOLAP, which allow the user to store large data volumes of detailed information and perform complex calculations within reasonable response time. ## 6 Quality Criteria The accuracy of data mining and analysis techniques depends on the quality of the DW. As mentioned in Adelman and Moss (2000) and Kimball and Ross (2013), to build an efficient ADW, the quality of the DW should meet the following important criteria: 1. 1. Making information easily accessible. 2. 2. Presenting consistent information. 3. 3. Integrating data correctly and completely. 4. 4. Adapting to change. 5. 5. Presenting and providing right information at the right time. 6. 6. Being a secure bastion that protects the information assets. 7. 7. Serving as the authoritative and trustworthy foundation for improved decision making. The analytics tools need to provide right information at the right time. 8. 8. Achieving benefits, both tangible and intangible. 9. 9. Being accepted by DW users. The above criteria must be formulated in a form of measurements. For example, with the 8th criterion, it needs to determine quality indicators about benefits, such as improved fertiliser management, cost containment, risk reduction, better or faster decision, and efficient information transaction. In the last criterion, a user satisfaction survey should be used to find out how a given DW satisfies its user’s expectations. ## 7 ADW Implementation Currently, there are many popular large-scale database types that can implement DWs. Redshift (Amazon document, 2018), Mesa (Gupta and et al., 2016), Cassandra (Hewitt and Carpenter, 2016; Neeraj, 2015), MongoDB (Chodorow, 2013; Hows and et al., 2015) and Hive (Du, 2018; Lam and et al., 2016). In Ngo and et al. (2019), the authors analysed the most popular no-sql databases, which fulfil most of the aforementioned criteria. The advantages, disadvantages, as well as similarities and differences between Cassandra, MongoDB and Hive were investigated carefully in the context of ADW. It was reported that Hive is a better choice as it can be paired with MongoDB to implement the proposed ADW for the following reasons: 1. 1. Hive is based on Hadoop which is the most powerful cloud computing platform for Big Data. Besides, HQL is similar to SQL which is popular for the majority of users. Hive supports well high storage capacity, business intelligent and data science more than MongoDB or Cassandra. These Hive features are useful to implement ADW. 2. 2. Hive does not have real-time performance so it needs to be combined with MongoDB or Cassandra to improve its performance. 3. 3. MongoDB is more suitable than Cassandra to complement Hive because: 1) MongoDB supports joint operation, full text search, ad-hoc query and second index which are helpful to interact with the users. Cassandra does not support these features; 2) MongoDB has the same master – slave structure with Hive that is easy to combine. While the structure of Cassandra is peer - to - peer; 3) Hive and MongoDB are more reliable and consistent. So the combination of both Hive and MongoDB adheres to the CAP theorem. Figure 7: Agricultural Data Warehouse Implementation The ADW implementation is illustrated in Figure 7 which contains three modules, namely Integrated Information, Products and Raw Data. The Integrated Information module includes two components; MongoDB and Hive. MongoDB receives real-time data; as user data, logs, sensor data or queries from Products module, such as web application, web portal or mobile app. Besides, some results which need to be obtained in real-time will be transferred from the MongoDB to Products. Hive stores the online data and sends the processed data to MongoDB. Some kinds of queries having complex calculations will be sent directly to Hive. In the Raw Data module, almost data in Operational Databases or External Data components, is loaded into Cassandra. It means that we use Cassandra to represent raw data storage. Hence, with the diverse formats of raw data; image, video, natural language and sql data, Cassandra is better to store them than SQL databases. In the idle times of the system, the updated raw data in Cassandra will be imported into Hive through the ELT tool. This improves the performance of ETL and helps us deploy ADW on cloud or distributed systems. ## 8 Performance Analysis The performance analysis was conducted using MySQL 5.7.22, JDK 1.8.0_171, Hadoop 2.6.5 and Hive 2.3.3 which run on Bash, on Ubuntu 16.04.2, and on Windows 10. All experiments were run on a desktop with an Intel Core i7 CPU (2.40 GHz) and 16 GB memory. We only evaluate the performance of reading operation as ADW is used for reporting and data analysis. The database of ADW is duplicated into MySQL to compare performance. By combining popular HQL/SQL commands, namely Where, Group by, Having, Left (right) Join, Union and Order by, we created 10 groups for testing. Every group has 5 queries and uses one, two or more commands (see Table 2). Moreover, every query uses operators; And, Or, $\geq$, Like, Max, Sum and Count, to express complex queries. Table 2: Command combinations of queries Group | Commands ---|--- $G_{1}$ | Where $G_{2}$ | Where, Group by $G_{3}$ | Where, Left (right) Join $G_{4}$ | Where, Union $G_{5}$ | Where, Order by $G_{6}$ | Where, Left (right) Join, Order by $G_{7}$ | Where, Group by, Having $G_{8}$ | Where, Group by, Having, Order by $G_{9}$ | Where, Group by, Having, Left (right) Join, | Order by $G_{10}$ | Where, Group by, Having, Union, Order by $0$$10$$20$$30$$40$$50$$0$$10$$20$$30$1Queries ($q_{i}$)Different times ($Times_{q_{i}}$)Group 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8Group 9Group 10 Figure 8: Different times between MySQL and ADW in runtime of every Query All queries were executed three times and we took the average value of the their execution timess. The difference in runtime between MySQL and ADW for a query $q_{i}$ is calculated as $Times_{q_{i}}=RT^{mysql}_{q_{i}}/RT^{ADW}_{q_{i}}$. Where, $RT^{mysql}_{q_{i}}$ and $RT^{ADW}_{q_{i}}$ are average runtimes of query $q_{i}$ on MySQL and ADW, respectively. Moreover, with each group $G_{i}$, the difference in runtime between MySQL and ADW is $Times_{G_{i}}=RT^{mysql}_{G_{i}}/RT^{ADW}_{G_{i}}$. Where, $RT_{G_{i}}=Average(RT_{q_{i}})$ is average runtime of group $G_{i}$ on MySQL or ADW. Figure 8 describes the time difference between MySQL and ADW for every query. Although running on one computer, but with large data volume, ADW is faster than MySQL on 46 out of 50 queries. MySQL is faster for three queries $12^{th}$, $13^{th}$ and $18^{th}$ belonging to groups $3^{rd}$ and $4^{th}$. The two systems returned the same time for query $24^{th}$ from group $5^{th}$. Within each query group, for fair performance comparison, the queries combine randomly fact tables and dimensional tables. This makes complex queries taking more time and the time difference is significant. When varying the sizes and structures of the tables, the difference is very significant; see Figure 8. $0$$2$$4$$6$$8$$10$$2$$4$$6$Mean$6.24$$2.92$$1.22$$2.86$$2.27$$4.66$$3.36$$4.63$$3.16$$1.56$$3.19$Groups ($G_{i}$)Different times ($Times_{G_{i}}$) Figure 9: Different times between MySQL and ADW in runtime of every group Beside comparing runtime in every query, we aslo compare runtime of every group presented in Figure 9. Comparing to MySQL, ADW is more than at most (6.24 times) at group $1^{st}$ which uses only Where command, and at least (1.22 times) at group $3^{rd}$ which uses Where and Joint commands. 12345678910Mean$0$$500$$1{,}000$$1{,}081.5$$599.7$$111.7$$790.4$$776.6$$1{,}109.2$$483$$1{,}057.3$$297.9$$571.1$$687.8$$173.4$$205.2$$91.2$$276.4$$342.8$$238$$143.7$$228.3$$94.2$$366.4$$216.1$Groups ($G_{i}$)Average runtimes (seconds)MySQLADW Figure 10: Average Runtimes of MySQL and ADW in every Groups Figure 10 presents the average runtime of the 10 query groups on MySQL and ADW. Mean, the run time of a reading query on MySQL and ADW is 687.8 seconds and 216.1 seconds, respectively. It means that ADW is faster 3.19 times. In the future, by deploying ADW solution on cloud or distributed systems, we believe that the performance will be even much better than MySQL. ## 9 Application for Decision Making The proposed ADW and study its performance on real agricultural data, we illustrated some queries examples to show how to extract information from ADW. These queries incorporate inputs on crop, yield, pest, soil, fertiliser, inspection, farmer, businessman and operation time to reduce labour and fertiliser inputs, farmer services, disease treatment and also increase yields. These query information could not be extracted if the Origin’s separate 29 datasets have not been integrated into ADW. The data integration through ADW is actually improve the value of a crop management data over time to better decision-making. Example 1: List fields, crops in the fields, yield and pest in the field with conditions: (1) the fields do not used ’urea’ fertilizer; (2) the crops has ’yellow rust’ or ’brown rust’ diseases; (3) the crops were grown in 2015. select CR.CropName, FI.FieldName, FF.Yield, PE.CommonName, FF.PestNumber, PE.Description from FieldFact FF, Crop CR, Field FI, Pest PE, Fertiliser FE, Inspection INS, OperationTime OP where FF.CropID = CR.CropID and FF.FieldID = FI.FieldID and FF.PestID = PE.PestID and FF.FertiliserID = FE.FertiliserID and CR.CropID = INS.CropID and FF.OperationTimeID = OP.OperationTimeID and FE.FertiliserName <> ’urea’ and (INS.Description = ’Yellow Rust’ or INS.Description = ’Brown Rust’) and Year(INS.Date) = ’2015’ and Year(OP.StartDate) = ’2015’ and Year(OP.EndDate) = ’2015’ Example 2: List farmers and their crop quantities were sold by Ori Agro company in 08/2016. select FA.FarmerID, FA.FarmerName, CR.CropName, SF.Unit, SUM(SF.Quantity) from Salefact SF, business BU, farmer FA, crop CR where SF.BusinessID = BU.BusinessID and SF.FarmerID = FA.FarmerID and SF.CropID = CR.CropID and Month(SF.SaleDate) = ’08’ and Year(SF.SaleDate) = ’2016’ and BU.BusinessName = ’Ori Agro’ group by CR.CropName Example 3: List Crops and their fertiliser and treatment information. In that, crops were cultivated and harvested in 2017, Yield $>$ 10 tons/ha and attached by ’black twitch’ pest. Besides, the soil in field has PH $>6$ and Silt $<=50$ mg/l. Select CR.CropName, FE.FertiliserName, FF.FertiliserQuantity, TR.TreatmentName, TR.Rate, TR.TreatmentComment From FieldFact FF, Crop CR, OperationTime OT, Soil SO, PEST PE, Fertiliser FE, Treatment TR Where FF.CropID = CR.CropID and FF.OperationTimeID = OT.OperationTimeID and FF.SoildID = SO.SoilID and FF.PestID = PE.PestID and FF.FertiliserID = FE.FertiliserID and FF.TreatmentID = TR.TreatmentID and Year(OT.StartDate) = ’2017’ and Year(OT.EndDate) = ’2017’ and FF.Yield > 10 and SO.PH > 6 and SO.Silt <= 50 and PE.CommonName = ’Black twitch’ Example 4: List crops, fertilisers, corresponding fertiliser quantities in spring, 2017 in every field and site of 10 farmers (crop companies) who used the large amount of $P_{2}O_{5}$ in winter, 2016. To execute this request, the query needs to exploit data in the FieldFact fact table and the six dimension tables, namely Crop, Field, Site, Farmer, Fertiliser and OperationTime. The query consists of two subqueries which return 10 farmers (crop companies) that used the largest amount of Urea in spring, 2016. Select FI.FieldName, SI.SiteName, FA.FarmerName, CR.CropName, FE.FertiliserName, FF.FertiliserQuantity, FE.Unit, OT.StartDate From FieldFact FF, Crop CR, Field FI, Site SI, Farmer FA, Fertiliser FE, Operationtime OT Where FF.CropID = CR.CropID and FF.FieldID = FI.FieldID and FF.FertiliserID = FE.FertiliserID and FF.OperationTimeID = OT.OperationTimeID and FI.SiteID = SI.SiteID and SI.FarmerID = FA.FarmerID and OT.Season = ’Spring’ and YEAR(OT.StartDate) = ’2017’ and FA.FarmerID IN( Select FarmerID From (Select SI.FarmerID as FarmerID, SUM(FF.FertiliserQuantity) as SumFertiliser From FieldFact FF, Field FI, Site SI, Fertiliser FE, OperationTime OT Where FF.FieldID = FI.FieldID and FF.FertiliserID = FE.FertiliserID and FF.OperationTimeID = OT.OperationTimeID and SI.SiteID = FI.SiteID and FE.FertiliserName = ’SO3’ and OT.Season = ’Spring’ and YEAR(OT.StartDate) = ’2016’ Group by SI.FarmerID Order by SumFertiliser DESC Limit 10 )AS Table1 ) ## 10 Conclusion and Future Work In this paper, we presented a schema herein optimised for the real agricultural datasets that were made available to us. The schema been designed as a constellation so it is flexible to adapt to other agricultural datasets and quality criteria of agricultural Big Data. Based on some existing popular open source DWs, We designed and implemented the agricultural DW by combining Hive, MongoDB and Cassandra DWs to exploit their advantages and overcome their limitations. ADW includes necessary modules to deal with large scale and efficient analytics for agricultural Big Data. Moreover, through particular reading queries using popular HQL/SQL commands, ADW storage outperforms MySQL by far. Finally, we outlined some complex HQL queries that enabled knowledge extraction from ADW to optimize of agricultural operations. In the future work, we shall pursue the deployment of ADW on a cloud system and implement more functionalities to exploit this DW. The future developments will include: (1) experimentation and analyzation the performance of MongoDB and the affectation between MongoDB and Hive; (2) The sophisticated the data mining and the spreading activation algorithms (Ngo, 2014) to determine crop data characteristics and combine with expected outputs to extract useful knowledge; (3) Predictive models based on machine learning algorithms; (4) An intelligent interface and graph representation (Helmer and et al., 2015) for data access; (5) Combination with the ontology to extract knowledge (Ngo and et al., 2011; Cao and et al., 2012). ## Appendix The followings are HQL/SQL scripts of 10 queries which are representative of 10 query groups. The average runtimes of these queries on MySQL and ADW are shown in Figure 11. 1) The query $5^{th}$ belongs to the group $1^{st}$: SELECT fieldfact.FieldID, crop.cropname, fieldfact.yield FROM fieldfact, crop WHERE fieldfact.cropid = crop.cropid and SprayQuantity = 7 and (crop.CropName like ’P\%’ or crop.CropName like ’R\%’ or crop.CropName like ’G\%’); 2) The query $10^{th}$ belongs to the group $2^{nd}$: SELECT soil.PH, count(*) FROM fieldfact, soil WHERE fieldfact.SoildID = soil.SoilID and fieldfact.sprayquantity = 2 GROUP by soil.PH; 5101520253035404550$0$$1{,}000$$2{,}000$$97.9$$754.8$$52.7$$2{,}297$$1{,}192$$2{,}188.4$$95.4$$265.9$$439.5$$892.4$$3$$233.2$$3.6$$479$$422.6$$226.7$$5.2$$7.6$$212.3$$472.1$Queries ($q_{i}$)Average runtimes (seconds)MySQLADW Figure 11: Average runtimes of MySQL and ADW in 10 typical queries 3) The query $15^{th}$ belongs to the group $3^{rd}$: SELECT fieldfact.yield, fertiliser.fertiliserName, fertiliser.fertiliserGroupName FROM fieldfact RIGHT JOIN fertiliser on fieldfact.fertiliserID = fertiliser.fertiliserID WHERE fieldfact.fertiliserQuantity = 10 and fertiliser.fertiliserName like ’%slurry%’; 4) The query $20^{th}$ belongs to the group $4^{th}$: SELECT sprayproductname FROM fieldfact, spray WHERE fieldfact.sprayid = spray.sprayid and fieldfact.watervolumn > 5 and fieldfact.watervolumn < 20 UNION SELECT productname FROM product, orderfact WHERE product.ProductID = orderfact.ProductID and (orderfact.Quantity = 5 or orderfact.Quantity = 6); 5) The query $25^{th}$ belongs to the group $5^{th}$: SELECT fieldfact.fieldID, field.FieldName, field.FieldGPS, spray.SprayProductName FROM fieldfact, field, spray WHERE fieldfact.FieldID = field.FieldID and fieldfact.SprayID = spray.SprayID and fieldfact.PestNumber = 6 ORDER BY field.FieldName; 6) The query $30^{th}$ belongs to the group $6^{th}$: SELECT fieldfact.FieldID, nutrient.NutrientName, nutrient.Quantity, nutrient.‘Year‘ FROM fieldfact RIGHT JOIN nutrient on fieldfact.NutrientID = nutrient.NutrientID WHERE fieldfact.NutrientQuantity = 3 and fieldfact.fertiliserquantity = 3 ORDER BY nutrient.NutrientName LIMIT 10000; 7) The query $35^{th}$ belongs to the group $7^{th}$: SELECT crop.cropname, sum(fieldfact.watervolumn) as sum1 FROM fieldfact, crop WHERE fieldfact.cropid = crop.cropid and fieldfact.sprayquantity = 8 and crop.EstYield >= 1 and crop.EstYield <=10 GROUP BY crop.cropname HAVING sum1 > 100; 8) The query $40^{th}$ belongs to the group $8^{th}$: SELECT crop.cropname, sum(fieldfact.fertiliserquantity) as sum1 FROM fieldfact, crop WHERE fieldfact.cropid = crop.cropid and fieldfact.nutrientquantity= 5 and crop.EstYield <=1 GROUP by crop.cropname HAVING sum1 > 30 ORDER BY crop.cropname; 9) The query $45^{th}$ belongs to the group $9^{th}$: SELECT nutrient.NutrientName, sum(nutrient.Quantity) as sum1 FROM fieldfact LEFT JOIN nutrient on fieldfact.NutrientID = nutrient.NutrientID WHERE nutrient.nutrientName like ’%tr%’ and (fieldfact.pestnumber = 16 or fieldfact.pestnumber = 15) GROUP by nutrient.NutrientName HAVING sum1 <300 ORDER BY nutrient.NutrientName; 10) The query $50^{th}$ belongs to the group $10^{th}$: SELECT sprayproductname as name1, sum(fieldfact.watervolumn) as sum1 FROM fieldfact, spray WHERE fieldfact.sprayid = spray.sprayid and fieldfact.Yield > 4 and fieldfact.Yield < 8 GROUP by sprayproductname HAVING sum1 > 210 UNION SELECT productname as name1, sum(orderfact.Quantity) as sum2 FROM product, orderfact WHERE product.ProductID = orderfact.ProductID and (orderfact.Quantity = 5 or orderfact.Quantity = 6) GROUP by productname HAVING sum2 > 50 ORDER BY name1; ## Acknowledgment This research is an extended work of Ngo and et al. 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2024-09-04T02:54:58.587189

2003.04472

arxiv-papers

# Entanglements and correlations of one-dimensional quantum spin-1/2 chain with anisotropic power-law long range interactions Jie Ren<EMAIL_ADDRESS>Department of Physics, Changshu Institute of Technology, Changshu 215500, China Wen-Long You College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China School of Physical Science and Technology, Soochow University, Suzhou, Jiangsu 215006, China Xiaoqun Wang<EMAIL_ADDRESS>Key Laboratory of Artificial Structures and Quantum Control of MOE, Shenyang National Laboratory for Materials Science, School of Physics and Astronomy, Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China Collaborative Innovation Center for Advanced Microstructures, Nanjing University, Nanjing 210093, China Beijing Computational Science Research Center, Beijing 100084, China ###### Abstract The correlations, entanglement entropy, and fidelity susceptibility are calculated for a one-dimensional spin-1/2 XXZ chain with anisotropic power-law long range interactions by employing the density matrix renormalization group method. In particular, this long-range interaction is assigned to ferromagnetic for transversal components, while it can be either ferro- or antiferromagnetic for the longitudinal spin component. Two ground-state phase diagrams are established versus the anisotropy of the interactions which not only changes the phase boundaries of the counterparts with short-range interactions, but also leads to the emergence of exotic phases. We found that the long-range interactions of the $z$-component results in a Wigner crystal phase, whereas the transversal one may break a continuous symmetry, resulting in a continuous symmetry breaking phase. ###### pacs: 03.67.-a,05.30.Jp ## I Introduction The quantum phase transition (QPT) and quantum critical phenomena are generally important in understanding novel properties involved in strongly correlated systems, such as quantum magnetic materials. Usually, short-range interactions, e.g., nearest neighbor and next nearest neighbor interactions, are considered to be sufficient for appropriate descriptions on the major magnetic properties of those systems Sachdev ; XWang2000 ; Luo2017 ; You19 ; WN19 ; Luo2019 . However, there actually exist several types of long range interactions such as the Coulomb interaction $1/r$ Saffman , the dipole-dipole interaction $1/r^{3}$ Lahaye ; Deng ; Yan , and the van der Waals interaction $1/r^{6}$ Saffman in some complicated compounds, where relevant electrons are in higher orbits of atoms with lower symmetries subject to crystal field effects. Moreover, in recent years, some long-range interactions have been generated in ultracold atomic systems with the optical lattices or trapped ions. For instance, a power-law Ising interaction $1/r^{\alpha}$ with an adjustable exponent $0<\alpha<3$ has been realized in trapped ions Britton ; Islam ; Gorshkov ; Jurcevic . This kind of experimental progress has greatly stimulated theoretical studies on possible novel effects particularly resulting from long-range interactions W ; Koffel ; Sun01 ; Zhu ; gong16 ; gong17 ; gong17L ; gong16R ; Frerot ; Vanderstraeten . In particular, a transition was revealed by the calculation of the entanglement for a long- range ($\sim r^{-\alpha}$) antiferromagnetic Ising chain Koffel , and is affirmed further by the fidelity susceptibility, being second-order for all $\alpha$ Sun01 ; Zhu . Moreover, by combining the linear spin-wave theory, field theory approach and density-matrix renormalization-group (DMRG) white ; KWHP ; U01 ; U02 ; McCulloch , effects of the long range interactions on local correlation functions, entanglement entropy and central charge are investigated for both spin-1/2 gong17 and spin-1 gong16 to await experimental observation. In addition, one also finds that long-range interactions and long-range hopping may lead to drastic effects on the many- body localization in a one-dimensional (1D) spinless fermion system Nag2019 , which essentially corresponds to a $XY$ type of long range spin interaction. In this regard, the anisotropic long-range spin interaction can be anticipated to give rise to more effects on quantum transitions. In this paper, we study a ID spin-1/2 XXZ system with anisotropic power-law long range interactions in terms of the entanglement entropy, fidelity susceptibility, and correlation functions by performing DMRG calculations. Phase diagrams are established with respect to the power exponents and the anisotropy of interactions. In the following, Sec II presents the Hamiltonian in our studies. The details on DMRG calculations and the definitions of those calculated quantities are discussed in Sec. III. Numerical results are shown in Sec IV with further discussions given in the last section. ## II Hamiltonian In the paper, we consider the following spin-$1/2$ chain with anisotropic long-range interactions, and its Hamiltonian is given by: $\displaystyle H=\sum_{j>i}\\{\frac{J_{xy}}{|i-j|^{\alpha}}(S^{x}_{i}S^{x}_{j}+S^{y}_{i}S^{y}_{j})+\frac{J_{z}}{|i-j|^{\beta}}S^{z}_{i}S^{z}_{j}\\},$ (1) where $i$ and $j$ are the sites of one dimensional lattice, and $S^{\gamma}=\sigma^{\gamma}/2$ with $\gamma=x,y$, or $z$, setting $\hbar=1$ and $\sigma^{\gamma}$ being the Pauli matrices. Interactions between two spins separated by a distance of $r=|i-j|$ decay as $r^{-\alpha}$ for both $x$ and $y$ components of spins, but as $r^{-\beta}$ for the $z$ direction. As usual, the parameters $\alpha,\beta$ are both taken positive, while $J_{xy}=-1$ is set up for the simplicity so that $J_{z}$ readily stands for an anisotropic parameter involved in the establishment of the phase diagram. For this system, in the limit of $\alpha,\beta\rightarrow+\infty$, the Hamiltonian is reduced to describe a spin-1/2 anisotropic chain with the nearest-neighbor interaction. It turns out that the system involves a ferromagnetic (FM) phase for $J_{z}<-1$, whereas a gapful antiferromagnetic (AFM) phase can be shown for $J_{z}>1$. Furthermore, in the region of $-1<J_{z}\leq 1$, the system displays an $XY$ phase where quantum fluctuations exclude the existence of any long-range order but correlation functions behave as a power-law decay of the distance characterized as in the Luttinger liquid. For more general values of $\alpha$ and $\beta$, long range interactions may result in different features for those phases, which are expected also to be properly characterized by long-distance correlation functions as exploited below. ## III Measurements and Method Thanks to the DMRG methodwhite ; KWHP ; U01 , the ground state properties of quasi-one-dimensional systems can be calculated with very high accuracy. For the present studies of Hamiltonian (1), we adopt both infinite-size DMRG (iDMRG) McCulloch and finite-size DMRG, which are based on matrix product states U02 . The number of eigenstates for the reduced matrix is kept up to $m=400$ in the truncation of bases, which allows the truncation error to be smaller than $10^{-9}$. In our calculations where finite-size DMRG algorithm, we handle the long range interaction with directly using as a summation over matrix product of operators (MPOs) rather than the summation of finite exponential terms with MPOs Vidal , which inevitably introduces additional systematic error otherwise. Our codes are mainly based on iTensor C++ library tesnor . Since the $z$-component of the total spins for the present system commutes with the Hamiltonian (1), the ground-state energy is obtained by comparing the lowest energies for each subspace of $S^{z}_{t}=\sum_{i=1}^{L}\langle S^{z}_{i}\rangle$. We found that the ground state resides in the sector of either $S^{z}_{t}=0$ or $S^{z}_{t}=L/2$. To examine the reliability of our numerics, we also perform the finite-size DMRG with varying the number of states in the truncated bases. Once the ground state energy and the corresponding ground state are identified accurately, the first excited state and the corresponding energy (gap) can be determined similarly as orthonormalized to the ground state. For a quantum many-body system, the entanglement entropy (EE) can be extracted from the ground state wavefunction $|\psi_{0}\rangle$ properly to characterize the quantum phase transition induced by the interaction or external fields. Usually, one may separate a given Hamiltonian into two subsystems $A$ and $B$, and compute the reduced density matrix for part $A$ by partially tracing over the degree of freedom of the subsystem $B$, which can be written formally as $\rho_{A}=\textrm{Tr}_{B}(|\psi_{0}\rangle\langle\psi_{0}|).$ Then, the entanglement entropy measuring the entanglement between parts $A$ and $B$ is given by $\displaystyle S_{A}=-\textrm{Tr}(\rho_{A}\ln\rho_{A}).$ (2) which is evaluated in terms of the eigenvalues of $\rho_{A}$ feasibly in DMRG calculations. For a one-dimensional short-range interacting system with an open boundary condition (OBC), the conformal field theory (CFT) suggests that the entanglement entropy for the subsystem $A$ with size $l$ possesses the following finite-size $L$ scaling behavior Cardy $\displaystyle S_{l}=\frac{c}{6}\ln[\frac{L}{\pi}\sin(\frac{\pi l}{L})]+S_{0},$ (3) where $c$ is the central charge which usually has different values for different phases and $S_{0}$ is a non-universal constant. This scaling behavior has been employed to explore the critical entanglement of defects Zhao2006 and Gaussian transitionHu2011 . In this paper, we will show that this scaling behavior is applicable to a case associated with long-range interactions. ## IV Results ### IV.1 $1/\alpha=0$ Now we first consider the case of $\alpha=\infty$, which implies that only the nearest-neighbor term of $xy-$interaction survives. It turn out that the long- range interaction for $z-$component governed by $\beta$ may result in novel properties in competition with the $xy-$components. In this case, Hamiltonian (1) can be recast to describe a one-dimensional interacting spinless fermionic chain via the Jordan-Wigner transformation: $\displaystyle S^{z}_{i}$ $\displaystyle=$ $\displaystyle\frac{1}{2}-c_{i}^{\dagger}c_{i},$ $\displaystyle S^{+}_{i}$ $\displaystyle=$ $\displaystyle e^{i\pi\sum_{j=1}^{i-1}c_{i}^{\dagger}c_{i}}c_{i},$ $\displaystyle S^{-}_{i}$ $\displaystyle=$ $\displaystyle e^{i\pi\sum_{j=1}^{i-1}c_{i}^{\dagger}c_{i}}c_{i}^{\dagger},$ where $S^{\pm}_{i}$=$S^{x}_{i}$ $\pm$ $iS^{y}_{i}$ are the raising and lowering spin operators. Subsequently, the ferromagnetic $J_{xy}-$term thus simply represents the hopping of fermions, while the $J_{z}-$term stands for the density-density interactions of fermions, which can be either attractive for $J_{z}<0$ or repulsive for $J_{z}>0$. One may expect that this density- density interaction results in quantum transitions for different $\alpha$ and $\beta$. To explore this, we compute the correlation functions between two spins at $i$ and $j$ with a distance of $r=|i-j|$ and for $\beta$ = 2 with using the iDMRG algorithm. Figure 1 shows results for $r=99$. One can see that when $J_{z}<-0.636$, the transverse correlation $\langle S^{+}_{i}S^{-}_{i+99}\rangle=0$ and the longitudinal correlation $\langle S^{z}_{i}S^{z}_{i+99}\rangle=1/4$, implying that the system is in the FM phase, and then $\langle S^{+}_{i}S^{-}_{i+99}\rangle$ suddenly jumps to a positive value at $J_{z}=-0.636$ and $\langle S^{z}_{i}S^{z}_{i+99}\rangle$ drops to zero simultaneously. This discontinuity indicates that the ground state undergoes a first order transition from the FM phase into the $XY$ phase. This discontinuous feature is thus utilized here to determine the critical values of $\beta$ and $J_{z}$ for the quantum phase transition between the $XY$ and FM phases. Figure 1: (Color online) Correlation functions $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ and $\langle S^{z}_{i}S^{z}_{i+r}\rangle$ are plotted as a function of $z$-component interaction $J_{z}$ for $\alpha=\infty$, $\beta=2$ and $r=99$. Inset: a log-log plot for $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ as a function of $r$ when $J_{z}=\pm 0.5$. Figure 2: (Color online) (a) Entanglement entropies are plotted as a function of $z$-component interaction $J_{z}$ for various system sizes with $\alpha=\infty$ and $\beta=2$. (b) The peak positions of $S_{L/2}$ versus system sizes $L$. Moreover, as $J_{z}$ further increases, the transverse correlation $\langle S^{+}_{i}S^{-}_{i+99}\rangle$ gradually reduce to zero, while the longitudinal correlation $\langle S^{z}_{i}S^{z}_{i+99}\rangle$ turns to negative for $J_{z}\gtrsim 3/2$, which signals that the system is driven into a AFM phase. A little scrutiny reveals that the transverse correlation $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ satisfies a power-law decay with the distance $r$ gong17 , as manifested in the inset of Fig. 1. To determine the critical point at the transition between the $XY$ phase and AFM phase more precisely, we also calculate the von Neumann entropy, i.e. entanglement entropy, for the right part apart from the rest for the chain with using the finite-size DMRG algorithm. The entanglement entropy is shown in Fig. 2 as a function of $J_{z}$ with $\beta=2$ for different sizes of the chain. With increasing $J_{z}$, the EE increases first and then declines. The peak becomes more pronounced for a larger size $L$ and the location of the peak moves to a lower value of $J_{z}$, characterizing a transition between the $XY$ phase and the AFM phase Wang . According to the finite-size scaling theory Fisher ; Barber83 , it is expected that the position of the pseudo-critical point for a finite- size system approaches the true critical point as $L$ $\to$ $\infty$. For relevant operators in the driving Hamiltonian on sufficiently large-size systems, i.e., $\nu$$d$$<$2, where $\nu$ is the critical exponent of the correlation length and $d$ the dimensionality of the system, the leading term in the expansion of pseudo-critical point obeys $\displaystyle|J_{z}^{c}(L)-J_{z}^{c}(\infty)|\propto L^{-1/\nu},$ (4) where $J_{z}^{c}(\infty)$ is the critical value for the thermodynamic limit. Such algebraic convergence can be accelerated considerably by some elaborated strategies Roncaglia . We obtain that $J_{z}^{c}=1.520$ and $\nu=1.695$ for the present case consistent with the inflection point of the correlations shown in Fig. 1. We note that the scaling behavior of Eq. (4) with $L$ is also valid for the maximum of fidelity susceptibility defined in Eq.(6) You2011 (see below). Figure 3: (Color online) Finite size scaling of the energy gap $\Delta$ with various $\beta$ and $J_{z}$. Symbols show numerical results obtained by DMRG calculations and solid lines are fits of the data by quadratic polynomials in $1/L$. The results for $J_{z}=0$ is also plotted as for comparison. Low-lying excitation energy often reveals perspective features of different phases in the quantum many-body interacting systems. As mentioned previously, the system involves the gapless $XY$ phase for $-1<J_{z}\leq 1$ in the limit of $\beta=\infty$, which has the central charge $c_{\rm eff}=1$ owing to the conformal symmetry Vidal03 . In the Jordan-Wigner representation of the Hamiltonian (1), the spinless interacting fermion has a linear $1/L-$dependence for the finite-size energy gap as a relativistic spectrum at the Fermi point or the low-lying property of the spectrum for the Luttinger liquid. When $\beta\neq\infty$, however, it is clearly of great interest whether such a $XY$ phase can be robust against a strong long-range repulsive interaction. For $J_{z}=1$ and $\beta=1$ Schulz ; Li , it was suggested that the ground state would be a quasi-Wigner crystal (WC), which results from the dominant long-range repulsive interaction over the kinetic energy. We calculated the finite-size gap energy $\Delta(L)$ between the ground state and the first excitation energies as a function of system sizes for various cases as illustrated in Fig.3, one can see that the energy gap $\Delta(\beta,J_{z})$ can be either zero, including the case of $J_{z}=1$ and $\beta=1$, or finite in the thermodynamic limit, which can be assigned to $XY$ and gapped quasi-WC phases, respectively. However, for given $J_{z}$, when $\beta$ approaches its critical values $\beta_{c}$ from either $XY$ phase or WC phase where $\Delta(L)=\Delta(\beta,J_{z})+A_{1}/L+O(1/L^{2})$ You14 , it becomes rather difficult to accurately determine the phase boundary between these two phases due to limited precisions on tiny values of $\Delta(L)$. Instead, we adopt the effective center charge $c_{\rm eff}$ deducted from the scaling behavior of the entanglement entropy given in Eq.(3) which enable us more accurately to allocate the phase boundary. We note that this scaling behavior is valid in the presence of the long range interaction as demonstrated numerically in Fig. 4, although it was originally derived for the short range interacting cases with conformal symmetries Cardy ; Laflorencie . Figure 4: (Color online) The Scaling behavior of entanglement entropy versus $\ln(x)=\ln[L/\pi\sin(\pi l/L)]$ for different values of $\beta^{-1}$ with $L=300$. Inset shows the fitted coefficients as a function of $\beta^{-1}$ for system sizes $L=200$ (square) and $L=300$ (circle). Figure 4 shows the entanglement entropy as a function of $\ln[L/\pi\sin(\pi l/L)]$ for various values of $\beta$ and positive $J_{z}$. It is instructive that the entanglement entropy still follows up the scaling behavior of Eq. (3), although conformal symmetries are not yet known here in general. Subsequently, the slope of the linear behavior gives rise to an effective central charge $c_{\rm eff}$ which varies with $\beta$ as illustrated for system sizes $L=200$ and $300$ at $J_{z}=1$ in the inset of Fig. 4. One can see that finite-size effects for small $1/\beta$ is small but still visible, resulting in the correction to $c^{0}_{\rm eff}=1$ for the thermodynamic limit, but diminishes with increasing $1/\beta$. The curves for these two sizes cross with a horizontal line corresponding to $c_{eff}=c^{0}_{\rm eff}$ at $1/\beta_{c}=0.756$, where irrelevant corrections vanish to Eq. (3). The finite-size effect then becomes negligible for $\beta\leq\beta_{c}$. This provides alternative way with higher accuracy to determine transition points between the $XY$ (critical) and WC (noncritical) phases Alet ; gong16 ; gong17 . Figure 5: (Color online) Phase diagram of Hamiltonian (1) as a functions of the interaction $J_{z}$ and $1/\beta$ with $\alpha\rightarrow+\infty$. In addition, we note that the FM phase is formed owing to the instability of effectively attractive density-density interaction for $J_{z}\leq 0$ upon changing $1/\beta$. Accordingly, the central charge is zero for the FM phase, but it has the value of 3/2 on its phase boundary with the $XY$ phase for the thermodynamic limitsChen ; Olalla ; Alba . To this end, the phase diagram is depicted in Fig. 5 for $\alpha=\infty$. One can see that the critical points between the $XY$ phase and the FM phase asymptotically approach $J_{z}=0$, while the critical points between the AFM phase and the $XY$ phase mounts up with increasing $1/\beta$. Moreover, it is worthwhile to mention that at $\beta=0$ with $J_{z}>0$, $J_{z}$ term effectively results in one sort of long-range frustrations and has the same strength for all the sites, among which diagonal elements cancel each other in the ground state in correspondence to $S^{z}_{total}=0$ subspaceZerobeta . In this case, the ground state again becomes gapless and the central charge equals to one. Particularly, the energy gap $\Delta(L)$ is scaled to zero in the limit of $L\rightarrow\infty$ independent of $J_{z}$ as illustrated for both $J_{z}=1$ and $J_{z}=2$ in Fig. 3. Moreover, the entanglement entropy behaves as same between $J_{z}=1,2$, resulting in $c_{\rm eff}\simeq 1.02$, as seen in Fig. 4. As connected to $J_{z}=0$, it is natural to consider that the system is indeed in the $XY$ phase, i.e. the transition between the FM and $XY$ phases takes place at $J_{z}=0$ for $1/\beta=\infty$. ### IV.2 $1/\beta=0$ In this section, we turn to the case of $\beta\rightarrow+\infty$. In this case, only the nearest neighbor interaction survives in the $J_{z}-$terms of the Hamiltonian Eq. (1). The exponent $\alpha$ of the $XY-$long range interaction can be considered a tunable parameter to explore the quantum phase transition for various values of $J_{z}$. Figure 6: (Color online) Correlation functions $\langle S^{+}_{i}S^{-}_{i+99}\rangle$ and $\langle S^{z}_{i}S^{z}_{i+99}\rangle$ are plotted as a function of the interaction $J_{z}$ for (a) $\alpha=4$ and (b) $\alpha=2$. Inset: A log-log plot of $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ as a function of the distance $r$ with $J_{z}=\pm 0.5$. Figure 6 shows the dependence of two-spin correlations on $J_{z}$ with a distance of $|i-j|=99$ for different $\alpha$, calculated by using the iDMRG algorithm. When $J_{z}$ is negatively large enough, $\langle S^{+}_{i}S^{-}_{i+99}\rangle=0$, $\langle S^{z}_{i}S^{z}_{i+99}\rangle=1/4$, suggesting that the system is in the FM phase. When $J_{z}$ is sufficiently large, the transverse correlations remain zero, whereas $\langle S^{z}_{i}S^{z}_{i+99}\rangle$ becomes negative so that the ground state is a AFM state. Analogous to the case of $\alpha=\infty$, here we again utilize the discontinuity of the correlation functions to allocate the critical points for $\alpha$ and $J_{z}$ at the boundary of the FM phase, while the boundary of the AFM phase is also determined in terms of the entanglement entropy (see below). In an intermediate range of $J_{z}$, one can further see that the transverse correlations $\langle S^{+}_{i}S^{-}_{i+99}\rangle$ is positive but longitudinal correlations $\langle S^{z}_{i}S^{z}_{i+99}\rangle$ vanish. Interestingly, we find that $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ is a concave function of $J_{z}$ for $\alpha=2$, but becomes a convex one for $\alpha=4$. Moreover, when $J_{z}=\pm 0.5$, $\langle S^{+}_{i}S^{-}_{i+r}\rangle$ behaves as a power-law of $1/r$, vanishing in the limit of $r\rightarrow\infty$ as illustrated for $\alpha=4$ in the inset of Fig. 6(a), but ${\lim_{r\to+\infty}}\langle S^{+}_{i}S^{-}_{i+r}\rangle$ approaches a finite constant as seen for $\alpha=2$ from the inset of Fig. 6(b). Therefore, the ground states for $\alpha=2$ in the intermediate range of $J_{z}$ is different that for $\alpha=4$. In this range of $J_{z}$, it is natural to assign the large$-\alpha$ phase to the $XY$ phase, since this phase contains a special case where $\alpha=\infty$ and $J_{z}=0$ such that the Hamiltonian (1) is reduced to describe a standard $XY$ chain, as already shown Fig. (5). Moreover, when $\alpha$ is small or even not too large, one can show that a $U(1)$ symmetry in the ground state is spontaneously broken at $J_{z}=0$ with using the conformal field analysis and perturbation calculationgong17 . It turns out that one can expect the emergence of a continuous symmetry breaking (CSB) phase with gapless excitations for a small$-\alpha$ phase. It has been shown that a Berezinskii- Kosterlitz-Thouless like transition happens between the CSB phase and the $XY$ phase at $1/\alpha_{c}\simeq 0.34$, at which the central charge is numerically increased by $4\%$ from unit. However, the criteria of the $4\%$ addition to the central charge might be invalid for the determination of the critical points with general values of $J_{z}$. To address this issue, we calculate the fidelity susceptibility which has been proposed for the identification of the critical points of continuous quantum phase transitionsGu2010 and even deconfined quantum critical points Sun19 , and successfully applied to various strongly correlated systems You15 ; You17 ; Ren18 ; Luo18 . As a quantum information metric Gu2010 ; You , the fidelity measures the similarity between the two closest ground states when the parameter $\alpha$ is tuned tiny for the Hamiltonian (1), which is defined as $F=|\langle\psi_{0}(\alpha)|\psi_{0}(\alpha+\delta\alpha)\rangle|,$ (5) where $\delta\alpha$ denotes a tiny deviation. Subsequently, we obtain the derivatives of interactions $\delta J_{i,j}=-\frac{J_{xy}}{|i-j|^{\alpha}}\ln|i-j|\delta\alpha$, where $J_{i,j}$ is the interaction strength between two spins at sites $i$ and $j$. The average derivatives of interactions per site are practically considered as an effective tuning parameter $\delta J=\frac{\sum_{i<j}\delta J_{i,j}}{L}$. Therefore, the fidelity susceptibility per site can be calculated numerically by $\chi=\lim_{\delta J\rightarrow 0}\frac{-2\textrm{ln}F}{L(\delta J)^{2}},$ (6) whose peak is thus used to identify the critical value of $\alpha$ and to separate the CSB phase from the $XY$ phase for each $J_{z}$. In our numerical calculations, we take $\delta\alpha=0.005$. For the case of $L=100$ and $\alpha=3$, the effective tuning parameter $\delta J\simeq 0.001$. The ground-state fidelity susceptibility per site $\chi$ is shown for $J_{z}=0,1$ as a function of the parameter $\alpha$ for different sizes in Fig. 7 (a) and (b), respectively. For each $J_{z}$, one can see that the peaks of $\chi$ grow with respect to increasing the system size so that a divergence peak would be expected for the $L\rightarrow\infty$ limit to signal the appearance of a quantum phase transition. In order to locate the quantum critical point $\alpha_{c}$ for the thermodynamic limit, we uses the finite- size scaling analysis to obtain $\alpha_{c}=2.83$ and $\nu=1$ at $J_{z}=0$ as seen in the inset of Fig. 7(a). This value of $\alpha_{c}$ is good consistent with that determined by the central charge and the perturbation theory calculation gong17 . Similarly, we can determine critical points at other values of $J_{z}$ for the boundary between the CSB and $XY$ pases. In particular, the critical value of $\alpha_{c}=2.45$ for $J_{z}=1.0$ is obtained from the results shown in Fig. 7(b). Figure 7: (Color online) Fidelity susceptibility per site is plotted as a function of parameter $\alpha$ for various system sizes with (a) $J_{z}=0$ and (b) $J_{z}=1.0$. Inset: Scaling behavior of the fidelity susceptibility peak points with respect to $1/L$. Now we turn to quantum phase transitions between the intermediate and AFM phases, which are characterized by the peaks of the entanglement entropies as demonstrated for $\alpha=2,4$ in Fig. 8. One can see that the peaks for both cases in (a) and (c) of Fig. 8 move to lower values of $J_{z}$ when $L$ increases. Fitting the locations of peaks with the formula (4) as shown in (b) and (d) of Fig. 8, one can obtain that $J_{z}^{c}=1.35$ and $2.21$, respectively. Such fitted results agree very well with the inflexion points of the correlations shown in Fig. 6. In the same manner, we allocate more critical values of $J_{z}$ and $\alpha$ for the boundary of the AFM phase with both $XY$ and CSB phases. Figure 8: (Color online) Entanglement entropy is plotted as a function of the interaction $J_{z}$ on different system sizes for (a) $\alpha=4$ and (c) $\alpha=2$. The peak positions of $S_{L/2}$ versus the system size $L$ for (b) $\alpha=4$ and (d) $\alpha=2$. Based on the above analysis on the properties of the correlation functions, the fidelity susceptibility and the entanglement entropy, we establish the ground-state phase diagram for the Hamiltonian (1) with $\alpha=\infty$ as shown in Fig. 9. Figure 9: (Color online) Phase diagram of Hamiltonian (1) as a functions of the interaction $J_{z}$ and $\alpha$ with $\beta\rightarrow+\infty$. ## V Discussion In this paper, we study quantum phase transitions for a quantum spin-$1/2$ chain with anisotropic power-law-decaying long-range interactions, which are characterized by exponent parameters $\alpha$ for $xy-$term and $\beta$ for $z-$term, by employing density-matrix renormalization-group method. With numerically analyzing the effects of $\alpha$ and $\beta$ on the spin-spin correlation functions, the entanglement entropy and the central charge, and the fidelity susceptibility, we establish two phase diagrams for $\alpha=\infty$ and $\beta=\infty$, respectively. Both cases involve a ferromagnetic phase and an antiferromagnetic phase corresponding to sufficiently negative and positive $J_{z}$, respectively. However, in the intermediate regime of $J_{z}$, the former involves not only a usual $XY$ phase effectively equivalent to a short range repulsive density- density interaction, but also a Wigner-crystal phase which essentially results from for a sufficient strong long-range $J_{z}$ term; for the later, the gapped Wigner crystal phase is replaced by a continuous $U(1)$ symmetry breaking phase. Moreover, it is interesting to notice that the WC and CSB phases actually reveal two different mechanisms, which intrinsically result from either two-body processes of the strong long-range repulsive interaction or one-body kinetic processes of the long-range hoping in the fermion representation. From this study, we found that the entanglement entropy and the central charge can be used efficiently to extract critical values of the quantum phase transition between two phases when one of them possesses a well-defined central charge but another one is gapful Luo2019 . However, when one is encountered with a quantum phase transition between two gapless phases, the fidelity susceptibility alternatively provides a more feasible way to allocate the critical points as applied to the transition between the $XY$ and continuous $U(1)$ symmetry breaking phases. We so far focus on the ground state phase diagrams only for $\alpha=\infty$ and $\beta=\infty$. There are actually a couple of important aspects beyond the above two cases for the Hamiltonian (1), such as ground phase diagrams with $\alpha=\beta$ and $J_{xy}>0$, extensions to two leg-ladders and even two dimensions, etc. The emergence of any non-trivial gapless phase, corresponding novel low-lying excitation spectra or exotic collective excitations with special symmetries, and thermodynamic and dynamic properties would be very interesting questions for the presence of long range interactions but are certainly open for further studies in the future. ###### Acknowledgements. This work is supported by the National Program on Key Research Project (Grant No. 2016YFA0300501) and the National Natural Science Foundation of China under Grants No. 11104021, 11474211, 61674110 and 11974244. W.L.Y is appreciative of support from the start-up fund of Nanjing University of Aeronautics and Astronautics. X.W. also acknowledges additional supports from a Shanghai talent program. ## References * (1) S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, England, 1999). * (2) Xiaoqun Wang, Mod. Phys. Lett. B 14, 327 (2000). * (3) Qiang Luo, Shijie Hu, Bin Xi, Jize Zhao, and Xiaoqun Wang, Phys. Rev. B 95, 165110 (2017). * (4) Qiang Luo, Jize Zhao and Xiaoqun Wang, Phys. Rev. B 100, 121111(R) (2019). * (5) T. C. Yi, W. L. You, N. Wu and A. M. Oleś, Phys. Rev. B 100, 024423 (2019). * (6) N. Wu and W. L. You, Phys. Rev. B 100, 085130 (2019). * (7) M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys. 82, 2313 (2010). * (8) T. Lahaye, C. Menotti, L. 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2024-09-04T02:54:58.599700

2003.04520

arxiv-papers

# Extensions of some matrix inequalities related to trace and partial traces††thanks: This paper was firstly announced in March, 2020, and was later published on Linear Algebra and its Applications 639 (2022) 205–224. See https://doi.org/10.1016/j.laa.2022.01.006. E-mail addresses: <EMAIL_ADDRESS>(Yǒngtāo Lǐ). Yongtao Li∗ School of Mathematics, Hunan University Changsha, Hunan, 410082, P.R. China ###### Abstract We first present a determinant inequality related to partial traces for positive semidefinite block matrices. Our result extends a result of Lin [Czech. Math. J. 66 (2016)] and improves a result of Kuai [Linear Multilinear Algebra 66 (2018)]. Moreover, we provide a unified treatment of a result of Ando [ILAS Conference (2014)] and a recent result of Li, Liu and Huang [Operators and Matrices 15 (2021)]. Furthermore, we also extend some determinant inequalities involving partial traces to a larger class of matrices whose numerical ranges are contained in a sector. In addition, some extensions on trace inequalities for positive semidefinite $2\times 2$ block matrices are also included. Dedicated to Prof. Weijun Liu on his 60th birthday Key words: Partial traces; Trace inequalities; Fiedler and Markham; Numerical range in a sector; 2010 Mathematics Subject Classification. 15A45, 15A60, 47B65. ## 1 Introduction Throughout the paper, we use the following standard notation. The set of $n\times n$ complex matrices is denoted by $\mathbb{M}_{n}(\mathbb{C})$, or simply by $\mathbb{M}_{n}$, and the identity matrix of order $n$ by $I_{n}$, or $I$ for short. We write $\lambda_{i}(A)$ and $\sigma_{i}(A)$ for the $i$-th largest eigenvalue and singular value of $A$, respectively. By convention, if $A\in\mathbb{M}_{n}$ is positive semidefinite, we write $A\geq 0$. For Hermitian matrices $A$ and $B$ with the same size, $A\geq B$ means that $A-B$ is positive semidefinite, i.e., $A-B\geq 0$. If $A=[a_{i,j}]$ is of order $m\times n$ and $B$ is of order $s\times t$, the tensor product of $A$ with $B$, denoted by $A\otimes B$, is an $ms\times nt$ matrix that partitioned into $m\times n$ block matrices with the $(i,j)$-block being the $s\times t$ matrix $a_{i,j}B$. In this paper, we are interested in complex block matrices. Let $\mathbb{M}_{n}(\mathbb{M}_{k})$ be the set of complex matrices partitioned into $n\times n$ blocks with each block being $k\times k$. The element of $\mathbb{M}_{n}(\mathbb{M}_{k})$ is usually written as ${H}=[H_{i,j}]_{i,j=1}^{n}$, where $H_{i,j}\in\mathbb{M}_{k}$ for all $i,j$. Now we introduce the definition of partial traces, which comes from Quantum Information Theory [32, p. 12]. For $H\in\mathbb{M}_{n}(\mathbb{M}_{k})$, the first partial trace (map) $H\mapsto\mathrm{tr}_{1}H\in\mathbb{M}_{k}$ is defined as the adjoint map of the embedding map $X\mapsto I_{n}\otimes X\in\mathbb{M}_{n}\otimes\mathbb{M}_{k}$. Correspondingly, the second partial trace (map) $H\mapsto\mathrm{tr}_{2}H\in\mathbb{M}_{n}$ is defined as the adjoint map of the embedding map $Y\mapsto Y\otimes I_{k}\in\mathbb{M}_{n}\otimes\mathbb{M}_{k}$. Therefore, we have $\langle I_{n}\otimes X,H\rangle=\langle X,\mathrm{tr}_{1}H\rangle,\quad\forall X\in\mathbb{M}_{k},$ (1) and $\langle Y\otimes I_{k},H\rangle=\langle Y,\mathrm{tr}_{2}H\rangle,\quad\forall Y\in\mathbb{M}_{n},$ where $\langle\cdot,\cdot\rangle$ stands for the Hilbert-Schmidt inner product, i.e., $\langle A,B\rangle={\rm tr}(A^{*}B)$. The above definition of partial traces is implicit. Assume that $H=[H_{i,j}]_{i,j=1}^{n}$ is an $n\times n$ block matrix with $H_{i,j}\in\mathbb{M}_{k}$, the visualized version of the partial traces is equivalently given in [4, pp. 120–123] as $\mathrm{tr}_{1}{H}=\sum\limits_{i=1}^{n}H_{i,i},$ (2) and $\mathrm{tr}_{2}{H}=\bigl{[}\mathrm{tr}H_{i,j}\bigr{]}_{i,j=1}^{n}.$ It is easy to see that both ${\rm tr}_{1}H$ and ${\rm tr}_{2}H$ are positive semidefinite whenever ${H}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ is positive semidefinite; see, e.g. [36, p. 237] or [37] for more details. The first or second partial trace is a source for matrix inequalities and extensively studied in recent years; see [2, 7, 11, 22, 29] for related topics. Let $A=[A_{i,j}]_{i,j=1}^{n}$ be an $n\times n$ block matrix with each block being a $k\times k$ matrix. The usual transpose of $A$ is defined as $A^{T}=[A_{j,i}^{T}]_{i,j=1}^{n}$. We define the partial transpose of $A$ by $A^{\tau}=[A_{j,i}]_{i,j=1}^{n}$, that is, the partial transpose of $A$ is the matrix obtained by transposing blocks of $A$ independently. More precisely, $A^{T}=\begin{bmatrix}A_{1,1}^{T}&\cdots&A_{n,1}^{T}\\\ \vdots&\ddots&\vdots\\\ A_{1,n}^{T}&\cdots&A_{n,n}^{T}\end{bmatrix}~{}~{}\text{and}~{}~{}A^{\tau}=\begin{bmatrix}A_{1,1}&\cdots&A_{n,1}\\\ \vdots&\ddots&\vdots\\\ A_{1,n}&\cdots&A_{n,n}\end{bmatrix}.$ Although $A$ and $A^{\tau}$ have the same trace, they may have different eigenvalues, so they are not necessarily similar. Moreover, it is known that $A\geq 0$ does not necessarily imply $A^{\tau}\geq 0$. For example, taking $A=\begin{bmatrix}A_{1,1}&A_{1,2}\\\ A_{2,1}&A_{2,2}\end{bmatrix}=\left[\begin{array}[]{cc;{2pt/2pt}cc}1&0&0&1\\\ 0&0&0&0\\\ \hdashline[2pt/2pt]0&0&0&0\\\ 1&0&0&1\end{array}\right].$ (3) We can see from the definition that $A^{\tau}=\begin{bmatrix}A_{1,1}&A_{2,1}\\\ A_{1,2}&A_{2,2}\end{bmatrix}=\left[\begin{array}[]{cc;{2pt/2pt}cc}1&0&0&0\\\ 0&0&1&0\\\ \hdashline[2pt/2pt]0&1&0&0\\\ 0&0&0&1\end{array}\right].$ One could easily observe that $A$ is positive semidefinite, but $A^{\tau}$ is not positive semidefinite since it contains a principal submatrix $\left[\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix}\right]\ngeq 0$. Moreover, the eigenvalues of $A$ are $2,0,0,0$, and the eigenvalues of $A^{\tau}$ are $1,1,1,-1$, so $A$ and $A^{\tau}$ are not similar. In addition, replacing $A_{1,1}$ in the above matrix by $\left[\begin{smallmatrix}1&0\\\ 0&1\end{smallmatrix}\right]$ also gives a well example. From this discussion, we say that $A$ is positive partial transpose (or PPT for short) if both $A$ and $A^{\tau}$ are positive semidefinite. We recommend [10, 19, 26, 27] for recent progress. The paper is organized as follows. In Section 2, we shall review some preliminaries for a class of matrices whose numerical ranges are contained in a sector (known as the sector matrices). This is a natural extension of the class of positive definite matrices. In Section 3, we shall study the recent results involving the Fiedler–Markham inequality. We provide an extension of a result of Lin [29], and our result is also an improvement of a result of Kuai [17]; see Theorem 3.5. Moreover, we shall extend a result of Choi [6] to the so-called sector matrices; see Theorem 3.7. In Section 4, we give a unified treatment of a result of Ando [2] (or see [30]) as well as a recent result of Li, Liu and Huang [22]. Our new treatment is more concise than original proof. Moreover, we also present some Ando type determinant inequalities for partial traces, and then we extend these inequalities to sector matrices; see Theorems 4.7 and 4.8. In Section 5, we shall prove some inequalities for positive semidefinite $2\times 2$ block matrices; see Theorems 5.2, 5.3 and 5.4. Our result extend slightly the recent elegant work on trace inequalities that proved by Kittaneh and Lin [18] and Lin [26] as well. ## 2 Preliminaries Recall that $\sigma_{i}(A)$ denotes $i$-th largest singular value of $A$. When $A$ is Hermitian, we know that all eigenvalues of $A$ are real numbers, and we write $\lambda_{i}(A)$ for the $i$-th largest eigenvalue. The numerical range of $A\in\mathbb{M}_{n}$ is defined by $W(A)=\\{x^{*}Ax:x\in\mathbb{C}^{n},x^{*}x=1\\}.$ For $\alpha\in[0,{\pi}/{2})$, let $S_{\alpha}$ be the sector on complex plane defined as $S_{\alpha}=\\{z\in\mathbb{C}:\Re z>0,|\Im z|\leq(\Re z)\tan\alpha\\}=\\{re^{i\theta}:r>0,|\theta|\leq\alpha\\}.$ For $A\in\mathbb{M}_{n}$, the Cartesian (Toeptliz) decomposition is given as $A=\Re A+i\cdot\Im A$, where $\Re A=\frac{1}{2}(A+A^{*})$ and $\Im A=\frac{1}{2i}(A-A^{*})$. We know from the definition that if $W(A)\subseteq S_{0}$, then $A$ is positive definite. Moreover, it is easy to verify that if $W(A)\subseteq S_{\alpha}$ for some $\alpha\in[0,{\pi}/{2})$, then $\Re(A)$ is positive definite. Such class of matrices whose numerical ranges are contained in a sector is called the sector matrices class. Clearly, the concept of sector matrices is an extension of positive definite matrices. Over the past few years, various studies on sector matrices have been obtained in the literature; see, e.g., [8, 16, 17, 28, 34, 38]. Before starting our results, we now summarise the following lemmas. ###### Lemma 2.1 [28] Let $0\leq\alpha<{\pi}/{2}$ and $A\in\mathbb{M}_{n}$ with $W(A)\subseteq S_{\alpha}$. Then $|\det A|\leq(\sec\alpha)^{n}\det(\Re A).$ ###### Lemma 2.2 [14, p. 510] Let $X$ be an $n$-square complex matrix. Then $\lambda_{i}(\Re X)\leq\sigma_{i}(X),\quad i=1,2,\ldots,n.$ Moreover, if $\Re X$ is positive definite, then $\det\Re X+|\det\Im X|\leq|\det X|.$ The following lemma is called the Fischer inequality, which gives an upper bound for the determinant of a positive semidefinite block matrix in terms of the determinants of its principal diagonal blocks. In particular, when all blocks have order $1\times 1$, this inequality is also known as the Hadamard inequality; see, e.g., [14, p. 506] and [36, p. 217]. ###### Lemma 2.3 Let $H=[H_{i,j}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be positive semidefinite. Then $\det H\leq\prod_{i=1}^{n}\det H_{i,i}.$ ###### Lemma 2.4 If $H\in\mathbb{M}_{n}(\mathbb{M}_{k})$ satisfies $W(H)\\!\subseteq S_{\alpha}$, then $W({\rm tr}_{1}H)\\!\subseteq S_{\alpha}$ and $W({\rm tr}_{2}H)\\!\subseteq S_{\alpha}$, i.e., if $H$ is a sector matrix with angle $\alpha\in[0,\pi/2)$, then so are ${\rm tr}_{1}H$ and ${\rm tr}_{2}H$. We remark that this lemma was partially proved in [17, Proposition 3.2] for the case ${\rm tr}_{2}H$. Motivated by [17], we here include a detailed proof for the remaining case ${\rm tr}_{1}H$. Proof. Consider the Cartesian decomposition $H=\Re H+i\cdot\Im H$, then ${\rm tr}_{1}H={\rm tr}_{1}(\Re H)+i\cdot{\rm tr}_{1}(\Im H).$ For every $x\in\mathbb{C}^{k}$ with $x^{*}x=1$, as $\Re H$ is positive definite, we get $\Re\bigl{(}x^{*}({\rm tr}_{1}H)x\bigr{)}=x^{*}\bigl{(}\Re({\rm tr}_{1}H)\bigr{)}x=x^{*}\bigl{(}{\rm tr}_{1}(\Re H)\bigr{)}x>0.$ On the other hand, by a direct computation, $\frac{\left|\Im\bigl{(}x^{*}({\rm tr}_{1}H)x\bigr{)}\right|}{\Re\bigl{(}x^{*}({\rm tr}_{1}H)x\bigr{)}}=\frac{\left|x^{*}({\rm tr}_{1}(\Im H))x\right|}{x^{*}({\rm tr}_{1}(\Re H))x}=\frac{\left|\langle xx^{*},{\rm tr}_{1}(\Im H)\rangle\right|}{\langle xx^{*},{\rm tr}_{1}(\Re H)\rangle}.$ Note that $I_{n}\otimes(xx^{*})$ is positive semidefinite. We consider the spectral decomposition $I_{n}\otimes(xx^{*})=\sum_{i=1}^{nk}\lambda_{i}u_{i}u_{i}^{*},$ where $\lambda_{i}\geq 0$ and $u_{i}$ are unit vectors in $\mathbb{C}^{nk}$. By the definition in (1), it follows that $\displaystyle\frac{\left|\langle xx^{*},{\rm tr}_{1}(\Im H)\rangle\right|}{\langle xx^{*},{\rm tr}_{1}(\Re H)\rangle}$ $\displaystyle=\frac{\left|\langle I_{n}\otimes(xx^{*}),\Im H\rangle\right|}{\langle I_{n}\otimes(xx^{*}),\Re H\rangle}=\frac{\left|\sum_{i=1}^{nk}\lambda_{i}\langle u_{i}u_{i}^{*},\Im H\rangle\right|}{\sum_{i=1}^{nk}\lambda_{i}\langle u_{i}u_{i}^{*},\Re H\rangle}$ $\displaystyle\leq\frac{\sum_{i=1}^{nk}\lambda_{i}\left|u_{i}^{*}(\Im H)u_{i}\right|}{\sum_{i=1}^{nk}\lambda_{i}u_{i}^{*}(\Re H)u_{i}}\leq\max_{1\leq i\leq nk}\frac{\left|u_{i}^{*}(\Im H)u_{i}\right|}{u_{i}^{*}(\Re H)u_{i}}=\max_{1\leq i\leq nk}\frac{\left|\Im(u_{i}^{*}Hu_{i})\right|}{\Re(u_{i}^{*}Hu_{i})}.$ This completes the proof. Remark. Based on the second equivalent definition (2), one could also give other ways to prove Lemma 2.4. We leave the details for the interested reader. ## 3 Extensions on Fiedler–Markham’s inequality Let ${H}=[H_{i,j}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be positive semidefinite. Recall that both ${\rm tr}_{1}H$ and ${\rm tr}_{2}H$ are positive semidefinite; see, e.g., [37]. In 1994, Fiedler and Markham [9, Corollary 1] proved a celebrated determinant inequality involving the second partial trace. ###### Theorem 3.1 [9] Let ${H}=[H_{i,j}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be positive semidefinite. Then $\left(\frac{\det\bigl{(}{\rm tr}_{2}H\bigr{)}}{k}\right)^{k}\geq\det{H}.$ In 2016, Lin [29] revisited this inequality using some terminology from quantum information theory, and gave an alternative proof of Theorem 3.1 by applying an important identity connecting ${\rm tr}_{2}H$ and $H$. Moreover, a natural question is that whether an analogous result corresponding to the Fiedler–Markham inequality holds for ${\rm tr}_{1}H$. Lin [29] answered this question and proved the following counterpart. ###### Theorem 3.2 [29] Let ${H}=[H_{ij}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be positive semidefinite. Then $\left(\frac{\det({\rm tr}_{1}H)}{n}\right)^{n}\geq\det H.$ It is clear that in the proof of both Theorem 3.1 and Theorem 3.2, Fiedler and Markham, and Lin used the superadditivity of determinant functional, which states that $\det\left(\sum_{i=1}^{n}H_{i,i}\right)\geq\sum_{i=1}^{n}\det H_{i,i}\geq n\left(\prod_{i=1}^{n}\det H_{i,i}\right)^{1/n}.$ This inequality can be improved by the Fan-Ky determinant inequality (see [14, p. 488]), i.e., the log-concavity of the determinant over the cone of positive semidefinite matrices: $\det\left(\frac{1}{n}\sum_{i=1}^{n}H_{i,i}\right)\geq\left(\prod_{i=1}^{n}\det H_{i,i}\right)^{1/n}.$ (4) In addition, we mention here that a careful examination of the new proof of Theorem 3.1 in [29] can also reveal this improvement. This improvement was also pointed out in [6, 31]. Next, we state the strong version of Theorem 3.1 and Theorem 3.2. ###### Theorem 3.3 Let ${H}=[H_{ij}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be positive semidefinite. Then $\left(\frac{\det\bigl{(}{\rm tr}_{2}H\bigr{)}}{k^{n}}\right)^{k}\geq\det{H},$ and $\left(\frac{\det({\rm tr}_{1}H)}{n^{k}}\right)^{n}\geq\det H.$ We observe in Theorem 3.3 that the second inequality seems easier to prove than the first inequality because it is more convenient to build inequalities on ${\rm tr}_{1}H=\sum_{i=1}^{n}H_{i,i}$. In [20], the authors showed that both inequalities can be deduced mutually. In 2018, Kuai [17] (or see [34]) further extended Theorem 3.3 to sector matrices and showed that if $0\leq\alpha<{\pi}/{2}$ and $H\in\mathbb{M}_{n}(\mathbb{M}_{k})$ satisfies $W(H)\subseteq S_{\alpha}$, then $\left|\frac{\det({\rm tr}_{2}H)}{k^{n}}\right|^{k}\geq(\cos\alpha)^{nk}|\det H|,$ (5) and $\left|\frac{\det({\rm tr}_{1}H)}{n}\right|^{n}\geq(\cos\alpha)^{(3n-2)k}|\det H|.$ (6) Our first goal in this section is to improve Kuai’s result (6). The key step in our improvement is the following identity connecting ${\rm tr}_{1}(H)$ and $H$, which has been applied to quantum information theory, such as the sub- additivity of $q$-entropies. This identity can be found in [15, eq.(26)] or [5, Lemma 2]. ###### Lemma 3.4 Let $X$ and $Y$ be generalized Pauli matrices on $\mathbb{C}^{n}$; these operators act as $Xe_{j}=e_{j+1}$ and $Ye_{j}=e^{2\pi j\sqrt{-1}/n}e_{j}$, where $e_{j}$ is the $j$-th column of the identity matrix $I_{n}$ and $e_{n+1}=e_{1}$. Then $\frac{1}{n}\sum_{l,j=1}^{n}(X^{l}Y^{j}\otimes I_{k})H(X^{l}Y^{j}\otimes I_{k})^{*}=I_{n}\otimes({\rm tr}_{1}H).$ Remark. The identity in this lemma can yield an alternative proof of Lemma 2.4. Moreover, the analogous identity for ${\rm tr}_{2}H$ can be seen in [15] or [33, eq.(14)]. Now, we are ready to present an improvement on inequality (6). ###### Theorem 3.5 Let $0\leq\alpha<{\pi}/{2}$ and $H\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be such that $W(H)\subseteq S_{\alpha}$. Then $\left|\frac{\det({\rm tr}_{1}H)}{n^{k}}\right|^{n}\geq(\cos\alpha)^{nk}|\det H|.$ Proof. Note that both $X$ and $Y$ in Lemma 3.4 are unitary, so are $X^{l}Y^{j}\otimes I_{k}$ for all $l,j$. Moreover, we have $\Re(UHU^{*})=U(\Re H)U^{*}$ for every unitary $U$. Thus, $\displaystyle|\det H|\\!\\!\\!\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\prod_{l,j=1}^{n}\left|\det(X^{l}Y^{j}\otimes I_{k})H(X^{l}Y^{j}\otimes I_{k})^{*}\right|^{1/n^{2}}$ (7) $\displaystyle\overset{\text{Lemma \ref{lem22}}}{\leq}$ $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!(\sec\alpha)^{nk}\prod_{l,j=1}^{n}\left(\det(X^{l}Y^{j}\otimes I_{k})(\Re H)(X^{l}Y^{j}\otimes I_{k})^{*}\right)^{1/n^{2}}$ $\displaystyle\overset{\text{Fan-Ky ineq.(\ref{eqfk})}}{\leq}$ $\displaystyle\\!\\!\\!\\!\\!\\!\\!(\sec\alpha)^{nk}\det\Bigg{(}\frac{1}{n^{2}}\sum_{l,j=1}^{n}(X^{l}Y^{j}\otimes I_{k})(\Re H)(X^{l}Y^{j}\otimes I_{k})^{*}\Bigg{)}$ $\displaystyle\overset{\text{Lemma \ref{lem24}}}{=}$ $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!(\sec\alpha)^{nk}\det\left(\frac{1}{n}\Bigl{(}I_{n}\otimes{\rm tr}_{1}(\Re H)\Bigr{)}\right)$ $\displaystyle=$ $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{(\sec\alpha)^{nk}}{n^{nk}}\det\Bigl{(}I_{n}\otimes{\rm tr}_{1}(\Re H)\Bigr{)}.$ Clearly, we have ${\rm tr}_{1}(\Re H)=\Re({\rm tr}_{1}H)$. For $X\in\mathbb{M}_{n}$ and $Y\in\mathbb{M}_{k}$, it is well-known that $\det(X\otimes Y)=(\det X)^{k}(\det Y)^{n}$; see, e.g., [35, Chapter 2]. It follows that $\det\Bigl{(}I_{n}\otimes{\rm tr}_{1}(\Re H)\Bigr{)}=(\det I_{n})^{k}\bigl{(}\det({\rm tr}_{1}\Re H)\bigr{)}^{n}=\bigl{(}\det\Re({\rm tr}_{1}H)\bigr{)}^{n}.$ By Proposition 2.4, we have $W({\rm tr}_{1}H)\subseteq S_{\alpha}$, which implies that $\Re({\rm tr}_{1}H)$ is positive definite. Therefore, by Lemma 2.2, we get $\bigl{(}\det\Re({\rm tr}_{1}H)\bigr{)}^{n}\leq\bigl{(}|\det({\rm tr}_{1}H)|-|\det\Im({\rm tr}_{1}H)|\bigr{)}^{n}\leq|\det({\rm tr}_{1}H)|^{n},$ which together with (7) yields the desired result. Remark. By applying the techniques from [20], we know that Kuai’s inequality (5) can also be deduced from the inequality in Theorem 3.5 and vice versa. In the sequel, we shall focus our attention on some recent results which are similar with the Fiedler–Markham inequality. Let ${H}=[H_{i,j}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be a block matrix with $H_{i,j}=[h_{l,m}^{i,j}]_{l,m=1}^{k}$. We define an $n\times n$ matrix $G_{l,m}$ as below. $G_{l,m}:=\bigl{[}h_{l,m}^{i,j}\bigr{]}_{i,j=1}^{n}\in\mathbb{M}_{n}.$ A direct computation yields ${\rm tr}_{1}H=\sum_{i=1}^{n}H_{i,i}=\sum_{i=1}^{n}\bigl{[}h_{l,m}^{i,i}\bigr{]}_{l,m=1}^{k}=\left[\begin{matrix}\sum\limits_{i=1}^{n}h_{l,m}^{i,i}\end{matrix}\right]_{l,m=1}^{k}=\bigl{[}{\rm tr}\,G_{l,m}\bigr{]}_{l,m=1}^{k}.$ For notational convenience, we denote $\widetilde{H}=\bigl{[}G_{l,m}\bigr{]}_{l,m=1}^{k}\in\mathbb{M}_{k}(\mathbb{M}_{n}).$ We can see that $\widetilde{H}$ is obtained from $H$ by rearranging the entries in an appropriate order. The above observation yields ${\rm tr}_{1}H={\rm tr}_{2}\widetilde{H}$. Moreover, it is not hard to check that $\widetilde{H}$ and $H$ are unitarily similar; see, e.g., [6, Theorem 7] or [20, Theorem 4]. Motivated by these relations, Choi [6] introduced recently the definition of partial determinants corresponding to partial traces. For $H=[H_{i,j}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{k})$, the partial determinants are defined as $\mathrm{det}_{1}H:=\bigl{[}\det G_{l,m}\bigr{]}_{l,m=1}^{k},$ and $\mathrm{det}_{2}H:=\bigl{[}\det H_{i,j}\bigr{]}_{i,j=1}^{n}.$ To some extent, the partial determinants share some common properties relative to partial traces. For instance, it is easy to see that if ${H}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ is positive semidefinite, then both $\mathrm{det}_{1}H$ and $\mathrm{det}_{2}H$ are positive semidefinite; see, e.g. [36, p. 221]. Moreover, it was proved in [6] that $\mathrm{det}({\rm tr}_{1}H)\geq{\rm tr}(\mathrm{det}_{2}H),$ and $\mathrm{det}({\rm tr}_{2}H)\geq{\rm tr}(\mathrm{det}_{1}H).$ Additionally, Choi [6] proved two analogues of Theorem 3.1 and Theorem 3.2 for partial determinants. ###### Theorem 3.6 [6] Let ${H}\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be positive semidefinite. Then $\left(\frac{{\rm tr}(\mathrm{det}_{1}H)}{k}\right)^{k}\geq\det H,$ and $\left(\frac{{\rm tr}(\mathrm{det}_{2}H)}{n}\right)^{n}\geq\det H.$ Next, we will extend Theorem 3.6 to sector matrices. We write $|A|$ for the nonnegative matrix whose entries are the absolute of the entries of $A$. This notation is only used in the following theorem. ###### Theorem 3.7 Let $0\leq\alpha<{\pi}/{2}$ and $H\in\mathbb{M}_{n}(\mathbb{M}_{k})$ be such that $W(H)\subseteq S_{\alpha}$. Then $\left(\frac{{\rm tr}|\mathrm{det}_{1}H|}{k}\right)^{k}\geq(\cos\alpha)^{nk}|\det H|,$ and $\left(\frac{{\rm tr}|\mathrm{det}_{2}H|}{n}\right)^{n}\geq(\cos\alpha)^{nk}|\det H|.$ Proof. First of all, we shall prove the second inequality. We observe that $\Re H_{1,1}$, $\ldots,\Re H_{n,n}$ are the diagonal block matrices of $\Re H$. By Lemma 2.1 and Lemma 2.3, we obtain $\displaystyle|\det H|$ $\displaystyle\leq(\sec\alpha)^{nk}\det(\Re H)\leq(\sec\alpha)^{nk}\prod_{i=1}^{n}\det(\Re H_{i,i})$ $\displaystyle\leq(\sec\alpha)^{nk}\prod_{i=1}^{n}|\det H_{i,i}|\leq(\sec\alpha)^{nk}\left(\frac{1}{n}\sum_{i=1}^{n}|\det H_{i,i}|\right)^{n},$ where the third inequality follows from Lemma 2.2 and the last one follows from the arithmetic mean-geometric mean inequality. We now prove the first desired inequality by employing the relations between $\mathrm{det}_{1}$ and $\mathrm{det}_{2}$. Recall that $\widetilde{H}=[G_{l,m}]_{l,m=1}^{k}\in\mathbb{M}_{k}(\mathbb{M}_{n})$ and $\mathrm{det}_{1}H=\mathrm{det}_{2}\widetilde{H}$. Since $\widetilde{H}$ and $H$ are unitarily similar, we can get $\det\widetilde{H}=\det H$ and $W(\widetilde{H})\subseteq S_{\alpha}$. Moreover, $\widetilde{H}$ is also positive semidefinite. By applying the second inequality to $\widetilde{H}$, we get $\left(\frac{{\rm tr}|\mathrm{det}_{1}H|}{k}\right)^{k}=\left(\frac{{\rm tr}|\mathrm{det}_{2}\widetilde{H}|}{k}\right)^{k}\geq(\cos\alpha)^{kn}|\det\widetilde{H}|=(\cos\alpha)^{kn}|\det H|.$ This completes the proof. ## 4 Extensions on Ando’s inequality To make our statements more transparent and compatible with the previous works in the literature. In this section, we assume that $A$ is an $m\times m$ block matrix with each block being an $n\times n$ matrix. Let ${A}=[A_{i,j}]_{i,j=1}^{m}\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. We know that both ${\rm tr}_{1}A$ and ${\rm tr}_{2}A$ are positive semidefinite; see, e.g., [36, p. 237] and [37, Theorem 2.1]. To some degree, these two partial traces are closely related and mutually affect each other. We write $\lVert A\lVert_{q}=\left(\sum_{i}\sigma_{i}(A)^{q}\right)^{1/q}$ for the Schatten $q$-norm of $A$. In 2007, Audenaert [1] proved the following norm inequality, ${\rm tr}\,A+\lVert A\lVert_{q}\geq\lVert{\rm tr}_{1}A\rVert_{q}+\lVert{\rm tr}_{2}A\rVert_{q}.$ (8) A straightforward argument exploiting Audenaert’s result leads to a proof of the subadditivity of $q$-entropies (Tsallis entropies) for finite-dimensional bipartite quantum states; see [1, 5] and references therein. In 2014, Ando [2] (or see [30, Proposition 2.2] for an alternative proof) established the following remarkable inequality in the sense of the Löwner ordering. ###### Theorem 4.1 [2, 30] Let $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. Then $({\rm tr}A)I_{mn}+A\geq I_{m}\otimes(\mathrm{tr}_{1}A)+(\mathrm{tr}_{2}A)\otimes I_{n}.$ Ando’s result reveals closely the interplay between the first and second partial trace. Equivalently, this inequality can be rewritten as $({\rm tr}A)I_{mn}-(\mathrm{tr}_{2}A)\otimes I_{n}\geq I_{m}\otimes(\mathrm{tr}_{1}A)-A.$ (9) We observe that the positivity of $A$, together with the identity ${\rm tr}\,A=\sum_{i=1}^{m}{\rm tr}A_{i,i}={\rm tr}({\rm tr}_{2}A)$, leads to $({\rm tr}A)I_{m}\geq\lambda_{\max}({\rm tr}_{2}A)I_{m}\geq{\rm tr}_{2}A$, which guarantees that in (9) the left hand side $({\rm tr}A)I_{mn}-(\mathrm{tr}_{2}A)\otimes I_{n}$ is positive semidefinite. However, the two matrices of the right hand side in (9) might be incomparable. For instance, the matrix $A$ in (3) gives an example. Motivated by this observation, Li, Liu and Huang [22] presented a further generalization. ###### Theorem 4.2 [22] Let $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. Then $({\rm tr}A)I_{mn}-({\rm tr}_{2}A)\otimes I_{n}\geq A-I_{m}\otimes({\rm tr}_{1}A),$ and $({\rm tr}A)I_{mn}+({\rm tr}_{2}A)\otimes I_{n}\geq A+I_{m}\otimes({\rm tr}_{1}A).$ A map (not necessarily linear) $\Phi:\mathbb{M}_{n}\to\mathbb{M}_{k}$ is called positive if it maps positive semidefinite matrices to positive semidefinite matrices. A map $\Phi:\mathbb{M}_{n}\to\mathbb{M}_{k}$ is said to be $m$-positive if for every $m\times m$ block matrix $[A_{i,j}]_{i,j=1}^{m}\in\mathbb{M}_{m}(\mathbb{M}_{n})$, $[A_{i,j}]_{i,j=1}^{m}\geq 0\Rightarrow[\Phi(A_{i,j})]_{i,j=1}^{m}\geq 0.$ Clearly, being $1$-positive is equivalent to being positive. The map $\Phi$ is said to be completely positive if it is $m$-positive for every integer $m\geq 1$. It is well-known that both the trace map and determinant map are completely positive; see, e.g., [36, p. 221, p. 237] or [37]. On the other hand, a map $\Phi$ is said to be $m$-copositive if for every $[A_{i,j}]_{i,j=1}^{m}\in\mathbb{M}_{m}(\mathbb{M}_{n})$, $[A_{i,j}]_{i,j=1}^{m}\geq 0\Rightarrow[\Phi(A_{j,i})]_{i,j=1}^{m}\geq 0,$ and $\Phi$ is said to be completely copositive if it is $m$-copositive for every integer $m\geq 1$. Furthermore, a map $\Phi$ is called completely PPT if it is both completely positive and completely copositive; see [26, 10, 39] for related topics. Both Theorem 4.1 and Theorem 4.2 illustrated the implicit interaction and connection between the first trace and second trace. The proof of Theorem 4.1 depends mainly on the 2-copositivity of $\Psi(X)=({\rm tr}X)I-X$; see e.g., [2] and [30] for more details. Correspondingly, the proof of Theorem 4.2 relies similarly on the 2-copositivity of $\Phi(X)=({\rm tr}X)I+X$; see [22]. For more application of these two maps, we refer readers to papers [26, 21]. In this section, we give a unified treatment of both Theorem 4.1 and Theorem 4.2. Our treatment is more concise than the original proof. We need to use a recent result of Choi [6, 7], which investigates more relations between the partial traces and the partial transpose. ###### Lemma 4.3 [6, 7] Let $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. Then $({\rm tr}_{2}A^{\tau})\otimes I_{n}\geq\pm A^{\tau}$, and $I_{m}\otimes{\rm tr}_{1}A^{\tau}\geq\pm A^{\tau}.$ Now, we present a unified treatment of Theorems 4.1 and 4.2 as well. New proof of Theorem 4.1. We define the map $\Phi:\mathbb{M}_{m}(\mathbb{M}_{n})\to\mathbb{M}_{m}(\mathbb{M}_{n})$ as $\Phi_{2}^{-}(X):=({\rm tr}_{2}X^{\tau})\otimes I_{n}-X^{\tau}.$ On the other hand, we define $\Phi_{1}^{-}(X):=I_{m}\otimes{\rm tr}_{1}X^{\tau}-X^{\tau}.$ Lemma 4.3 implies that both $\Phi_{2}^{-}$ and $\Phi_{1}^{-}$ are positive linear maps on $\mathbb{M}_{m}(\mathbb{M}_{n})$. Let $A$ be a positive semidefinite block matrix. Thus, we have $\Phi_{2}^{-}(A)=({\rm tr}_{2}A^{\tau})\otimes I_{n}-A^{\tau}\geq 0.$ Acting the map $\Phi_{1}^{-}$ to the matrix $\Phi_{2}^{-}(A)$, we can obtain $\Phi_{1}^{-}\bigl{(}\Phi_{2}^{-}(A)\bigr{)}=I_{m}\otimes{\rm tr}_{1}{\Phi_{2}^{-}(A)}^{\tau}-{\Phi_{2}^{-}(A)}^{\tau}\geq 0.$ (10) By a directed computation, we can get ${\Phi_{2}^{-}(A)}^{\tau}=({\rm tr}_{2}A)\otimes I_{n}-A$ and ${\rm tr}_{1}{\Phi_{2}^{-}(A)}^{\tau}={\rm tr}_{1}\bigl{(}({\rm tr}_{2}A)\otimes I_{n}-A\bigr{)}=\sum_{i=1}^{m}({\rm tr}A_{i,i})I_{n}-{\rm tr}_{1}A=({\rm tr}A)I_{n}-{\rm tr}_{1}A.$ Therefore, inequality (10) yields the desired result in Theorem 4.1. $\blacksquare$ Remarks. In the above proof, we can see that Theorem 4.1 is just a direct consequence of Lemma 4.3. To our surprise, Theorem 4.1 can also be proved by using the positivity of $\Phi_{1}^{-}$ first, and then applying the positivity of $\Phi_{2}^{-}$ later. More precisely, we first derive ${\Phi_{1}^{-}(A)}\geq 0$, and then we have $\Phi_{2}^{-}(\Phi_{1}^{-}(A))\geq 0$. Upon simplification, one can immediately get Theorem 4.1 again. We summarize this observation as the following proposition. ###### Proposition 4.4 For every $X\in\mathbb{M}_{m}(\mathbb{M}_{n})$, we have $\Phi_{1}^{-}(\Phi_{2}^{-}(X))=\Phi_{2}^{-}(\Phi_{1}^{-}(X))$. Correspondingly, we can present an alternative proof of Theorem 4.2 similarly. New proof of Theorem 4.2. We define the maps $\Phi_{2}^{+}$ and $\Phi_{1}^{+}$ on $\mathbb{M}_{m}(\mathbb{M}_{n})$ as $\Phi_{2}^{+}(X):=({\rm tr}_{2}X^{\tau})\otimes I_{n}+X^{\tau},$ and $\Phi_{1}^{+}(X):=I_{m}\otimes{\rm tr}_{1}X^{\tau}+X^{\tau}.$ We can see from Lemma 4.3 that both $\Phi_{2}^{+}$ and $\Phi_{1}^{+}$ are positive linear maps. Similar to the lines of the previous proof, we get $\Phi_{1}^{-}(\Phi_{2}^{+}(A))=\Phi_{2}^{+}(\Phi_{1}^{-}(A))\geq 0$, which leads to $({\rm tr}A)I_{mn}-({\rm tr}_{2}A)\otimes I_{n}\geq A-I_{m}\otimes({\rm tr}_{1}A).$ Moreover, we have $\Phi_{1}^{+}(\Phi_{2}^{-}(A))=\Phi_{2}^{-}(\Phi_{1}^{+}(A))\geq 0$. It follows that $({\rm tr}A)I_{mn}+({\rm tr}_{2}A)\otimes I_{n}\geq A+I_{m}\otimes({\rm tr}_{1}A).$ We mention that the positivity of $\Phi_{1}^{+}(\Phi_{2}^{+}(A))$ yields a trivial result. $\blacksquare$ In the remaining of this section, we shall pay attention to determinant inequalities of sector matrices involving partial traces. Motivated by Audenaert’s result (8), Lin [29] recently obtained a determinantal inequality for partial traces, which states that if $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ is positive semidefinite, then $({\rm tr}A)^{mn}+\det A\geq\det({\rm tr}_{1}A)^{m}+\det({\rm tr}_{2}A)^{n}.$ (11) We remark here that Fu, Lau and Tam [11, Corollary 2.2] recently improved (11) when $A$ is a density matrix, i.e., a positive semidefinite matrix with trace equal to $1$. The key step in the proof of (11) attributes to Theorem 4.1 together with the following interesting lemma. It is worth noting that Lemma 4.5 is graceful and useful in deriving matrix inequalities; see, e.g., [23, 24, 25] for applications on Oppenheim type inequalities. ###### Lemma 4.5 [30] Let $X,Y,W$ and $Z$ be positive semidefinite matrices of the same order. If $X\geq W,X\geq Z$ and $X+Y\geq W+Z$, then $\det X+\det Y\geq\det W+\det Z.$ Remark. We observe that Lemma 4.5 implies the determinant inequality: $\det(A+B+C)+\det C\geq\det(A+C)+\det(B+C),$ where $A,B$ and $C$ are positive semidefinite matrices. With the help of Lemma 4.5, we can easily present two analogues of (11). ###### Proposition 4.6 Let $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. Then $({\rm tr}A)^{mn}+\det({\rm tr}_{1}A)^{m}\geq\det A+\det({\rm tr}_{2}A)^{n},$ and $({\rm tr}A)^{mn}+\det({\rm tr}_{2}A)^{n}\geq\det A+\det({\rm tr}_{1}A)^{m}.$ Proof. We prove the first inequality only, since the second one can be proved in exactly the same way. Let $X=({\rm tr}A)I_{mn},Y=I_{m}\otimes({\rm tr}_{1}A),W=A$ and $Z=({\rm tr}_{2}A)\otimes I_{n}$. It is easy to see that $({\rm tr}A)I_{m}=\sum_{i=1}^{m}({\rm tr}A_{i,i})I_{m}=\bigl{(}{\rm tr}({\rm tr}_{2}A)\bigr{)}I_{m}\geq\lambda_{\max}({\rm tr}_{2}A)I_{m}\geq{\rm tr}_{2}A,$ which implies that $X\geq Z\geq 0$, and clearly $X\geq W\geq 0$. Moreover, Theorem 4.2 says that $X+Y\geq W+Z$. That is, all conditions in Lemma 4.5 are satisfied. Therefore, $\displaystyle({\rm tr}A)^{mn}+\det\bigl{(}I_{m}\otimes({\rm tr}_{1}A)\bigr{)}\geq\det A+\det\bigl{(}({\rm tr}_{2}A)\otimes I_{n}\bigr{)}.$ It is well-known [35, p. 37] that for every $X\in\mathbb{M}_{m}$ and $Y\in\mathbb{M}_{n}$, $\det(X\otimes Y)=(\det X)^{n}(\det Y)^{m}.$ Thus, we complete the proof of the required result. We next give an improvement on Proposition 4.6. ###### Theorem 4.7 Let $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. Then $({\rm tr}A)^{mn}+\det({\rm tr}_{1}A)^{m}\geq m^{nm}\bigl{(}\det A+\det({\rm tr}_{2}A)^{n}\bigr{)},$ and $({\rm tr}A)^{mn}+\det({\rm tr}_{2}A)^{n}\geq n^{mn}\bigl{(}\det A+\det({\rm tr}_{1}A)^{m}\bigr{)}.$ Proof. We only prove the second inequality. Invoking Theorem 3.3, we get $\left(\frac{\det({\rm tr}_{2}A)}{n^{m}}\right)^{n}\geq\det A.$ Equivalently, we have $\det({\rm tr}_{2}A)^{n}\geq n^{mn}\det A$. It suffices to show that $({\rm tr}A)^{n}\geq n^{n}\det({\rm tr}_{1}A).$ Note that ${\rm tr}A=\sum_{i=1}^{m}{\rm tr}(A_{i,i})={\rm tr}\left(\sum_{i=1}^{m}A_{i,i}\right)={\rm tr}({\rm tr}_{1}A).$ We denote $X:={\rm tr}_{1}A$, which is a positive semidefinite matrix of order $n$. So we need to prove that $({\rm tr}X)^{n}\geq n^{n}\det X$. This is equivalent to showing $\left(\sum_{i=1}^{n}\lambda_{i}(X)\right)^{n}\geq n^{n}\prod_{i=1}^{n}\lambda_{i}(X),$ which is a direct consequence of the AM-GM inequality. Surprisingly, the proof of Theorem 4.7 seems simpler than that of Proposition 4.6 since it does not rely on Theorem 4.2 and Lemma 4.5. However, it allows us to provide a great improvement on Proposition 4.6 whenever $m,n$ are large integers. In the sequel, we shall denote $|A|=(A^{*}A)^{1/2}$, which is called the modulus of $A$. We remark that this notation is different from that in Theorem 3.7. Note that $|A|$ is positive semidefinite, and the eigenvalues of $|A|$ are called the singular values of $A$. In 2019, Yang, Lu and Chen [34] extended (11) to sector matrices. $({\rm tr}|A|)^{mn}+\det|A|\geq(\cos\alpha)^{mn}|\det({\rm tr}_{1}A)|^{m}+(\cos\alpha)^{mn}|\det({\rm tr}_{2}A)|^{n}.$ Now, we are ready to present an extension on Theorem 4.7. ###### Theorem 4.8 Let $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be such that $W(A)\subseteq S_{\alpha}$. Then $({\rm tr}|A|)^{mn}+\left|{\det({\rm tr}_{1}A)}\right|^{m}\geq(m\cos\alpha)^{mn}\bigl{(}\det|A|+|\det({\rm tr}_{2}A)|^{n}\bigr{)},$ and $({\rm tr}|A|)^{mn}+\left|{\det({\rm tr}_{2}A)}\right|^{n}\geq(n\cos\alpha)^{mn}\bigl{(}\det|A|+|\det({\rm tr}_{1}A)|^{m}\bigr{)}.$ Proof. We only prove the first inequality. According to the definition of $S_{\alpha}$, if $W(A)\subseteq S_{\alpha}$, then $\Re A$ is positive definite and its trace is positive. By Lemma 2.2, we have ${\rm tr}|A|=\sum_{i=1}^{mn}\sigma_{i}(A)\geq\sum_{i=1}^{mn}\lambda_{i}(\Re A)={\rm tr}(\Re A)\geq 0.$ It is noteworthy by Lemma 2.4 that $W({\rm tr}_{1}A)\subseteq S_{\alpha}$ and $W({\rm tr}_{2}A)\subseteq S_{\alpha}$. Clearly, we have $\Re({\rm tr}_{1}A)={\rm tr}_{1}(\Re A)$ and $\Re({\rm tr}_{2}A)={\rm tr}_{2}(\Re A)$. By setting $X={\rm tr}_{1}A$ in Lemma 2.2, we get $|\det({\rm tr}_{1}A)|\geq\det\bigl{(}\Re({\rm tr}_{1}A)\bigr{)}=\det\bigl{(}{\rm tr}_{1}(\Re A)\bigr{)}.$ Note that $\Re A$ is positive semidefinite. By applying Theorem 4.7, we can obtain $\displaystyle({\rm tr}|A|)^{mn}+\left|{\det({\rm tr}_{1}A)}\right|^{m}$ $\displaystyle\geq({\rm tr}\,\Re A)^{mn}+\bigl{(}{\det{\rm tr}_{1}(\Re A)}\bigr{)}^{m}$ $\displaystyle\geq m^{nm}\bigl{(}\det(\Re A)+\bigl{(}\det\Re({\rm tr}_{2}A)\bigr{)}^{n}\bigr{)}$ $\displaystyle\geq(m\cos\alpha)^{mn}|\det A|+(m\cos\alpha)^{mn}|\det({\rm tr}_{2}A)|^{n},$ where the last inequality holds from Lemma 2.2 by setting $X=A$ and ${\rm tr}_{2}A$ respectively. ## 5 Trace inequalities for two by two block matrices Positive semidefinite $2\times 2$ block matrices are extensively studied, such a partition yields a great deal of versatile and elegant matrix inequalities; see, e.g., [13, 18, 21, 12] for details. Recently, Kittaneh and Lin [18] (or see [26]) proved the following trace inequalities. ###### Theorem 5.1 [18, 26] Let $\begin{bmatrix}A&B\\\ B^{*}&C\end{bmatrix}\in\mathbb{M}_{2}(\mathbb{M}_{k})$ be positive semidefinite. Then ${\rm tr}A\,{\rm tr}C-{\rm tr}B^{*}\,{\rm tr}B\geq\bigl{|}{\rm tr}AC-{\rm tr}B^{*}B\bigr{|},$ and ${\rm tr}A\,{\rm tr}C+{\rm tr}B^{*}\,{\rm tr}B\geq{\rm tr}AC+{\rm tr}B^{*}B.$ In this section, we present some inequalities related to trace for $2\times 2$ block matrices, which are slight extensions of the result of Kittaneh and Lin. We now need to introduce some notations. Let $\otimes^{r}A:=A\otimes\cdots\otimes A$ be the $r$-fold tensor power of $A$. ###### Theorem 5.2 Let $\begin{bmatrix}A&B\\\ B^{*}&C\end{bmatrix}\in\mathbb{M}_{2}(\mathbb{M}_{k})$ be positive semidefinite. Then for $r\in\mathbb{N}^{*}$, $(\mathrm{tr}A\,\mathrm{tr}C)^{r}-({\rm tr}B^{*}\,\mathrm{tr}B)^{r}\geq\bigl{|}(\mathrm{tr}AC)^{r}-(\mathrm{tr}B^{*}B)^{r}\bigr{|},$ and $(\mathrm{tr}A\,\mathrm{tr}C)^{r}+({\rm tr}B^{*}\,\mathrm{tr}B)^{r}\geq(\mathrm{tr}AC)^{r}+(\mathrm{tr}B^{*}B)^{r}.$ Proof. Note that $\begin{bmatrix}\\!\\!\\!\otimes^{r}A&\otimes^{r}B\\\ \otimes^{r}B^{*}&\otimes^{r}C\end{bmatrix}$ is a principal submatrix of $\otimes^{r}\begin{bmatrix}A&B\\\ B^{*}&C\end{bmatrix}$. Thus $\begin{bmatrix}\\!\\!\\!\otimes^{r}A&\otimes^{r}B\\\ \otimes^{r}B^{*}&\otimes^{r}C\end{bmatrix}$ is again positive semidefinite. By applying Theorem 5.1 to this block matrix, we get $\bigl{|}{\rm tr}(\otimes^{r}A)(\otimes^{r}C)-{\rm tr}(\otimes^{r}B^{*})(\otimes^{r}B)\bigr{|}\leq{\rm tr}\otimes^{r}\\!\\!A\,{\rm tr}\otimes^{r}\\!\\!C-{\rm tr}\otimes^{r}\\!B^{*}{\rm tr}\otimes^{r}\\!\\!B,$ and ${\rm tr}(\otimes^{r}A)(\otimes^{r}C)+{\rm tr}(\otimes^{r}B^{*})(\otimes^{r}B)\leq{\rm tr}\otimes^{r}\\!\\!A\,{\rm tr}\otimes^{r}\\!\\!C+{\rm tr}\otimes^{r}\\!B^{*}{\rm tr}\otimes^{r}\\!\\!B.$ Invoking the well-known facts [35, Chapter 2]: $(\otimes^{r}X)(\otimes^{r}Y)=\otimes^{r}(XY)$ and ${\rm tr}(\otimes^{r}X)=({\rm tr}\,X)^{r}$, the desired inequalities follow immediately. Remark. Theorem 5.2 was proved in the first version of our manuscript (announced on March 10, 2020, arXiv: 2003.04520v1). We remark that this result was recently and independently rediscovered by Fu and Gumus in [12] using a quite different method. Let $e_{t}(X)$ denote the $t$-th elementary symmetric function of the eigenvalues of the square matrix $X$. $e_{t}(X):=\sum_{1\leq i_{1}<i_{2}<\cdots<i_{t}\leq n}\prod_{j=1}^{t}\lambda_{i_{j}}(X).$ In particular, we know that $e_{1}(X)={\rm tr}(X)$. We can get the following theorem. ###### Theorem 5.3 Let $\begin{bmatrix}A&B\\\ B^{*}&C\end{bmatrix}\in\mathbb{M}_{2}(\mathbb{M}_{k})$ be positive semidefinite. Then for $t\in\\{1,2,\ldots,k\\}$, $e_{t}(A)e_{t}(C)-e_{t}(B^{*})e_{t}(B)\geq|e_{t}(AC)-e_{t}(B^{*}B)|,$ and $e_{t}(A)e_{t}(C)+e_{t}(B^{*})e_{t}(B)\geq e_{t}(AC)+e_{t}(B^{*}B).$ Proof. The first inequality can be found in [18, Corollary 2.7]. We next give the outline of the proof of the second one. Note that $\begin{bmatrix}\\!\\!\\!\otimes^{t}A&\otimes^{t}B\\\ \otimes^{t}B^{*}&\otimes^{t}C\end{bmatrix}$ is positive semidefinite. By restricting this block matrix to the symmetric class of tensor product (see, e.g., [3, pp. 16–20]), we know that $\begin{bmatrix}\\!\\!\\!\wedge^{t}A&\wedge^{t}B\\\ \wedge^{t}B^{*}&\wedge^{t}C\end{bmatrix}$ is still positive semidefinite. Note that $e_{t}(X)={\rm tr}(\wedge^{t}X)$ and $(\wedge^{t}X)(\wedge^{t}Y)=\wedge^{t}(XY)$. Applying Theorem 5.1 to this block matrix yields the required result. Let $s_{t}(X)$ be the $t$-th complete symmetric polynomial of eigenvalues of $X$, i.e., $s_{t}(X):=\sum_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{t}\leq n}\prod_{j=1}^{t}\lambda_{i_{j}}(X).$ Clearly, we have $s_{1}(X)={\rm tr}(X)$. We can get the following slight extension similarly. ###### Theorem 5.4 Let $\begin{bmatrix}A&B\\\ B^{*}&C\end{bmatrix}\in\mathbb{M}_{2}(\mathbb{M}_{k})$ be positive semidefinite. Then for $t\in\\{1,2,\ldots,k\\}$, $s_{t}(A)s_{t}(C)-s_{t}(B^{*})s_{t}(B)\geq|s_{t}(AC)-s_{t}(B^{*}B)|,$ and $s_{t}(A)s_{t}(C)+s_{t}(B^{*})s_{t}(B)\geq s_{t}(AC)+s_{t}(B^{*}B).$ Proof. Note that $\begin{bmatrix}\\!\\!\\!\otimes^{t}A&\otimes^{t}B\\\ \otimes^{t}B^{*}&\otimes^{t}C\end{bmatrix}$ is positive semidefinite. By restricting this block matrix to the symmetric class of tensor product (see, e.g., [3, pp. 16–20]), we know that $\begin{bmatrix}\\!\\!\\!\vee^{t}A&\vee^{t}B\\\ \vee^{t}B^{*}&\vee^{t}C\end{bmatrix}$ is still positive semidefinite. Similarly, we know that ${\rm tr}(\vee^{t}X)=s_{t}(X)$ and $(\vee^{t}X)(\vee^{t}Y)=\vee^{t}(XY)$. Applying Theorem 5.1 to this block matrix leads to the desired result. ## Acknowledgments This paper is dedicated to Prof. Weijun Liu (Central South University) on his 60th birthday, October 22 of the lunar calendar in 2021. I would like to thank Prof. Yuejian Peng for reading carefully through an earlier version of this paper. 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2024-09-04T02:54:58.633886

2003.04628

arxiv-papers

# Large-Scale Evaluation of Keyphrase Extraction Models Ygor Gallina<EMAIL_ADDRESS>LS2N, Université de NantesNantesFrance , Florian Boudin<EMAIL_ADDRESS>LS2N, Université de NantesNantesFrance and Béatrice Daille beatrice.daille@univ- nantes.fr LS2N, Université de NantesNantesFrance (2020) ###### Abstract. Keyphrase extraction models are usually evaluated under different, not directly comparable, experimental setups. As a result, it remains unclear how well proposed models actually perform, and how they compare to each other. In this work, we address this issue by presenting a systematic large-scale analysis of state-of-the-art keyphrase extraction models involving multiple benchmark datasets from various sources and domains. Our main results reveal that state-of-the-art models are in fact still challenged by simple baselines on some datasets. We also present new insights about the impact of using author- or reader-assigned keyphrases as a proxy for gold standard, and give recommendations for strong baselines and reliable benchmark datasets. Keyphrase generation, natural language processing, evaluation ††copyright: acmcopyright††journalyear: 2020††doi: 10.1145/1122445.1122456††conference: JCDL ’20: ACM/IEEE Joint Conference on Digital Libraries; August 01–05, 2020; Xi’an, Shaanxi, China††booktitle: JCDL ’20: ACM/IEEE Joint Conference on Digital Libraries, August 01–05, 2020, Xi’an, Shaanxi, China††isbn: 978-1-4503-XXXX-X/20/06††ccs: Information systems Digital libraries and archives††ccs: Information systems Information retrieval††ccs: Computing methodologies Information extraction ## 1\. Introduction Keyphrases are single or multi-word lexical units that represent the main concepts in a document (Evans and Zhai, 1996). They are particularly useful for indexing, searching and browsing digital libraries (Barker et al., 1972; Zhai, 1997; Gutwin et al., 1999; Witten et al., 2009), and have proven themselves as effective features in many downstream natural language processing tasks (Hulth and Megyesi, 2006; Litvak and Last, 2008; Berend, 2011). Still, most documents do not have assigned keyphrases, and manual annotation is simply not a feasible option (Mao and Lu, 2017). There is therefore a great need for automated methods to assign relevant keyphrases to documents. Automatic keyphrase extraction111Also referred to as keyphrase generation or keyphrase annotation. – that is, the task of extracting keyphrases either from the content of the document or from a controlled vocabulary – has received much attention from the research community (Kim et al., 2010; Gollapalli et al., 2015; Augenstein et al., 2017). Thus, many keyphrase extraction models were proposed over the last years, ranging from early statistics-based models (Witten et al., 1999), to popular graph-based ranking models (Mihalcea and Tarau, 2004), and recent neural models (Meng et al., 2017). However, because of the great discrepancies in experimental setups among past studies, it is very difficult to compare and contrast the effectiveness of these models, and even more so to assess the progress of the field as a whole. More specifically, we observe striking differences in how models are parameterized, evaluated and compared in previous work. To name just a few examples, experiments are most often conducted on different benchmark datasets, all of which differ in domain, size, language or quality of the gold standard (that is, reference keyphrases supplied by authors, readers or professional indexers). This not only makes the reported results hard to contrast, but also has a profound impact on trained model performance (Gallina et al., 2019). In addition, and since there is no consensus as to which evaluation metric is most reliable for keyphrase extraction (Zesch and Gurevych, 2009; Hussey et al., 2012; Hasan and Ng, 2014), diverse measures are commonly seen in the literature, thus preventing any further direct comparisons. Moreover, the evaluation of missing keyphrases – that is, gold keyphrases that do not occur in the content of the document – is still an open question and there is little agreement on whether they should be included or not (Kim et al., 2010). We strongly believe that this lack of empirical rigor is a real hindrance to progress on keyphrase extraction, and that a systematic comparison of existing models under the same conditions is needed to fully understand how they actually perform. In this work, we resolve this issue by conducting the first large-scale study on automatic keyphrase extraction. More precisely, we present an extensive comparative analysis of state-of-the-art keyphrase extraction models involving 9 benchmark datasets from various domains. To ensure controlled, fair and reliable experiments, we embarked upon the difficult process of re-implementing all of the models presented in this paper222Link to the code will appear here after the review period. and pre- processing the datasets in a unified and systematic way333Link to the datasets will appear here after the review period.. Using these new large-scale experimental results, we seek to better understand how well state-of-the-art models perform across sources, domains and languages. We also go further than prior work and investigate the following research questions: 1. (1) How much progress have we made on keyphrase extraction since early models? 2. (2) What is the impact of using non-expert gold standards, that is, author- or reader-assigned keyphrases, when training and evaluating keyphrase extraction models? 3. (3) Which baselines and benchmark datasets should be included in future work for a better understanding of the pros and cons of a newly proposed model? ## 2\. Benchmark Datasets Benchmark datasets for evaluating automatic keyphrase extraction cover a wide range of sources ranging from scientific articles and web pages to twitter and email messages. We collected 9 of the most widely used datasets which we believe are representative of the different sources and domains found in previous work. Detailed statistics for each selected dataset are shown in Table 2. They are grouped into three categories that are outlined below: Scientific articles: Among the selected datasets, three are composed of full-text scientific publications: ACM (Krapivin et al., 2009) and SemEval (Kim et al., 2010) about computer science, and PubMed (Schutz, 2008) from the medical domain. Not surprisingly, they contain only a small number of documents due to copyright reasons. These datasets provide author-assigned keyphrases which serve as a reasonable, but far from perfect, proxy for expert annotations. In the case of SemEval, student annotators were hired to extend gold annotation labels. Paper abstracts: Scientific abstracts, often referred to as bibliographic records, are arguably the most prevalent documents for benchmarking keyphrase extraction. They are readily available in great quantities and come with author-assigned keyphrases that can be used as gold standard. We gathered three datasets, all dealing with the computer science domain: Inspec (Hulth, 2003), WWW (Caragea et al., 2014) and KP20k (Meng et al., 2017). It is worth noting that with more than half a million documents, KP20k is the largest dataset to date and one of the few that is large enough to train neural models. News articles: News texts are the last source of documents present among the collected datasets. Similar to paper abstracts, online news are available in large quantities and can be easily mined from the internet. We selected the following three datasets: DUC-2001 (Wan and Xiao, 2008), 500N-KPCrowd (Marujo et al., 2012) and KPTimes (Gallina et al., 2019). The first two datasets provide reader-assigned keyphrases, while KPTimes supplies indexer-assigned key-phrases extracted from metadata and initially intended for search engines. It is interesting to observe that only two datasets in our study, namely Inspec and KPTimes, provide gold keyphrases annotated by professional indexers. Dataset | Ann. | Train | Test | #words | #kp | %abs ---|---|---|---|---|---|--- 5pt. PubMed (Schutz, 2008) | $A$ | - | 1 320 | 5 323 | 5.4 | 16.9 ACM (Krapivin et al., 2009) | $A$ | - | 2 304 | 9 198 | 5.3 | 16.3 SemEval (Kim et al., 2010) | $A\cup R$ | 144 | 100 | 7 961 | 14.7 | 19.7 Scientific articles (avg.) | 7 494 | 8.5 | 17.6 5pt. Inspec (Hulth, 2003) | $I$ | 1 000 | 500 | 135 | 9.8 | 22.4 WWW (Caragea et al., 2014) | $A$ | - | 1 330 | 164 | 4.8 | 52.0 KP20k (Meng et al., 2017) | $A$ | 530K | 20K | 176 | 5.3 | 42.6 Paper abstracts (avg.) | 158 | 6.6 | 39.0 5pt. DUC-2001 (Wan and Xiao, 2008) | $R$ | - | 308 | 847 | 8.1 | 3.7 KPCrowd (Marujo et al., 2012) | $R$ | 450 | 50 | 465 | 46.2 | 11.2 KPTimes (Gallina et al., 2019) | $I$ | 260K | 10K | 921 | 5.0 | 54.7 News articles (avg.) | 744 | 19.8 | 23.2 Table 1. Statistics of the datasets. Gold annotation is supplied by authors ($A$), readers ($R$) or professional indexers ($I$). The number of documents in the training and testing splits are shown. The average number of keyphrases (#kp) and words (#words) per document, and the ratio of missing keyphrases (%abs) are computed on the test set. Datasets containing scientific articles or abstracts rely primarily on author- assigned keyphrases as gold standard. They therefore exhibit similar properties for the average number of ground truth keyphrases per document ($\approx 5$). On the other hand, articles are on average significantly longer than abstracts ($\approx 7500$ words vs. $\approx 160$ words respectively) and consequently reveal a much smaller fraction of missing keyphrases ($\approx 18\%$ vs. $\approx 39\%$ respectively). Datasets with reader-assigned keyphrases exhibit the lowest numbers of missing keyphrases, which can be explained by the fact that readers appear to produce gold-standard annotations in an extractive fashion (Wang et al., 2015). We also confirmed this empirically by computing the ratio of missing keyphrases in the author- assigned ($24\%$) and reader-assigned ($17.5\%$) gold annotations of the SemEval dataset. In contrast, the opposite trend is observed for KPTimes that comes with gold standards annotated by professional indexers and that shows the highest percentage of missing keyphrases ($54.7\%$). This indicates the the more abstractive nature of indexer-assigned keyphrases. Put differently, it is known that non-expert annotations are less constrained and may include seldom- used variants or misspellings (Sood et al., 2007), whereas indexers strive to rely on a consistent terminology and assign the same keyphrase to all documents for a given topic, even when it does not occur in these documents. To investigate this further, we looked at how many variants of an index term, in this case “artificial neural network”, could be found in the author- assigned keyphrases of KP20k. All in all, we found dozens of variants for this term, including “neural network”, “neural network (nns)”, “neural net”, “artificial neural net” or “nn”. This apparent lack of annotation consistency intuitively has two consequences: 1) it makes it harder for supervised approaches to learn a good model, 2) it makes automatic evaluation much less reliable as it is based on exact string matching. It is important to stress that datasets containing scientific articles may contain noisy texts. Indeed, most articles were automatically converted from PDF format to plain text and thus are likely to contain irrelevant pieces of text (e.g. muddled sentences, equations). Previous work show that noisy inputs undermine the overall performance of keyphrase extraction models (Boudin et al., 2016). In this study, we do not insist on a perfect input and we are aware that reported results may be improved with an increase in pre-processing effort. ## 3\. Models Roughly speaking, previous works on keyphrase extraction can be divided into two groups depending on whether they adopt a supervised learning procedure or not. This section starts by introducing the baselines we will use in our experiments, and then proceeds to describe the state-of-the-art keyphrase extraction models we re-implemented sorted into the aforementioned two groups. ### 3.1. Baselines Having strong baselines to compare with is a prerequisite for contrasting the results of proposed models. In previous studies, various baselines were considered, complicating the analysis and interpretation of the reported results. Our stance here is to establish three baselines, each associated with a particular feature that is commonly used in keyphrase extraction models. All baselines are also unsupervised, allowing their use and performance analysis on any of the benchmark datasets Keyphrase position is a strong signal for both unsupervised and supervised models, simply because texts are usually written so that the most important ideas go first (Marcu, 1997). In single document summarization for example, the lead baseline –that is, the first sentences from the document–, while incredibly simple, is still a competitive baseline (Kedzie et al., 2018). Similar to the lead baseline, we propose the FirstPhrases baseline that extracts the first $N$ keyphrase candidates from a document. We are not aware of any previous work reporting that baseline, yet, as we will see in §5, it achieves remarkably good results. Graph-based ranking models for keyphrase extraction are, perhaps, the most popular models in the literature. Therefore, as a second baseline, we use TextRank (Mihalcea and Tarau, 2004), which weights keyphrase candidates using a random walk over a word-graph representation of the document. In a nutshell, TextRank defines the importance of a word in terms of how it relates to other words in the document, and ranks candidates according to the words they contain. The third baseline, TF$\times$IDF (Salton and Buckley, 1988), have been repeatedly used in previous comparative studies (Kim et al., 2010; Meng et al., 2017, inter alia). In contrast with the other two baselines that do no require any resources whatsoever (beyond the document itself), TF$\times$IDF makes use of the statistics collected from unlabelled data to weight keyphrase candidates. As such, it often gives better results, in some cases even on par with state-of-the-art models (Ye and Wang, 2018). ### 3.2. Unsupervised models Annotated data are not always available or easy to obtain, which motivates the further development of unsupervised models for keyphrase extraction. Besides, looking back at previous work, most attempts to address this problem employ unsupervised approaches. In this study, we selected three recent state-of-the- art models based on their reported performance. The first model we investigate is PositionRank (Florescu and Caragea, 2017), a graph-based model that incorporates two features (position and frequency) into a biased PageRank algorithm. This model operates at the word level, and assigns a score to each candidate using the sum of its individual word scores. As such, it suffers from over-generation errors444These errors occur when a model correctly outputs a keyphrase because it contains an important word, but at the same time erroneously predicts other keyphrases because they contain the same word. (Hasan and Ng, 2014), but still achieves good performance on short texts. The second model we consider, MPRank (Boudin, 2018), relies on a multipartite graph representation to enforce topical diversity while ranking keyphrase candidates. It includes a mechanism to incorporate keyphrase selection preferences in order to introduce a bias towards candidates occurring first in the document. MultipartiteRank was shown to consistently outperform other unsupervised graph-based ranking models. Both aforementioned models only exploit the document itself to extract keyphrases. The third model we include, EmbedRank (Bennani-Smires et al., 2018), leverages sentence embeddings for ranking keyphrase candidates. Candidates are weighted according to their cosine distance to the document embedding, while diversity in the selected keyphrases is promoted using Maximal Marginal Relevance (MMR) (Goldstein and Carbonell, 1998). Despite its simplicity, this model was shown to outperform other unsupervised models on short texts (abstracts and news). ### 3.3. Supervised models Supervised models can be further divided into two categories, depending on whether they rely on a neural network or not. Traditional supervised models treat the keyphrase extraction problem as a binary classification task. Here, we include such a model, namely Kea (Witten et al., 1999), in order to precisely quantify the performance gap with recent neural-based models. KEA uses a Naive Bayes classifier trained on a set of only two handcrafted features we have elected as baseline features: the TF$\times$IDF score of the candidate and the normalized position of its first occurrence in the document. Previous work has reported confusing and conflicting results555On SemEval, (Meng et al., 2017) report an F@10 score of $2.6$ while (Boudin, 2016) report a score of $19.3$. for Kea, raising questions about how it actually performs. Neural models for keyphrase extraction rely on an encoder-decoder architecture (Cho et al., 2014; Sutskever et al., 2014) with an attention mechanism (Bahdanau et al., 2014; Luong et al., 2015). Training these models require large amounts of annotated training data, and is therefore only possible on the KP20k and KPTimes datasets. The second supervised model we include in this study is CopyRNN (Meng et al., 2017), an encoder-decoder model that incorporates a copying mechanism (Gu et al., 2016) in order to be able to predict phrases that rarely occur. When properly trained, this model was shown to be very effective in extracting keyphrases from scientific abstracts. The third supervised model we use, CorrRNN (Chen et al., 2018), extends the aforementioned model by introducing correlation constraints. It employs a coverage mechanism (Tu et al., 2016) that diversifies attention distributions to increase topic coverage, and a review mechanism to avoid generating duplicates. As such, it produces more diverse and less redundant keyphrases. Note that only neural models have the ability to generate missing keyphrases, which in theory gives them a clear advantage over the other models. ## 4\. Experimental settings In addition to the variation in the choice of benchmark datasets and baselines, there are also major discrepancies in parameter settings and evaluation metrics between previous studies. For example, there is no point in contrasting the results in (Meng et al., 2017), (Florescu and Caragea, 2017) and (Teneva and Cheng, 2017), three papers about keyphrase extraction published in the same year at ACL, since neither benchmark datasets, parameter settings nor evaluation metrics are comparable. To address this problem, we use the same pre-processing tools, parameter settings and evaluation procedure across all our experiments. | Scientific articles | Paper abstracts | News articles ---|---|---|--- | PubMed | ACM | SemEval | Inspec | WWW | KP20k | DUC-2001 | KPCrowd | KPTimes Model | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP | $\text{F}@10$ | MAP FirstPhrases | 15.4 | 14.7 | 13.6 | 13.5 | 13.8 | 10.5 | 29.3 | 27.9 | 10.2 | 09.8 | 13.5 | 12.6 | 24.6 | 22.3 | 17.1 | 16.5 | 09.2 | 08.4 TextRank | 01.8 | 01.8 | 02.5 | 02.4 | 03.5 | 02.3 | 35.8 | 31.4 | 08.4 | 05.6 | 10.2 | 07.4 | 21.5 | 19.4 | 07.1 | 09.5 | 02.7 | 02.5 TF$\times$IDF | 16.7 | 16.9 | 12.1 | 11.4 | 17.7 | 12.7 | 36.5 | 34.4 | 09.3 | 10.1 | 11.6 | 12.3 | 23.3 | 21.6 | 16.9 | 15.8 | 09.6 | 09.4 PositionRank | 04.9 | 04.6 | 05.7 | 04.9 | 06.8 | 04.1 | 34.2 | 32.2 | 11.6† | 08.4 | 14.1† | 11.2 | 28.6† | 28.0† | 13.4 | 12.7 | 08.5 | 06.6 MPRank | 15.8 | 15.0 | 11.6 | 11.0 | 14.3 | 10.6 | 30.5 | 29.0 | 10.8† | 10.4 | 13.6† | 13.3† | 25.6 | 24.9† | 18.2 | 17.0 | 11.2† | 10.1† EmbedRank | 03.7 | 03.2 | 02.1 | 02.1 | 02.5 | 02.0 | 35.6 | 32.5 | 10.7† | 07.7 | 12.4 | 10.0 | 29.5† | 27.5† | 12.4 | 12.4 | 04.0 | 03.3 Kea | 18.6† | 18.6† | 14.2† | 13.3 | 19.5† | 14.7† | 34.5 | 33.2 | 11.0† | 10.9† | 14.0† | 13.8† | 26.5† | 24.5† | 17.3 | 16.7 | 11.0† | 10.8† CopyRNN | 24.2† | 25.4† | 24.4† | 26.3† | 20.3† | 13.8 | 28.2 | 26.4 | 22.2† | 24.9† | 25.4† | 28.7† | 10.5 | 07.2 | 08.4 | 04.2 | 39.3† | 50.9† CorrRNN | 20.8† | 19.4† | 21.1† | 20.5† | 19.4 | 10.9 | 27.9 | 23.6 | 19.9† | 20.3† | 21.8† | 22.7 | 10.5 | 06.5 | 07.8 | 03.2 | 20.5† | 20.3† Table 2. Performance of keyphrase extraction models. † indicates significance over the baselines. ### 4.1. Parameter settings We pre-process all the texts using the Stanford CoreNLP suite (Manning et al., 2014) for tokenization, sentence splitting and part-of-speech (POS) tagging. All non-neural models operate on a set of keyphrase candidates, extracted from the input document. Selecting appropriate candidates is particularly important since it determines the upper bound on recall, and the amount of irrelevant candidates that models will have to deal with. For a fair and meaningful comparison, we use the same candidate selection heuristic across models. We follow the recommendation by Wang et al. (2014) and select the sequences of adjacent nouns with one or more preceding adjectives of length up to five words. Candidates are further filtered by removing those shorter than 3 characters or containing non-alphanumeric symbols. We implemented the neural models in PyTorch (Paszke et al., 2017) using AllenNLP (Gardner et al., 2018), and the non-neural models using the pke toolkit (Boudin, 2016). As neural models require large amounts of annotated data to be trained, we trained our models on the KP20k dataset for both scientific papers and abstracts, and on KPTimes for news texts. We compute Document Frequency (DF) counts and learn Kea models on training sets. For datasets without training splits, we apply a leave-one-out cross-validation procedure on the test sets for calculating DF counts and training models. We use the optimal parameters suggested by the authors for each model, and leverage pre-trained sentence embeddings666https://github.com/epfml/sent2vec for EmbedRank. We also found out that the training set of KP20k contains a non-negligible number of documents from the test sets of other datasets. We removed those documents prior to training. ### 4.2. Evaluation metrics Although there is no consensus as to which metric is the most reliable for keyphrase extraction, a popular evaluation strategy is to compare the top $k$ extracted keyphrases against the gold standard. We adopt this strategy and report the f-measure at the top 10 extracted keyphrases. In previous work, we often see differences in how gold standards are handled during evaluation. For example, some studies evaluate their models on the present and missing portions of the gold standard separately (Meng et al., 2017; Ye and Wang, 2018; Chen et al., 2018, inter alia), whereas other work use the entire gold standard (Florescu and Caragea, 2017; Boudin, 2018, inter alia). We chose the latter because recent models, in addition to extracting keyphrases from the content of the document, are able to generate missing keyphrases. Following common practice, gold standard and output keyphrases are stemmed to reduce the number of mismatches. One issue with the f-measure is that the ranks of the correct keyphrases are not taken into account. To evaluate the overall ranking performance of the models, we also report the Mean Average Precision (MAP) scores of the ranked lists of keyphrases. We use the Student’s paired t-test to assess statistical significance at the $0.05$ level. ### 4.3. Replicability of results In Table 3, we compare the results of our re-implementations against those reported in the original papers. We note that all models show comparable results. We observe the largest differences with original scores for CopyRNN ($+2$) and CorrRNN ($-4.3$) that can be easily explained by minor differences in training parameters. Model | Dataset (metric) | Orig. | Ours ---|---|---|--- PositionRank | WWW (F$@$8) | 12.3 | 11.7 MPRank | SemEval-2010 (F$@$10) | 14.5 | 14.3 EmbedRank | Inspec (F$@$10) | 37.1 | 35.6 CopyRNN | KP20k (F$@$10 on present) | 26.2 | 28.2 CorrRNN | ACM (F$@$10 on present) | 27.8 | 23.5 Table 3. Original vs. re-implementation scores. ## 5\. Results Results are presented in Table 2. First of all, we notice that no model significantly outperforms the baselines on all datasets. This is rather surprising, as one would expect that neural models would be consistently better than a simple TF$\times$IDF model for example. Rather, we see that the TF$\times$IDF baseline is very competitive on long documents, while the FirstPhrases baseline performs remarkably well, especially on news texts. Still, overall, CopyRNN achieves the best performance with, in the case of KPTimes, MAP scores exceeding 50%. When we look at only unsupervised models, MPRank achieves the best results across datasets. Also, it comes as no surprise that Kea exhibits strong performance across datasets because it combines two effective features, as demonstrated by the results of the TF$\times$IDF and FirstPhrases baselines. Conversely, despite the addition of mechanisms for promoting diversity in the output, CorrRNN is almost always outperformed by CopyRNN, suggesting that the added correlation constraints are not effective at filtering out spurious keyphrases. In light of the above, we can now answer the following question: “How much progress have we made since early models?”. It is clear that neural-based models are the new state-of-the-art for keyphrase extraction, achieving F@10 scores up to three times that of previous models. That being said, CopyRNN, which is the best overall model, fails to consistently outperform the baselines on all datasets. One reason for that is the limited generalization ability of neural-based models (Meng et al., 2017; Chen et al., 2018; Gallina et al., 2019), which means that their performance degrades on documents that differ from the ones encountered during training. This is besides confirmed by the extremely low performance of these models on DUC-2001 and KPCrowd. Much more work needs to be done in tackling this issue if neural models are to substitute for older supervised models. Perhaps most disappointing is the fact that state-of-the-art unsupervised models are still challenged by the TF$\times$IDF baseline. Here, we suspect the reasons are twofold. First, the models we have investigated do not use in-domain data which may not only limit their performance, but also, as in the case of EmbedRank that uses out-of- domain (Wikipedia) data, be detrimental to their performance. Second, unlike neural generative models, they are not able to produce keyphrases that do not occur in the source document, further limiting their potential effectiveness. As outlined in §2, gold standards provided by lay annotators, such as authors and readers, exhibit strong inconsistency issues. One might therefore wonder “What is the impact of non-expert annotations on training and evaluating keyphrase extraction models?”. Intuitively, models evaluated against these annotations are likely to receive lower scores because they make training more difficult (that is, assigning different keyphrases to documents about the same topic may confuse the model) while increasing the number of false negatives during evaluation. This is exactly what we observe in Table 2 where the best scores for Inspec and KPTimes, whose gold standards are provided by professional indexers, are higher in magnitude than those of the other datasets. Precisely quantifying how much impact lay annotations have on performance is no easy task as it implies a double-annotation process by both expert and non-expert annotators. Luckily enough, a small sample of documents from Inspec are also found in KP20k, allowing us to compare the performance of keyphrases models between both annotation types. Results are shown in Table 4. First, we see that overall performance is nearly cut in half when evaluating against author-provided gold standard, suggesting that reported scores in previous studies are arguably underestimated. Second, neural models again do not show their superiority against indexer-assigned keyphrases, which advocates the need for more experiments on datasets that include expert annotations. | $\text{F}@10$ | MAP ---|---|--- Model | I | A | I | A FirstPhrases | 25.8 | 13.7 | 26.1 | 13.2 TextRank | 33.4 | 12.2 | 29.6 | 09.3 TF$\times$IDF | 34.6 | 14.2 | 33.3 | 16.1 PositionRank | 32.9 | 15.9 | 31.0 | 13.0 MPRank | 26.4 | 13.8 | 27.6 | 13.6 EmbedRank | 34.3 | 15.3 | 31.3 | 11.5 Kea | 32.5 | 15.2 | 31.9 | 15.9 CopyRNN | 33.7 | 28.9‡ | 29.8 | 33.8‡ CorrRNN | 28.6 | 25.3 | 24.2 | 28.2 Avg. | 31.3 | 17.2 | 29.4 | 17.2 Table 4. Results on a subset of 55 documents from Inspec for indexer (I) and author (A) gold annotations. ‡ indicates significance over every other model. Figure 1. Average number of keyphrases in common between model outputs. The third question we want to address in this study is “Which baselines and benchmark datasets should be included in future work for a better understanding of the pros and cons of a newly proposed model?”. Having strong baselines to compare with is of utmost importance, and our results give an indication of which model is relevant. When properly trained, neural models drastically outperform all other models and represent the state-of-the-art. Since CopyRNN achieve the best results, it should be included in future work for comparison. In an unsupervised setting, or in a data-sparse scenario where neural models can not be applied, the picture is less clear. To help us understand which model is worth investigating, we conducted an additional set of experiments aimed at comparing the outputs from all models in a pairwise manner. The motivation behind these experiments is that including multiple models that behave similarly is of limited interest. Similarities between model outputs, viewed in terms of the number of keyphrases in common, are graphed as a heatmap in Figure 1. Overall, we observe different patterns for each source of documents. The shorter the document is, the more similar outputs are, which is mostly due to a smaller search space (that is, a smaller number of keyphrase candidates). We note that the three best unsupervised models, namely FirstPhrases, MPRank and TF$\times$IDF, generate very similar keyphrases (up to 42% identical). Considering this, and given their reported performances (Table 2), we argue that TF$\times$IDF (or KEA if seed training data is available) should be considered as strong unsupervised baseline in subsequent work. These recommendations of baselines also affect the choice of which benchmark datasets one has to use. As neural models are data-hungry, KP20k and KPTimes are the default options for paper abstracts and news articles. For scientific articles, we recommend using SemEval for two reasons: 1) it is widely used by existing studies; and 2) it provides a double-annotated gold standard (author- and reader-assigned keyphrases) that alleviates annotation inconsistencies to some extent. Our experiments highlight several issues in evaluating keyphrase extraction models with existing benchmark datasets. Another way of assessing the effectiveness of these models would be to explore their impact on other tasks as an extrinsic evaluation. To the best of our knowledge, there is no previously published research on that matter despite many downstream tasks that already benefit from keyphrase information such as article recommendation (Collins and Beel, 2019) or browsing interfaces (Gutwin et al., 1999) in digital libraries. This points to an interesting future direction that allows for a deeper understanding of the limitations of current models. ## 6\. Conclusion This paper presents a large scale evaluation of keyphrase extraction models conducted on multiple benchmark datasets from different sources and domains. Results indicate that keyphrase extraction is still an open research question, with state-of-the-art neural-based models still challenged by simple baselines on some datasets. We hope that this work will serve as a point of departure for more rigorous analysis and evaluation of proposed keyphrase extraction models. We provide all the code and data on a public repository777Link to the repository will appear here after the review period., as well as a public leaderboard to facilitate the comparison between models. ## References * (1) * Augenstein et al. (2017) Isabelle Augenstein, Mrinal Das, Sebastian Riedel, Lakshmi Vikraman, and Andrew McCallum. 2017\. 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2024-09-04T02:54:58.652265

2003.04654

arxiv-papers

Parton models and frame independence of high-energy cross-sections. O.V. Kancheli 111 Email<EMAIL_ADDRESS> Institute for Theoretical and Experimental Physics, 117218, Moscow, Russia ###### Abstract We describe some ambiguities which take place when on calculates the cross- sections in parton models at high energies and the connected limitations on the asymptotic of high energy amplitudes that follows from the conditions of boost-invariance of cross-sections. It turns out that the resulting constraints are of the same type as the following from the t-channel unitarity conditions. So that on can suppose that this similarity, by their nature, has much more general grounds. ## 1 Introduction There are two main theoretical approaches to a study of the behavior of high energy amplitudes and cross-sections. In one approach, we directly calculate the amplitudes by summing the contributions of the Feynman diagrams of the corresponding field theory or use some effective theory like reggeon diagrams or various string-like dual models. In the other - parton like approach to high-energy collisions 222 A few useful reviews ( [1]\- [7]) we usually consider separately three main stages of the system evolution in the process of particles collision. Firstly, one constructs the quantum states $\Psi(P)$ of high energy particle with momentum $P\gg m$ in terms of superposition $\Psi(P)~{}=~{}\sum_{n}\int_{\\{k_{i}\\}}~{}f_{n}(P,\\{k_{i}\\})~{}|n,\\{k_{i}\\}>$ (1) of the n-particle states $|n,\\{k_{i}\\}>$ of some “primary” constituents - partons with 3-momenta $\\{k_{i}\\}$. The “choice” of these partons is not unique, and partons can be bare point like particles, particles with varying virtuality, QCD color dipoles, fast string configurations, distributions of the Coulomb-like fields, etc. The state $\Psi(\vec{P})$ must fulfill the Schroedinger equation $\hat{H}~{}\Psi(\vec{P})=\sqrt{\vec{P}^{2}+m^{2}}~{}~{}\Psi(\vec{P})~{},$ where the Hamiltonian $\hat{H}$ is the function of parton fields, so that $\Psi(\vec{P})$ is the eigenfunction of $\hat{H}$ with eigenvalues defining the particles physical mass $m$. After that one can use such a state $\Psi(P_{1})$ to calculate its interaction with some low energy target or with other fast particle in the state $\Psi(P_{2})$ in terms of “simple” amplitudes of parton interaction. There is also the third stage corresponding to an evolution in the final state when moving away partons transform and combine into physical particles (hadronization…). But often this stage is not very restrictive, especially when we calculate various integrated cross-sections. And we will not consider it in this article. It is essential that with the energy growing in most parton descriptions the structure of the parton state becomes more and more complicated for all the theories containing vector (like QCD) and tensor fields (gravity) and mean parton number in states $\Psi(P)$and the average transverse size of the region they occupy grow with $P$. When we consider the collision of two fast particles in the parton states $\Psi(P_{1})$ and $\Psi(P_{2})$ at some large $s=(P_{1}+P_{2})^{2}\gg m^{2}$ we can choose for this any longitudinal Lorentz system. But the resulting values of cross-sections of various processes must not depend from this choice of frame. And this is nontrivial condition in parton approach, because in different longitudinal systems (that is for various $P_{1}$ and $P_{2}$ at the same value of $s$) the different parton configurations firstly meet one another at the moment of particles collision. And, moreover, by choosing a different system we also can move the dynamics, from stage one to two and vice versa. If we make all calculation precisely - with hermitian Hamiltonian we probably can be sure that all restrictions coming from Lorentz-invariance and the unitarity conditions will be satisfied. But if we make some approximations, especially dictated by phenomenological or pictorial arguments, the unitarity conditions itself can probably be the only general way to check that the results are not contradictory. Various restrictions from the t-channel unitarity are very essential for the amplitudes describing high energy hadron interactions, and they are directly taken into account in reggeon amplitudes [10]. But in parton approaches it is not evident how to take them into account. In the reggeon field theory and in the dual (string) models the t-unitarity conditions are automatically fulfilled. But at high reggeon (pomeron) density such un approach can become unreliable. The parton approach has no problems with high parton density, but here there is no direct way how to control possible restrictions coming from the t-unitarity. One can hope that the longitudinal Lorentz (boost) invariance of all cross- sections calculated in a parton approach is in some sense equivalent to the mean form of the t-unitarity for multiparticle amplitudes. So, if we calculate any cross-section using the partonic wave functions $\Psi(P_{a})$ and $\Psi(P_{b})$ of fast colliding hadrons with momenta $P_{a},$ $P_{b}$ then we expect that this cross-section must be the same in all longitudinal Lorentz frames \- that is if we calculate the cross-sections using $\Psi(L(\vartheta)P_{a})$ and $\Psi(L^{-1}(\vartheta)P_{b})$, where $L(\vartheta)$ is a longitudinal boost. It is essential, that in a parton picture such boosts $L(\vartheta)$ act on hadrons Fock state very nontrivial changing the number of partons, etc. No precise arguments for such general propositions (the boost invariance for parton cross-sections $\simeq$ t-unitarity) are known. Although it is by itself natural that the calculations of cross-sections in the parton picture must give a frame independent answer. Also this is, in particular, confirmed in if we give the partonic interpretation to reggeon diagrams, by t-cutting them at various intermediate rapidities, as if we calculate various multiparticle inclusive cross-sections. In this article 333The material of this paper partially intercepts with the article of the author [8]. we consider some examples illustrating how the requirement of boost-invariance essentially restricts the structure of high energy collision dynamics. We see that it restricts in the same way as it follows from the conditions of t-unitarity. ## 2 Restrictions on a parton states from the boost invariance of high-energy collision cross-sections. Simple Examples In this section we illustrate how the requirement of the frame independence (boost-invariance - BI) restricts the behavior of high-energy cross-sections calculated in the parton approach. We suppose that partons are point like particles with perturbative interaction and consider here some examples which show how BI condition works. Also we choose very high energy interactions, where the mean number of parton in HE state is large, so one can consider firstly only states with mean number of partons and only after that take into account corrections from other components of the Fock wave function of a fast particle. So, the picture of interaction is almost quasiclassical. We consider the behavior at a boost-transformation of the inelastic cross- sections $\sigma_{in}$ or of the connected quantity - the transparency $T=1-\sigma_{in}=|S|^{2}$, which is often more sensitive to the breaking of BI. We choose some frame where the colliding particles have rapidities $y_{1}=y$ and $y_{2}=Y-y$, where $Y=\ln(s/m^{2})$, and require that calculated cross-sections do not depend on $y$ We begin from the simplest parton models of a fast hadron - the rare parton gas state and of the black disk state. ### 2.1 Collision of a rare gas like parton states Let us consider the collision of two particles which can be represented as the partonic clouds that are in a state of a very rare gas. This is the case usually described by reggeon diagrams, that, by their construction, include t-unitarity requirements. Let the mean number of partons in colliding hadrons be $n(y)$, $n(Y-y)$ and the mean transverse radii of regions occupied by these partons are $R(y)$, $R(Y-y)$, respectively. Then the total inelastic cross- section can be expressed as: $\displaystyle\sigma_{in}(Y)~{}=~{}\sigma_{0}~{}n(y)~{}n(Y-y)~{}~{}-~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ $\displaystyle~{}~{}~{}-~{}c_{1}~{}\sigma_{0}~{}n(y)n(Y-y)~{}\Big{(}~{}\frac{\sigma_{0}~{}n(y)}{R^{2}(y)}~{}+~{}\frac{\sigma_{0}~{}n(Y-y)}{R^{2}(Y-y)}~{}+~{}$ (2) $\displaystyle~{}+~{}\frac{\sigma_{0}~{}n(y)~{}n(Y-y)}{R^{2}(y)+R^{2}(Y-y)}~{}\Big{)}~{}+...~{}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ where $\sigma_{0}$ is the parton-parton cross-section, $c_{1}\sim 1$. The first term in (2.1) corresponds to a collision of at least one pair of partons. The next terms describe corrections from screening and multiple collisions 444The cross-sections of local interactions of point-like particles decrease as a function of their relative energy as $\sigma_{0}(s)\sim 1/s$. As a result in (2.1) enter, in fact, only the numbers of low energy partons $n(y),n(Y-y)$ of the colliding particles in this coordinate system. . For the rare parton gas one can at first approximation neglect multiple collisions and screening, that is to leave only the first term in (2.1). Then, from the requirement of the independence of $\sigma_{0}~{}n(y)n(Y-y)$ on $y$ follows the unique solution for $n(y)=~{}n_{0}e^{y\Delta_{0}}$ (3) with some real constants $n_{0}$, $\Delta_{0}$. The following from (3) behavior of $\sigma_{in}(Y)=\sigma_{0}~{}n_{0}^{2}~{}e^{Y\Delta_{0}}$ (4) in the elastic amplitude corresponds to a regge pole in the complex angular momentum plane (and not to a cut or some more complicated regge singularity ). And this condition follows [9] in a relativistic Regge approach only from the 2-particle t-unitarity of the elastic amplitude. Note that the coefficient in (4) is in fact factorized - for the collision of different particles a+b one must $n^{2}\rightarrow n_{a}n_{b}$ . This factorization in regge approach also follows from t-unitarity. Moreover, it is interesting to consider [6] the behavior cross-section $\sigma_{in}$ with the definite impact parameter $B$, normalized so, that $\sigma_{in}(Y)~{}=~{}\int~{}d^{2}B~{}\sigma_{in}(Y,y,B)~{}~{}.$ (5) In this case the analog of the first term in (2.1) can be represented as $\sigma_{in}(Y,y,B)~{}=~{}\sigma_{0}\int d^{2}x_{\bot}~{}\rho(y,|x_{\bot}|)~{}\rho(Y-y,|B-x_{\bot}|)~{}~{}~{}~{},~{}~{}~{}~{}$ (6) where $\rho(y,x_{\bot})$ is the transverse parton density $(~{}n(y)=\int d^{2}x_{\bot}\rho(y,x_{\bot})~{})$ . Then from the frame independence of the $\sigma_{in}(Y,y,B)$ the form of transverse parton density $n(y,x_{\perp})$ can be essentially restricted. The condition of y-independence can be writhen as $\frac{\partial}{\partial y}~{}\sigma_{in}(Y,y,B)~{}=~{}0~{}.$ (7) Going here to conjugate to $x_{\bot}$ variable $\rho(y,x_{\bot})=\int d^{2}k\cdot e^{ikx_{\bot}}~{}\tilde{\rho}(y,k)$ we come from (7) to the equation $\frac{\partial}{\partial y}~{}\big{(}~{}\tilde{\rho}(y,k)~{}\tilde{\rho}(Y-y,k)~{}\big{)}~{}=~{}0~{},$ which has the solution $\tilde{\rho}(y,k)=f_{1}(k)\cdot e^{yf_{2}(k)}$ and then as a result $\rho(y,x_{\bot})\sim\int d^{2}k~{}e^{ikx_{\bot}}~{}f_{1}(k)~{}e^{yf_{2}(k)}~{},$ (8) where $f_{1}$, $f_{2}$ are arbitrary functions of $k$. For $y\rightarrow\infty$ the integral in (8) can be taken by the steepest decent method, so that only the neighborhoods of zeros of $\partial f_{2}(k)/\partial k$ are essential. Then from the positivity of the parton density $\rho$ it follows that $f_{2}$ is positive and so the dominant contribution must come from the region $k\sim 0$, otherwise $\rho(y,x_{\bot})$ will oscillate in $x_{\bot}$. So in the essential region $f_{2}(k)\simeq c_{1}-c_{2}k^{2},~{}~{}c_{2}>0$, and estimating the integral (8) we come to the expression for the density of low energy partons $\rho(y,x_{\bot})~{}\sim~{}y^{-1}~{}e^{\big{(}c_{1}y-x_{\bot}^{2}/4c_{2}r_{0}^{2}y\big{)}}~{}~{},~{}~{}~{}c_{2}>0~{}.$ (9) The expression (9) corresponds to the Gauss form of parton distribution in $x_{\bot}$ which usually results from the diffusion of partons in $x_{\bot}$ plane during the parton cascading. The mean radius of a low energy parton cloud $R(y)~{}\sim r_{0}\sqrt{~{}y}$ is also fixed here only from the condition of the frame independence. In the elastic amplitude the Eq. (9) corresponds to the contribution of the regge pole with the trajectory $\alpha(t)=1+\Delta+\alpha^{\prime}t$ , where $\Delta=c_{1},~{}\alpha^{\prime}=c_{2}$. If we make the next step and impose the condition of $y$ independence on the sum of two terms in the right side of (2.1) and assume that the correction to (3) is small, we become instead of (3) the corrected expression $n(y)=n_{0}~{}e^{\Delta_{0}y}-~{}a_{2}~{}n_{0}^{2}\frac{\sigma_{0}}{R^{2}(y)}~{}e^{2\Delta_{0}y}~{}+...~{}~{}~{}~{}$ (10) From here it is simple to conclude that $\sigma_{in}(Y)=n_{0}^{2}~{}\sigma_{0}e^{(\Delta_{0}Y)}~{}-~{}~{}n_{0}^{4}\frac{a_{2}\sigma_{0}^{2}}{R^{2}(Y)}~{}e^{2(\Delta_{0}Y)}~{}~{}.$ (11) The second term in (11) corresponds to the the contribution of two reggeon cuts, whose structure is almost complectly fixed here from the boost- invariance. The arbitrary coefficient $a_{2}>1$, depends on the weight of the diffractive amplitudes entering in the two regeon emission vertex. The possible next terms in (10), corresponding to higher regge cuts, can be found in the same way by iterative applying the boost-invariance condition to the combinations of screening terms in the expression (2.1) for $\sigma_{in}$. Thus, it can be seen that for rare parton states we come to the restrictions on there structure that arise from the reggeon diagrams and are defined by the t-unitarity. At the end of this section note that at at all currently available energies the dominant high energy hadron interactions are well described by the regge approach with the soft pomeron exchange and the respective cuts. This directly corresponds to the Gauss-like parton distribution consistent with the parton frame independence. ### 2.2 Collision of a black disks Now let us consider the opposite limiting case of colliding parton clouds, when the mean parton density is very high and partons fill a transverse disk with the radius $R(y)$ depending on particles energy $E=me^{y}$. Then the total inelastic cross-section can be determined from purely geometrical conditions - it is defined by the area of an impact parameter space, corresponding to the overlapping of the colliding black disks : $\sigma_{in}(Y)~{}=~{}\pi~{}\Big{(}R(y)+R(Y-y)\Big{)}^{2}~{}.$ (12) From the condition of independence of the right side of Eq.(12) on $y$ evidently follows the unique solution for $R(y)~{}=~{}r_{0}\cdot y+r_{1}$ (13) It is interesting that in this case we immediately come directly to asymptotically constant cross-sections (when $r_{1}=0$), or to the Froissart type behavior of cross-sections $\sigma_{in}(Y)\simeq\pi r_{0}^{2}Y^{2}+\pi r_{0}r_{1}Y+\pi r_{1}^{2}~{}~{}.$ (14) Here, in the Froissart case the elastic cross-section is diffractive and $\sigma_{el}=\sigma_{in}$. Also the terms $~{}\pi r_{0}r_{1}Y$ in (14) correspond to a diffraction generation as is natural in the Froissart case. ### 2.3 Collision of the parton grey disks The real parton disk (even at $Y\gg 1$) cannot be absolutely black because the parton density at every particles energy is finite. Besides that the local parton density fluctuations also can lower the parton density in the individual events and this leads to the grow of the locale transparency of such disks. For such parton disks the conditions of BI can lead to rather strong restrictions on the structure of “grey” parton states and their interactions. Firstly, consider the collisions of grey disks with some constant grayness - when the mean transverse parton density is stabilized at some fixed value and do not grow with energy i.e. the local disk transparency also does not change with energy 555One can expect this type of the behavior in the (2+1)D QCD, which is soft and if here the parton saturation takes place [8].. Then it is easy to see that the condition of the boost invariance can at all not be fulfilled for such models. In the lab.frame of one particle the transparency $T_{lab}(Y,B)=const(Y)~{}~{}~{}~{}~{}at~{}~{}~{}~{}~{}Y\rightarrow\infty~{}~{},$ because at all $B$ only a finite number ($\sim 1$) of partons must penetrate through the grey parton disk of the other fast particle. And in the center of the mass system at the same impact parameter the large number of partons $N_{12}$ must penetrate. For the grey disk $N_{12}\sim S_{12}(y,Y,B)$ \- the transverse area of two disks overlapping region. And for growing with $Y$ disk radius the $S_{12}$ also grows. For the Froisart type growth we have $S_{12}\sim Y^{2}$ at $B\ll R(Y)$. Then in systems close to the center of mass $T_{scm}(Y,B)\sim e^{(-cN_{12})}\sim\exp{(-cY^{2})}\rightarrow 0~{}~{}~{}~{},~{}~{}~{}~{}~{}~{}~{}~{}c\sim 1~{}~{}~{}.$ Therefore, the case of a grey disk with a constant (or a slowly ($\sim 1$) varying ) parton density should be probably excluded. In all more or less realistic situations, such, for example that one can expect in the QCD, the parton disk can have a grey parton border, even when the inner parts of the disk become almost black. In this case the average parton density can be roughly represented as $\rho(y,x_{\bot})~{}\simeq~{}\rho_{d}(x_{\bot})~{}\theta(R(y)-x_{\bot})~{}+~{}\rho_{0}~{}\theta(x_{\bot}-R(y))f(x_{\bot})~{}~{},$ (15) where $\rho_{d}(x_{\bot})$ describes the behavior of the parton density in the inner part of the disk, and where the grey border has the width $\lambda(y)\ll R(y)$. In this border the parton density varies from the high value (almost black) to small one. For example, it can have the form $f(x_{\bot})~{}\simeq~{}e^{-(x_{\bot}-R(y))/\lambda(y)}~{}~{}.$ (16) For collisions with an impact parameter $B<R(Y)$, when colliding disks stuck with their almost black parts, we possibly can have a boost-invariant behavior of $\sigma_{in}$. But for collisions with $B-R(Y)\sim\lambda(Y))$, when the discs collide with their grey edges, the situation is different. In the Lab frame of one of particle the transparency $T_{lab}=const\sim 1$, because here only some ($\sim 1$) partons must penetrate without interaction through the grey edge of the large disc. And in the arbitrary system at the same impact parameter the transparency is $T(y,Y-y,B)\sim e^{-N_{12}(y,Y-y,B)}\sim e^{-S_{12}(y,Y-y,B)/r_{0}^{2}}~{}~{},$ where $N_{12}(y,Y-y,B)$ is the average number of parton interactions during the collision and $S_{12}(y,Y-y,B)$ is the area of the two disks intersection region. This region has a form of elongated figure whose width is $\sim\lambda(y)$ and the length $l(y)\sim\sqrt{R(y)\lambda(y)}$ for $y\lesssim Y-y$. So, for such B the two disks intersection area is $S_{12}(y,Y-y,B\simeq R(Y)+\lambda)~{}~{}\sim~{}~{}R(y)^{1/2}*\lambda(y)^{3/2}~{}~{}.$ In the center of mass system this gives for $Y\gg 1$ $T_{scm}\sim\exp(-c(R(y)/r_{0})^{1/2})\rightarrow 0$ even for parton disks with $\lambda\sim const(Y)$, although the width of the grey border can also grows together with the disk size 666One can expect [8] that for a realistic parton disc due to border shape fluctuations the mean width of the grey zone grows with Y as $\lambda\sim\sqrt{Y}$. Therefore, for the border collisions with such $B$ and $Y$ we have no boost-invariance of T , and this conclusion in fact almost does not depend on the explicit form of the border distribution $f(x_{\bot})$. Probably, the only exception is the Gauss type distribution of the parton density when the whole disk has the “structure of border”. ### 2.4 Particle to heavy nuclei interaction A slightly different type of restriction on the parton structure follows from the boost invariance if we consider the high energy collision of a particle p (for example a proton, a pion or any test color dipole) with heavy nucleus $(A\gg 1)$. To see this we compare the estimate of transparency T in Lab frame of nuclei and the Lab frame of p. Also we choose Y not very large, but so that $Y\gg\ln A$, and consider a collision at $B=0$. In fact, in this case we have a collision of p with a long ($\sim A^{1/3}$) tube of nucleons, and we want to calculate the probability of the passage of p through A without interaction. First, consider the p $\bigotimes$ A collision in the Lab frame of p. For such an Y due to the Lorentz contraction of the moving nuclei all soft partons of the fast nuclei are placed in a tiny transverse region of the longitudinal size $\sim 1/m$. And if the parton saturation take place, the number of soft partons $N_{A}$ interacting with p should almost not depend on A, because all “additional” soft partons coming from different nucleons in the A-tube are absorbed one by another. Therefore, one can expect that the transparency in the p-lab. frame is $T_{p}~{}\sim~{}e^{-N_{p}(Y)}~{}~{}.$ (17) On the other hand, to calculate T in the Lab frame of A at the same B and Y we must find the probability that fast particle p penetrate without interaction trough the $A^{1/3}$ long tube of nucleons. Here one can expect that $T_{A}~{}\sim~{}e^{-c(Y)A^{1/3}}$ (18) Because in such a “thought experiment” we can arbitrary choose Y and A and the distance between the nucleons in the tube, we come to an apparent contradiction with the frame independence. This means that some constrains must be imposed on the parton dynamics. The simplest way is to suppose that there is almost no parton saturation in the A-tube. Or on the contrary - that some kind of the mechanism works, which makes the interaction of a fast p particle with the nucleus otherwise dependent on A. Possibly some indications on the causes of this inconsistency can be found if we consider the regge description of this reaction, where we can calculate $\sigma_{in}(Y,B)=1-T$ for a large A and not to a large Y. If we take for a single pomeron exchange in the p $\bigotimes$ A reaction the amplitude $v(y,b)\sim ig^{2}A^{1/3}\exp{(\Delta y-b^{2}/4\alpha^{\prime}y)}$ and consider firstly the simple eiconal case which corresponds to a situation without parton saturation we become for the corresponding S-matrix $S(Y,B)=\exp{(iv(Y,B))}$, and this gives for the transparency $T(Y,B=0)~{}=~{}|~{}S(Y,B=0~{}|^{2}~{}\sim~{}\exp{\Big{(}-2g^{2}A^{1/3}e^{\Delta Y}\Big{)}}~{}~{}.$ (19) The simplest way to take into account something similar to the parton saturation is to include into the single pomeron exchange amplitude the pomeron cascading from the side of A-vertices. So that from the p-side the pomeron line joins to p and from the A side the pomeron line branching joins to many nucleons. This corresponds to the new amplitude $v~{}\rightarrow~{}\tilde{v}~{}=~{}\frac{v}{1-i\frac{r}{g\Delta}v}~{}~{},$ (20) were $r$ is the 3-pomeron vertex. In this case for large $A^{1/3}$ and $B~{}=~{}0$ the amplitude $\tilde{v}$ is stabilized at the value $|\tilde{v}|=g\Delta/r$. And, therefore, the corresponding transparency approaches to $T(Y,B=0)~{}=~{}\exp{\big{(}-2|\tilde{v}|~{}\big{)}}~{}=~{}\exp{\Big{(}-2g\Delta/r\Big{)}}~{}~{}.$ (21) Comparing the expressions ( 17 ) with (21) and (18) with (19) we see their similarity, but this unfortunately does not help to find the right answer, because the simple expression like (20) dos not take into account various pomeron interactions in the $\tilde{v}$-cascade and also the other pomeron interactions in eiconal multipomeron diagrams 777 Note that approximately the same inconsistency appears if we consider the heavy A $\bigotimes$ A interaction end compare the estimates of T in the Lab frame and in the CM system . ### 2.5 Possible boost-invariant parton density distributions in a grey disk In fact, in the case of asymptotically growing cross-section all parton distributions corresponding to real theories like QCD will, probably, lead to the grey dick. And it is interesting to find the sensible examples of parton distributions that correspond to the boost-invariant T. Let us consider collisions of particles with some parton distribution $\rho(y,x_{\bot})$ and try to find the minimal conditions on the form of $\rho(y,x_{\bot})$ for which cross-sections are boost-invariant With the exponential precision the transparency can be expressed as: $T(Y,y,B)~{}\sim~{}\exp\Big{(}~{}-N(y,Y-y,B)~{}\Big{)}~{}~{}~{},$ (22) where $N(y,Y-y,B)=\sigma_{0}\cdot\int d^{2}b\cdot\rho(y,|b|)\cdot\rho(Y-y,|B-b|)$ (23) is proportional to the mean number of the parton scattering when two $\cal F$ d penetrate one through another during their collision at the impact parameter B. Because the expression (23) has the same structure as (6) one can repeat here the calculation given above. Then we find that the expression ( 23 ) can be boost invariant only for some very special Gaussian form of parton density $\rho$ inside the disk : $\rho(y,x_{\bot})~{}\sim~{}\rho_{0}~{}\frac{1}{y}~{}e^{\Delta y-x_{\bot}^{2}/yr_{0}^{2}}~{}~{},~{}~{}$ (24) This corresponds to the distribution arising in the parton cascade when partons only split and do not join. The same answer (Eq (9)) for $\rho(y,x_{\bot})$ was found for the rare parton systems - but here the density can be arbitrary high. In the connected elastic amplitude it corresponds to a regge pole exchange with the intercept $\Delta$. In fact, the expression (24) for $\Delta>0$ corresponds again to almost black disk (but without parton saturation !) of the radius $r_{0}\Delta~{}y$ with a thin grey border, because the parton density changes here very fast from a small to a big values at the distances $\delta x_{\bot}\sim r_{0}/\sqrt{\Delta}$. In general case one must take into account that partons in the colliding disks can have different virtualities $u\sim\ln k^{2}_{\bot}/m^{2}$ , where the parton density $\rho(y,b,u)$ has now nontrivial dependence on $u$. Partons with large $u$ are more strongly localized in transverse coordinates and their interaction cross-sections $\sigma(u_{1},u_{2})$ usually decrease for large $u_{i}$. The expression for the transparency in the process of collision of two parton disks has again the form (22), where the mean number of parton interactions during the collision is given by the following generalization of (23) $N(y,Y-y,B)=\int d^{2}b\int du_{1}du_{2}~{}\sigma(u_{1},u_{2})\cdot\rho(y,|b|,u_{1})\rho(Y-y,|B-b|,u_{2})~{}.$ (25) In this case the restrictions on the form of $\rho(y,b,u)$ coming from the frame independence condition $(\partial/\partial y)N(y,Y-y,B)=0$ are not so strong as for (23 \- 24). If the parton cross-sections that enter (25) can be approximately factorized as $\sigma(u_{1},u_{2})\sim\ell(u_{1})\cdot\ell(u_{2})~{}~{},$ then the condition for the boost invariance of $N$ can be reduced to the more simple equation $\int du~{}\ell(u)\rho(y,b,u)~{}=\rho_{0}\frac{1}{y}~{}e^{\Delta y-b^{2}/yr_{0}^{2}}$ (26) In this case the form of $\rho(y,b,u)$ for some interesting models are again almost completely fixed. For example, so is the superposition of grey saturated disks with different virtualities $\rho(y,b,u)\sim\varphi(y,u)~{}\theta(r_{1}\chi(y)-bu^{a})~{},$ so that the mean radii of these disks $r_{1}\chi(y)/u^{a}$ decrease 888In QCD the radii of hard subdisks can grow as $\sim y/\sqrt{u}$, and this corresponds to $a=2$ with growth of u. Here, from equation(26), one can find the explicit expression for $\varphi(y,u)~{}=~{}\varphi_{0}~{}\frac{e^{\Delta y}}{\ell(u)~{}u^{2a}~{}y}~{}\exp{\Big{(}-c_{2}\frac{\chi^{2}(y)}{y~{}u^{2a}}~{}~{},\Big{)}}$ (27) where $\varphi_{0},~{}a,~{}\Delta,~{}c_{2}$ and functions $\ell(u),~{}\chi(y)$ can be chosen arbitrary. If we choose $\chi(y)=~{}\chi_{0}y$ so to have the Froissart type of the growth of disk radius we will have from (27) for the disk density $\varphi(y,u)~{}\sim~{}~{}\frac{e^{\Delta y}}{\ell(u)u^{2a}y}~{}\exp{\Big{(}-\tilde{c_{2}}\frac{y}{u^{2a}}~{}~{}.\Big{)}}$ (28) For large $u$ it is natural to expect that $\ell(u)\sim e^{-cu}$ and therefore the mean density of hard subdisks will grow with u and y. ### 2.6 Corrections to the mean picture from a big fluctuations in the colliding states To discuss if the boost-invariance can be somehow restored also when the mean parton density $\rho(y,x_{\bot})$ is not of the Gauss form (24) on must take into account all essential parton configurations, and also these ones that are very far from the mean one. In this case, one can hope that in different frames the main contribution into cross-sections comes from some different parton components so to compensate the variation of the contribution of the mean states. Here especially interesting can be the rare components of $\Psi(P)$. In the Fock state of a fast particle such rare parton configurations contain a relatively small number of partons and therefore it can give large contribution to the transparency and compensate the boost non- invariance of $T$ and other quantities in the mean density states. Such configurations can mainly arise due to large fluctuations in the initial stages of the patron cascade. CM one can ask for such a parton component $|~{}bare>$ for a fast hadron that does not contain a black disk at all and interacts slowly (or does not interact at all). We can schematically represent such a state of fast particle : $\Psi(P\rightarrow\infty)~{}\simeq~{}f_{d}~{}|disk>+~{}f_{b}~{}|~{}bare>~{},~{}~{}~{}f_{d}\gg f_{b}~{},$ where $f_{b}$ is the amplitude of the rare component $|bare>$ and $|disk>$ is the superposition of “big” parton components that gives the main contributions in a various cross-sections. The probability for a fast hadron to be in the rare state is $w(y)\sim|f_{b}|^{2}$. In this case the expression for the transparency can be generalized to : $\displaystyle T(y,Y-y)~{}~{}\simeq~{}~{}T_{mean}(y,Y-y)+~{}\tau_{bd}\cdot\big{(}w(y)+w(Y-y)\big{)}~{}~{}+$ $\displaystyle+~{}\tau_{bb}\cdot w(Y-y))\cdot w(y)~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (29) where the transparencies of the rare component $\tau_{bd}$ and $\tau_{bb}$ can be finite and do not decrease with growth of $y$. The term $T_{mean}(y,Y-y)$ coming from the $|disk>\bigotimes|disk>$ interaction is not boost-invariant and it can be $const(Y\rightarrow\infty$) in the lab.frame for a saturated (grey) disk, and very small in csm. The two last terms in (2.6) coming from the $|bare>\cdot~{}|disk>$ and $|bare>\cdot~{}|bare>$ components can dominate and so can make $T$ boost invariant. But it is possible only if $w(y)$ is approximately constant for high $y$. Various estimates of $w(y)$ lead to a decreasing function of the type $w(y)\sim\exp(-\gamma\cdot y)$ for the case of the growing total cross-section. It corresponds to the choice at every rapidity stage of such an evolution direction, that does not increase the parton number 999Such a behavior of $w(y)$ can also be found from the boost-invariance condition applied to the behavior of some hard cross- sections. See Eq.(31 \- 32) . Such a behavior of $w$ leads to the expression $T(y,Y-y)~{}\sim~{}\tau_{bd}\cdot\big{(}~{}e^{-\gamma~{}(Y-y)}~{}+~{}e^{-\gamma~{}y}~{}\big{)}+\tau_{bb}\cdot e^{-\gamma~{}Y}~{},$ (30) corresponding to the collision of the rare state $|~{}bare>$ with other particle. Such contribution to $T$ is $y$ dependent, and therefore on this way the frame indeprndence can also not be restored. ### 2.7 Collision of parton disks in the case of particles moving in the same direction When we impose the condition of the independence of the cross-sections $\sigma_{in}(Y,y,b)$ on the choice of system (i.e. on $y$), we can choose the values of $y$ not only in the interval $0<y<Y$, i.e. between Lab and center of mass ( CM ) systems. But let us also consider systems with $y<0$ and $y>Y$ , when both parton disks move in one direction. This, in principle, can lead to additional constraints on the amplitudes. But in this case, at first glance, paradoxes may also arise when estimating the probability of the interaction. Especially this is seen in the case of the growing cross-section. To illustrate this let us consider the case of colliding Froissart type disks when their radii $R(y)$ and $R(Y-y)$ grow with the particles rapidity as $R(y)=r_{0}y$ and $R(Y-y)=r_{0}(Y-y)$, and estimate the behavior of the inelastic cross-section with the definite impact parameter $\sigma_{in}(Y,y,B)$ . We chose $B>r_{0}Y$. In this case, when $0<y<Y$, the parton disks pass one by another without interaction. But this is only if they move towards each other because here $B>R(y)+R(Y-y)$ and therefore $\sigma_{in}=0$. But if we at the same $B$ choose the system so that disks move in the same direction and so that $y\gg Y\gg 1$ then disks will overlap when one disk will go through another. And therefore partons from one disk can interact with partons from another disk. But it is essential that in such disk interaction no new particles can be created. Indeed, if in this case a particles can be created, their momenta will be small ($\sim m$) in this system. And the creation of such a particle in CM system would correspond to a creation of a particle with energy $\sim me^{y}$, where $y\gg Y$ and this is forbidden by the energy-momentum conservation ; so $\sigma_{in}=0$. From the other hand, the exchange of particles between these discs with an approximate momentum conservation (or, with the exchange of small transverse momenta) can give a contribution to their elastic scattering and which comes here also from large transverse distances ( $B>R(Y)$ ). The parton wave functions of these “ intersecting disks” can be entangled one by another by such a mechanism, and also the conversion of a pure state to a mixed one for every disk can in principal take place. There is here probably no contradiction with the parton picture, since there is no way to distinguish between low energy partons in the wave function (1) and the close energy partons from vacuum fluctuations. The entanglement between states of disks in such a collisions is proportional to their area. This suggests that these discs have entropy $\sim$ their area ($\sim$ the number of low energy partons), i.e. $\sim y^{2}$ in this case of the Froissart type growth of cross-sections. ### 2.8 Limitations on the dynamic of a hard elastic scattering In the field theory the high energy hard elastic scattering of point-like particles leads usually to the power behavior of elastic cross sections $d\sigma_{1}^{el}(s,t\simeq-s/2)/dt\sim 1/s^{a}~{},~{}~{}~{}~{}~{}$ For the scattering of particles composed from n constituents with approximately equal momenta we have $d\sigma_{n}^{el}(s,t\sim-s/2)/dt\sim\mu^{-4(n-1)}(d\sigma_{1}(s/n^{2},t\sim-s/n^{2})/dt)^{n}~{}~{}.$ But the mean state can contain the growing number of partons and the direct application of this expression leads to a small contribution. In this case the main contribution to $d\sigma/dt$ can come from the rare parton configurations containing the minimal number of partons (when both particles are in a “bare” state). Then, the cross-section of particles in the system, where $~{}s=m^{2}e^{Y}$ and the energies of colliding particles are $me^{y},~{}me^{Y-u}$, can be represented as $d\sigma^{el}(s,t\sim-s/2)/dt\sim\big{(}~{}d\sigma_{0}(s,t\sim-s/2)/dt~{}\big{)}^{n_{0}}~{}w(y)w(Y-y)~{}~{},$ (31) where $w(y)$ is the probability that particle with energy $me^{y}$ is in the bare state, and $n_{0}$ \- the number of “valent” components in the bare state ($n_{0}\simeq 2\div 3$ for meson $\div$ baryon). It follows from the boost- invariance of (31) that $w(y)\sim e^{-2cy}$ (32) This condition essentially restricts the behavior of the asymptotic of hard scattering and, in particular, gives the information about the amplitude ( $\sim\sqrt{w(y)}$ ) of the bare component of $\Psi(P)$. The similar limitation follows from the consideration of the asymptotic cross- sections of two particle reactions with exchange of quantum numbers (such as $\pi^{-}+p~{}\rightarrow~{}\pi^{0}+n$). Here again, the dominant parton configuration contributing to such reactions must contain the minimum number of partons. So again, we have the factor $w(y)w(Y-y)$ in the cross-section. Additionally, there is the factor of type $e^{-2gy}$ connected with the probability that this parton configuration contains also the small energy parton with “needed” quantum numbers. Therefore, from the frame independence of amplitudes of such reactions we also come to the condition (32). And, if interpreting in terms of the exchange of some nonvacuum reggeon we come to estimate their intercept as $\alpha(0)\simeq 1-c-g$ . ## 3 Summary The main aim of this note was to illustrate that the condition of boost- invariants essentially restricts the behavior of high energy cross-sections calculated in parton approaches. And the form of resulting constrains is of the same type as coming from the t-channel unitarity condition. So that one can suppose that this similarity, by their nature, has much more general grounds. Such a condition works especially effectively in the case of growing with energy cross-section, that is, just when the t-unitarity conditions for amplitudes is complicated to apply - because here the multiparticle exchange becomes important. In this case the resulting restrictions on the asymptotic behavior are rather strong and can, in principle, exclude some popular models. ## References * [1] V.N. Gribov, arXiv:hep-ph/0006158 * [2] S.Brodsky, H-C Pauli, S.Pinsky, 9705477, Phys.Rept 301 (1998) 299 * [3] M.Perry, arXiv:hep-ph/9612244 * [4] T.Heinzl, arXiv hep-th/0008096 * [5] A.Harindranath, arXiv:hep-ph/9612244 * [6] A.B.Kaidalov, ITEP School of Physics 1983 * [7] Y. Kovchegov, E. Levin , Quantum Chromodynamics at High Energy, Cambridge University Press, 2012 * [8] O.V. Kancheli arXiv 1609.07657 * [9] V.N. Gribov, I.Ya. Pomeranchuk, Phys.Rev.Lett. 8,343,412 * [10] V.N. Gribov, I.Ya. Pomeranchuk and K.A. Ter-Martirosyan, Phys. Rev. 139B (1965) 184 ; V.N. Gribov, Soviet Phys. JETP 26, 414, (1968) .

2024-09-04T02:54:58.663665

2003.04662

arxiv-papers

# The intra-cluster light as a tracer of the total matter density distribution: a view from simulations Isaac Alonso Asensio,1,2 Claudio Dalla Vecchia,1,2 Yannick M. Bahé,3David J. Barnes4 and Scott T. Kay5 1Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, E-38205 La Laguna, Tenerife, Spain 2Departamento de Astrofísica, Universidad de La Laguna, Av. Astrofísico Francisco Sánchez s/n, E-38206 La Laguna, Tenerife, Spain 3Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands 4Department of Physics, Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 5Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Manchester M13 9PL, UK E-mail<EMAIL_ADDRESS>(IAA) (Accepted XXX. Received YYY; in original form ZZZ) ###### Abstract By using deep observations of clusters of galaxies, it has been recently found that the projected stellar mass density closely follows the projected total (dark and baryonic) mass density within the innermost $\sim 140$ kpc. In this work, we aim to test these observations using the Cluster-EAGLE simulations, comparing the projected densities inferred directly from the simulations. We compare the iso-density contours using the procedure of Montes & Trujillo (2019), and find that the shape of the stellar mass distribution follows that of the total matter even more closely than observed, although their radial profiles differ substantially. The ratio between stellar and total matter density profiles in circular apertures, shows a slope close to $-1$, with a small dependence on the cluster’s total mass. We propose an indirect method to calculate the halo mass and mass density profile from the radial profile of the intra-cluster stellar mass density. ###### keywords: galaxies: clusters: general – methods: numerical ††pubyear: 2020††pagerange: The intra-cluster light as a tracer of the total matter density distribution: a view from simulations–References ## 1 Introduction Ten to more than thirty percent of the stellar light of clusters of galaxies comes from a diffuse distribution of stars emitting the so called intra- cluster light (ICL), the inferred fraction depending on the definition of the border between the brightest central galaxy and the diffuse stellar component, the radial extent at which the stellar mass distribution is integrated and the relaxation state of the clusters (e.g., Krick & Bernstein, 2007; Gonzalez et al., 2013; Mihos et al., 2017; Jiménez-Teja et al., 2018; Zhang et al., 2019). This distribution is produced by the stripping of stars from galaxies undergoing mergers and tidal interactions during their evolution in the cluster environment (see Mihos, 2015, for a review). Due to its low surface brightness, the observational study of the stellar population producing the intra-cluster light has been challenging. An increasing effort towards deep imaging of clusters of galaxies in the recent years, both through individual cluster imaging, up to $z\simeq 1.5$, (e.g., Mihos et al., 2005; Montes & Trujillo, 2014; Burke et al., 2015; Morishita et al., 2017; Ko & Jee, 2018; Jiménez-Teja et al., 2018; Montes & Trujillo, 2018; DeMaio et al., 2018; DeMaio et al., 2020), and by stacking observations of multiple clusters (e.g., Zibetti et al., 2005; Zhang et al., 2019) has allowed new insights into the ICL. Figure 1: Projected density of stars (top) and matter (bottom) for three different cluster of the C-EAGLE simulations at $z=0.352$. The secondary density peak in the Cluster CE-21 (middle panel) can be due to a recent major merger event which has stripped stars from the interacting galaxies, or numerical artifacts produced by SUBFIND not being able to correctly assign stellar particles to substructures. One recent, remarkable result achieved with deep imaging is the tight correlation between the distribution of the stellar surface density, inferred from its surface brightness, and the surface density of the total mass, measured by modelling the gravitational lensing signal (Montes & Trujillo, 2019, hereafter MT19). MT19 proposed that the surface density of the stellar mass not bound to galaxies should settle in the potential well of the cluster similarly to the dark matter. This could be used to trace the total matter distribution of clusters within a cluster-centric distance set by the depth of the observations. They also compared their result with total mass surface densities inferred from the X-ray emission of the intra-cluster medium, and concluded that this method is limited by the misalignment of the gaseous component with respect to the dark matter and stellar mass in non-relaxed clusters. Their quantitative analysis made use of the Modified Haussdorf Distance (MHD) (Dubuisson & Jain, 1994) to quantify the deviation between iso- density contours of stars and total matter. They found that, in general, the stellar surface density has smaller MHD values than that of the intra-cluster medium where both are compared with the iso-density contours of total mass. In this Letter, we test this observational result with state-of-the-art cosmological, hydrodynamic simulations of the Cluster-EAGLE project (C-EAGLE, Barnes et al., 2017; Bahé et al., 2017). We give a brief description of the simulations in the next section and a description of the analysis in section 3. The main results of this work are shown in section 4 and discussed in section 5, along with some concluding remarks. ## 2 Simulations We have used the set of 30 zoom-in cluster simulations performed within the C-EAGLE project. The simulated clusters are uniformly distributed in the mass range $10^{14}<M_{200}/\mathrm{M}_{\odot}<10^{15.4}$, where $M_{200}$ is the halo mass.111$M_{200}$ is the mass enclosed in a sphere of radius $r_{200}$, whose mean density equals 200 times the critical density of the Universe. The simulations were performed with the EAGLE model for galaxy formation and evolution, with the AGNdT9 calibration (Schaye et al., 2015). They provide a physical spatial resolution of $\epsilon=0.7~{}\mathrm{kpc}$ (at $z<2.8$) and baryonic mass resolution of $m_{\mathrm{gas}}\approx 1.81\times 10^{6}~{}\mathrm{M}_{\odot}$. For more information on the EAGLE model and its comparison with global relations of the observed galaxy population, the reader is referred to Schaye et al. (2015) and Crain et al. (2015). For more details on the numerical algorithms describing photo-ionization equilibrium cooling, star formation, stellar evolution, stellar feedback, black hole growth and feedback, and the hydrodynamic scheme we refer the reader to Wiersma et al. (2009a), Schaye & Dalla Vecchia (2007), Wiersma et al. (2009b), Dalla Vecchia & Schaye (2012), Rosas-Guevara et al. (2015), and Schaller et al. (2015), respectively. Figure 2: Fraction of stellar mass contributing to the ICL, $f_{\mathrm{ICL}}$, as function of halo mass, $M_{200}$, for all clusters in the sample. There is no evidence for a correlation with halo mass. The solid line marks the average value of $f_{\mathrm{ICL}}$, and the dashed lines the spread around it. For the results presented here, we have used the particle data, friends-of- friends and SUBFIND (Dolag et al., 2009) groups at $z=0.352$ to match the average redshift of the Hubble Frontier-Fields clusters (Lotz et al., 2017). Furthermore, the same analysis was performed at $z=0$, and we found no significant difference. Throughout the paper we assume the cosmological parameters of the C-EAGLE simulations, $(\Omega_{0},\Omega_{\Lambda},h,n_{\mathrm{s}},\sigma_{8})=(0.307,0.693,0.6777,0.961,0.8288)$ (Planck Collaboration et al., 2014), where $\Omega_{0}$ and $\Omega_{\Lambda}$ are the matter and dark energy fractions, $h$ is the Hubble constant in units of $100~{}\mathrm{km}\,\mathrm{Mpc}^{-1}\,\mathrm{s}^{-1}$, $n_{\mathrm{s}}$ and $\sigma_{8}$ are the spectral index and the power spectrum normalisation used to generate the initial conditions. In the analysis, we have used all particles belonging to the main halo of the largest friends-of-friends group in each simulation, i.e., we excluded all particles bound to satellite galaxies and substructures within the same friends-of-friends group. Maps of projected stellar and total matter density were produced with a spatial resolution of $5~{}\mathrm{kpc}$, in order to mimic the spatial resolution employed in the analysis of the observational data ($3\times 3~{}\mathrm{arcsec}^{2}$ at $z\simeq 0.35$). We have repeated the analysis with higher ($3.75~{}\mathrm{kpc}$) and lower ($7.5~{}\mathrm{kpc}$) resolution without finding any remarkable difference. The main advantage with respect to observations is that there is no need of masking the light of satellite galaxies. However, debris from tidal interactions between galaxies will be included in the projected matter density. Furthermore, there are biases due to SUBFIND failing to assign stellar particles to satellites (Bahé et al., in prep). Figure 3: Isodensity contours of the inner ($R\leq 140~{}\mathrm{kpc}$) and outer ($R>140~{}\mathrm{kpc}$) regions (top and bottom, respectively) of total matter (blue dotted lines) and stars (red dashed lines) for three different clusters. Lighter colours indicate larger distances (lower densities) from the centre. Examples of the projected stellar and total mass density are shown in Fig. 1 for three simulated clusters of increasing virial mass. The top row corresponds to the projected density of stars, while the bottom row shows the density of total matter (dark and baryonic). Uncertainties on the amount of ICL mass produced and its radial distribution may arise from the modelling of the star formation rate and the spatial and mass resolution of numerical simulations. The EAGLE model matches quite accurately the observed stellar mass and luminosity functions (Schaye et al., 2015; Trayford et al., 2015). Moreover, it reproduces the evolution of the stellar mass function and the observationally inferred density of stars in the universe up to high redshift ($z=7$) (Furlong et al., 2015). However, while the reference simulation matches the observed sizes of galaxies over several decades in stellar mass, the AGNdT9 calibration yields an offset in the relation towards more compact galaxies (Schaye et al., 2015). This last point seems to be relevant in the interpretation of the ICL mass fractions described in the next section, where the inferred values are on the low side of the distribution of those derived from observations (see references in section 1): compact galaxies are less prone to stripping. On the other hand, Henden et al. (2019) noted that having too large in size galaxies in their simulations boosts the effect of tidal stripping, increasing the fraction of stellar mass in the ICL, and that uncertainties in galaxy sizes are the major contributors to the uncertainty in the determining the fraction of mass in the ICL in simulations. ## 3 Analysis Before describing the methodology used in the analysis of the simulation data, we briefly discuss a consistency check for the simulated clusters. We computed the fraction of stellar mass in the ICL, $f_{\mathrm{ICL}}$, and compared it with expected observational and theoretical values. For the sake of ease, we adopted the methodology of (Rudick et al., 2011). The mass fraction has been computed as the stellar mass with projected stellar density below some threshold surface brightness, $\mu$, with respect to the total stellar mass within $r_{200}$. As in (Rudick et al., 2011), we have converted the stellar surface density into surface brightness assuming a constant mass-to-light ratio of $5~{}\mathrm{M}_{\odot}\,\mathrm{L}^{-1}_{\odot}$, and set $\mu=26.5~{}\mathrm{mag}\,\mathrm{arcsec}^{-2}$ as the threshold. We show in figure 2 the computed $f_{\mathrm{ICL}}$ as function of halo mass. We find that $f_{\mathrm{ICL}}=0.091\pm 0.013$ (solid and dashed lines), with no significant correlation with the total mass of the clusters (the Pearson correlation coefficient is $0.0063$). The result is consistent with that of (Rudick et al., 2011). Although the range of halo masses in our sample is rather narrower, similar fractions and the lack of correlation have been reported by (Pillepich et al., 2018), when using a definition of the ICL related to the size of the central galaxy. The result is consistent with previous simulations (Rudick et al., 2011; Contini et al., 2014), where they applied semi-analytical models to N-body simulations, and hydrodynamical cosmological simulations (Pillepich et al., 2018; Henden et al., 2019). Finally, observations using similar thresholds have reported as well similar mass fractions (Krick & Bernstein, 2007; Montes & Trujillo, 2014). We have followed a methodology similar to MT19 to extract iso-density contours. We computed circularly averaged radial profiles of the density of the stellar and total mass. For this, we take the position of the minimum of the potential energy as centre of the cluster (McAlpine et al., 2016). The projected densities for drawing the contours222We used the contour function of matplotlib to compute the contours. were selected interpolating the profiles at radii of 50, 75, 100, 125, $140~{}\mathrm{kpc}$ (the distances used by MT19) for the inner part, and of 170, 220, 300, 460, 620, 780, 940, $1100~{}\mathrm{kpc}$ for the outer regions, and only up to $r_{200}$. At large distance from the centre of the clusters ($r>140~{}\mathrm{kpc}$), we down-sample the images merging $4\times 4$ pixels, thus degrading the spatial resolution to $20~{}\mathrm{kpc}$, to smooth the otherwise very noisy contours. The contours of the projected densities are shown in Fig. 3, for the same three clusters as depicted in Fig. 1. The projected total mass density contours are drawn with blue dotted lines, and the projected stellar density contours with red dashed lines, where a darker colour indicates a smaller radius. The top row is a close-up view of the contours near the centre of the clusters, out to $140~{}\mathrm{kpc}$, whereas the contours at larger distances are shown in the bottom row. We measured projected radial distances from the centre of the cluster instead of elliptical distances to the centre of the brightest central galaxy, as usually done in observations. This simplification is not crucial to derive the iso-density contours, as it only changes the values of density at which the contours will be drawn. In practice, this means that the distances we use are systematically different from those of MT19, the difference depending on the eccentricity of the brightest central galaxy, or the presence of more than one central galaxy, that we excluded from the analysis, or both. As this is only an exploratory analysis we ignore these differences. As in MT19, to compare the shape of the contours, we estimated the Modified Hausdorff distance (MHD) defined by Dubuisson & Jain (1994): $d_{\mathrm{MH}}(X,Y)=\max\left(d(X,Y),d(Y,X)\right),\\\ $ (1) where $d(X,Y)=\frac{1}{N_{X}}\sum_{\textbf{x}\in X}\min_{\textbf{y}\in Y}\|\textbf{x}-\textbf{y}\|.$ (2) The two samples, $X\equiv\\{\textbf{x}_{1},\textbf{x}_{2},\dots,\textbf{x}_{N_{x}}\\}$ and $Y\equiv\\{\textbf{y}_{1},\textbf{y}_{2},\dots,\textbf{y}_{N_{y}}\\}$, contains the points defining two contours, and $\|\cdot\|$ is the Euclidean norm. As we may have different closed contours for the same density value, we select for each distance the contour composed by the largest number of segments. The selected contours are shown in Fig. 3. Figure 4: Left panel. Comparison of the MHD of MT19 (in green, signle measurements with error bars) and that from the C-EAGLE simulations (in blue, solid line). The shadows indicate the 1-$\sigma$ region for each method. A small scatter in the radial distance of the MT19 data has been added for clarity. Right panel. Histogram of $\zeta$ computed from all the contours taken inside the virial radius of each C-EAGLE cluster. The vertical (green) solid line represents the mean value of $\zeta$ obtained by MT19, embedded in its 1-$\sigma$ region. The dotted, vertical line indicates their lowest value. When measuring the MHD close to the virial radius of the clusters, we would expect an increase of its value, as the outskirts of clusters are not dynamically relaxed and fewer stellar particles are populating it, producing noisier contours. In order to compare the MHD across different distances, we define the relative MHD as $\zeta=\frac{d_{\mathrm{MH}}(r)}{r}\,,$ (3) where $r$ is the distance at which the iso-density contours have been computed. This way, we are measuring deviations as fraction of the distance. We find that this definition removes almost entirely the correlation with distance. ## 4 Results In the left panel of Fig. 4, we show with the blue, solid line the mean value of $d_{\mathrm{MH}}$, the shaded area depicts the 1-$\sigma$ confidence interval. We overplotted the MHDs calculated by MT19, as well as their 1-$\sigma$ area, in green. For sake of clarity, observational points for individual cluster are slightly displaced along the x-axis. From that panel, we can highlight that: 1. 1. the $d_{\mathrm{MH}}$ from both simulations and observations are of the same order of magnitude; 2. 2. they show the same trends with radius; 3. 3. and simulations have a $\sim 50\%$ lower $d_{\mathrm{MH}}$ than observations, with smaller scatter. As $d_{\mathrm{MH}}$ increases monotonically with the distance at which it is computed, we introduced the relative MHD, $\zeta$, to obtain a distance-free similarity measurement. We show in Fig. 4 (right panel) the distribution of $\zeta$ for all contours and clusters, in blue, and the $\zeta$ extracted from MT19’s data, in green. Most of the values of $\zeta$ are lower than those observed: 96 percent of the relative MHDs are below the mean observed value. The shape of the distribution is remarkably close to a Gaussian distribution in logarithmic space, with mean $\langle\zeta\rangle=0.107$ and dispersion $\sigma_{\zeta}=0.080$, indicating that $\zeta$ is a solid, scale-free estimate of the similarity of contours at any cluster-centric distance. Figure 5: Left panel. Stellar (solid lines) and total matter(dashed lines) surface density profiles from the particles of the main halo of the cluster. We consider only the ICL mass (see text), including the particles not bounded to any substructure. The dashed line is the threshold used for computing the ICL mass fraction (i.e., $\mu=26.5$ mag arcsec-2, or $\Sigma_{*}\approx 1.4\times 10^{6}~{}\mathrm{M}_{\odot}\,\mathrm{kpc}^{-2}$). Right panel. The ratio between the stellar and total matter density profiles for all the clusters. The red, dashed line is the best-fit power law given in equation 4. We would like to highlight two relevant issues with the definition of $d_{\mathrm{MH}}$ that can bias the observational values towards higher values. The $d_{\mathrm{MH}}$ is defined based on points and not continuous segments. This obviously simplifies the computation, but it has to be taken into account when dealing with coarse datasets, as two similar shapes can have a non-negligible $d_{\mathrm{MH}}$. Second, each point’s contribution is defined positive and with respect to the other set of points. This provides a distance that increases monotonically with noise (Dubuisson & Jain, 1994), thus special care must be taken when dealing with data with low signal-to- noise or large uncertainties. Both these points could be driving the observed $d_{\mathrm{MH}}$ towards higher values, as masking galaxies introduces non- continuous contours and the spatial resolution of the lensing models is limited. In addition to the study of the similarity between the total matter and stellar mass distribution, we have also compared the density profiles of the stellar component. In Fig. 5 (left panel) we show circularly averaged density profiles of the stellar particles. They follow a power-law behaviour up to $\sim 500~{}\mathrm{kpc}$ for the lightest halos, and $\sim 1~{}\mathrm{Mpc}$ for the more massive ones. At such distances, the interactions between substructures are weaker, and fewer particles get ejected to the intra-cluster medium, thus they can no longer successfully trace the potential well. In the right panel of Fig. 5 we show the ratio between the stellar and total matter density profiles. This ratio is close to a power law with scatter of $0.1~{}\mathrm{dex}$ and a slope of about $-1$. We have performed a fit to all the profiles at once, with and without normalising the radial distance using $r_{200}$, yielding the relations: $\displaystyle\log_{10}\Sigma_{\mathrm{tot}}=$ $\displaystyle\log_{10}\Sigma_{*}+$ $\displaystyle(1.115\pm 0.005)\log_{10}r-(0.25\pm 0.01)\,,$ (4) $\displaystyle\log_{10}\Sigma_{\mathrm{tot}}=$ $\displaystyle\log_{10}\Sigma_{*}+$ $\displaystyle(1.085\pm 0.004)\log_{10}(r/r_{200})+(3.144\pm 0.005)\,.$ (5) The residuals of both fits have a similar scatter: $0.147$ and $0.127$ dex for equations 4 and 5, respectively. We recall that the AGNdT9 feedback calibration, used in the C-EAGLE simulations, yields more compact galaxies than the reference model for stellar masses $M_{\star}>10^{10}~{}\mathrm{M}_{\odot}$. The less efficient tidal stripping may therefore deposit more stellar mass closer to the centre of the cluster, resulting in a steeper density profile. However, this bias may be of secondary importance, at least within the central $100~{}\mathrm{kpc}$ (Bahé et al., in prep.). We propose a new, indirect way of measuring a cluster’s mass knowing its stellar density profile in the innermost region. First, via deep imaging as that performed by MT19, the stellar density profile can be obtained and extrapolated up to $r_{200}$ assuming a power law. Then, using equation 4 or 5, the total mass density profile can be computed. This profile can be integrated to obtain an estimation of the cluster’s total mass. This procedure would be similar to that proposed by Pillepich et al. (2018). In that case, however, only the power law slope of the 3D stellar mass density profile was used to infer the total halo mass, in our case we use more information (the 2D stellar density profile and equation 4 or 5), expecting less scatter in the mass estimate. ## 5 Discussion & Conclusions We have studied the similarity of the projected stellar and total matter distributions in the halos of massive galaxy clusters using the C-EAGLE set of 30 zoom-in simulations of clusters of galaxies. In the analysis, we considered as constituents of the diffuse distribution of stellar mass only particles in the friends-of-friends group that were not assigned to any substructure by the SUBFIND algorithm. We can summarise our results as follows: 1. 1. we confirm the finding of MT19: the projected distribution of stars closely follow the projected distribution of the total mass, although their radial profiles differ substantially; 2. 2. the ICL, approximated as those stars in the region where $\mu>26.5~{}\mathrm{mag}\,\mathrm{arcsec}^{-2}$ ($\Sigma_{*}\approx 1.4\times 10^{6}~{}\mathrm{M}_{\odot}$), accounts for $\sim 10$ percent of the stellar content of the cluster within $r_{200}$; this fraction does not show any correlation with the mass of the cluster; 3. 3. the ratio between the surface density profiles of the stellar to the total matter follows a simple power-law up to the virial radius, equations 4 and 5; as the slope and amplitude of the stellar surface density profile can be extracted from observation, we proposed a method to estimate the total mass surface density profile, thus the mass of the halo; 4. 4. the similarity between the stellar and total matter distributions in the cluster halo is even higher in the simulations than that observed by MT19 (Fig. 4); This indicates that stars closely trace the underlying gravitational potential; 5. 5. in order to show any self-similarity, we have introduced the relative measure, $\zeta=d_{\mathrm{MH}}/r$, whose distribution resembles a log normal when using all the clusters and contours pairs; the parameter $\zeta$ could be used to study the relaxation state of a cluster; the maximum of this distribution is located at $\zeta\sim 0.1$, thus the typical $d_{\mathrm{MH}}$ is about $10\%$ of the distance at which it is computed. The study of the spatial distribution of the ICL can be used to infer, in high detail, the distribution of the underlying dark matter in clusters of galaxies. Moreover, the average density profile of total matter can be extracted, and extrapolated up to the virial radius, only by measuring the slope of the stellar mass density profile and its normalisation close to the centre of the cluster. This is complementary to the study of Pillepich et al. (2018), where only the total halo mass was given as function of the slope of the 3D stellar density profile, with larger uncertainty. ## Acknowledgements We are very grateful to Ignacio Trujillo and Mireia Montes for supporting this work with useful ideas and discussions. CDV acknowledges the support of the Spanish Ministry of Science, Innovation and Universities (MCIU) through grants RYC-2015-18078 and PGC2018-094975-B-C22. 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2024-09-04T02:54:58.674905

2003.04720

arxiv-papers

# The mean and variance in coupons required to complete a collection Rohit Pandey ###### Abstract This paper is about the Coupon collector’s problem. There are some coupons, or baseball cards, or other plastic knick-knacks that are put into bags of chips or under soda bottles, etc. A collector starts collecting these trinkets and wants to form a complete collection of all possible ones. Every time they buy the product however, they don’t know which coupon they will “collect” until they open the product. How many coupons do they need to collect before they complete the collection? In this paper, we explore the mean and variance of this random variable, $N$ using various methods. Some of them work only for the special case with the coupons having equal probabilities of being collected, while others generalize to the case where the coupons are collected with unequal probabilities (which is closer to a real world scenario). ## Problems and expressions ### Problems There are $n$ coupons in a collection. A collector has the ability to purchase a coupon, but can’t choose the coupon he purchases. Instead, the coupon is revealed to be coupon $i$ with probability $p_{i}=\frac{1}{n}$. Let $N$ be the number of coupons he’ll need to collect before he has at least one coupon of each type. Let’s call this random variable $N$. Now, we want to solve the following problems: P1 The expected value of $N$ when the coupons have equal probabilities of being collected. P2 The expected value of $N$ when the coupons have unequal probabilities of being collected. P3 The variance of $N$ when the coupons have equal probabilities of being collected. P4 The variance of $N$ when the coupons have unequal probabilities of being collected. P5 The density function of $N$ (meaning the entire distribution) when the coupons have equal probabilities. P6 The density function of $N$ (meaning the entire distribution) when the coupons have unequal probabilities. This paper will go over various solutions, some more powerful (can answer more of the above questions) than others. It’s also clear that if we can solve the even numbered problems (2,4,6) we can simply substitute $p_{i}=\frac{1}{n}\;\;\forall i$ and solve the corresponding odd numbered problems (1,3,5) respectively. ### Expressions In this section, we provide the solutions to the problems, P1 through P6 and devote the rest of the paper to their derivations. ###### Theorem 1 (Expression for P1). The expected number of coupons a collector will need to complete the collection when the probabilities of collecting each of the $n$ coupons is $\frac{1}{n}$ is: $E(N)=n\sum\limits_{m=1}^{n}\frac{1}{m}$ ###### Theorem 2 (Expression for P2). The variance in the number of coupons a collector will need to complete the collection when the probabilities of collecting each of the $n$ coupons is $\frac{1}{n}$ is: $V(N)=n^{2}\sum\limits_{i=1}^{n}\frac{1}{i^{2}}-n\sum\limits_{k=1}^{n}\frac{1}{k}$ ###### Theorem 3 (Expression for P3). The expected number of coupons a collector will need to complete the collection when the probabilities of collecting coupon $i$ is $p_{i}$ ($\sum\limits_{i=1}^{n}p_{i}=1$) is: $E(N)=\sum\limits_{j}\frac{1}{p}_{j}-\sum\limits_{i<j}\frac{1}{p_{i}+p_{j}}+\dots+(-1)^{m-1}\frac{1}{p_{1}+\dots+p_{m}}$ ###### Theorem 4 (Expression for P4). The variance in the number of coupons a collector will need to complete the collection when the probabilities of collecting coupon $i$ is $p_{i}$ ($\sum\limits_{i=1}^{n}p_{i}=1$) is: $V(N)=\left(\sum\frac{1}{p_{j}^{2}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})^{2}}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})^{2}}\right)-\\\ \left(\sum\frac{1}{p_{j}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})}\right)^{2}-\\\ \left(\sum\frac{1}{p_{j}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})}\right)$ ## 1 A sum of geometric random variables ### 1.1 Proof 1 of theorem 1 Consider a state where the collector has already collected $m$ coupons. How many coupons does he need to collect to get to $m+1$? Let this be represented by the random variable, $N_{m}$. Then, if the total coupons needed is $N$, we have: $N=\sum\limits_{m=1}^{n}N_{m}$ Every coupon collected from here is like a coin toss where with probability $\frac{m}{n}$, the collector hits a coupon he already has and makes no progress. With probability $\frac{n-m}{n}$, he collects a new coupon. So, this becomes a geometric random variable with $p=\frac{n-m}{n}$. We know that a geometric random variable has a mean $\frac{1}{p}$ and variance $\frac{1-p}{p^{2}}$. Hence, $E(N_{m})=\frac{n}{n-m}$ Taking expectation of equation (1) and substituting we have: $E(N)=E(N_{m})=\sum\limits_{m=1}^{n}\frac{n}{n-m}=n\sum\limits_{m=1}^{n}\frac{1}{n-m}$ Substituting $m=n-m$ we get: $E(N)=n\sum\limits_{m=1}^{n}\frac{1}{m}$ ### 1.2 Proof 1 of theorem 3 Since the random variables $N_{m}$ are independent, the variance of their sum is equal to the sum of their variances. So, proceeding similarly to section 1.1 the variance, $V(N)$ can be calculated. $V(N)=n^{2}\sum\limits_{i=1}^{n}\frac{1}{i^{2}}-n\sum\limits_{k=1}^{n}\frac{1}{k}$ ## 2 Maximum of minimums identity With this approach, we can prove theorems 1 and 2 ### 2.1 Proof 1 of theorem 3 Let $N_{j}$ be the number of coupons to be collected before we see the first coupon of type $j$ and $N$ the number of coupons until all are collected. We have: $N=\max_{1\leq j\leq n}N_{j}$ In conjunction with the maximum of minimums identity we get: $N=\sum N_{j}-\sum_{1\leq j\leq k\leq n}\min N_{j},N_{k}+\sum_{1\leq j\leq k\leq i\leq n}\min N_{j},N_{k},N_{i}-\dots$ (1) and the fact that $\min_{1\leq j\leq m}N_{j}$ is a geometric random variable with parameter $p=\sum\limits_{j=1}^{m}p_{j}$ lead to the result of theorem 3 and from there, we can substitute $p_{j}=\frac{1}{n}\forall j$ to get the result of theorem 1 $E(N)=n\sum\limits_{k=1}^{n}\frac{1}{k}$ Note that it’s not easy to get the variance, $V(N)$ with this approach because the terms in equation 1 are not independent. ## 3 A recurrence With this approach, we can prove theorems 1 and 3. Consider a state where the collector has $m$ coupons in his collection. Let $T_{m}$ be the number of coupons needed to complete the collection. If the total coupons he needs to collect to complete the collection is $N$, we then have: $N=T_{0}$ Now, we could observe that (the $N_{m}$ are the variables defined in section 1): $N_{m}=T_{m+1}-T_{m}$ and summing over all $m$ (and noting that $T_{n}=0$) leads us to: $T_{0}=\sum_{m}N_{m}$ and this leads to the approach in section 1 which makes the problem much easier to solve. Alternately, we can continue working with the $T_{m}$’s and construct a recurrence. Consider what happens when the collector has $m$ coupons and he collects one more. With probability $\frac{m}{n}$, he fails to add a new coupon and is back to where he started, making no progress. Let $I(\frac{n}{m})$ be a Bernoulli random variable with $p=\frac{n}{m}$. We then have the expression: $T_{m}=1+I\left(\frac{m}{n}\right)T_{m}^{\prime}+\left(1-I\left(\frac{m}{n}\right)\right)T_{m+1}$ (2) Where $T_{m}^{\prime}$ is i.i.d with $T_{m}$. ### 3.1 Proof 2 of theorem 1 Taking expectation to both sides, $E(T_{m})=1+\frac{m}{n}E(T_{m})+\frac{n-m}{n}T_{m+1}$ $E(T_{m})\left(1-\frac{m}{n}\right)=1+\left(1-\frac{m}{n}\right)T_{m+1}$ $E(T_{m})-E(T_{m+1})=\frac{n}{n-m}$ As noted before, the L.H.S is simply $E(N_{m})$ as defined in A1. In general we have, $\sum\limits_{m=k}^{n-1}E(T_{m})-\sum\limits_{m=k}^{n-1}E(T_{m+1})=\sum\limits_{m=k}^{n-1}\frac{n}{n-m}$ Noting that $T_{n}=0$ we have, $E(T_{k})=\sum\limits_{m=k}^{n-1}\frac{n}{n-m}$ And letting $m=n-k$ $E(T_{n-m})=n\sum\limits_{k=1}^{m}\frac{1}{k}$ We’re interested in $T_{0}$, so let’s substitute $m=n$ in equation (3). $E(T_{0})=n\sum\limits_{k=1}^{n}\frac{1}{k}$ ### 3.2 Proof 2 of theorem 3 Now, let’s try and find the variance, $V(N)=V(T_{0})$. Let’s square both sides of equation (1). To make the algebra easier, let’s re-arrange and note that $I(\frac{m}{n})(1-I(\frac{m}{n}))=I(\frac{m}{n})-I(\frac{m}{n})^{2}=0$. $=>(T_{m}-1)^{2}=I\left(\frac{m}{n}\right)^{2}T_{m}^{\prime 2}+(1+I\left(\frac{m}{n}\right)^{2}-2I\left(\frac{m}{n}\right))T_{m+1}^{2}$ Now, note the following property of Bernoulli random variables: $I(\frac{m}{n})^{2}=I(\frac{m}{n})$. This means: $T_{m}^{2}-2T_{m}+1=I\left(\frac{m}{n}\right)T_{m}^{\prime 2}+(1-I\left(\frac{m}{n}\right))T_{m+1}^{2}$ We have to be careful here to note which random variables are i.i.d. and which are identical. See here. Taking expectation and doing some algebra gives us, $\left(1-\frac{m}{n}\right)E(T_{m}^{2})=2E(T_{m})+\left(1-\frac{m}{n}\right)E(T_{m+1}^{2})-1$ $=>E(T_{m}^{2})-E(T_{m+1}^{2})=2E(T_{m})\frac{n}{n-m}-\frac{n}{n-m}$ $=>\sum\limits_{m=0}^{n-1}E(T_{m}^{2})-\sum\limits_{m=0}^{n-1}E(T_{m+1}^{2})=\sum\limits_{m=0}^{n-1}2E(T_{m})\frac{n}{n-m}-\sum\limits_{m=0}^{n-1}\frac{n}{n-m}$ $=>E(T_{0}^{2})-E(T_{n}^{2})=\sum\limits_{m=0}^{n-1}2E(T_{m})\frac{n}{n-m}-\sum\limits_{m=0}^{n-1}\frac{n}{n-m}$ But, $T_{n}=0$ and from equation (3), $E(T_{m})=n\sum\limits_{k=1}^{n-m}\frac{1}{k}$. So we get: $E(T_{0}^{2})=\sum\limits_{m=0}^{n-1}2E(T_{m})\frac{n}{n-m}-\sum\limits_{m=0}^{n-1}\frac{n}{n-m}$ $=>E(T_{0}^{2})=2n^{2}\sum\limits_{m=0}^{n-1}\frac{1}{n-m}\sum\limits_{k=1}^{n-m}\frac{1}{k}-n\sum\limits_{m=0}^{n-1}\frac{1}{n-m}$ Now, change variables $j=n-m$ $=>E(T_{0}^{2})=2n^{2}\sum\limits_{j=n}^{1}\frac{1}{j}\sum\limits_{k=1}^{j}\frac{1}{k}-n\sum\limits_{j=n}^{1}\frac{1}{j}$ $=>E(T_{0}^{2})=2n^{2}\sum\limits_{1\leq k\leq j\leq n}\frac{1}{jk}-E(T_{0})$ This can be used in conjunction with the result of theorem 1 to get the variance. $V(T_{0})=2n^{2}\sum\limits_{1\leq k\leq j\leq n}\frac{1}{jk}-E(T_{0})-E(T_{0})^{2}$ Substituting the result of theorem 1, $V(T_{0})=2n^{2}\sum\limits_{1\leq k\leq j\leq n}\frac{1}{jk}-n\sum\limits_{i=1}^{n}\frac{1}{i}-\left(n\sum\limits_{i=1}^{n}\frac{1}{i}\right)^{2}$ (3) Comparing equation 3 above with the result of theorem 3 we get the easily verifiable identity: $2\sum_{1\leq j\leq k\leq n}\frac{1}{jk}=\sum\limits_{i=1}^{n}\frac{1}{i^{2}}+\left(\sum\limits_{i=1}^{n}\frac{1}{i}\right)^{2}$ ## 4 Using a Poisson process to make dependence disappear Using the Poisson process to magically concoct independent random variables. This is the most powerful of all approaches since it’s the only one that allows us to solve for both mean and variance for the coupon collector’s problem for the general case of coupons having unequal probabilities (and higher moments as well). It is hence able to solve problems P1 through P4. In example 5.17 of [1], the Coupon collector’s problem is tackled for the general case where the probability of drawing coupon $j$ is given by $p_{j}$ and of course, $\sum\limits_{j}p_{j}=1$. Now, he imagines that the collector collects the coupons in accordance to a Poisson process with rate $\lambda=1$. Furthermore, every coupon that arrives is of type $j$ with probability $p_{j}$. Now, he defines $X_{j}$ as the first time a coupon of type $j$ is observed, if the $j$th coupon arrives in accordance to a Poisson process with rate $p_{j}$. We’re interested in the time it takes to collect all coupons, $X$ (for now, eventually, we’re interested in the number of coupons to be collected, $N$). So we get: $X=\max_{1\leq j\leq m}X_{j}$ Note that if we denote $N_{j}$ as the number of coupons to be collected before the first coupon of type $j$ is seen, we also have for the number needed to collect all coupons, $N$: $N=\max_{1\leq j\leq m}N_{j}$ This equation is less useful since the $N_{j}$ are not independent. It can still be used to get the mean (see section 2), but trying to get the variance with this approach gets considerably more challenging due to this lack of independence of the underlying random variables (the are positively correlated). But, the incredible fact that the $X_{j}$ are independent (discussion on that here), allows us to get: $F_{X}(t)=P(X<t)=P(X_{j}<t\;\forall\;j)=\prod\limits_{j=1}^{m}(1-e^{-p_{j}t})$ (4) ### 4.1 Proof 2 of theorem 2 Now, Ross uses the expression: $E(X)=\int\limits_{0}^{\infty}S_{X}(t)dt$, where $S_{X}(t)$ is the survival function to get: $E(X)=\int\limits_{0}^{\infty}\left(1-\prod\limits_{j=1}^{m}(1-e^{-p_{j}t})\right)dt$ $=\sum\limits_{j}\frac{1}{p}_{j}-\sum\limits_{i<j}\frac{1}{p_{i}+p_{j}}+\dots+(-1)^{m-1}\frac{1}{p_{1}+\dots+p_{m}}$ and this proves the result of theorem 2. ### 4.2 Proof 4 of theorem 1 In the special case of all coupons having equal probabilities of being collected we have: $p_{j}=\frac{1}{n}\forall j$ Substituting in the equation above we get: $E(X)=\sum\limits_{k=1}^{n}(-1)^{k}\frac{{n\choose k}}{k}$ (5) Let’s solve a general version of the binomial sum in equation 5. ###### Proposition 5. We have the following binomial sum: $\sum_{k=1}^{n}(-1)^{k-1}\frac{{n\choose k}}{k^{r}}=\sum_{i_{1}<i_{2}<\dots<i_{r}}\frac{1}{i_{1}i_{2}\dots i_{r}}$ ###### Proof. Using the Binomial theorem: $\frac{1-(1-t)^{n}}{t}=\sum\limits_{k=1}^{n}(-1)^{k-1}{{n\choose k}}{t^{k-1}}$ Integrate both sides from $0$ to $x$. $\int\limits_{0}^{x}\frac{1-(1-t)^{n}}{t}dx=\sum\limits_{k=1}^{n}(-1)^{k-1}{{n\choose k}}\frac{x^{k}}{k}$ For the LHS, let $1-t=u$ $\int\limits_{1}^{1-x}\frac{1-(u)^{n}}{1-u}(-du)=\sum\limits_{k=1}^{n}(-1)^{k-1}{{n\choose k}}\frac{x^{k}}{k}$ $\frac{\sum\limits_{k=1}^{n}\frac{1-(1-x)^{k}}{k}}{x}=\sum\limits_{k=1}^{n}(-1)^{k-1}\frac{{n\choose k}}{k}x^{k-1}$ Integrate both sides from $0$ to $1$, we get: $\sum\limits_{k=1}^{n}\frac{1}{k}\int\limits_{0}^{1}\frac{1-(1-x)^{k}}{x}dx=\sum\frac{{n\choose k}}{k^{2}}(-1)^{k-1}$ Substituting $1-x=t$ in the integral and expanding the geometric series we get: $\sum\limits_{k=1}^{n}\frac{1}{k}\sum\limits_{j=1}^{k}\frac{1}{j}=\sum\frac{{n\choose k}}{k^{2}}(-1)^{k-1}=\sum\limits_{k=1}^{n}\sum\limits_{j=1}^{k}\frac{1}{jk}$ This can very easily be extended to $k^{r}$ in the denominator: $\sum_{k=1}^{n}(-1)^{k-1}\frac{{n\choose k}}{k^{r}}=\sum_{i_{1}<i_{2}<\dots<i_{r}}\frac{1}{i_{1}i_{2}\dots i_{r}}$ (6) ∎ Substituting $r=1$ in equation 6 and equation 5 we have, $E(X)=n\sum\limits_{k=1}^{n}\frac{1}{k}$ Further, Ross shows that $E(N)=E(X)$ using the law of total expectation. First, he notes, $E(X|N=n)=nE(T_{i})$ where $T_{i}$ are the inter-arrival times for coupon arrivals. Since these are assume to be exponential with rate 1, $E(X|N)=N$ Taking expectations on both sides and using the law of total expectation we get: $E(X)=E(N)$ ### 4.3 Proof 1 of theorem 4 This approach can easily be extended to find $V(N)$, the variance (not covered by Ross). We can use the following expression to get $E(X^{2})$: $E(X^{2})=\int\limits_{0}^{\infty}2tP(X>t)dt=\int\limits_{0}^{\infty}2t\left(1-\prod\limits_{j=1}^{n}(1-e^{-p_{j}t})\right)dt$ Using the fact that $\int\limits_{0}^{\infty}te^{-pt}=\frac{1}{p^{2}}$ and the same algebra as for $E(X)$ we get: $\frac{E(X^{2})}{2}=\sum\frac{1}{p_{j}^{2}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})^{2}}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})^{2}}$ (7) Equation 7 has given us $E(X^{2})$ but remember that we’re interested in finding $E(N^{2})$ and from there, $V(N)$. So, we need to relate the variances of the two random variables. Using the law of total variance we get: $V(X)=E(V(X|N))+V(E(X|N))$ So per equation (3) we have: $V(X)=E(V(X|N))+V(N)$ Now, $V(X|N)=NV(T_{i})$ And since $T_{i}\sim Exp(1)$, we have $V(T_{i})=1$ meaning, $V(X|N)=N$. Substituting into (2), $V(X)=E(N)+V(N)$ So, $V(N)=E(X^{2})-E(N)-E(N)^{2}$ (8) Substituting equation 7 and the result of theorem 2 into equation 8 we get: $V(N)=\left(\sum\frac{1}{p_{j}^{2}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})^{2}}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})^{2}}\right)-\\\ \left(\sum\frac{1}{p_{j}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})}\right)^{2}-\\\ \left(\sum\frac{1}{p_{j}}-\sum_{i<j}\frac{1}{(p_{i}+p_{j})}+\dots+(-1)^{n-1}\frac{1}{(p_{1}+\dots+p_{n})}\right)$ (9) ### 4.4 Proof 3 of theorem 2 Now, let’s consider the special case where all coupons have an equal probability of being selected. In other words, $p_{j}=\frac{1}{n}\;\forall\;j$. We get: $\frac{E(X^{2})}{2}=n^{2}\left(\sum\limits_{k=1}^{n}(-1)^{k-1}\frac{{n\choose k}}{k^{2}}\right)$ (10) We now solve a general version of the binomial summation in equation 10 above. Using equations 6 and 10 we get: $E(X^{2})=2n^{2}\left(\sum_{j=1}^{n}\sum_{k=1}^{j}\frac{1}{jk}\right)$ (11) Using equations 11 and 8, we get the same result we got from the recurrence in section 3, equation 3. ## Acknowledgements I’d like to thank mathexchange user, Simon for encouraging me to convert the Q&A page on this into a paper. ## References * [1] Ross, S. (2010). Introduction to Probability Models, 10th ed. Elsevier.

2024-09-04T02:54:58.689254

2003.04730

arxiv-papers

# Strategy Logic with Imperfect Information Raphaël Berthon École Normale Supérieure de RennesComputer Science and TelecommunicationRennesFrance<EMAIL_ADDRESS>, Bastien Maubert 0000-0002-9081-2920 Università degli Studi di Napoli “Federico II”DIETINaplesItaly<EMAIL_ADDRESS>, Aniello Murano Università degli Studi di Napoli “Federico II”DIETINaplesItaly<EMAIL_ADDRESS>, Sasha Rubin Università degli Studi di Napoli “Federico II”DIETINaplesItaly <EMAIL_ADDRESS>and Moshe Y. Vardi Rice UniversityHoustonTexasUSA <EMAIL_ADDRESS> (September 2018) ###### Abstract. We introduce an extension of Strategy Logic for the imperfect-information setting, called $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, and study its model-checking problem. As this logic naturally captures multi-player games with imperfect information, this problem is undecidable; but we introduce a syntactical class of “hierarchical instances” for which, intuitively, as one goes down the syntactic tree of the formula, strategy quantifications are concerned with finer observations of the model, and we prove that model-checking $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ restricted to hierarchical instances is decidable. This result, because it allows for complex patterns of existential and universal quantification on strategies, greatly generalises the decidability of distributed synthesis for systems with hierarchical information. It allows us to easily derive new decidability results concerning strategic problems under imperfect information such as the existence of Nash equilibria, or rational synthesis. To establish this result we go through an intermediary, “low-level” logic much more adapted to automata techniques. $\textnormal{{QCTL}}^{*}$ is an extension of $\textnormal{{CTL}}^{*}$ with second-order quantification over atomic propositions that has been used to study strategic logics with perfect information. We extend it to the imperfect information setting by parameterising second-order quantifiers with observations. The simple syntax of the resulting logic, $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, allows us to provide a conceptually neat reduction of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ to $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ that separates concerns, allowing one to forget about strategies and players and focus solely on second-order quantification. While the model-checking problem of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is, in general, undecidable, we identify a syntactic fragment of hierarchical formulas and prove, using an automata-theoretic approach, that it is decidable. We apply our result to solve complex strategic problems in the imperfect- information setting. We first show that the existence of Nash equilibria for deterministic strategies is decidable in games with hierarchical information. We also introduce distributed rational synthesis, a generalisation of rational synthesis to the imperfect-information setting. Because it can easily be expressed in our logic, our main result provides a solution to this problem in the case of hierarchical information. strategic reasoning, imperfect information, perfect recall, distributed synthesis, hierarchical information, Nash equilibria, rational synthesis ††journal: TOCL††copyright: acmlicensed††doi: 0000001.0000001††ccs: Theory of computation Logic and verification††ccs: Theory of computation Modal and temporal logics††ccs: Theory of computation Automata over infinite objects ## 1\. Introduction Temporal logics such as LTL (Pnueli, 1977) or $\textnormal{{CTL}}^{*}$ (Emerson and Halpern, 1986) are extremely successful logics that have been studied in great detail and extended in many directions along the past decades, notably in relation with the development of the model-checking approach to program verification (Clarke et al., 1999). When considering systems with multiple components such as multi-agent systems or distributed programs, popular extensions of temporal logics are the family of so-called _logics for strategic reasoning_ , or _strategic logics_ , which introduce operators that can express the existence of strategies for components to ensure that the system’s executions satisfy certain temporal properties. A fundational logic in this family is Alternating-time Temporal Logic (ATL) (Alur et al., 2002). It extends $\textnormal{{CTL}}^{*}$ with a coalition operator $\langle A\rangle\varphi$, where $A$ is a subset of components/agents of the system, which reads as “coalition $A$ has a strategy to enforce property $\varphi$ no matter what the other components/agents do”. This logic is thus quite expressive, as it allows for instance to express the existence of winning strategies in games played on graphs. However it is not well suited to reason about other important solution concepts in game theory, such as Nash equilibria. To address this problem Strategy Logic (SL) was introduced (Chatterjee et al., 2010a; Mogavero et al., 2014). In SL strategies are treated as first-order objects, thanks to strategy variables $x$ that can be quantified upon and bound to players: $\langle\\!\langle x\rangle\\!\rangle$ reads as “there exists a strategy $x$”, and $(a,x)$ reads as “strategy $x$ is assigned to player $a$”. This leads to a very expressive logic that can express many solution concepts from game-theory such as best response, existence of Nash equilibria or subgame-perfect equilibria. Imperfect information. An essential property of realistic multi-player games is that players often have a limited view of the system. Such imperfect information, or partial observation, is usually captured by equipping the models with equivalence relations $o$ (called _observations_) over the state space, that specify indistinguishable states. Strategies are then required to be _uniform_ , i.e., they cannot assign different moves to indistinguishable situations. Imperfect information is known to make games computationally harder to solve. For two-player reachability games, Reif showed in (Reif, 1984) that deciding the existence of winning strategies is Exptime -complete for imperfect information, while it is in Ptime for perfect information. This result has later been generalised to omega-regular objectives (Berwanger et al., 2010; Doyen and Raskin, 2011), and adapted to the setting of program synthesis from temporal specifications (Pnueli and Rosner, 1989; Kupferman and Vardi, 1999). In the case of multiple players/components/agents, which interests us here, the situation is even worse: the existence of distributed winning strategies is undecidable already for two players with incomparable observation trying to enforce some reachability objective in the presence of an adversarial third player (Peterson and Reif, 1979), and a similar result was also proved in the framework of distributed synthesis (Pnueli and Rosner, 1990). Since then, the formal-methods community has spent much effort finding restrictions and variations that ensure decidability (Kupferman and Vardi, 2001; Pnueli and Rosner, 1990; Gastin et al., 2009; Peterson et al., 2002; Finkbeiner and Schewe, 2005; Pinchinat and Riedweg, 2005; Schewe and Finkbeiner, 2007; Berwanger et al., 2018). The common thread in these approaches is hierarchical information: players can be totally ordered according to how well they observe the game. Another line of works establishes that decidability can be retained by forbidding private communication, i.e., by considering variants around the idea that all new information should be public (van der Meyden and Vardi, 1998; van der Meyden and Wilke, 2005; Ramanujam and Simon, 2010; Belardinelli et al., 2017b, a; Bouyer, 2018). Strategy Logic with imperfect information. We propose an extension of Strategy Logic to the imperfect-information setting, which we call $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. The first step is to choose how to introduce imperfect information in the logic. In the formal-methods literature it is typical to associate observations to players. In $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, instead, we associate observations to strategies: the strategy quantifier $\langle\\!\langle x\rangle\\!\rangle{}$ from SL is now parameterised by observation $o$, written $\langle\\!\langle x\rangle\\!\rangle^{o}$. This novelty allows one to express, in the logic, that a player’s observation changes over time, to capture for instance the loss of a sensor resulting in a diminished observation power. We also add to our logic $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ the outcome quantifier ${\bf A}$ from Branching-time Strategy Logic (BSL) (Knight and Maubert, 2019), which quantifies on outcomes of strategies currently used by the agents, and the unbinding operator $(a,\operatorname{?})$, which frees an agent from her current strategy. This does not increase the expressivity of the logic but presents advantages that we discuss in Section 2.2. For instance it allows us to naturally consider nondeterministic strategies (Strategy Logic only considers deterministic ones), which in turn allows us to capture module checking, the extension of model checking to open systems (Kupferman et al., 2001; Jamroga and Murano, 2014, 2015). The logic $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ is very powerful: it is an extension of SL (which considers perfect information), and of the imperfect-information strategic logics $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$ (Bulling and Jamroga, 2014) and $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc,i}}$ (Laroussinie et al., 2015). As already mentioned, $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ can express the distributed synthesis problem (Pnueli and Rosner, 1990). This problem asks whether there are strategies for components $a_{1},\dots,a_{n}$ of a distributed system to enforce some property given as an LTL formula $\psi$ against all behaviours of the environment. This can be expressed by the $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula $\Phi_{\textsc{Synth}}:=\langle\\!\langle x_{1}\rangle\\!\rangle^{o_{1}}\dots\langle\\!\langle x_{n}\rangle\\!\rangle^{o_{n}}(a_{1},x_{1})\dots(a_{n},x_{n}){\bf A}\psi$, where $o_{i}$ represents the local view of component $a_{i}$. Also, $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ can express more complicated specifications by alternating quantifiers, binding the same strategy to different agents and rebinding (these are inherited from SL), as well as changing observations. For instance, it can express the existence of Nash equilibria. Main result. Of course, the high expressivity of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ comes at a cost from a computational complexity point of view. Its satisfiability problem is undecidable (this is already true of SL), and so is its model-checking problem (this is already true of $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$ even for the single formula $\langle\\{a,b\\}\rangle{\bf F}p$ (Dima and Tiplea, 2011), which means that agents $a$ and $b$ have a strategy profile to reach a situation where $p$ holds). We mentioned that the two main settings in which decidability is retrieved for distributed synthesis are hierarchical information and public actions. We extend the first approach to the setting of strategic logics by introducing a syntactic class of “hierarchical instances” of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, i.e., formula/model pairs, and proving that the model-checking problem on this class of instances is decidable. Intuitively, an instance of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ is hierarchical if, as one goes down the syntactic tree of the formula, the observations annotating strategy quantifications can only become finer. Although the class of hierarchical instances refers not only to the syntax of the logic but also to the model, the class is syntactical in the sense that it depends only on the structure of the formula and the observations in the model. Moreover, it is straightforward to check (in linear time) whether an instance is hierarchical or not. Applications. Because the syntax of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ allows for arbitrary alternations of quantifiers in the formulas, our decidability result for hierarchical instances allows one to decide strategic problems more involved than module checking and distributed synthesis. For instance, we show in Section 7 how one can apply our result to establish that the existence of Nash equilibria is decidable in games with imperfect information, in the case of hierarchical observations and deterministic strategies. This problem is relevant as Nash equilibria do not always exist in games with imperfect information (Filiot et al., 2018). We then consider the problem of rational synthesis (Fisman et al., 2010; Kupferman et al., 2016; Condurache et al., 2016; Filiot et al., 2018), both in its cooperative and non-cooperative variants. We introduce the generalisations of these problems to the case of imperfect information, and call them cooperative and non-cooperative _rational distributed synthesis_. We then apply again our main result to establish that they are decidable in hierarchical systems for deterministic strategies. For the non-cooperative variant, we need the additional assumption that the environment is at least as informed as the system. This is the case for example when one ignores the actual observation power of the environment, and considers that it plays with perfect information. Doing so yields systems that are robust to any observation power the environment may have. As Reif puts it, this amounts to synthesising strategies that are winning even if the opponent “cheats” and uses information it is not supposed to have access to (Reif, 1984). Approach. In order to solve the model-checking problem for $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ we introduce an intermediate logic $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, an extension to the imperfect-information setting of $\textnormal{{QCTL}}^{*}$ (Laroussinie and Markey, 2014), itself an extension of $\textnormal{{CTL}}^{*}$ by second- order quantifiers over atoms. This is a low-level logic that does not mention strategies and into which one can effectively compile instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. States of the models of the logic $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ have internal structure, much like the multi-player game structures from (Peterson et al., 2001) and distributed systems (Halpern and Vardi, 1989). Model-checking $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is also undecidable (indeed, we show how to reduce from the MSO-theory of the binary tree extended with the equal-length predicate, known to be undecidable (Läuchli and Savioz, 1987)). We introduce the syntactical class $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ of hierarchical formulas as those in which innermost quantifiers observe more than outermost quantifiers, and prove that model-checking is decidable using an extension of the automata-theoretic approach for branching-time logics. We provide a reduction from model checking $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ to model checking $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ that preserves being hierarchical, thus establishing our main contribution, i.e., that model checking the hierarchical instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ is decidable. Complexity. To establish the precise complexity of the problems we solve, we introduce a new measure on formulas called _simulation depth_. This measure resembles the notion of alternation depth (see, e.g., (Mogavero et al., 2014)), which counts alternations between existential and universal strategy (or second-order) quantifications. But instead of merely counting alternations between such operators, simulation depth reflects the underlying automata operations required to treat formulas, while remaining a purely syntactical notion. We prove that the model-checking problem for the hierarchical fragment of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ and $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ are both $(k+1)$-Exptime -complete for formulas of simulation depth at most $k$. Already for the perfect-information fragment, this result is more precise than what was previously known. Indeed, precise upper bounds based on alternation depth were known for syntactic fragments of SL but not for the full logic (Mogavero et al., 2014). Related work. The literature on imperfect information in formal methods and artificial intelligence is very vast. Imperfect information has been considered in two-player games (Reif, 1984; Doyen and Raskin, 2011; Berwanger et al., 2010), module checking (Kupferman et al., 2001; Jamroga and Murano, 2015), distributed synthesis of reactive systems (Pnueli and Rosner, 1990; Kupferman and Vardi, 2001; Finkbeiner and Schewe, 2005) and strategies in multiplayer games (Peterson and Reif, 1979; Peterson et al., 2002; Berwanger et al., 2018), Nash equilibria (Ramanujam and Simon, 2010; Bouyer et al., 2017; Bouyer, 2018), rational synthesis (Filiot et al., 2018; Gutierrez et al., 2018), doomsday equilibria (Chatterjee et al., 2017), admissible strategies (Brenguier et al., 2017), quantitative objectives (Degorre et al., 2010; Pérez, 2017), and more, some of which we detail below. Limited alternation of strategy quantification was studied in (Chatterjee and Doyen, 2014a), in which several decidability results are proved for two and three alternations of existential and universal quantifiers. Except for one where the first player has perfect information, all the problems solved in this work are hierarchical instances, and are thus particular cases of our main result. Quantified $\mu$-Calculus with partial observation is studied in (Pinchinat and Riedweg, 2005), where the model-checking problem is solved by considering a syntactic constraint based on hierarchical information, as we do for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$. However they consider asynchronous perfect recall, and the automata techniques they use to deal with imperfect information cannot be used in the synchronous perfect-recall setting that we consider in this work. Similarly the narrowing operation on tree automata (see Section 4.1), which is crucial in our model-checking procedure, considers synchronous perfect recall and does not seem easy to adapt to the asynchronous setting. A number of works have considered strategic logics with imperfect information. Various semantics for ATL with imperfect information have been studied in, e.g., (Jamroga and Bulling, 2011; Jamroga and van der Hoek, 2004). The model- checking problem for these logics, which is undecidable for agents with perfect recall (Dima and Tiplea, 2011), has been studied for agents with bounded memory, for which decidability is recovered (Schobbens, 2004; Lomuscio and Raimondi, 2006). An epistemic strategic logic with original operators different from those of ATL and SL is proposed in (Huang and Van Der Meyden, 2014). It considers imperfect information strategies, but only for agents without memory. Concerning perfect recall, which interest us in this work, decidability results have also been obtained for ATL (Guelev et al., 2011) and ATL with strategy context (Laroussinie et al., 2015) when agents have the same information. In (Knight and Maubert, 2019), a branching-time variant of SL is extended with epistemic operators and agents with perfect recall. Strategies are not required to be uniform in the semantics, but this requirement can be expressed in the language. However no decidability result is provided. Another variant of SL extended with epistemic operators and imperfect-information, perfect- recall strategies is presented in (Belardinelli, 2015), but model checking is not studied. The latter logic is extended in (Belardinelli et al., 2017a), in which its model-checking problem is solved on the class of systems where all agents’ actions are public, which is an assumption orthogonal to hierarchical information. The work closest to ours is (Finkbeiner and Schewe, 2010) which introduces a logic CL in which one can encode many distributed synthesis problems. In this logic, hierarchical information is a necessary consequence of the syntax and semantics, and as a result its model-checking problem is decidable. However, CL is close in spirit to our $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$, and its semantics is less intuitive than that of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. Furthermore, by means of a natural translation we derive that CL is strictly included in the hierarchical instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ (Section 6.2). In particular, hierarchical instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ can express non-observable goals, while CL cannot. When considering players that choose their own goals it may be natural to assume that they can observe the facts that define whether their objectives are satisfied or not. But when synthesising programs for instance, it may be enough that their behaviours enforce the desired properties, without them having the knowledge that it is enforced. Such non- observable winning conditions have been studied in, e.g., (Chatterjee and Doyen, 2010; Degorre et al., 2010; Berwanger et al., 2018). Outline. In Section 2 we define $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ and hierarchical instances, and present some examples. In Section 3 we define $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ and its hierarchical fragment $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$. The proof that model checking $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ is decidable, including the required automata preliminaries, is in Section 4. The hierarchy-preserving translation of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ into $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is in Section 5. In Section 6 we compare $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ with related logics, and in Section 7 we apply our main result to obtain decidability results for various strategic problems under imperfect information. Finally we conclude and discuss future work in Section 8. ## 2\. SL with imperfect information In this section we introduce $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, an extension of SL to the imperfect-information setting with synchronous perfect-recall. Our logic presents several original features compared to SL, which we discuss in detail in Section 2.3: we introduce an _outcome quantifier_ akin to the path quantifier in branching-time temporal logics, we allow for nondeterministic strategies and unbinding agents from their strategies, and we annotate strategy quantifiers with observation symbols which denote the information available to strategies. We first fix some basic notations. ### 2.1. Notations Let $\Sigma$ be an alphabet. A _finite_ (resp. _infinite_) _word_ over $\Sigma$ is an element of $\Sigma^{*}$ (resp. $\Sigma^{\omega}$). Words are written $w=w_{0}w_{1}w_{2}\ldots$, i.e., indexing begins with $0$. The _length_ of a finite word $w=w_{0}w_{1}\ldots w_{n}$ is $|w|:=n+1$, and $\mbox{last}(w):=w_{n}$ is its last letter. Given a finite (resp. infinite) word $w$ and $0\leq i<|w|$ (resp. $i\in\mathbb{N}$), we let $w_{i}$ be the letter at position $i$ in $w$, $w_{\leq i}$ is the prefix of $w$ that ends at position $i$ and $w_{\geq i}$ is the suffix of $w$ that starts at position $i$. We write $w\preccurlyeq w^{\prime}$ if $w$ is a prefix of $w^{\prime}$, and $\textit{pref}\,(w)$ is the set of finite prefixes of word $w$. Finally, the domain of a mapping $f$ is written $\textit{dom}(f)$, its codomain $\textit{codom}(f)$, and for $n\in\mathbb{N}$ we let $[n]:=\\{i\in\mathbb{N}:1\leq i\leq n\\}$. ### 2.2. Syntax For the rest of the paper, for convenience we fix a number of parameters for our logics and models: AP is a finite non-empty set of _atomic propositions_ , Ag is a finite non-empty set of _agents_ or _players_ , and Var is a finite non-empty set of _variables_. The main novelty of our logic is that we specify which information is available to a strategy, by annotating strategy quantifiers $\langle\\!\langle x\rangle\\!\rangle$ with _observation symbols_ $o$ from a finite set Obs, that we also fix for the rest of the paper. When we consider model-checking problems, these data are implicitly part of the input. ###### Definition 2.1 ($\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ Syntax). The syntax of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ is defined by the following grammar: $\displaystyle\varphi:=$ $\displaystyle\;p\mid\neg\varphi\mid\varphi\vee\varphi\mid\langle\\!\langle x\rangle\\!\rangle^{o}\varphi\mid(a,x)\varphi\mid(a,\operatorname{?})\varphi\mid{\bf E}\psi$ $\displaystyle\psi:=$ $\displaystyle\;\varphi\mid\neg\psi\mid\psi\vee\psi\mid{\bf X}\psi\mid\psi{\bf U}\psi$ where $p\in\textnormal{AP}$, $x\in\textnormal{Var}$, $o\in\textnormal{Obs}$ and $a\in\textnormal{Ag}$. Formulas of type $\varphi$ are called _state formulas_ , those of type $\psi$ are called _path formulas_ , and $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ consists of all the state formulas defined by the grammar. Boolean operators and temporal operators, ${\bf X}$ (read “next”) and ${\bf U}$ (read “until”), have the usual meaning. The _strategy quantifier_ $\langle\\!\langle x\rangle\\!\rangle^{o}$ is a first-order-like quantification on strategies: $\langle\\!\langle x\rangle\\!\rangle^{o}\varphi$ reads as “there exists a strategy $x$ that takes decisions based on observation $o$ such that $\varphi$ holds”, where $x$ is a strategy variable. The _binding operator_ $(a,x)$ assigns a strategy to an agent, and $(a,x)\varphi$ reads as “when agent $a$ plays strategy $x$, $\varphi$ holds”. The _unbinding operator_ $(a,\operatorname{?})$ instead releases agent $a$ from her current strategy, if she has one, and $(a,\operatorname{?})\varphi$ reads as “when agent $a$ is not assigned any strategy, $\varphi$ holds”. Finally, the _outcome quantifier_ ${\bf E}$ quantifies on outcomes of strategies currently in use: ${\bf E}\psi$ reads as “$\psi$ holds in some outcome of the strategies currently used by the players”. We use abbreviations $\top:=p\vee\neg p$, $\perp:=\neg\top$, $\varphi\to\varphi^{\prime}:=\neg\varphi\vee\varphi^{\prime}$, $\varphi\leftrightarrow\varphi^{\prime}:=\varphi\to\varphi^{\prime}\wedge\varphi^{\prime}\to\varphi$ for boolean connectives, ${\bf F}\varphi:=\top{\bf U}\varphi$ (read “eventually $\varphi$”), ${\bf G}\varphi:=\neg{\bf F}\neg\varphi$ (read “globally $\varphi$”) for temporal operators, $[\\![x]\\!]^{o}\varphi:=\neg\langle\\!\langle x\rangle\\!\rangle^{o}\neg\varphi$ (read “for all strategies $x$ based on observation $o$, $\varphi$ holds”) and ${\bf A}\psi:=\neg{\bf E}\neg\psi$ (read “all outcomes of the current strategies satisfy $\psi$”). For every formula $\varphi\in\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, we let $\textit{free}\,(\varphi)$ be the set of variables that appear free in $\varphi$, i.e., that appear out of the scope of a strategy quantifier. A formula $\varphi$ is a _sentence_ if $\textit{free}\,(\varphi)$ is empty. Finally, we let the _size_ $|\varphi|$ of a formula $\varphi$ be the number of symbols in $\varphi$. ### 2.3. Discussion on the syntax We discuss the syntactic differences between our logic and usual Strategy Logic. Outcome quantifier. This quantifier was introduced in Branching-time Strategy Logic (BSL) (Knight and Maubert, 2019), which corresponds to the perfect- information fragment of the logic we define here. It removes a quirk of previous definitions, in which temporal operators could only be evaluated in contexts where all agents were assigned a strategy. The outcome quantifier, instead, allows for evaluation of temporal properties on partial assignments. As a result, the notions of free agents and agent-complete assignments from previous definitions of Strategy Logic are no longer needed (see, e.g., (Mogavero et al., 2014)). In addition, the outcome quantifier highlights the inherent branching-time nature of Strategy Logic: indeed, in SL, branching- time properties can be expressed by resorting to artificial strategy quantifications for all agents. It will also make the correspondence with $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ tighter, which will allow us to establish the precise complexity of the problem we solve, while the exact complexity of model checking classic SL with perfect information is still not known. Finally, since the usual definition of SL requires that the current strategies define a unique outcome on which linear-time temporal operators are evaluated, only deterministic strategies were considered. The introduction of the outcome quantifier allows us to consider nondeterministic strategies. Unbinding. With the possibility to evaluate temporal operators even when some agents are not bound to any strategy, it becomes interesting to include the unbinding operator $(a,\operatorname{?})$, introduced in (Laroussinie and Markey, 2015) for ATL with strategy context and also present in BSL. Note that the outcome quantifier and unbinding operator do not increase the expressivity of SL, at the level of sentences (Knight and Maubert, 2019). Observations. In games with imperfect information and ATL-like logics with imperfect information, a strategy is always bound to some player, and thus it is clear with regards to what observations it should be defined. In SL on the other hand, strategy quantification and binding are separate. This adds expressive power with regards to ATL by allowing, for instance, to assign the same strategy to two different players, but it also entails that when a quantification is made on a strategy, one does not know with regards to which observation this strategy should be defined. We know of three ways to solve this. One is the approach followed here, which consists in associating with strategy quantifiers an observation power. The second solution is to abandon the separation between quantification and binding and to use instead quantifiers of the form $\exists_{a}$, meaning “there exists a strategy for player $a$”, like in (Chatterjee et al., 2010b; Belardinelli, 2014): with this operator, the strategy is immediately bound to player $a$, which indicates with regards to which observation the strategy should be compatible. The third one, adopted in (Belardinelli et al., 2017a), consists in requiring that a strategy be uniform for all agents to whom it will be bound in the formula. We chose to adopt the first solution for its simplicity and expressiveness. Indeed the second solution limits expressiveness by disallowing, for instance, binding the same strategy to different agents. The third solution leads to a logic that is more expressive than the second one, but less than the first one. Indeed, the logic that we study here can capture the logic from (Belardinelli et al., 2017a) (assuming that models contain observations corresponding to unions of individual observations), and in addition $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ can express changes of agents’ observation power. ### 2.4. Semantics The models of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ are classic concurrent game structures extended by an interpretation for observation symbols in Obs. ###### Definition 2.2 ($\textrm{CGS}_{\textnormal{ii}}$ ). A _concurrent game structure with imperfect information_ (or $\textrm{CGS}_{\textnormal{ii}}$ for short) is a tuple $\mathcal{G}=(\textnormal{Ac},V,E,\ell,v_{\iota},\mathcal{O})$ where * • Ac is a finite non-empty set of _actions_ , * • $V$ is a finite non-empty set of _positions_ , * • $E:V\times\textnormal{Ac}^{\textnormal{Ag}}\to V$ is a _transition function_ , * • $\ell:V\to 2^{\textnormal{AP}}$ is a _labelling function_ , * • $v_{\iota}\in V$ is an _initial position_ , and * • $\mathcal{O}:\textnormal{Obs}\to 2^{V\times V}$ is an _observation interpretation_. For $o\in\textnormal{Obs}$, $\mathcal{O}(o)$ is an equivalence relation on positions, that we may write $\sim_{o}$. It represents what a strategy with observation $o$ can see: $\mathcal{O}(o)$-equivalent positions are indistinguishable to such a strategy. Also, $\ell(v)$ is the set of atomic propositions that hold in position $v$. We define the size $|\mathcal{G}|$ of a $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}=(\textnormal{Ac},V,E,\ell,v_{\iota},\mathcal{O})$ as the size of an explicit encoding of the transition function: $|\mathcal{G}|:=|V|\times|\textnormal{Ac}|^{|\textnormal{Ag}|}\times\lceil\log(|V|)\rceil$. We may write $v\in\mathcal{G}$ for $v\in V$. We now introduce a number of notions involved in the semantics of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. Consider a $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}=(\textnormal{Ac},V,E,\ell,v_{\iota},\mathcal{O})$. Joint actions. In a position $v\in V$, each player $a$ chooses an action $c_{a}\in\textnormal{Ac}$, and the game proceeds to position $E(v,\bm{c})$, where $\bm{c}\in\textnormal{Ac}^{\textnormal{Ag}}$ stands for the _joint action_ $(c_{a})_{a\in\textnormal{Ag}}$. Given a joint action $\bm{c}=(c_{a})_{a\in\textnormal{Ag}}$ and $a\in\textnormal{Ag}$, we let $\bm{c}_{a}$ denote $c_{a}$. Plays. A _finite_ (resp. _infinite_) _play_ is a finite (resp. infinite) word $\rho=v_{0}\ldots v_{n}$ (resp. $\pi=v_{0}v_{1}\ldots$) such that $v_{0}=v_{\iota}$ and for every $i$ such that $0\leq i<|\rho|-1$ (resp. $i\geq 0$), there exists a joint action $\bm{c}$ such that $E(v_{i},\bm{c})=v_{i+1}$. Strategies. A (nondeterministic) _strategy_ is a function $\sigma:V^{+}\to 2^{\textnormal{Ac}}\setminus\emptyset$ that maps each finite play to a nonempty finite set of actions that the player may play. A strategy $\sigma$ is _deterministic_ if for all $\rho$, $\sigma(\rho)$ is a singleton. We let Str denote the set of all strategies. Assignments. An _assignment_ is a partial function $\chi:\textnormal{Ag}\cup\textnormal{Var}\rightharpoonup\mbox{\emph{Str}}$, assigning to each player and variable in its domain a strategy. For an assignment $\chi$, a player $a$ and a strategy $\sigma$, $\chi[a\mapsto\sigma]$ is the assignment of domain $\textit{dom}(\chi)\cup\\{a\\}$ that maps $a$ to $\sigma$ and is equal to $\chi$ on the rest of its domain, and $\chi[x\mapsto\sigma]$ is defined similarly, where $x$ is a variable; also, $\chi[a\mapsto\operatorname{?}]$ is the restriction of $\chi$ to domain $\textit{dom}(\chi)\setminus\\{a\\}$. In addition, given a formula $\varphi\in\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, an assignment is _variable-complete for $\varphi$_ if its domain contains all free variables of $\varphi$. Outcomes. For an assignment $\chi$ and a finite play $\rho$, we let $\textnormal{Out}(\chi,\rho)$ be the set of infinite plays that start with $\rho$ and are then extended by letting players follow the strategies assigned by $\chi$. Formally, $\textnormal{Out}(\chi,\rho)$ is the set of plays of the form $\rho\cdot v_{1}v_{2}\ldots$ such that for all $i\geq 0$, there exists $\bm{c}$ such that for all $a\in\textit{dom}(\chi)\cap\textnormal{Ag}$, $\bm{c}_{a}\in\chi(a)(\rho\cdot v_{1}\ldots v_{i})$ and $v_{i+1}=E(v_{i},\bm{c})$, with $v_{0}=\mbox{last}(\rho)$. Synchronous perfect recall. In this work we consider players with _synchronous perfect recall_ , meaning that each player remembers the whole history of a play, a classic assumption in games with imperfect information and logics of knowledge and time. Each observation relation is thus extended to finite plays as follows: $\rho\sim_{o}\rho^{\prime}$ if $|\rho|=|\rho^{\prime}|$ and $\rho_{i}\sim_{o}\rho^{\prime}_{i}$ for every $i\in\\{0,\ldots,|\rho|-1\\}$. Imperfect-information strategies. For $o\in\textnormal{Obs}$, a strategy $\sigma$ is an _$o$ -strategy_ if $\sigma(\rho)=\sigma(\rho^{\prime})$ whenever $\rho\sim_{o}\rho^{\prime}$. The latter constraint captures the essence of imperfect information, which is that players can base their strategic choices only on the information available to them. For $o\in\textnormal{Obs}$ we let $\mbox{\emph{Str}}_{o}$ be the set of all $o$-strategies. ###### Definition 2.3 ($\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ semantics). The semantics of a state formula is defined on a $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$, an assignment $\chi$ that is variable-complete for $\varphi$, and a finite play $\rho$. For a path formula $\psi$, the finite play is replaced with an infinite play $\pi$ and an index $i\in\mathbb{N}$. The definition by mutual induction is as follows: $\begin{array}[]{lcl}\mathcal{G},\chi,\rho\models p&\text{ if }&p\in\ell(\mbox{last}(\rho))\\\\[1.0pt] \mathcal{G},\chi,\rho\models\neg\varphi&\text{ if }&\mathcal{G},\chi,\rho\not\models\varphi\\\\[1.0pt] \mathcal{G},\chi,\rho\models\varphi\vee\varphi^{\prime}&\text{ if }&\mathcal{G},\chi,\rho\models\varphi\;\text{ or }\;\mathcal{G},\chi,\rho\models\varphi^{\prime}\\\\[1.0pt] \mathcal{G},\chi,\rho\models\langle\\!\langle x\rangle\\!\rangle^{o}\varphi&\text{ if }&\exists\,\sigma\in\mbox{\emph{Str}}_{o}\;\text{ s.t. }\;\mathcal{G},\chi[x\mapsto\sigma],\rho\models\varphi\\\\[1.0pt] \mathcal{G},\chi,\rho\models(a,x)\varphi&\text{ if }&\mathcal{G},\chi[a\mapsto\chi(x)],\rho\models\varphi\\\\[1.0pt] \mathcal{G},\chi,\rho\models(a,\operatorname{?})\varphi&\text{ if }&\mathcal{G},\chi[a\mapsto\operatorname{?}],\rho\models\varphi\\\\[1.0pt] \mathcal{G},\chi,\rho\models{\bf E}\psi&\text{ if }&\text{there exists }\pi\in\textnormal{Out}(\chi,\rho)\text{ such that }\mathcal{G},\chi,\pi,|\rho|-1\models\psi\\\\[5.0pt] \mathcal{G},\chi,\pi,i\models\varphi&\text{ if }&\mathcal{G},\chi,\pi_{\leq i}\models\varphi\\\\[1.0pt] \mathcal{G},\chi,\pi,i\models\neg\psi&\text{ if }&\mathcal{G},\chi,\pi,i\not\models\psi\\\\[1.0pt] \mathcal{G},\chi,\pi,i\models\psi\vee\psi^{\prime}&\text{ if }&\mathcal{G},\chi,\pi,i\models\psi\;\text{ or }\;\mathcal{G},\chi,\pi,i\models\psi^{\prime}\\\\[1.0pt] \mathcal{G},\chi,\pi,i\models{\bf X}\psi&\text{ if }&\mathcal{G},\chi,\pi,i+1\models\psi\\\\[1.0pt] \mathcal{G},\chi,\pi,i\models\psi{\bf U}\psi^{\prime}&\text{ if }&\exists\,j\geq i\mbox{ s.t. }\mathcal{G},\chi,\pi,j\models\psi^{\prime}\\\ &&\text{ and }\forall\,k\text{ s.t. }i\leq k<j,\;\mathcal{G},\chi,\pi,k\models\psi\end{array}$ ###### Remark 1. Observe that because of the semantics of the outcome quantifier, and unlike usual definitions of SL, the meaning of an $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ sentence depends on the assignment in which it is evaluated. For instance the $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula ${\bf A}{\bf F}p$ is clearly a sentence, but whether $\mathcal{G},\chi,\rho\models{\bf A}{\bf F}p$ holds or not depends on which agents are bound to a strategy in $\chi$ and what these strategies are. However, as usual, a sentence does not require an assignment to be evaluated, and for an $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ sentence $\varphi$ we let $\mathcal{G},\rho\models\varphi$ if $\mathcal{G},\emptyset,\rho\models\varphi$ for the empty assignment $\emptyset$, and we write $\mathcal{G}\models\varphi$ if $\mathcal{G},v_{\iota}\models\varphi$. SL is the fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ obtained by interpreting all observation symbols as the identity relation (which models perfect information), restricting to deterministic strategies, and considering only assignments in which each agent has a strategy (in this case the outcome of an assignment consists of a single play; one can thus get rid of the outcome quantifier and evaluate temporal operators in the unique outcome of the current assignment, as usually done in SL). Also, $\textnormal{{CTL}}^{*}$ is the fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ which uses no binding, unbinding or strategy quantification. ### 2.5. Discussion on the semantics We now discuss some aspects of the semantics. Evaluation on finite plays. Unlike previous definitions of Strategy Logic, we evaluate formulas on finite plays (instead of positions), where the finite play represents the whole history starting from the initial position of the $\textrm{CGS}_{\textnormal{ii}}$ in which the formula is evaluated. There are several reasons to do so. First, it allows us to define the semantics more simply without having to resort to the notion of assignment translations. Second, it makes it easier to see the correctness of the reduction to $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, that we present in Section 5. In SL, a strategy only has access to the history of the game starting from the point where the strategy quantifier from which it arises has been evaluated. In contrast, in $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ strategies have access to the whole history, starting from the initial position. However this does not affect the semantics, in the sense that the perfect-information fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ with deterministic strategies corresponds to SL. Indeed, when agents have perfect information, having access to the past or not does not affect the existence of strategies to enforce temporal properties that only concern the future. Players not remembering their actions. Our definition of synchronous perfect recall only considers the sequence of positions in finite plays, and forgets about actions taken by players. In particular, it is possible in this definition that a player cannot distinguish between two finite plays in which she plays different actions. This definition is standard in games with imperfect information (van der Meyden and Wilke, 2005; Berwanger et al., 2010; Doyen and Raskin, 2011; Berwanger et al., 2018), since remembering one’s actions or not is indifferent for the existence of distributed winning strategies or Nash equilibria. However it makes a difference for some more involved solution concepts that are expressible in strategic logics such as $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. For instance it is observed in (Bouyer, 2017, Appendix A) that some games admit subgame-perfect equilibria only if agents remember their own past actions. Nonetheless we consider the setting where agents do not remember their actions, as it is the most general. Indeed, as noted in (Chatterjee and Doyen, 2014b, Remark 2.1, p.8), one can simulate agents that remember their own actions by storing in positions of the game the information of the last joint move played (this may create $|\textnormal{Ac}|^{|\textnormal{Ag}|}$ copies of each position, but the branching degree is unchanged). One can then adapt indistinguishability relations to take actions into account. For instance, for an observation symbol $o$ and an agent $a$, one could consider a new observation symbol $o_{a}$ that would be interpreted in the enriched game structure as the refinement of $\sim_{o}$ that considers two positions indistinguishable if they are indistinguishable for $\sim_{o}$ and contain the same last action for agent $a$. Binding agent $a$ only to strategies that use observation of the form $o_{a}$ for some $o$ captures the fact that agent $a$ remembers her actions. Agents changing observation. In $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ observations are not bound to agents but to strategies. And because agents can change their strategy thanks to the binding operator, it follows that they can change observation, or more precisely they can successively play with strategies that have different observations. For instance consider a controller that observes a system through a set of $n$ sensors $S=\\{s_{1},\ldots,s_{n}\\}$ as in, e.g., (Bittner et al., 2012). Let $o_{i}$ be the observation power provided by the set of sensors $S\setminus\\{s_{i}\\}$ (one can think of a system where states are tuples of local states, each sensor observing one component). Also let $o$ be the observation power provided by the full set $S$ of sensors, and let atom $\text{fault}_{i}$ represent the fact that a fault occurs on sensor $s_{i}$. The formula $\varphi:=\langle\\!\langle x\rangle\\!\rangle^{o}(a,x){\bf A}{\bf G}\left(\text{safe}\wedge\bigwedge_{i=1}^{n}\text{fault}_{i}\to\langle\\!\langle x\rangle\\!\rangle^{o_{i}}(a,x){\bf A}{\bf G}\text{\,safe}_{i}\right)$ expresses that the controller $a$ has a strategy (which uses all sensors in $S$) to maintain the system safe, and if a sensor is lost, it can respond by switching to a strategy using the remaining sensors to maintain some alternative, possibly weaker, security requirement $\text{safe}_{i}$. ### 2.6. Model checking and hierarchical instances We now introduce the main decision problem of this paper, which is the model- checking problem for $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. An _$\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ -instance_ is a model together with a formula, i.e., it is a pair $(\mathcal{G},\Phi)$ where $\mathcal{G}$ is a $\textrm{CGS}_{\textnormal{ii}}$ and $\Phi\in\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. ###### Definition 2.4 (Model checking $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$). The _model-checking problem_ for $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ is the decision problem that, given an $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instance $(\mathcal{G},\Phi)$, returns ‘Yes’ if $\mathcal{G}\models\Phi$, and ‘No’ otherwise. It is well known that deciding the existence of winning strategies in multi- player games with imperfect information is undecidable for reachability objectives (Peterson et al., 2001). Since this problem is easily reduced to the model-checking problem for $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, we get the following result. ###### Theorem 2.5. The model-checking problem for $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ is undecidable. Hierarchical instances. We now isolate a sub-problem obtained by restricting attention to _hierarchical instances_. Intuitively, an $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instance $(\mathcal{G},\Phi)$ is hierarchical if, as one goes down a path in the syntactic tree of $\Phi$, the observations tied to quantifications become finer. ###### Definition 2.6 (Hierarchical instances). An $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instance $(\mathcal{G},\Phi)$ is _hierarchical_ if for every subformula $\varphi_{1}=\langle\\!\langle y\rangle\\!\rangle^{o_{1}}\varphi^{\prime}_{1}$ of $\Phi$ and subformula $\varphi_{2}=\langle\\!\langle x\rangle\\!\rangle^{o_{2}}\varphi^{\prime}_{2}$ of $\varphi^{\prime}_{1}$, it holds that $\mathcal{O}(o_{2})\subseteq\mathcal{O}(o_{1})$. If $\mathcal{O}(o_{2})\subseteq\mathcal{O}(o_{1})$ we say that $o_{2}$ is _finer_ than $o_{1}$ in $\mathcal{G}$, and that $o_{1}$ is _coarser_ than $o_{2}$ in $\mathcal{G}$. Intuitively, this means that a player with observation $o_{2}$ observes game $\mathcal{G}$ no worse than, i.e., knows at least as much as a player with observation $o_{1}$. ###### Remark 2. If one uses the trick described in Section 2.5 to model agents that remember their own actions, then for an agent $a$ to know at least as much as another agent $b$ it needs to be the case that, in particular, agent $a$ observes all actions played by agent $b$. ###### Example 2.7 (Fault-tolerant diagnosibility). Consider the following formula from Section 2.5: $\varphi:=\langle\\!\langle x\rangle\\!\rangle^{o}(a,x){\bf A}{\bf G}\left(\text{safe}\wedge\bigwedge_{i=1}^{n}\text{fault}_{i}\to\langle\\!\langle x\rangle\\!\rangle^{o_{i}}(a,x){\bf A}{\bf G}\text{\,safe}_{i}\right)$ As already discussed, it expresses that the controller can react to the loss of a sensor to keep ensuring some property of the system. Clearly, the controller’s observation $o_{i}$ after the loss of sensor $i$ is coarser than its original observation $o$, and thus formula $\varphi$ in such a system does not form a hierarchical instance. We now give an example of scenario where hierarchical instances occur naturally. ###### Example 2.8 (Security levels). Consider a system with different “security levels”, where higher levels have access to more data (i.e., can observe more). Assume that the $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$ is such that $\mathcal{O}(o_{n})\subseteq\mathcal{O}(o_{n-1})\subseteq\ldots\subseteq\mathcal{O}(o_{1})$: in other words, level $n$ has the highest security clearance, while level $1$ has the lowest. Consider that agent $a$ wants to reach some objective marked by atom “goal”, that it starts with the lowest observation clearance $o_{1}$, and that atomic formula “$\text{promote}_{i}$” means that the agent is granted access to level $i$ (observe that whenever we have $\text{promote}_{i}$, we should also have $\text{promote}_{j}$ for all $j<i$). For every $i$ we let $\varphi_{i}(\varphi^{\prime}):=\text{goal}\vee(\text{promote}_{i}\wedge\langle\\!\langle x\rangle\\!\rangle^{o_{i}}(a,x){\bf A}{\bf F}\varphi^{\prime})$ Now the formula $\varphi:=\varphi_{1}(\varphi_{2}(\ldots\varphi_{n-1}(\varphi_{n}(\text{goal}))\ldots))$ means that agent $a$ can enforce her goal, possibly by first getting access to higher security levels and using this additional observation power to reach the goal. Because the strategy quantifications that are deeper in the formula have access to more information, this formula forms a hierarchical instance in $\mathcal{G}$. Here is the main contribution of this work: ###### Theorem 2.9. The model-checking problem for $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ restricted to the class of hierarchical instances is decidable. We prove this result in Section 5 by reducing it to the model-checking problem for the hierarchical fragment of a logic called $\textnormal{{QCTL}}^{*}$ with imperfect information, which we now introduce and study in order to use it as an intermediate, “low-level” logic between tree automata and $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. We then discuss some applications of this theorem in Section 7. ## 3\. $\textnormal{{QCTL}}^{*}$ with imperfect information In this section we introduce an imperfect-information extension of $\textnormal{{QCTL}}^{*}$ (Sistla, 1983; Kupferman, 1999; Kupferman et al., 2000a; French, 2001; Laroussinie and Markey, 2014), which is an extension of $\textnormal{{CTL}}^{*}$ with second-order quantification on atomic propositions. In order to introduce imperfect information, instead of considering equivalence relations between states as in concurrent game structures, we will enrich Kripke structures by giving internal structure to their states, i.e., we see states as $n$-tuples of local states. This way of modelling imperfect information is inspired from Reif’s multi-player game structures (Peterson et al., 2001) and distributed systems (Halpern and Vardi, 1989), and we find it very suitable to application of automata techniques, as discussed in Section 3.3. The syntax of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is similar to that of $\textnormal{{QCTL}}^{*}$, except that we annotate second- order quantifiers by subsets $\textnormal{{o}}\subseteq[n]$. The idea is that quantifiers annotated by o can only “observe” the local states indexed by $i\in\textnormal{{o}}$. We define the tree-semantics of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$: this means that we interpret formulas on trees that are the unfoldings of Kripke structures (this will capture the fact that players in $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ have synchronous perfect recall). We then define the syntactic class of _hierarchical formulas_ and prove, using an automata-theoretic approach, that model checking this class of formulas is decidable. For the rest of the section we fix some natural number $n\in\mathbb{N}$ which parameterises the logic $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, and which is the number of components in states of the models. ### 3.1. $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ Syntax The syntax of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is very similar to that of $\textnormal{{QCTL}}^{*}$: the only difference is that we annotate quantifiers by a set of indices that defines the “observation” of that quantifier. Concrete observations. A set $\textnormal{{o}}\subseteq[n]$ is called a _concrete observation_ (to distinguish it from observations $o$ in the definitions of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$). ###### Definition 3.1 ($\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ Syntax). The syntax of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is defined by the following grammar: $\displaystyle\varphi:=$ $\displaystyle\;p\mid\neg\varphi\mid\varphi\vee\varphi\mid{\bf E}\psi\mid\exists^{\textnormal{{o}}}p.\,\varphi$ $\displaystyle\psi:=$ $\displaystyle\;\varphi\mid\neg\psi\mid\psi\vee\psi\mid{\bf X}\psi\mid\psi{\bf U}\psi$ where $p\in\textnormal{AP}$ and $\textnormal{{o}}\subseteq[n]$. Formulas of type $\varphi$ are called _state formulas_ , those of type $\psi$ are called _path formulas_ , and $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ consists of all the state formulas defined by the grammar. We use standard abbreviation ${\bf A}\psi:=\neg{\bf E}\neg\psi$. We also use $\exists p.\,\varphi$ as a shorthand for $\exists^{[n]}p.\,\varphi$, and we let $\forall p.\,\varphi:=\neg\exists p.\,\neg\varphi$. Given a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $\varphi$, we define the set of _quantified propositions_ ${\textnormal{AP}_{\exists}}(\varphi)\subseteq\textnormal{AP}$ as the set of atomic propositions $p$ such that $\varphi$ has a subformula of the form $\exists^{\textnormal{{o}}}p.\,\varphi$. We also define the set of _free propositions_ $\textnormal{AP}_{f}(\varphi)\subseteq\textnormal{AP}$ as the set of atomic propositions that have an occurrence which is not under the scope of any quantifier of the form $\exists^{\textnormal{{o}}}p.\,$ Observe that ${\textnormal{AP}_{\exists}}(\varphi)\cap\textnormal{AP}_{f}(\varphi)$ may not be empty, i.e., a proposition may appear both free and quantified in (different places of) a formula. ### 3.2. $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ semantics Several semantics have been considered for $\textnormal{{QCTL}}^{*}$, the two most studied being the _structure semantics_ and the _tree semantics_ (see (Laroussinie and Markey, 2014) for more details). For the semantics of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ we adapt the tree semantics, and we explain the reasons for doing so in Section 3.3. As already mentioned, for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ we consider structures whose states are tuples of local states. We now define these structures and related notions. ###### Definition 3.2 (Compound Kripke structures). A _compound Kripke structure_ , or CKS , over AP is a tuple $\mathcal{S}=(S,R,\ell,s_{\iota})$ where * • $S\subseteq\prod_{i\in[n]}L_{i}$ is a set of _states_ , with $\\{L_{i}\\}_{i\in[n]}$ a family of $n$ disjoint finite sets of _local states_ , * • $R\subseteq S\times S$ is a left-total111i.e., for all $s\in S$, there exists $s^{\prime}$ such that $(s,s^{\prime})\in R$. _transition relation_ , * • $\ell:S\to 2^{\textnormal{AP}}$ is a _labelling function_ and * • $s_{\iota}\in S$ is an _initial state_. A _path_ in $\mathcal{S}$ is an infinite sequence of states $\lambda=s_{0}s_{1}\ldots$ such that for all $i\in\mathbb{N}$, $(s_{i},s_{i+1})\in R$. A _finite path_ is a finite non-empty prefix of a path. We may write $s\in\mathcal{S}$ for $s\in S$, and we define the _size_ $|\mathcal{S}|$ of a CKS $\mathcal{S}=(S,R,s_{\iota},\ell)$ as its number of states: $|\mathcal{S}|:=|S|$. Since we will interpret $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ on unfoldings of CKS , we now define infinite trees. Trees. In many works, trees are defined as prefix-closed sets of words with the empty word $\epsilon$ as root. Here trees represent unfoldings of Kripke structures, and we find it more convenient to see a node $u$ as a sequence of states and the root as the initial state. Let $X$ be a finite set of _directions_ (typically a set of states). An _$X$ -tree_ $\tau$ is a nonempty set of words $\tau\subseteq X^{+}$ such that: * • there exists $r\in X$, called the _root_ of $\tau$, such that each $u\in\tau$ starts with $r$ ($r\preccurlyeq u$); * • if $u\cdot x\in\tau$ and $u\cdot x\neq r$, then $u\in\tau$, * • if $u\in\tau$ then there exists $x\in X$ such that $u\cdot x\in\tau$. The elements of a tree $\tau$ are called _nodes_. If $u\cdot x\in\tau$, we say that $u\cdot x$ is a _child_ of $u$. The _depth_ of a node $u$ is $|u|$. An $X$-tree $\tau$ is _complete_ if for every $u\in\tau$ and $x\in X$, $u\cdot x\in\tau$. A _path_ in $\tau$ is an infinite sequence of nodes $\lambda=u_{0}u_{1}\ldots$ such that for all $i\in\mathbb{N}$, $u_{i+1}$ is a child of $u_{i}$, and $Paths(u)$ is the set of paths that start in node $u$. Labellings. An _AP -labelled $X$-tree_, or _$(\textnormal{AP},X)$ -tree_ for short, is a pair $t=(\tau,\ell)$, where $\tau$ is an $X$-tree called the _domain_ of $t$ and $\ell:\tau\rightarrow 2^{\textnormal{AP}}$ is a _labelling_ , which maps each node to the set of propositions that hold there. For $p\in\textnormal{AP}$, a _$p$ -labelling_ for a tree is a mapping $\ell_{p}:\tau\to\\{0,1\\}$ that indicates in which nodes $p$ holds, and for a labelled tree $t=(\tau,\ell)$, the $p$-labelling of $t$ is the $p$-labelling $u\mapsto 1$ if $p\in\ell(u)$, 0 otherwise. The composition of a labelled tree $t=(\tau,\ell)$ with a $p$-labelling $\ell_{p}$ for $\tau$ is defined as $t\otimes\ell_{p}:=(\tau,\ell^{\prime})$, where $\ell^{\prime}(u)=\ell(u)\cup\\{p\\}$ if $\ell_{p}(u)=1$, and $\ell(u)\setminus\\{p\\}$ otherwise. A $p$-labelling for a labelled tree $t=(\tau,\ell)$ is a $p$-labelling for its domain $\tau$. A _pointed labelled tree_ is a pair $(t,u)$ where $u$ is a node of $t$. If $u=w\cdot x$, the _subtree_ $t_{u}$ of $t=(\tau,\ell)$ is defined as $t_{u}:=(\tau_{u},\ell_{u})$ with $\tau_{u}=\\{x\cdot w^{\prime}\mid w\cdot x\cdot w^{\prime}\in\tau\\}$, and $\ell_{u}(x\cdot w^{\prime})=\ell(w\cdot x\cdot w^{\prime})$. A labelled tree is _regular_ if it has finitely many disctinct subtrees. In the tree semantics of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ that we consider here, formulas are evaluated on tree unfoldings of CKS , which we now define. Tree unfoldings. Let $\mathcal{S}=(S,R,\ell,s_{\iota})$ be a compound Kripke structure over AP. The _tree-unfolding of $\mathcal{S}$_ is the $(\textnormal{AP},S)$-tree $t_{\mathcal{S}}:=(\tau,\ell^{\prime})$, where $\tau$ is the set of all finite paths that start in $s_{\iota}$, and for every $u\in\tau$, $\ell^{\prime}(u):=\ell(\mbox{last}(u))$. Note that a labelled tree is regular if and only if it is the unfolding of some finite Kripke structure. Narrowing. Let $X$ and $Y$ be two finite sets, and let $(x,y)\in X\times Y$. The _$X$ -narrowing_ of $(x,y)$ is ${(x,y)\\!\downarrow_{X}}:=x$. This definition extends naturally to words and trees over $X\times Y$ (point-wise). Given a family of (disjoint) sets of local states $\\{L_{i}\\}_{i\in[n]}$ and a subset $I\subseteq[n]$, we let $L_{I}:=\prod_{i\in I}L_{i}$ if $I\neq\emptyset$ and $L_{\emptyset}:=\\{\mathbf{0}\\}$, where $\mathbf{0}$ is a special symbol. For $I,J\subseteq[n]$ and $z\in L_{I}$, we also define ${z\\!\downarrow_{J}}:=z\\!\downarrow_{L_{I\cap J}}$, where $z$ is seen as a pair $z=(x,y)\in L_{I\cap J}\times L_{I\setminus J}$, i.e., we apply the above definition with $X=L_{I\cap J}$ and $Y=L_{I\setminus J}$. This is well defined because having taken sets $L_{i}$ to be disjoint, the ordering of local states in $z$ is indifferent. We also extend this definition to words and trees. In particular, for every $L_{I}$-tree $\tau$, $\tau\\!\downarrow_{\emptyset}$ is the only $L_{\emptyset}$-tree, $\mathbf{0}^{\omega}$. Quantification and uniformity. In $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ $\exists^{\textnormal{{o}}}p.\,\varphi$ holds in a tree $t$ if there is some o-uniform $p$-labelling of $t$ such that $t$ with this $p$-labelling satisfies $\varphi$. Intuitively, a $p$-labelling of a tree is o-uniform if every two nodes that are indistinguishable for observation o agree on $p$. ###### Definition 3.3 (o-indistinguishability and o-uniformity in $p$). Fix $\textnormal{{o}}\subseteq[n]$ and $I\subseteq[n]$. * • Two tuples $x,x^{\prime}\in L_{I}$ are _o -indistinguishable_, written $x\approx_{\textnormal{{o}}}x^{\prime}$, if $x\\!\downarrow_{\textnormal{{o}}}=x^{\prime}\\!\downarrow_{\textnormal{{o}}}$. * • Two words $u=u_{0}\ldots u_{i}$ and $u^{\prime}=u^{\prime}_{0}\ldots u^{\prime}_{j}$ over alphabet $L_{I}$ are _o -indistinguishable_, written $u\approx_{\textnormal{{o}}}u^{\prime}$, if $i=j$ and for all $k\in\\{0,\ldots,i\\}$ we have $u_{k}\approx_{\textnormal{{o}}}u^{\prime}_{k}$. * • A $p$-labelling for a tree $\tau$ is _o -uniform_ if for all $u,u^{\prime}\in\tau$, $u\approx_{\textnormal{{o}}}u^{\prime}$ implies $\ell_{p}(u)=\ell_{p}(u^{\prime})$. ###### Definition 3.4 ($\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ semantics). We define by induction the satisfaction relation $\models$ of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$. Let $t=(\tau,\ell)$ be an AP-labelled $L_{I}$-tree, $u$ a node and $\lambda$ a path in $\tau$: $\displaystyle t,u\models$ $\displaystyle\,p$ if $\displaystyle\quad p\in\ell(u)$ $\displaystyle t,u\models$ $\displaystyle\,\neg\varphi$ if $\displaystyle\quad t,u\not\models\varphi$ $\displaystyle t,u\models$ $\displaystyle\,\varphi\vee\varphi^{\prime}$ if $\displaystyle\quad t,u\models\varphi\mbox{ or }t,u\models\varphi^{\prime}$ $\displaystyle t,u\models$ $\displaystyle\,{\bf E}\psi$ if $\displaystyle\quad\exists\,\lambda\in Paths(u)\mbox{ s.t. }t,\lambda\models\psi$ $\displaystyle t,u\models$ $\displaystyle\,\exists^{\textnormal{{o}}}p.\,\varphi$ if $\displaystyle\quad\exists\,\ell_{p}\mbox{ a $\textnormal{{o}}$-uniform $p$-labelling for $t$ such that }t\otimes\ell_{p},u\models\varphi$ $\displaystyle t,\lambda\models$ $\displaystyle\,\varphi$ if $\displaystyle\quad t,\lambda_{0}\models\varphi$ $\displaystyle t,\lambda\models$ $\displaystyle\,\neg\psi$ if $\displaystyle\quad t,\lambda\not\models\psi$ $\displaystyle t,\lambda\models$ $\displaystyle\,\psi\vee\psi^{\prime}\quad$ if $\displaystyle\quad t,\lambda\models\psi\mbox{ or }t,\lambda\models\psi^{\prime}$ $\displaystyle t,\lambda\models$ $\displaystyle\,{\bf X}\psi$ if $\displaystyle\quad t,\lambda_{\geq 1}\models\psi$ $\displaystyle t,\lambda\models$ $\displaystyle\,\psi{\bf U}\psi^{\prime}$ if $\displaystyle\quad\exists\,i\geq 0\mbox{ s.t. }t,\lambda_{\geq i}\models\psi^{\prime}\text{ and }\forall j\text{ s.t. }0\leq j<i,\;t,\lambda_{\geq j}\models\psi$ We write $t\models\varphi$ for $t,r\models\varphi$, where $r$ is the root of $t$. Given a CKS $\mathcal{S}$ and a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $\varphi$, we also write $\mathcal{S}\models\varphi$ if $\mathcal{S},s_{\iota}\models\varphi$. ###### Example 3.5. Consider the following CTL formula: $\mathbf{border}(p):={\bf A}{\bf F}p\wedge{\bf A}{\bf G}(p\rightarrow{\bf A}{\bf X}{\bf A}{\bf G}\neg p).$ This formula holds in a labelled tree if and only if each path contains exactly one node labelled with $p$. Now, consider the following $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula: $\mathbf{level}(p):=\exists^{\emptyset}p.\,\mathbf{border}(p).$ For a blind quantifier, two nodes of a tree are indistinguishable if and only if they have same depth. Therefore, this formula holds on a tree iff the $p$’s label all and only the nodes at some fixed depth. This formula can thus be used to capture the equal level predicate on trees. Actually, just as $\textnormal{{QCTL}}^{*}$ captures MSO, one can prove that $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ with tree semantics subsumes MSO with equal level (Elgot and Rabin, 1966; Läuchli and Savioz, 1987; Thomas, 1992). In Theorem 3.7 we make use of a similar observation to prove that model-checking $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is undecidable. ### 3.3. Discussion on the definition of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ We now motivate in detail some aspects of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$. Modelling of imperfect information. We model imperfect information by means of local states (rather than equivalence relations) because this greatly facilitates the use of automata techniques. More precisely, in our decision procedure of Section 4 we use an operation on tree automata called _narrowing_ , which was introduced in (Kupferman and Vardi, 1999) to deal with imperfect- information in the context of distributed synthesis for temporal specifications. Given an automaton $\mathcal{A}$ that works on $X\times Y$-trees, where $X$ and $Y$ are two finite sets, and assuming that we want to model an operation performed on trees while observing only the $X$ component of each node, this narrowing operation allows one to build from $\mathcal{A}$ an automaton $\mathcal{A}^{\prime}$ that works on $X$-trees, such that $\mathcal{A}^{\prime}$ accepts an $X$-tree if and only if $\mathcal{A}$ accepts its widening to $X\times Y$ (intuitively, this widening is the $X\times Y$-tree in which each node is labelled as its projection on the original $X$-tree; see Section 4 for details). With our definition of compound Kripke structures, their unfoldings are trees over the Cartesian product $L_{[n]}$. To model a quantification $\exists^{\textnormal{{o}}}p$ with observation $\textnormal{{o}}\subseteq[n]$, we can thus use the narrowing operation to forget about components $L_{i}$, for $i\in[n]\setminus\textnormal{{o}}$. We then use the classic projection of nondeterministic tree automata to perform existential quantification on atomic proposition $p$. Since the choice of the $p$-labelling is made directly on $L_{\textnormal{{o}}}$-trees, it is necessarily o-uniform. Choice of the tree semantics. The two most studied semantics for $\textnormal{{QCTL}}^{*}$ are the _structure semantics_ , in which formulas are evaluated directly on Kripke structures, and the _tree semantics_ , in which Kripke structures are first unfolded into infinite trees. Tree semantics thus allows quantifiers to choose the value of a quantified atomic proposition in each _finite path_ of the model, while in structure semantics the choice is only made in each state. When $\textnormal{{QCTL}}^{*}$ is used to express existence of strategies, existential quantification on atomic propositions labels the structure with strategic choices; in this kind of application, structure semantics reflects so-called _positional_ or _memoryless_ strategies, while tree semantics captures _perfect-recall_ or _memoryful_ strategies. Since in this work we are interested in perfect-recall strategies, we only consider the tree semantics. ### 3.4. Model checking $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ We now define the model-checking problem studied in the rest of this section. ###### Definition 3.6 (Model checking $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$). The _model-checking problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$_ is the following decision problem: given an instance $(\mathcal{S},\Phi)$ where $\mathcal{S}$ is a CKS, and $\Phi$ is a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula, return ‘Yes’ if $\mathcal{S}\models\Phi$ and ‘No’ otherwise. We now prove that the model-checking problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is undecidable. This comes as no surprise since, as we will show in Section 5, $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ can express the existence of distributed winning strategies in imperfect-information games. However we propose a proof that shows the connection between $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ and MSO with equal- level predicate (Elgot and Rabin, 1966; Läuchli and Savioz, 1987; Thomas, 1992). This proof also has the benefit of showing that $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is undecidable already for formulas that involve only propositional quantifiers that observe either everything or nothing. ###### Theorem 3.7. The model-checking problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is undecidable. ###### Proof. Let $\textnormal{{MSO}}_{\textnormal{eq}}$ denote the extension of the logic MSO (without unary predicates) by a binary predicate symbol eq. $\textnormal{{MSO}}_{\textnormal{eq}}$ is interpreted on the full binary tree, and the semantics of $\text{eq}(x,y)$ is that $x$ and $y$ have the same depth in the tree. We show how to effectively translate $\textnormal{{MSO}}_{\textnormal{eq}}$ into $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, and our result follows since the $\textnormal{{MSO}}_{\textnormal{eq}}$-theory of the binary tree is undecidable (Läuchli and Savioz, 1987). The translation from $\textnormal{{MSO}}_{\textnormal{eq}}$ to $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is obtained by extending that from MSO to QCTL (Laroussinie and Markey, 2014), using the formula $\mathbf{level}(\cdot)$ from Example 3.5 to help capture the equal- length predicate. We define a translation $\widehat{\quad}$ from $\textnormal{{MSO}}_{\textnormal{eq}}$ to $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ such that for every tree $t$ with root $r$, nodes $u_{1},\ldots,u_{i}\in t$ and sets of nodes $U_{1},\ldots,U_{j}\subseteq t$, and every $\textnormal{{MSO}}_{\textnormal{eq}}$ formula ${\varphi(x,x_{1},\ldots,x_{i},X_{1},\ldots,X_{j})}$, we have that (1) $t,r,u_{1},\ldots,u_{i},U_{1},\ldots,U_{j}\models\varphi(x,x_{1},\ldots,x_{i},X_{1},\ldots,X_{j})\text{\quad if and only if \quad}\widehat{t},r\models\widehat{\varphi}$ where $\widehat{t}$ is obtained from $t$ by defining the labelling for fresh atomic propositions $p_{x_{k}}$ and $p_{X_{k}}$, with $k\in[i]$, as follows: $p_{x_{k}}\in\widehat{\ell}(u)$ if $u=u_{k}$ and $p_{X_{k}}\in\widehat{\ell}(u)$ if $u\in U_{k}$. The translation of MSO to $\textnormal{{QCTL}}^{*}$ from (Laroussinie and Markey, 2014) can be extended to one from $\textnormal{{MSO}}_{\textnormal{eq}}$ to $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ by adding rules for the equal level predicate. Indeed, for $\varphi(x,x_{1},\ldots,x_{i},X_{1},\ldots,X_{j})\in\textnormal{{MSO}}_{\textnormal{eq}}$, we inductively define the $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $\widehat{\varphi}$ as follows, where $k\in[i]$: $\begin{array}[]{rclcrcl}\widehat{x=x_{k}}&:=&p_{x_{k}}&&\widehat{x_{k}=x_{l}}&:=&{\bf E}{\bf F}(p_{x_{k}}\wedge p_{x_{l}})\\\\[5.0pt] \widehat{x\in X_{k}}&:=&p_{X_{k}}&&\widehat{x_{k}\in X_{l}}&:=&{\bf E}{\bf F}(p_{x_{k}}\wedge p_{X_{l}})\\\\[5.0pt] \widehat{\neg\varphi^{\prime}}&:=&\neg\widehat{\varphi^{\prime}}&&\widehat{\varphi_{1}\vee\varphi_{2}}&:=&\widehat{\varphi_{1}}\vee\widehat{\varphi_{2}}\\\\[5.0pt] \widehat{\exists x_{k}.\varphi^{\prime}}&:=&\lx@intercol\exists <EMAIL_ADDRESS>\widehat{\exists X_{k}.\varphi^{\prime}}&:=&\lx@intercol\exists <EMAIL_ADDRESS>\widehat{S(x,x_{k})}&:=&{\bf E}{\bf X}p_{x_{k}}&&\widehat{S(x_{k},x)}&:=&\perp\\\\[5.0pt] \widehat{S(x_{k},x_{l})}&:=&\lx@intercol{\bf E}{\bf F}(p_{x_{k}}\wedge{\bf E}{\bf X}p_{x_{l}})\hfil\lx@intercol\end{array}$ where $\mathrm{uniq}(p):={\bf E}{\bf F}p\wedge\forall q.\;\left({\bf E}{\bf F}(p\wedge q)\rightarrow{\bf A}{\bf G}(p\rightarrow q)\right)$ holds in a tree iff it has exactly one node labelled with $p$. To understand the $x=x_{k}$ and $x\in X_{k}$ cases, consider that $x$ will be interpreted as the root. For the $S(x_{k},x)$ case, observe that $x$ has no incoming edge since it is interpreted as the root. Second-order quantification $\exists X_{k}$ is translated into quantification on atomic proposition $p_{X_{k}}$, and first- order quantification $\exists x_{k}$ is treated similarly, with the additional constraint that quantification is limited to $p_{x_{k}}$-labellings that set $p_{x_{k}}$ to true in one and only one node of the tree. The rules for eq are as follows: $\displaystyle\widehat{\text{eq}(x,x_{k})}$ $\displaystyle:=p_{x_{k}}$ $\displaystyle\widehat{\text{eq}(x_{k},x_{l})}$ $\displaystyle:=\exists^{\emptyset}p.\,\mathbf{border}(p)\wedge{\bf A}{\bf G}(p_{x_{k}}\rightarrow p\wedge p_{x_{l}}\rightarrow p)$ To understand the first case, observe that since $x$ is interpreted as the root, $x_{k}$ is on the same level as $x$ if and only if it is also assigned the root. For the second case, recall from Example 3.5 that the $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $\exists^{\emptyset}p.\,\mathbf{border}(p)$ places one unique horizontal line of $p$’s in the tree, and thus requiring that $x_{k}$ and $x_{l}$ be both on this line ensures that they are on the same level. The correctness of the translation follows from (1), which is proven by induction. Now take an instance $(t,\varphi(x))$ of the model-checking problem for $\textnormal{{MSO}}_{\textnormal{eq}}$ on the full binary tree $t$. Let $\mathcal{S}$ be a CKS with two states $s_{0}$ and $s_{1}$ (local states are irrelevant here), whose transition relation is the complete relation, and with empty labelling function. Clearly, $t_{\mathcal{S}}=t$, and applying (1) we get: $t,s_{0}\models\varphi(x)\text{\quad iff\quad}\widehat{t},s_{0}\models\widehat{\varphi}.$ Observe that in the previous line, because there are no free variables besides $x$, which stands for the root, we have that $\widehat{t}=t=t_{\mathcal{S}}$, hence we have indeed produced an instance of the model-checking problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$. ∎ ## 4\. A decidable fragment of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$: hierarchy on observations The main result of this section is the identification of an important decidable fragment of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$. ###### Definition 4.1 (Hierarchical formulas). A $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $\varphi$ is _hierarchical_ if for all subformulas $\varphi_{1}=\exists^{\textnormal{{o}}_{1}}p_{1}.\,\varphi^{\prime}_{1}$ and $\varphi_{2}=\exists^{\textnormal{{o}}_{2}}p_{2}.\,\varphi^{\prime}_{2}$ of $\varphi$ where $\varphi_{2}$ is a subformula of $\varphi^{\prime}_{1}$, we have $\textnormal{{o}}_{1}\subseteq\textnormal{{o}}_{2}$. In other words, a formula is hierarchical if innermore propositional quantifiers observe at least as much as outermore ones. ###### Example 4.2. Formula $\exists^{\\{1,2\\}}p.\,\exists^{\\{1,2,4\\}}q.\,{\bf A}{\bf G}(p\vee q)$ is hierarchical because $\\{1,2\\}\subseteq\\{1,2,4\\}$. On the other hand, formula $\exists^{\\{1,2\\}}p.\,\big{(}\exists^{\\{1,2,4\\}}q.\,{\bf A}{\bf G}(p\vee q)\wedge\exists^{\\{3\\}}q^{\prime}.\,{\bf E}{\bf F}(p\wedge q^{\prime})\big{)}$ is not, because $\\{1,2\\}\not\subseteq\\{3\\}$. Note that neither is it the case that $\\{3\\}\subseteq\\{1,2\\}$: the observation power of quantifiers $\exists^{\\{1,2\\}}p.\,$ and $\exists^{\\{3\\}}q^{\prime}.\,$ are incomparable. Finally, formula $\forall^{\\{1,2,3\\}}p.\,\exists^{\\{1,2\\}}q.\,.{\bf A}{\bf G}(p\vee q)$ is not hierarchical even though $\\{1,2\\}\subseteq\\{1,2,3\\}$, as the quantifier that observes best is _higher_ in the syntactic tree. We let $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ be the set of hierarchical $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formulas. ###### Theorem 4.3. Model checking $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ is non-elementary decidable. Since our decision procedure for the hierarchical fragment of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is based on an automata-theoretic approach, we recall some definitions and results for alternating tree automata. ### 4.1. Alternating parity tree automata We recall alternating parity tree automata. Because their semantics is defined via acceptance games, we start with basic definitions for two-player turn- based parity games, or simply parity games. Parity games. A _parity game_ is a structure $\mathcal{G}=(V,E,v_{\iota},C)$, where $V=V_{E}\uplus V_{A}$ is a set of _positions_ partitioned between positions of Eve ($V_{E}$) and those of Adam ($V_{A}$), $E\subseteq V\times V$ is a set of _moves_ , $v_{\iota}$ is an initial position and $C:V\to\mathbb{N}$ is a colouring function of finite codomain. In positions $V_{E}$, Eve chooses the next position, while Adam chooses in positions $V_{A}$. A play is an infinite sequence of positions $v_{0}v_{1}v_{2}\ldots$ such that $v_{0}=v_{\iota}$ and for all $i\geq 0$, $(v_{i},v_{i+1})\in E$ (written $v_{i}\to v_{i+1}$). We assume that for every $v\in V$ there exists $v^{\prime}\in V$ such that $v\to v^{\prime}$. A strategy for Eve is a partial function $V^{*}\rightharpoonup V$ that maps each finite prefix of a play ending in a position $v\in V_{E}$ to a next position $v^{\prime}$ such that $v\to v^{\prime}$. A play $v_{0}v_{1}v_{2}\ldots$ _follows_ a strategy $\sigma$ of Eve if for every $i\geq 0$ such that $v_{i}\in V_{E}$, $v_{i+1}=\sigma(v_{0}\ldots v_{i})$. A strategy $\sigma$ is winning if every play that follows it satisfies the parity condition, i.e., the least colour seen infinitely often along the play is even. Parity tree automata. Because it is sufficient for our needs and simplifies definitions, we assume that all input trees are complete trees. For a set $Z$, $\mathbb{B}^{+}(Z)$ is the set of formulas built from the elements of $Z$ as atomic propositions using the connectives $\vee$ and $\wedge$, and with $\top,\perp\in\mathbb{B}^{+}(Z)$. An _alternating tree automaton (ATA ) on $(\textnormal{AP},X)$-trees_ is a structure $\mathcal{A}=(Q,\delta,q_{{\iota}},C)$ where $Q$ is a finite set of states, $q_{{\iota}}\in Q$ is an initial state, $\delta:Q\times 2^{\textnormal{AP}}\rightarrow\mathbb{B}^{+}(X\times Q)$ is a transition function, and $C:Q\to\mathbb{N}$ is a colouring function. To ease reading we shall write atoms in $\mathbb{B}^{+}(X\times Q)$ between brackets, such as $[x,q]$. A _nondeterministic tree automaton (NTA ) on $(\textnormal{AP},X)$-trees_ is an ATA $\mathcal{A}=(Q,\delta,q_{{\iota}},C)$ such that for every $q\in Q$ and $a\in 2^{\textnormal{AP}}$, $\delta(q,a)$ is written in disjunctive normal form and for every direction $x\in X$ each disjunct contains exactly one element of $\\{x\\}\times Q$. An NTA is _deterministic_ if for each $q\in Q$ and $a\in 2^{\textnormal{AP}}$, $\delta(q,a)$ consists of a single disjunct. Acceptance of a pointed labelled tree $(t,u_{\iota})$, where $t=(\tau,\ell)$, by an ATA $\mathcal{A}=(Q,\delta,q_{\iota},C)$ is defined via the parity game $\mathcal{G}(\mathcal{A},t,u_{\iota})=(V,E,v_{\iota},C^{\prime})$ where $V=\tau\times Q\times\mathbb{B}^{+}(X\times Q)$, position $(u,q,\alpha)$ belongs to Eve if $\alpha$ is of the form $\alpha_{1}\vee\alpha_{2}$ or $[x,q^{\prime}]$, and to Adam otherwise, $v_{{\iota}}=(u_{\iota},q_{\iota},\delta(q_{\iota},u_{\iota}))$, and $C^{\prime}(u,q,\alpha)=C(q)$. Moves in $\mathcal{G}(\mathcal{A},t,u_{\iota})$ are defined by the following rules: $\begin{array}[]{ll}(u,q,\alpha_{1}\;\mbox{$\dagger$}\;\alpha_{2})\rightarrow(u,q,\alpha_{i})&\mbox{where }\mbox{$\dagger$}\in\\{\vee,\wedge\\}\mbox{ and }i\in\\{1,2\\},\\\ \lx@intercol(u,q,[x,q^{\prime}])\rightarrow(u\cdot x,q^{\prime},\delta(q^{\prime},\ell(u\cdot x)))\hfil\lx@intercol\end{array}$ Positions of the form $(u,q,\top)$ and $(u,q,\perp)$ are sinks, winning for Eve and Adam respectively. A pointed labelled tree $(t,u)$ is _accepted_ by $\mathcal{A}$ if Eve has a winning strategy in $\mathcal{G}(\mathcal{A},t,u)$, and the _language_ of $\mathcal{A}$ is the set of pointed labelled trees accepted by $\mathcal{A}$, written $\mathcal{L}(\mathcal{A})$. We write $t\in\mathcal{L}(\mathcal{A})$ if $(t,r)\in\mathcal{L}(\mathcal{A})$, where $r$ is the root of $t$. Finally, the _size_ $|\mathcal{A}|$ of an ATA $\mathcal{A}$ is its number of states plus the sum of the sizes of all formulas appearing in the transition function. Word automata. When the set of directions $X$ is a singleton, directions can be forgotten and infinite trees can be identified with infinite words. We thus call _parity word automaton_ a parity tree automaton on $(\textnormal{AP},X)$-trees where $X$ is a singleton. In the case of a nondeterministic parity word automaton, transitions can be represented as usual as a mapping $\Delta:Q\times 2^{\textnormal{AP}}\to 2^{Q}$ which, in a state $q\in Q$, reading the label $a\in 2^{\textnormal{AP}}$ of the current position in the word, indicates a set of states $\Delta(q,a)$ from which Eve can choose to send in the next position of the word. We recall four classic operations on tree automata. Complementation. Given an ATA $\mathcal{A}=(Q,\delta,q_{{\iota}},C)$, we define its _dual_ $\overline{\mathcal{A}}=(Q,\overline{\delta},q_{{\iota}},\overline{C})$ where, for each $q\in Q$ and $a\in 2^{\textnormal{AP}}$, $\overline{\delta}(q,a)$ is the dual of $\delta(q,a)$, i.e., conjunctions become disjunctions and vice versa, and $C(q):=C(q)+1$. ###### Theorem 4.4 (Complementation (Muller and Schupp, 1995)). For every labelled tree $t$ and node $u$ in $t$, $(t,u)\in\mathcal{L}(\overline{\mathcal{A}})\mbox{ if, and only if, }(t,u)\notin\mathcal{L}(\mathcal{A}).$ Projection. The second construction is a projection operation, used by Rabin to deal with second-order monadic quantification: ###### Theorem 4.5 (Projection (Rabin, 1969)). Given an NTA $\mathcal{N}$ on $(\textnormal{AP},X)$-trees and an atomic proposition $p\in\textnormal{AP}$, one can build in linear time an NTA $\mathcal{N}\\!\Downarrow_{p}$ on $(\textnormal{AP}\setminus\\{p\\},X)$-trees such that $(t,u)\in\mathcal{L}(\mathcal{N}\\!\Downarrow_{p})\mbox{\;\;\;iff\;\;\;}\mbox{ there exists a $p$-labelling $\ell_{p}$ for $t$ s.t. }(t\otimes\ell_{p},u)\in\mathcal{L}(\mathcal{N}).$ Intuitively, ${\mathcal{N}\\!\Downarrow_{p}}$ is automaton $\mathcal{N}$ with the only difference that when it reads the label of a node, it can choose to run as if $p$ was either true or false: if $\delta$ is the transition function of $\mathcal{N}$, that of ${\mathcal{N}\\!\Downarrow_{p}}$ is $\delta^{\prime}(q,a)=\delta(q,a\cup\\{p\\})\vee\delta(q,a\setminus\\{p\\})$, for any state $q$ and label $a\in 2^{\textnormal{AP}}$. Another way of seeing it is that $\mathcal{N}\\!\Downarrow_{p}$ guesses a $p$-labelling for the input tree, and simulates $\mathcal{N}$ on this modified input. Simulation. To prevent $\mathcal{N}\\!\Downarrow_{p}$ from guessing different labels for a same node in different executions, it is crucial that $\mathcal{N}$ be nondeterministic, which is the reason why we need the following result: ###### Theorem 4.6 (Simulation (Muller and Schupp, 1995)). Given an ATA $\mathcal{A}$, one can build in exponential time an NTA $\mathcal{N}$ such that $\mathcal{L}(\mathcal{N})=\mathcal{L}(\mathcal{A})$. The last construction was introduced by Kupferman and Vardi to deal with imperfect information aspects in distributed synthesis. To describe it we need to define a widening operation on trees which expands the directions in a tree. Tree widening. We generalise the widening operation defined in (Kupferman and Vardi, 1999). In the following definitions we fix a CKS $\mathcal{S}=(S,R,s_{\iota},\ell)$, and for $I\subseteq[n]$ we let $S_{I}:=\\{s\\!\downarrow_{I}\,\mid s\in S\\}\subseteq L_{I}$ (recall that $L_{I}=\prod_{i\in I}L_{i}$). Let $J\subseteq I\subseteq[n]$. For every $S_{J}$-tree $\tau$ rooted in $s_{J}$ and $s_{I}\in S_{I}$ such that $s_{I}\\!\downarrow_{J}=s_{J}$, we define the _$I$ -widening_ of $\tau$ as the $S_{I}$-tree $\tau\\!\uparrow^{I}_{s_{I}}:=\\{u\in s_{I}\cdot S_{I}^{*}\mid u\\!\downarrow_{J}\in\tau\\}.$ For an $(\textnormal{AP},S_{J})$-tree $t=(\tau,\ell)$ rooted in $s_{J}$ and $s_{I}\in S_{I}$ such that $s_{I}\\!\downarrow_{J}=s_{J}$, we let $t\\!\uparrow^{I}_{s_{I}}:=(\tau\\!\uparrow^{I}_{s_{I}},\ell^{\prime}),\mbox{ where }\ell^{\prime}(u):=\ell(u\\!\downarrow_{J}).$ When clear from the context we may omit the subscript $s_{I}$. It is the case in particular when referring to _pointed_ widenings of trees: $(t\\!\uparrow^{I},u)$ stands for $(t\\!\uparrow^{I}_{u_{0}},u)$. Narrowing. We now state a result from (Kupferman and Vardi, 1999) in our slightly more general setting (the proof can be adapted straightforwardly). The rough idea of this narrowing operation on ATA is that, if one just observes $S_{J}$, uniform $p$-labellings on $S_{I}$-trees can be obtained by choosing the labellings directly on $S_{J}$-trees, and then lifting them to $S_{I}$. ###### Theorem 4.7 (Narrowing (Kupferman and Vardi, 1999)). Given an ATA $\mathcal{A}$ on $S_{I}$-trees one can build in linear time an ATA ${\mathcal{A}\\!\downarrow_{J}}$ on $S_{J}$-trees such that for every pointed $(\textnormal{AP},S_{J})$-tree $(t,u)$ and every $u^{\prime}\in S_{I}^{+}$ such that $u^{\prime}\\!\downarrow_{J}=u$, $(t,u)\in\mathcal{L}(\mathcal{A}\\!\downarrow_{J})\mbox{ iff }(t\\!\uparrow^{I},u^{\prime})\in\mathcal{L}(\mathcal{A}).$ ### 4.2. Translating $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ to ATA In order to prove Theorem 4.3 we need some more notations and a technical lemma that contains the automata construction. ###### Definition 4.8. For every $\varphi\in\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, we let $I_{\varphi}:=\bigcap_{\textnormal{{o}}\in\textnormal{Obs}(\varphi)}\textnormal{{o}}\subseteq[n],$ where $\textnormal{Obs}(\varphi)$ is the set of concrete observations that occur in $\varphi$, with the intersection over the empty set defined as $[n]$. For a CKS $\mathcal{S}$ with state set $S\subseteq\prod_{i\in[n]}L_{i}$ we also let $S_{\varphi}:=\\{s\\!\downarrow_{I_{\varphi}}\mid s\in S\\}$. Elements of $S_{\varphi}$ will be the possible directions used by the automaton we build for $\varphi$. In other words, the automaton for $\varphi$ will work on $S_{\varphi}$-trees. The intuition is that the observations in $\varphi$ determine which components of the model’s states can be observed by the automaton. Our construction, that transforms a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ formula $\varphi$ and a CKS $\mathcal{S}$ into an ATA, builds upon the classic construction from (Kupferman et al., 2000b), which builds ATA for $\textnormal{{CTL}}^{*}$ formulas. In addition, we use projection of automata to treat second-order quantification, and to deal with imperfect information we resort to automata narrowing. Moreover, we use tree automata in an original way that allows us to deal with non-observable atomic propositions, which in turn makes it possible to consider non-observable winning conditions in our decidable fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. The classical approach to model checking via tree automata is to build an automaton that accepts all tree models of the input formula, and check whether it accepts the unfolding of the model (Kupferman et al., 2000b). We instead encode the model in the automata, using the input tree only to guess labellings for quantified propositions. Encoding the model in the automaton. Quantification on atomic propositions is classically performed by means of automata projection (see Theorem 4.5). But in order to obtain a labelling that is uniform with regards to the observation of the quantifier, we need to make use of the narrowing operation (see Theorem 4.7). Intuitively, to check that a formula $\exists^{\textnormal{{o}}}p.\,\varphi$ holds in a tree $t$, we would like to work on its narrowing $t^{\prime}:=t\\!\downarrow_{\textnormal{{o}}}$, guess a labelling for $p$ on this tree thanks to automata projection, thus obtaining a tree $t^{\prime}_{p}$, take its widening $t_{p}^{\prime\prime}:=t^{\prime}_{p}\\!\uparrow^{[n]}$, obtaining a tree with an o-uniform labelling for $p$, and then check that $\varphi$ holds on $t_{p}^{\prime\prime}$. The problem is that unless $t=(\tau,\ell)$ is o-uniform in every atomic proposition in AP, there is no way to define the labelling of $\tau\\!\downarrow_{\textnormal{{o}}}$ without losing information. This implies that, unless we restrict to models where all atomic propositions are observable for all observations o, we cannot pass the model as input to our automata, which will work on narrowings of trees. Therefore, to model check a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $\varphi$ on a CKS $\mathcal{S}$, each state of the automaton that we build for $\varphi$ will contain a state of $\mathcal{S}$. The automaton can thus guess paths in $\mathcal{S}$, and evaluate free occurrences of atomic propositions in $\mathcal{S}$ without reading the input tree. The input tree no longer represents the model, but we use it to carry labellings for quantified atomic propositions in ${\textnormal{AP}_{\exists}}(\varphi)$: we provide the automaton with an input tree whose labelling is initially empty, and the automaton, through successive narrowing and projection operations, decorates it with uniform labellings for quantified atomic propositions. We remark that this technique allows one to go beyond Coordination Logic (Finkbeiner and Schewe, 2010): by separating between quantified atomic propositions (that need to be uniform and are carried by the input tree) and free atomic propositions (that state facts about the model and are coded in the automaton), we manage to remove the restriction present in CL, that requires all facts about the model to be known to every strategy (see Proposition 6.3 in Section 6.2). To do this we assume without loss of generality that propositions that are quantified in $\varphi$ do not appear free in $\varphi$, i.e., ${\textnormal{AP}_{\exists}}(\varphi)\cap\textnormal{AP}_{f}(\varphi)=\emptyset$. Finally, given a formula $\varphi$, a CKS $\mathcal{S}$ and a state $s\in\mathcal{S}$, the truth value of $\varphi$ in $(\mathcal{S},s)$ does not depend on the labelling of $\mathcal{S}$ for atoms in ${\textnormal{AP}_{\exists}}(\varphi)$, which can thus be forgotten. Thus, from now on we will assume that an instance $(\mathcal{S},\Phi)$ of the model- checking problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ is such that ${\textnormal{AP}_{\exists}}(\Phi)\cap\textnormal{AP}_{f}(\Phi)=\emptyset$ and $\mathcal{S}$ is a CKS over $\textnormal{AP}_{f}(\Phi)$. Merging the decorated input tree and the model. To state the correctness of our construction, we will need to merge the labels for quantified propositions, carried by the input tree, with those for free propositions, carried by CKS $\mathcal{S}$. Because, through successive widenings, the input tree (represented by $t$ in the definition below) will necessarily be a complete tree, its domain will always contain the domain of the unfolding of $\mathcal{S}$ (represented by $t^{\prime}$ below), hence the following definition. ###### Definition 4.9 (Merge). Let $t=(\tau,\ell)$ be a complete $(\textnormal{AP},X)$-tree and $t^{\prime}=(\tau^{\prime},\ell^{\prime})$ an $(\textnormal{AP}\,^{\prime},X)$-tree with same root as $t$, where $\textnormal{AP}\cap\textnormal{AP}\,^{\prime}=\emptyset$. We define the _merge_ of $t$ and $t^{\prime}$ as the $(\textnormal{AP}\cup\textnormal{AP}\,^{\prime},X)$-tree $t\merge t^{\prime}:=(\tau\cap\tau^{\prime}=\tau^{\prime},\ell^{\prime\prime}),$ where $\ell^{\prime\prime}(u)=\ell(u)\cup\ell^{\prime}(u)$. We now describe our automata construction. Let $(\mathcal{S},\Phi)$ be an instance of the model-checking problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$, where $\mathcal{S}=(S,R,\ell_{\mathcal{S}},s_{\iota})$. ###### Lemma 4.10 (Translation). For every subformula $\varphi$ of $\Phi$ and state $s$ of $\mathcal{S}$, one can build an ATA $\mathcal{A}_{s}^{\varphi}$ on $({\textnormal{AP}_{\exists}}(\Phi),S_{\varphi})$-trees such that for every $({\textnormal{AP}_{\exists}}(\Phi),S_{\varphi})$-tree $t$ rooted in $s_{\iota}\\!\downarrow_{I_{\varphi}}$, every $u\in t_{\mathcal{S}}$ ending in $s$, it holds that $(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi})\mbox{\;\;\;iff\;\;\;}t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi.$ ###### Proof. Let ${\textnormal{AP}_{\exists}}={\textnormal{AP}_{\exists}}(\Phi)$ and $\textnormal{AP}_{f}=\textnormal{AP}_{f}(\Phi)$, and recall that $\mathcal{S}$ is labelled over $\textnormal{AP}_{f}$. For each state $s\in S$ and each subformula $\varphi$ of $\Phi$ (note that all subformulas of $\Phi$ are also hierarchical), we define by induction on $\varphi$ the ATA $\mathcal{A}_{s}^{\varphi}$ on $({\textnormal{AP}_{\exists}},S_{\varphi})$-trees. $\bm{\varphi=p:}$ First, by Definition 4.8, $S_{\varphi}=S_{[n]}=S$. We let $\mathcal{A}_{s}^{p}$ be the ATA over $S$-trees with one unique state $q_{\iota}$, with transition function defined as follows: $\delta(q_{\iota},a)=\begin{cases}\top&\mbox{if }\begin{array}[]{c}p\in\textnormal{AP}_{f}\mbox{ and }p\in\ell_{\mathcal{S}}(s)\\\ \mbox{ or }\\\ p\in{\textnormal{AP}_{\exists}}\mbox{ and }p\in a\end{array}\\\ \perp&\mbox{if }\begin{array}[]{c}p\in\textnormal{AP}_{f}\mbox{ and }p\notin\ell_{\mathcal{S}}(s)\\\ \mbox{ or }\\\ p\in{\textnormal{AP}_{\exists}}\mbox{ and }p\notin a\end{array}\end{cases}$ $\bm{\varphi=\neg\varphi^{\prime}:}$ We let $\mathcal{A}_{s}^{\varphi}:=\overline{\mathcal{A}_{s}^{\varphi^{\prime}}}$. $\bm{\varphi=\varphi_{1}\vee\varphi_{2}:}$ Because $I_{\varphi}=I_{\varphi_{1}}\cap I_{\varphi_{2}}$, and each $\mathcal{A}_{s}^{\varphi_{i}}$ for $i\in\\{1,2\\}$ works on $L_{\varphi_{i}}$-trees, we first narrow them so that they work on $L_{\varphi}$-trees: for $i\in\\{1,2\\}$, we let $\mathcal{A}_{i}:={\mathcal{A}_{s}^{\varphi_{i}}\\!\downarrow_{I_{\varphi}}}=(Q^{i},\delta^{i},q_{\iota}^{i},C^{i})$. Letting $q_{\iota}$ be a fresh initial state we define $\mathcal{A}_{s}^{\varphi}:=(\\{q_{\iota}\\}\cup Q^{1}\cup Q^{2},\delta,q_{\iota},C)$, where $\delta$ and $C$ agree with $\delta^{i}$ and $C^{i}$, respectively, on states from $Q^{i}$, and $\delta(q_{\iota},a)=\delta^{1}(q_{\iota}^{1},a)\vee\delta^{2}(q_{\iota}^{2},a)$. The colour of $q_{\iota}$ does not matter. $\bm{\varphi={\bf E}\psi:}$ Let $\max(\psi)=\\{\varphi_{1},\ldots,\varphi_{k}\\}$ be the set of maximal state subformulas of $\psi$. In a first step we see these maximal state subformulas as atomic propositions, we see $\psi$ as an LTL formula over $\max(\psi)$, and we build a nondeterministic parity word automaton $\mathcal{W}^{\psi}=(Q^{\psi},\Delta^{\psi},q^{\psi}_{\iota},C^{\psi})$ over alphabet $2^{\max(\psi)}$ that accepts exactly the models of $\psi$ (and uses two colours) (Vardi and Wolper, 1994). We define the ATA $\mathcal{A}$ that, given as input a $(\max(\psi),S_{\varphi})$-tree $t$, nondeterministically guesses a path $\lambda$ in $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$, or equivalently a path in $\mathcal{S}$ starting from $s$, and simulates $\mathcal{W}^{\psi}$ on it, assuming that the labels it reads while following $\lambda\\!\downarrow_{I_{\varphi}}$ in its input $t$ correctly represent the truth value of formulas in $\max(\psi)$ along $\lambda$. Recall that $\mathcal{S}=(S,R,s_{\iota},\ell_{\mathcal{S}})$; we define $\mathcal{A}:=(Q,\delta,q_{{\iota}},C)$, where * • $Q=Q^{\psi}\times S$, * • $q_{{\iota}}=(q^{\psi}_{{\iota}},s)$, * • for each $(q^{\psi},s^{\prime})\in Q$, $C(q^{\psi},s^{\prime})=C^{\psi}(q^{\psi})$, and * • for each $(q^{\psi},s^{\prime})\in Q$ and $a\in 2^{\max(\psi)}$, $\delta((q^{\psi},s^{\prime}),a)=\bigvee_{q^{\prime}\in\Delta^{\psi}(q^{\psi},a)}\bigvee_{s^{\prime\prime}\in R(s^{\prime})}[s^{\prime\prime}\\!\downarrow_{I_{\varphi}},\left(q^{\prime},s^{\prime\prime}\right)].$ The intuition is that $\mathcal{A}$ reads the current label in $2^{\max(\psi)}$, chooses nondeterministically a transition in $\mathcal{W}^{\psi}$, chooses a next state $s^{\prime\prime}$ in $S$ and proceeds in the corresponding direction $s^{\prime\prime}\\!\downarrow_{I_{\varphi}}\in S_{\varphi}$. Now from $\mathcal{A}$ we build the automaton $\mathcal{A}_{s}^{\varphi}$ over $S_{\varphi}$-trees labelled with “real” atomic propositions in ${\textnormal{AP}_{\exists}}$. Intuitively, in each node it visits, $\mathcal{A}_{s}^{\varphi}$ guesses what should be its labelling over $\max(\psi)$, it simulates $\mathcal{A}$ accordingly, and checks that the guess it made is correct. If, after having guessed a finite path $u\in t_{\mathcal{S}}$ ending in state $s^{\prime}$, $\mathcal{A}_{s}^{\varphi}$ guesses that $\varphi_{i}$ holds, it checks this guess by starting a copy of automaton $\mathcal{A}_{s^{\prime}}^{\varphi_{i}}$ from node $v=u\\!\downarrow_{I_{\varphi}}$ in its input $t$. Formally, for each $s^{\prime}\in\mathcal{S}$ and each $\varphi_{i}\in\max(\psi)$ we first build $\mathcal{A}_{s^{\prime}}^{\varphi_{i}}$, which works on $S_{\varphi_{i}}$-trees. Observe that $I_{\varphi}=\cap_{i=1}^{k}I_{\varphi_{i}}$, so that we need to narrow down these automata222In the conference version of this work (Berthon et al., 2017) we made a mistake here: we wrote that $I_{\varphi}=I_{\varphi_{i}}$, which is not the case in general. As a consequence we do need to narrow down automata, unlike what was written in the conference version.: We let $\mathcal{A}^{i}_{s^{\prime}}:=\mathcal{A}_{s^{\prime}}^{\varphi_{i}}\\!\downarrow_{I_{\varphi}}=(Q^{i}_{s^{\prime}},\delta^{i}_{s^{\prime}},q^{i}_{s^{\prime}},C^{i}_{s^{\prime}})$. We also let $\overline{\mathcal{A}^{i}_{s^{\prime}}}=(\overline{Q^{i}_{s^{\prime}}},\overline{\delta^{i}_{s^{\prime}}},\overline{q^{i}_{s^{\prime}}},\overline{C^{i}_{s^{\prime}}})$ be the dualisation of $\mathcal{A}^{i}_{s^{\prime}}$, and we assume without loss of generality all the state sets are pairwise disjoint. We define the ATA $\mathcal{A}_{s}^{\varphi}=(Q\cup\bigcup_{i,s^{\prime}}Q^{i}_{s^{\prime}}\cup\overline{Q^{i}_{s^{\prime}}},\delta^{\prime},q_{{\iota}},C^{\prime}),$ where the colours of states are left as they were in their original automaton, and $\delta^{\prime}$ is defined as follows. For states in $Q^{i}_{s^{\prime}}$ (resp. $\overline{Q^{i}_{s^{\prime}}}$), $\delta^{\prime}$ agrees with $\delta^{i}_{s^{\prime}}$ (resp. $\overline{\delta^{i}_{s^{\prime}}}$), and for $(q^{\psi},s^{\prime})\in Q$ and $a\in 2^{{\textnormal{AP}_{\exists}}}$ we let $\delta^{\prime}((q^{\psi},s^{\prime}),a)$ be the disjunction over ${a^{\prime}\in 2^{\max(\psi)}}$ of (2) $\displaystyle\Bigg{(}\delta\left((q^{\psi},s^{\prime}),a^{\prime}\right)\wedge\bigwedge_{\varphi_{i}\in a^{\prime}}\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a)\;\wedge\bigwedge_{\varphi_{i}\notin a^{\prime}}\overline{\delta^{i}_{s^{\prime}}}(\overline{q^{i}_{s^{\prime}}},a)\Bigg{)}.$ Note that in general it is not possible to define a $\max(\psi)$-labelling of $t$ that faithfully represents the truth values of formulas in $\max(\psi)$ for all nodes in $t_{\mathcal{S}}$, because a node in $t$ may correspond to different nodes in $t_{\mathcal{S}}$ that have same projection on $S_{\varphi}$ but satisfy different formulas of $\max(\psi)$. However this is not a problem because different copies of $\mathcal{A}_{s}^{\varphi}$ that visit the same node can guess different labellings, depending on the actual state of $\mathcal{S}$ (which is part of the state of $\mathcal{A}_{s}^{\varphi}$). $\bm{\varphi=\exists}^{\bm{\textnormal{{o}}}}\bm{p.\,\varphi^{\prime}:}$ We build automaton $\mathcal{A}_{s}^{\varphi^{\prime}}$ that works on $S_{\varphi^{\prime}}$-trees; because $\varphi$ is hierarchical, we have that $\textnormal{{o}}\subseteq I_{\varphi^{\prime}}$ and we can narrow down $\mathcal{A}_{s}^{\varphi^{\prime}}$ to work on $S_{\textnormal{{o}}}$-trees and obtain $\mathcal{A}_{1}:={\mathcal{A}_{s}^{\varphi^{\prime}}\\!\downarrow_{\textnormal{{o}}}}$. By Theorem 4.6 we can nondeterminise it to get $\mathcal{A}_{2}$, which by Theorem 4.5 we can project with respect to $p$, finally obtaining $\mathcal{A}_{s}^{\varphi}:=\mathcal{A}_{2}\\!\Downarrow_{p}$. Correctness. We now prove by induction on $\varphi$ that the construction is correct. In each case, we let $t=(\tau,\ell)$ be a complete $({\textnormal{AP}_{\exists}},S_{\varphi})$-tree rooted in $s_{\iota}\\!\downarrow_{I_{\varphi}}$. $\bm{\varphi=p:}$ First, note that $I_{p}=[n]$, so that $t$ is rooted in $s_{\iota}\\!\downarrow_{I_{\varphi}}=s_{\iota}$, and $u\\!\downarrow_{I_{\varphi}}=u$. Also recall that $u$ ends in $s$. Let us consider first the case where $p\in\textnormal{AP}_{f}$. By definition of $\mathcal{A}_{s}^{p}$, we have that $(t,u)\in\mathcal{L}(\mathcal{A}_{s}^{p})$ if and only if $p\in\ell_{\mathcal{S}}(s)$. We also have $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models p$ if and only if $p\in\ell^{\prime}(u)$, where $\ell^{\prime}$ is the labelling of tree $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$. By definition of unfolding and merge, we have that $\ell^{\prime}(u)=\ell_{\mathcal{S}}(s)$, which concludes this direction. Now if $p\in{\textnormal{AP}_{\exists}}$: by definition of $\mathcal{A}_{s}^{p}$, we have $(t,u)\in\mathcal{L}(\mathcal{A}_{s}^{p})$ if and only if $p\in\ell(u)$; also, by definition of the merge and unfolding, we have that $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models p$ if and only if $p\in\ell(u)$, and we are done. $\bm{\varphi=\neg\varphi^{\prime}:}$ Correctness follows from the induction hypothesis and Theorem 4.4. $\bm{\varphi_{1}\vee\varphi_{2}:}$ We have $\mathcal{A}_{i}=\mathcal{A}_{s}^{\varphi_{i}}\\!\downarrow_{I_{\varphi}}$, so by Theorem 4.7 we have $(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{i})$ if and only if $(t\\!\uparrow^{I_{\varphi_{i}}},u\\!\downarrow_{I_{\varphi_{i}}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi_{i}})$, which by induction hypothesis holds if and only if $(t\\!\uparrow^{I_{\varphi_{i}}})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi_{i}$, i.e., $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi_{i}$. We conclude by observing that $\mathcal{L}(\mathcal{A}_{s}^{\varphi})=\mathcal{L}(\mathcal{A}_{1})\cup\mathcal{L}(\mathcal{A}_{2})$. $\bm{\varphi={\bf E}\psi:}$ Suppose that $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models{\bf E}\psi$. There exists an infinite path $\lambda$ in $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$ starting at $u$ such that $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},\lambda\models\psi$. Again, let $\max(\psi)$ be the set of maximal state subformulas of $\varphi$, and let $w$ be the infinite word over $2^{\max(\psi)}$ that agrees with $\lambda$ on the state formulas in $\max(\psi)$, i.e., for each node $\lambda_{k}$ of $\lambda$ and formula $\varphi_{i}\in\max(\psi)$, it holds that $\varphi_{i}\in w_{k}$ if and only if $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},\lambda_{k}\models\varphi_{i}$. To show that $(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi})$ we show that Eve can win the acceptance game $\mathcal{G}(\mathcal{A}_{s}^{\varphi},t,u\\!\downarrow_{I_{\varphi}})$. In this game, Eve can guess the path $\lambda$ while the automaton follows $\lambda\\!\downarrow_{I_{\varphi}}$ in its input $t$, and she can also guess the corresponding word $w$ on $2^{\max(\psi)}$. By construction of $\mathcal{W}^{\psi}$, Eve has a winning strategy $\sigma_{\psi}$ in the acceptance game of $\mathcal{W}^{\psi}$ on $w$. From $\lambda$, $w$ and $\sigma_{\psi}$ we can easily define a strategy for Eve in $\mathcal{G}(\mathcal{A}_{s}^{\varphi},t,u\\!\downarrow_{I_{\varphi}})$ on all positions that can be reached while Adam does not choose to challenge her on a guess she made for the truth value of some maximal state subformula, and on such plays this strategy is winning because $\sigma_{\psi}$ is winning. Now if Adam challenges her on one of these guesses: Let $\lambda_{k}\in t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$ be a node along $\lambda$, let $s^{\prime}$ be its last direction and let $\lambda_{k}^{\prime}=\lambda_{k}\\!\downarrow_{I_{\varphi}}\in t$. Assume that in node $\lambda^{\prime}_{k}$ of the input tree, in a state $(q^{\psi},s^{\prime})\in Q$, Adam challenges Eve on some $\varphi_{i}\in\max(\psi)$ that she assumes to be true in $\lambda^{\prime}_{k}$, i.e., such that $\varphi_{i}\in w_{k}$. Formally, in the evaluation game this means that Adam chooses the conjunct $\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a)$ in transition formula 2, where $a=\ell(\lambda^{\prime}_{k})$, thus moving to position $(\lambda^{\prime}_{k},(q^{\psi},s^{\prime}),\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a))$. We want to show that Eve wins from this position. To do so we first show that $(t,\lambda^{\prime}_{k})\in\mathcal{L}(\mathcal{A}^{i}_{s^{\prime}})$. First, since $\mathcal{A}^{i}_{s^{\prime}}=\mathcal{A}_{s^{\prime}}^{\varphi_{i}}\\!\downarrow_{I_{\varphi}}$, by Theorem 4.7, $(t,\lambda^{\prime}_{k})\in\mathcal{L}(\mathcal{A}^{i}_{s^{\prime}})$ if and only if $(t\\!\uparrow^{I_{\varphi_{i}}},\lambda_{k}\\!\downarrow_{I_{\varphi_{i}}})\in\mathcal{L}(\mathcal{A}_{s^{\prime}}^{\varphi_{i}})$. Next, by applying the induction hypothesis we get that $(t\\!\uparrow^{I_{\varphi_{i}}},\lambda_{k}\\!\downarrow_{I_{\varphi_{i}}})\in\mathcal{L}(\mathcal{A}_{s^{\prime}}^{\varphi_{i}})$ if and only if $t\\!\uparrow^{I_{\varphi_{i}}}\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},\lambda_{k}\models\varphi_{i}$, i.e., $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},\lambda_{k}\models\varphi_{i}$. The latter holds because $\varphi_{i}\in w_{k}$, and by assumption $w_{k}$ agrees with $\lambda_{k}$ on $\varphi_{i}$. Thus $(t,\lambda^{\prime}_{k})\in\mathcal{L}(\mathcal{A}^{i}_{s^{\prime}})$. This means that Eve has a winning strategy from the initial position $(\lambda^{\prime}_{k},q^{i}_{s^{\prime}},\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a))$ of the acceptance game of $\mathcal{A}^{i}_{s^{\prime}}$ on $(t,\lambda^{\prime}_{k})$. Since $(\lambda^{\prime}_{k},q^{i}_{s^{\prime}},\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a))$ and $(\lambda^{\prime}_{k},(q^{\psi},s^{\prime}),\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a))$ contain the same node $\lambda^{\prime}_{k}$ and transition formula $\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a)$, the subgames that start in these positions are isomorphic and a winning strategy in one of these positions induces a winning strategy in the other, and therefore Eve wins Adam’s challenge (recall that positional strategies are sufficient in parity games (Zielonka, 1998)). With a similar argument, we get that also when Adam challenges Eve on some $\varphi_{i}\in\max(\psi)$ assumed not to be true in node $\lambda_{k}$, Eve wins the challenge. Finally, Eve wins the acceptance game of $\mathcal{A}_{s}^{\varphi}$ on $(t,u\\!\downarrow_{I_{\varphi}})$, and thus $(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi})$. For the other direction, assume that $(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi})$, i.e., Eve wins the evaluation game of $\mathcal{A}_{s}^{\varphi}$ on $(t,u\\!\downarrow_{I_{\varphi}})$. A winning strategy for Eve describes a path $\lambda$ in $t_{\mathcal{S}}$ from $s$, which is also a path in $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$ from $u$. This winning strategy also defines an infinite word $w$ over $2^{\max(\psi)}$ such that $w$ agrees with $\lambda$ on the formulas in $\max(\psi)$, and it also describes a winning strategy for Eve in the acceptance game of $\mathcal{W}^{\psi}$ on $w$. Hence $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},\lambda\models\psi$, and $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi$. $\bm{\varphi=\exists}^{\bm{\textnormal{{o}}}}\bm{p.\,\varphi^{\prime}:}$ First, by definition we have $I_{\varphi}=\textnormal{{o}}\cap I_{\varphi^{\prime}}$. Because $\varphi$ is hierarchical, $\textnormal{{o}}\subseteq\textnormal{{o}}^{\prime}$ for every $\textnormal{{o}}^{\prime}$ that occurs in $\varphi^{\prime}$, and thus $\textnormal{{o}}\subseteq I_{\varphi^{\prime}}$. It follows that $I_{\varphi}=\textnormal{{o}}$. Next, by Theorem 4.5 we have that (3) $(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi})\mbox{\;\;\;iff\;\;\;}\exists\,\ell_{p}\mbox{ a $p$-labelling for $t$ such that }(t\otimes\ell_{p},u)\in\mathcal{L}(\mathcal{A}_{2}).$ By Theorem 4.6, $\mathcal{L}(\mathcal{A}_{2})=\mathcal{L}(\mathcal{A}_{1})$, and since $\mathcal{A}_{1}=\mathcal{A}_{s}^{\varphi^{\prime}}\\!\downarrow_{\textnormal{{o}}}=\mathcal{A}_{s}^{\varphi^{\prime}}\\!\downarrow_{I_{\varphi}}$ we get by Theorem 4.7 that (4) $(t\otimes\ell_{p},u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{2})\mbox{\;\;\;iff\;\;\;}((t\otimes\ell_{p})\\!\uparrow^{L_{\varphi^{\prime}}},u\\!\downarrow_{I_{\varphi^{\prime}}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi^{\prime}}).$ By induction hypothesis, (5) $((t\otimes\ell_{p})\\!\uparrow^{L_{\varphi^{\prime}}},u\\!\downarrow_{I_{\varphi^{\prime}}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi^{\prime}})\mbox{\;\;\;iff\;\;\;}(t\otimes\ell_{p})\\!\uparrow^{L_{\varphi^{\prime}}}\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi^{\prime}.$ Now, by points (3), (4) and (5) and the fact that $(t\otimes\ell_{p})\\!\uparrow^{I_{\varphi^{\prime}}}\\!\uparrow^{[n]}=(t\otimes\ell_{p})\\!\uparrow^{[n]}$, we get that (6) $(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s}^{\varphi})\mbox{\;\;\;iff\;\;\;}\exists\,\ell_{p}\mbox{ a $p$-labelling for $t$ such that }(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi^{\prime}.$ We now prove the following equation which, together with point (6), concludes the proof: (7) $\begin{array}[]{c}\exists\,\ell_{p}\mbox{ a $p$-labelling for $t$ such that }(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi^{\prime}\\\ \mbox{\;\;\;iff\;\;\;}\\\ t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\exists^{\textnormal{{o}}}p.\,\varphi^{\prime}\end{array}$ Assume that there exists a $p$-labelling $\ell_{p}$ for $t$ such that $(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi^{\prime}$. Let $\ell_{p}^{\prime}$ be the $p$-labelling of $(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$. By definition of the merge, $\ell_{p}^{\prime}$ is equal to the $p$-labelling of $(t\otimes\ell_{p})\\!\uparrow^{[n]}$, which by definition of the widening is $I_{\varphi}$-uniform, i.e., it is o-uniform. In addition, it is clear that $(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}=(t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}})\otimes\ell_{p}^{\prime}$, which concludes this direction. For the other direction, assume that $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\exists^{\textnormal{{o}}}p.\,\varphi^{\prime}$: there exists a o-uniform $p$-labelling $\ell_{p}^{\prime}$ for $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$ such that $(t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}})\otimes\ell_{p}^{\prime},u\models\varphi^{\prime}$. We define a $p$-labelling $\ell_{p}$ for $t$ such that $(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\varphi^{\prime}$. First, let us write $t^{\prime}=t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}=(\tau^{\prime},\ell^{\prime})$. For each node $u$ of $t$, let $\ell_{p}(u)=\begin{cases}\ell_{p}^{\prime}(u^{\prime})&\mbox{if there exists }u^{\prime}\in\tau^{\prime}\mbox{ such that }u^{\prime}\\!\downarrow_{\textnormal{{o}}}=u,\\\ 0&\mbox{otherwise.}\end{cases}$ This is well defined because $\ell_{p}^{\prime}$ is o-uniform in $p$, so that if two nodes $u^{\prime},v^{\prime}$ project on $u$, we have $u^{\prime}\approx_{\textnormal{{o}}}v^{\prime}$ and thus $\ell_{p}^{\prime}(u^{\prime})=\ell_{p}^{\prime}(v^{\prime})$. In case there is no $u^{\prime}\in\tau^{\prime}$ such that $u^{\prime}\\!\downarrow_{I_{\varphi}}=u$, the value of $\ell_{p}(u)$ has no impact on $(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}$. Finally, $(t\otimes\ell_{p})\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}=(t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}})\otimes\ell_{p}^{\prime}$, hence the result. ∎ ### 4.3. Proof of Theorem 4.3 We now prove Theorem 4.3. Let $\mathcal{S}$ be a CKS with initial state $s_{\iota}$, and let $\Phi\in\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$. By Lemma 4.10 one can build an ATA $\mathcal{A}_{s_{\iota}}^{\Phi}$ such that for every labelled $S_{\varphi}$-tree $t$ rooted in $s_{\iota}\\!\downarrow_{I_{\varphi}}$, and every node $u\in t_{\mathcal{S}}$, $(t,u\\!\downarrow_{I_{\varphi}})\in\mathcal{L}(\mathcal{A}_{s_{\iota}}^{\varphi})$ if, and only if, $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}},u\models\Phi$. Let $\tau$ be the full $S_{\varphi}$-tree rooted in $s_{\iota}\\!\downarrow_{I_{\varphi}}$, and let $t=(\tau,\ell_{\emptyset})$, where $\ell_{\emptyset}$ is the empty labelling. Clearly, $t\\!\uparrow^{[n]}\merge\;t_{\mathcal{S}}=t_{\mathcal{S}}$, and because $t$ is rooted in $s_{\iota}\\!\downarrow_{I_{\varphi}}$, we get that $t\in\mathcal{L}(\mathcal{A}_{s_{\iota}}^{\varphi})$ if, and only if $t_{\mathcal{S}}\models\Phi$, i.e., $\mathcal{S}\models\Phi$. It remains to check whether tree $t$, which is regular, is accepted by $\mathcal{A}_{s_{\iota}}^{\Phi}$. This can be done by solving a parity game built from the product of $\mathcal{A}_{s_{\iota}}^{\Phi}$ with a finite Kripke structure representing $t$ (Löding, 2011). ### 4.4. Complexity To state a precise upper bound on the complexity of our procedure, we first introduce a syntactic notion of _simulation depth_ for formulas of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. While alternation depth (see, e.g., (Mogavero et al., 2014)) simply counts the number of alternations between existential and universal strategy quantifications, simulation depth reflects automata operations required to treat a formula, and counts the maximum number of nested simulations of alternating tree automata that need to be performed when applying our automata construction. However, like alternation depth, it is a purely syntactic notion. Formally we define a function $\mbox{sd}:\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}\to\mathbb{N}\times\\{\mbox{nd},\mbox{alt}\\}$ which returns, for each formula $\varphi$, a pair $\mbox{sd}(\varphi)=(k,x)$ where $k$ is the simulation depth of $\varphi$, and $x\in\\{\mbox{nd},\mbox{alt}\\}$ indicates whether the automaton $\mathcal{A}_{s}^{\varphi}$ built from $\varphi$ and a state $s$ of a CKS $\mathcal{S}$ is nondeterministic (nd) or alternating (alt). If $\mbox{sd}(\varphi)=(k,x)$ we shall denote $k$ by $\mbox{sd}_{k}(\varphi)$ and $x$ by $\mbox{sd}_{x}(\varphi)$. The inductive definition for state formulas is as follows: $\begin{array}[]{l}\mbox{sd}(p):=(0,\mbox{nd})\\\\[5.0pt] \mbox{sd}(\neg\varphi):=(\mbox{sd}_{k}(\varphi),\mbox{alt})\\\\[5.0pt] \mbox{sd}(\varphi_{1}\vee\varphi_{2}):=\left(\max_{i\in\\{1,2\\}}\mbox{sd}_{k}(\varphi_{i}),x\right),\\\ \hfill\mbox{where }x=\begin{cases}\mbox{nd}&\mbox{if }\mbox{sd}_{x}(\varphi_{1})=\mbox{sd}_{x}(\varphi_{2})=\mbox{nd}\\\ \mbox{alt}&\mbox{otherwise}\end{cases}\\\\[15.0pt] \mbox{sd}({\bf E}\psi):=\begin{cases}(0,\mbox{nd})&\mbox{if }\psi\in\textnormal{LTL}\\\ (\max_{\varphi\in\max(\psi)}\mbox{sd}_{k}(\varphi),\mbox{alt})&\mbox{otherwise}\end{cases}\\\\[15.0pt] \mbox{sd}(\exists^{\textnormal{{o}}}p.\,\varphi):=(k,\mbox{nd}),\\\ \hfill\quad\quad\quad\quad\quad\mbox{where }k=\begin{cases}\mbox{sd}_{k}(\varphi)&\mbox{if }\mbox{sd}_{x}(\varphi)=\mbox{nd}\mbox{ and }\textnormal{{o}}=I_{\varphi}\quad\mbox{(recall Definition~{}\ref{def- Iphi})}\\\ \mbox{sd}_{k}(\varphi)+1&\mbox{otherwise}\end{cases}\end{array}$ We explain each case. For an atomic proposition $p$, the automaton $\mathcal{A}_{s}^{p}$ is clearly nondeterministic and no simulation is involved in its construction. For a formula $\neg\varphi$, the automaton $\mathcal{A}_{s}^{\neg\varphi}$ is obtained by dualising $\mathcal{A}_{s}^{\varphi}$, an operation that in general does not return a nondeterministic automaton but an alternating one; also this dualisation does not involve any simulation, hence the definition of the first component. Now for the disjunction, the first component should be clear; for the second one, observe that by construction of $\mathcal{A}_{s}^{\varphi_{1}\vee\varphi_{2}}$, if both $\mathcal{A}_{s}^{\varphi_{1}}$ and $\mathcal{A}_{s}^{\varphi_{2}}$ are nondeterministic, then so is $\mathcal{A}_{s}^{\varphi_{1}\vee\varphi_{2}}$; otherwise, it is alternating. For the path quantifier, by construction $\mathcal{A}_{s}^{{\bf E}\psi}$ is alternating in the general case as it starts copies of automata for each maximal state subformula in $\psi$; for the first component, we recall that $\max(\psi)$ denotes the set of these maximal state subformulas and we observe that no additional simulation is performed to build $\mathcal{A}_{s}^{{\bf E}\psi}$ besides those needed to construct the automata for the maximal state subformulas. If $\psi$ is an LTL formula, then one can build the nondeterministic word automaton $\mathcal{W}^{\psi}$ directly working on “real” atomic propositions in ${\textnormal{AP}_{\exists}}\cup\textnormal{AP}_{f}$. The automaton $\mathcal{A}$ can then be built working directly on ${\textnormal{AP}_{\exists}}$, with $\mathcal{W}^{\psi}$ reading valuations for ${\textnormal{AP}_{\exists}}$ in the input tree and those for atoms in $\textnormal{AP}_{f}$ in the current state of $\mathcal{S}$. Because we do not need to guess valuations of maximal state subformulas and launch additional automata to check that these guesses are correct, we obtain a nondeterministic automaton. Finally, for a formula of the form $\exists^{\textnormal{{o}}}p.\,\varphi$, to build automaton $\mathcal{A}_{s}^{\exists^{\textnormal{{o}}}p.\,\varphi}$ we first build $\mathcal{A}_{s}^{\varphi}$, which we then narrow down to work on $L_{\textnormal{{o}}}$-trees. Since the narrowing operation introduces alternation, we need to nondeterminise the resulting automaton before projecting it with respect to $p$. Now observe that if $I_{\varphi}=\textnormal{{o}}$ we do not need to perform this narrowing, and thus if $\mathcal{A}_{s}^{\varphi}$ is a nondeterministic automaton we can directly perform the projection. This justifies the definition of the first component; for the second one, observe that the projection of a nondeterministic automaton is also nondeterministic. ###### Example 4.11. Assume that $n=3$, i.e., states of CKS have three components (recall that $[3]=\\{1,2,3\\}$). Let us consider formula $\varphi=\forall^{\\{1,3\\}}p.\,\forall^{[3]}q.\,\exists^{[3]}r.\,{\bf E}{\bf G}(p\wedge q\vee r)$. We describe how its simulation depth is computed. First, let us rewrite $\varphi=\neg\exists^{\\{1,3\\}}p.\,\exists^{[3]}q.\,\neg\exists^{[3]}r.\,{\bf E}{\bf G}(p\wedge q\vee r)$. Since ${\bf G}(p\wedge q\vee r)$ is an LTL formula, $\mbox{sd}({\bf E}{\bf G}(p\wedge q\vee r))=(0,\mbox{nd})$. Next, because $I_{{\bf E}{\bf G}(p\wedge q\vee r)}=[3]$, it follows that $\mbox{sd}(\exists^{[3]}r.\,{\bf E}{\bf G}(p\wedge q\vee r))=(0,\mbox{nd})$, and $\mbox{sd}(\neg\exists^{[3]}r.\,{\bf E}{\bf G}(p\wedge q\vee r))=(0,\mbox{alt})$. Next we have that $\mbox{sd}(\exists^{[3]}q.\,\neg\exists^{[3]}r.\,{\bf E}{\bf G}(p\wedge q\vee r))=(1,\mbox{nd})$. This reflects the fact that the automaton obtained for formula $\neg\exists^{[3]}r.\,{\bf E}{\bf G}(p\wedge q\vee r)$, which is alternating because of complementation, needs to be simulated before projecting it over $q$. Then, because $\\{1,3\\}\neq[3]$, it holds that $\mbox{sd}(\exists^{\\{1,3\\}}p.\,\exists^{[3]}q.\,\neg\exists^{[3]}r.\,{\bf E}{\bf G}(p\wedge q\vee r))=(2,\mbox{nd})$: to project over $p$ we first need to narrow down the previous automaton to make it see only components 1 and 3, and because the narrowing operation introduces alternation, the resulting automaton needs to be simulated before projecting it. Finally, we get that $\mbox{sd}(\varphi)=(2,\mbox{alt})$ We now introduce two additional depth measures on $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formulas, which help us establish more precise upper bounds on the sizes of the automata we build. For every $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $\varphi$, we let ${\bf E}\mathrm{d}(\varphi)$ be the maximum number of nested path quantifiers ${\bf E}$ in $\varphi$, and $\exists\mathrm{d}(\varphi)$ is the maximum number of nested second-order quantifiers $\exists$ in $\varphi$. We also inductively define the function $\mathrm{exp}\big{(}k\mid n\big{)}$, for $k,n\in\mathbb{N}$, as follows: $\mathrm{exp}\big{(}0\mid n\big{)}:=n$ and $\mathrm{exp}\big{(}k+1\mid n\big{)}:=2^{\mathrm{exp}\big{(}k\mid n\big{)}}$. ###### Proposition 4.12. Let $\Phi$ be a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ formula, $\mathcal{S}$ a CKS and $s\in\mathcal{S}$ a state. * • If $\mbox{sd}_{k}(\Phi)=0$, $\mathcal{A}_{s}^{\Phi}$ has at most $f_{\mathcal{S}}^{\Phi}$ states and 2 colours, and * • if $\mbox{sd}_{k}(\Phi)\geq 1$, $\mathcal{A}_{s}^{\Phi}$ has at most $\mathrm{exp}\big{(}\mbox{sd}_{k}(\Phi)\mid f_{\mathcal{S}}^{\Phi}\log f_{\mathcal{S}}^{\Phi}\big{)}$ states and its number of colours is at most $\mathrm{exp}\big{(}\mbox{sd}_{k}(\Phi)-1\mid f_{\mathcal{S}}^{\Phi}\log f_{\mathcal{S}}^{\Phi}\big{)}$, where $f_{\mathcal{S}}^{\Phi}=m_{1}^{\exists\mathrm{d}(\Phi)}|\Phi||\mathcal{S}|^{{\bf E}\mathrm{d}(\Phi)}2^{m_{2}|\Phi|{\bf E}\mathrm{d}(\Phi)}$, with $m_{1},m_{2}\in\mathbb{N}$ constants. Also, if $\mathcal{A}_{s}^{\varphi}$ has state set $Q$ then for each $q\in Q$ and $a\in 2^{{\textnormal{AP}_{\exists}}(\Phi)}$ we have $|\delta(q,a)|\leq|\mathcal{S}||Q|^{|\mathcal{S}|}2^{H|\varphi|}$, where $H=1+{\bf E}\mathrm{d}(\varphi)$. Constants $m_{1}$ and $m_{2}$ are derived from constants in the complexity of, respectively, the simulation procedure, and the procedure that builds a nondeterministic word automaton for an LTL formula. For more detail, see the proof of Proposition 4.12 in Appendix A. From this we get the following complexity result. ###### Proposition 4.13. The model-checking problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ formulas of simulation depth at most $k$ is $(k+1)$-Exptime -complete. ###### Proof. We start with the upper bounds. For an instance $(\Phi,\mathcal{S})$, our decision procedure in Section 4.3 first builds automaton $\mathcal{A}_{s_{\iota}}^{\Phi}$, and concludes by testing whether the full $S_{\Phi}$-tree with empty labelling $t$ is accepted by $\mathcal{A}_{s_{\iota}}^{\Phi}$. This can be done in time $O((|\mathcal{A}_{s_{\iota}}^{\Phi}|\cdot|t|)^{l})$, where $|t|$ is the size of a smallest Kripke structure representing the regular tree $t$, $|\mathcal{A}_{s_{\iota}}^{\Phi}|$ is the sum of the number of states and sizes of formulas in the transition function of $\mathcal{A}_{s_{\iota}}^{\Phi}$, and $l$ the number of colours it uses (Löding, 2011). Clearly $t$ can be represented by a Kripke structure of size $|S_{\Phi}|$, so that $|t|\leq|S_{\Phi}|\leq|\mathcal{S}|$. By Proposition 4.12, each formula in the transition function of $\mathcal{A}_{s_{\iota}}^{\Phi}$ is of size at most $|\mathcal{S}||Q|^{|\mathcal{S}|}2^{H|\Phi|}$, where $Q$ is the set of states in $\mathcal{A}_{s_{\iota}}^{\Phi}$ and $H=1+{\bf E}\mathrm{d}(\Phi)$. There are at most $|Q|2^{|{\textnormal{AP}_{\exists}}(\Phi)|}$ such formulas333In fact the final automaton $\mathcal{A}_{s_{\iota}}^{\Phi}$ does not read anything in its input, hence the alphabet could be considered to be a singleton. We thus have only $|Q|$ different formulas in the transition function, at most. and $|{\textnormal{AP}_{\exists}}(\Phi)|\leq|\Phi|$, so that $|\mathcal{A}_{s_{\iota}}^{\Phi}|\leq|Q|+|Q|2^{|{\textnormal{AP}_{\exists}}(\Phi)|}|\mathcal{S}||Q|^{|\mathcal{S}|}2^{H|\Phi|}\leq 2|\mathcal{S}||Q|^{|\mathcal{S}|+1}2^{(H+1)|\Phi|}$. Also $H+1\leq|\Phi|$, so we finally have $|\mathcal{A}_{s_{\iota}}^{\Phi}|\leq 2|\mathcal{S}||Q|^{|\mathcal{S}|+1}2^{|\Phi|^{2}}$. If $k=0$, by Proposition 4.12 $\mathcal{A}_{s_{\iota}}^{\Phi}$ has at most $f_{\mathcal{S}}^{\Phi}$ states and 2 colours, and $f_{\mathcal{S}}^{\Phi}$ is polynomial in $|\mathcal{S}|$ but exponential in $|\Phi|$. Therefore $|\mathcal{A}_{s_{\iota}}^{\Phi}|$ is exponential in $|\Phi|$ and in $|\mathcal{S}|$, and so is the complexity of checking that $t$ is accepted by $\mathcal{A}_{s_{\iota}}^{\Phi}$. If $k\geq 1$, by Proposition 4.12, $|Q|$ is $k$-exponential in $f_{\mathcal{S}}^{\Phi}\log f_{\mathcal{S}}^{\Phi}$, and $f_{\mathcal{S}}^{\Phi}\log f_{\mathcal{S}}^{\Phi}$ itself is polynomial in $|\mathcal{S}|$ but exponential in $|\Phi|$. As a result, $|\mathcal{A}_{s_{\iota}}^{\Phi}|$ is $(k+1)$-exponential in $|\Phi|$ and $k$-exponential in $|\mathcal{S}|$. Finally, still by Proposition 4.12, the number of colours $l$ is $(k-1)$-exponential in $f_{\mathcal{S}}^{\Phi}\log f_{\mathcal{S}}^{\Phi}$, hence $k$-exponential in $|\Phi|$. Checking that $t$ is accepted by $\mathcal{A}_{s_{\iota}}^{\Phi}$ can thus be done in time $(k+1)$-exponential in $|\Phi|$, and $k$-exponential in $|\mathcal{S}|$, which finishes to establish the upper bounds. For the lower bounds, consider the fragment $\textnormal{{EQ}}^{k}\textnormal{{CTL}}^{*}$ of $\textnormal{{QCTL}}^{*}$ (with perfect information) which consists in formulas in prenex normal form, i.e., with all second-order quantifications at the beginning, with at most $k$ alternations between existential and universal quantifiers, counting the first quantifier as one alternation (see (Laroussinie and Markey, 2014, p.8) for a formal definition). Clearly, $\textnormal{{EQ}}^{k}\textnormal{{CTL}}^{*}$ is a fragment of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ (with $n=1$), and formulas of $\textnormal{{EQ}}^{k}\textnormal{{CTL}}^{*}$ have simulation depth at most $k$. It is proved in (Laroussinie and Markey, 2014) that model checking $\textnormal{{EQ}}^{k}\textnormal{{CTL}}^{*}$ is $(k+1)$-Exptime -hard. ∎ ###### Remark 3. One may wonder why we do not get our lower bounds from the distributed synthesis problem in systems with hierarchical information. The reason is that this problem is $k$-Exptime -complete for LTL or $\textnormal{{CTL}}^{*}$ specifications (Pnueli and Rosner, 1990; Kupferman and Vardi, 2001) and can be expressed with formulas of simulation depth $k$, and thus would only provide $k$-Exptime lower-bounds for simulation depth $k$, while our problem is $k+1$-Exptime -complete. This may seem surprising, but we point out that thanks to alternation of existential and universal quantifiers, $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formulas with simulation depth $k$ can express more complex problems than classic distributed synthesis, such as existence of Nash equilibria (see Section 7.1). Improved upper bound. We now refine the previous result by observing that some subformulas can be model-checked independently in a bottom-up labelling algorithm which uses the above model-checking procedure as a subroutine. The height of exponential of the overall procedure for a formula $\Phi$ is thus determined by the maximal simulation-depth of the successive independent subformulas $\varphi$ treated by the labelling algorithm, instead of the simulation depth of the full formula $\Phi$. To make this precise we define the _simulation number_ of a sentence, akin to the alternation number introduced in (Mogavero et al., 2014). Let $\Phi\in\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, and assume without loss of generality that ${\textnormal{AP}_{\exists}}(\Phi)\cap\textnormal{AP}_{f}(\Phi)=\emptyset$. A state subformula $\varphi$ of $\Phi$ is a _subsentence_ if no atom quantified in $\Phi$ appears free in $\varphi$, i.e., $\varphi$ is a subsentence of $\Phi$ if ${\textnormal{AP}_{\exists}}(\Phi)\cap\textnormal{AP}_{f}(\varphi)=\emptyset$.444Observe that since we always assume that ${\textnormal{AP}_{\exists}}(\Phi)\cap\textnormal{AP}_{f}(\Phi)=\emptyset$, $\Phi$ is a subsentence of itself. The _simulation number_ $\mbox{sn}(\Phi)$ of a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $\Phi$ is the maximal simulation depth of $\Phi$’s subsentences, where the simulation depth is computed by considering strict subsentences as atoms. Note that because temporal operators of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ can only talk about the future, the truth value of a subsentence in a node $u$ of an unfolding $t_{\mathcal{S}}$ only depends on the current state $\mbox{last}(u)$. The bottom-up labelling algorithm for an instance $(\Phi,\mathcal{S})$ thus consists in iteratively model checking innermore subsentences of $\Phi$ in all states of $\mathcal{S}$, marking the states where they hold with fresh atomic propositions with which the corresponding subsentences are replaced in $\Phi$. ###### Proposition 4.14. The model-checking problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ formulas of simulation number at most $k$ is $(k+1)$-Exptime -complete. ## 5\. Model-checking hierarchical instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ In this section we establish that the model-checking problem for $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ restricted to the class of hierarchical instances is decidable (Theorem 2.9). ### 5.1. Reduction to $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ We build upon the proof in (Laroussinie and Markey, 2015) that establishes the decidability of the model-checking problem for $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc}}$ by reduction to the model-checking problem for $\textnormal{{QCTL}}^{*}$. The main difference is that we reduce to the model-checking problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ instead, using quantifiers on atomic propositions parameterised with observations that reflect the ones used in the $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ model-checking instance. Let $(\mathcal{G},\Phi)$ be a hierarchical instance of the $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ model-checking problem, and assume without loss of generality that each strategy variable is quantified at most once in $\Phi$. We define an equivalent instance of the model-checking problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$. Constructing the CKS $\mathcal{S}_{\mathcal{G}}$. We define $\mathcal{S}_{\mathcal{G}}$ so that (indistinguishable) nodes in its tree- unfolding correspond to (indistinguishable) finite plays in $\mathcal{G}$. The CKS will make use of atomic propositions $\textnormal{AP}_{v}:=\\{p_{v}\mid v\in V\\}$ (that we assume to be disjoint from AP). The idea is that $p_{v}$ allows the $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $(\Phi)_{s}^{\,\emptyset}$ to refer to the current position $v$ in $\mathcal{G}$. Later we will see that $(\Phi)_{s}^{\,\emptyset}$ will also make use of atomic propositions $\textnormal{AP}_{c}:=\\{p_{c}^{x}\mid c\in\textnormal{Ac}\mbox{ and }x\in\textnormal{Var}\\}$ that we assume, again, are disjoint from $\textnormal{AP}\cup\textnormal{AP}_{v}$. This allows the formula to use $p_{c}^{x}$ to refer to the actions $c$ advised by strategies $x$. Suppose $\textnormal{Obs}=\\{o_{1},\ldots,o_{n}\\}$, and let $\mathcal{G}=(\textnormal{Ac},V,E,\ell,v_{\iota},\mathcal{O})$. For $i\in[n]$, define the local states $L_{i}:=\\{[v]_{o_{i}}\mid v\in V\\}$ where $[v]_{o}$ is the equivalence class of $v$ for relation $\sim_{o}$. Since we need to know the actual position of the $\textrm{CGS}_{\textnormal{ii}}$ to define the dynamics, we also let $L_{n+1}:=V$. Define the CKS $\mathcal{S}_{\mathcal{G}}:=(S,R,s_{{\iota}},\ell^{\prime})$ where * • $S:=\\{s_{v}\mid v\in V\\}$, * • $R:=\\{(s_{v},s_{v^{\prime}})\mid\exists\bm{c}\in\textnormal{Ac}^{\textnormal{Ag}}\mbox{ s.t. }E(v,\bm{c})=v^{\prime}\\}\subseteq S^{2}$, * • $s_{{\iota}}:=s_{v_{{\iota}}}$, * • $\ell^{\prime}(s_{v}):=\ell(v)\cup\\{p_{v}\\}\subseteq\textnormal{AP}\cup\textnormal{AP}_{v}$, and $s_{v}:=([v]_{o_{1}},\ldots,[v]_{o_{n}},v)\in\prod_{i\in[n+1]}L_{i}$. For every finite play $\rho=v_{0}\ldots v_{k}$, define the node $u_{\rho}:=s_{v_{0}}\ldots s_{v_{k}}$ in $t_{\mathcal{S}_{\mathcal{G}}}$ (which exists, by definition of $\mathcal{S}_{\mathcal{G}}$ and of tree unfoldings). Note that the mapping $\rho\mapsto u_{\rho}$ defines a bijection between the set of finite plays and the set of nodes in $t_{\mathcal{S}_{\mathcal{G}}}$. Constructing the $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ formulas $(\varphi)_{s}^{\,f}$. We now describe how to transform an $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula $\varphi$ and a partial function $f:\textnormal{Ag}\rightharpoonup\textnormal{Var}$ into a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $(\varphi)_{s}^{\,f}$ (that will also depend on $\mathcal{G}$). Suppose that $\textnormal{Ac}=\\{c_{1},\ldots,c_{l}\\}$, and define $(\varphi)_{s}^{\,f}$ and $(\psi)_{p}^{\,f}$ by mutual induction on state and path formulas. The base cases are as follows: $(p)_{s}^{\,f}:=p$ and $(\varphi)_{p}^{\,f}:=(\varphi)_{s}^{\,f}$. Boolean and temporal operators are simply obtained by distributing the translation: $(\neg\varphi)_{s}^{\,f}:=\neg(\varphi)_{s}^{\,f}$, $(\neg\psi)_{p}^{\,f}:=\neg(\psi)_{p}^{\,f}$, $(\varphi_{1}\vee\varphi_{2})_{s}^{\,f}:=(\varphi_{1})_{s}^{\,f}\vee(\varphi_{2})_{s}^{\,f}$, $(\psi_{1}\vee\psi_{2})_{p}^{\,f}:=(\psi_{1})_{p}^{\,f}\vee(\psi_{2})_{p}^{\,f}$, $({\bf X}\psi)_{p}^{\,f}:={\bf X}(\psi)_{p}^{\,f}$ and $(\psi_{1}{\bf U}\psi_{2})_{p}^{\,f}:=(\psi_{1})_{p}^{\,f}{\bf U}(\psi_{2})_{p}^{\,f}$. We continue with the case of the strategy quantifier: $\begin{array}[]{lrl}&(\langle\\!\langle x\rangle\\!\rangle^{o}\varphi)_{s}^{\,f}&:=\exists^{\widetilde{o}}p_{c_{1}}^{x}\ldots\exists^{\widetilde{o}}p_{c_{l}}^{x}.\varphi_{\text{str}}(x)\wedge(\varphi)_{s}^{\,f}\\\\[5.0pt] \mbox{where}&\varphi_{\text{str}}(x)&:={\bf A}{\bf G}\bigvee_{c\in\textnormal{Ac}}p_{c}^{x}\\\\[5.0pt] \mbox{and}&\widetilde{o_{i}}&:=\\{j\mid\mathcal{O}(o_{i})\subseteq\mathcal{O}(o_{j})\\}.\end{array}$ The intuition is that for each possible action $c\in\textnormal{Ac}$, an existential quantification on the atomic proposition $p_{c}^{x}$ “chooses” for each node $u_{\rho}$ of the tree $t_{\mathcal{S}_{\mathcal{G}}}$ whether strategy $x$ allows action $c$ in $\rho$ or not, and it does so uniformly with regards to observation $\widetilde{o}$. $\varphi_{\text{str}}(x)$ checks that at least one action is allowed in each node, and thus that atomic propositions $p_{c}^{x}$ indeed define a strategy. We define $\widetilde{o_{i}}$ as $\\{j\mid\mathcal{O}(o_{i})\subseteq\mathcal{O}(o_{j})\\}$ instead of $\\{i\\}$ in order to obtain a hierarchical instance. Note that including all coarser observations does not increase the information accessible to the quantifier: indeed, two nodes are $\\{i\\}$-indistinguishable if and only if they are $\widetilde{o_{i}}$-indistinguishable. Here are the remaining cases: $\begin{array}[]{lrl}&((a,x)\varphi)_{s}^{\,f}&:=(\varphi)_{s}^{\,f[a\mapsto x]}\quad\quad\text{for }x\in\textnormal{Var}\cup\\{\operatorname{?}\\}\\\\[5.0pt] \mbox{and}&({\bf E}\psi)_{s}^{\,f}&:={\bf E}\,(\psi_{\text{out}}^{\,f}\wedge(\psi)_{p}^{\,f})\\\\[5.0pt] \mbox{where}&\psi_{\text{out}}^{\,f}&:={\bf G}\bigvee_{v\in V}\left(p_{v}\wedge\bigvee_{\bm{c}\in\textnormal{Ac}^{\textnormal{Ag}}}\bigwedge_{a\in\textit{dom}(f)}p_{\bm{c}_{a}}^{f(a)}\wedge{\bf X}\,p_{E(v,\bm{c})}\right).\end{array}$ $\psi_{\text{out}}^{\,f}$ checks that each player $a$ in the domain of $f$ follows the strategy coded by the $p_{c}^{f(a)}$. ###### Remark 4. If we consider the fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ that only allows for deterministic strategies, the translation can be adapted by simply replacing formula $\varphi_{\text{str}}(x)$ above with its deterministic variant $\varphi_{\text{str}}^{\text{det}}(x):={\bf A}{\bf G}\bigvee_{c\in\textnormal{Ac}}(p_{c}^{x}\wedge\bigwedge_{c^{\prime}\neq c}\neg p_{c^{\prime}}^{x}),$ which ensures that _exactly one_ action is chosen for strategy $x$ in each finite play, and thus that atomic propositions $p_{c}^{x}$ characterise a deterministic strategy. To prove correctness of the translation, given a strategy $\sigma$ and a strategy variable $x$ we let $\ell_{\sigma}^{x}:=\\{\ell_{p_{c}^{x}}\mid c\in\textnormal{Ac}\\}$ be the family of $p_{c}^{x}$-labellings for tree $t_{\mathcal{S}_{\mathcal{G}}}$ defined as follows: for each finite play $\rho$ in $\mathcal{G}$ and $c\in\textnormal{Ac}$, we let $\ell_{p_{c}^{x}}(u_{\rho}):=1$ if $c\in\sigma(\rho)$, 0 otherwise. For a labelled tree $t$ with same domain as $t_{\mathcal{S}_{\mathcal{G}}}$ we write $t\otimes\ell_{\sigma}^{x}$ for $t\otimes\ell_{p_{c_{1}}^{x}}\otimes\ldots\otimes\ell_{p_{c_{l}}^{x}}$. Given an infinite play $\pi$ and a point $i\in\mathbb{N}$, we also let $\lambda_{\pi,i}$ be the infinite path in $t_{\mathcal{S}_{\mathcal{G}}}$ that starts in node $u_{\pi_{\leq i}}$ and is defined as $\lambda_{\pi,i}:=u_{\pi_{\leq i}}u_{\pi_{\leq i+1}}u_{\pi_{\leq i+2}}\ldots$ Finally, for an assignment $\chi$ and a partial function $f:\textnormal{Ag}\rightharpoonup\textnormal{Var}$, we say that $f$ is _compatible_ with $\chi$ if $\textit{dom}(\chi)\cap\textnormal{Ag}=\textit{dom}(f)$ and for all $a\in\textit{dom}(f)$, $\chi(a)=\chi(f(a))$. ###### Proposition 5.1. For every state subformula $\varphi$ and path subformula $\psi$ of $\Phi$, finite play $\rho$, infinite play $\pi$, point $i\in\mathbb{N}$, for every assignment $\chi$ variable-complete for $\varphi$ (resp. $\psi$) and partial function $f:\textnormal{Ag}\rightharpoonup\textnormal{Var}$ compatible with $\chi$, assuming also that no $x_{i}$ in $\textit{dom}(\chi)\cap\textnormal{Var}=\\{x_{1},\ldots,x_{k}\\}$ is quantified in $\varphi$ or $\psi$, we have $\displaystyle\mathcal{G},\chi,{\rho}\models\varphi$ if and only if $\displaystyle t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},u_{\rho}\models(\varphi)_{s}^{\,f}$ $\displaystyle\mathcal{G},\chi,{\pi},i\models\psi$ if and only if $\displaystyle t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},\lambda_{\pi,i}\models(\psi)_{p}^{\,f}$ In addition, $\mathcal{S}_{\mathcal{G}}$ is of size linear in $|\mathcal{G}|$, and $(\varphi)_{s}^{\,f}$ and $(\psi)_{p}^{\,f}$ are of size linear in $|\mathcal{G}|^{2}+|\varphi|$. ###### Proof. The proof is by induction on $\varphi$. We detail the cases for binding, strategy quantification and outcome quantification, the others follow simply by definition of $\mathcal{S}_{\mathcal{G}}$ for atomic propositions and induction hypothesis for remaining cases. For $\varphi=(a,x)\varphi^{\prime}$, we have $\mathcal{G},\chi,{\rho}\models(a,x)\varphi^{\prime}$ if and only if $\mathcal{G},\chi[a\mapsto\chi(x)],{\rho}\models\varphi^{\prime}$. The result follows by using the induction hypothesis with assignment $\chi[a\mapsto x]$ and function $f[a\mapsto x]$. This is possible because $f[a\mapsto x]$ is compatible with $\chi[a\mapsto x]$: indeed $\textit{dom}(\chi[a\mapsto x])\cap\textnormal{Ag}$ is equal to $\textit{dom}(\chi)\cap\textnormal{Ag}\cup\\{a\\}$ which, by assumption, is equal to $\textit{dom}(f)\cup\\{a\\}=\textit{dom}(f[a\mapsto x])$. Also by assumption, for all $a^{\prime}\in\textit{dom}(f)$, $\chi(a^{\prime})=\chi(f(a^{\prime}))$, and by definition $\chi[a\mapsto\chi(x)](a)=\chi(x)=\chi(f[a\mapsto x](a))$. For $\varphi=\langle\\!\langle x\rangle\\!\rangle^{o}\varphi^{\prime}$, assume first that $\mathcal{G},\chi,{\rho}\models\langle\\!\langle x\rangle\\!\rangle^{o}\varphi^{\prime}$. There exists an $o$-uniform strategy $\sigma$ such that $\mathcal{G},\chi[x\mapsto\sigma],\rho\models\varphi^{\prime}.$ Since $f$ is compatible with $\chi$, it is also compatible with assignment $\chi^{\prime}=\chi[x\mapsto\sigma]$. By assumption, no variable in $\\{x_{1},\ldots,x_{k}\\}$ is quantified in $\varphi$, so that $x\neq x_{i}$ for all $i$, and thus $\chi^{\prime}(x_{i})=\chi(x_{i})$ for all $i$; and because no strategy variable is quantified twice in a same formula, $x$ is not quantified in $\varphi^{\prime}$, so that no variable in $\\{x_{1},\ldots,x_{k},x\\}$ is quantified in $\varphi^{\prime}$. By induction hypothesis $t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi^{\prime}(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi^{\prime}(x_{k})}^{x_{k}}\otimes\ell_{\chi^{\prime}(x)}^{x},u_{\rho}\models(\varphi^{\prime})_{s}^{\,f}.$ Because $\sigma$ is $o$-uniform, each $\ell_{p_{c}^{x}}\in\ell_{\sigma}^{x}=\ell_{\chi^{\prime}(x)}^{x}$ is $\widetilde{o}$-uniform, and it follows that $t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi^{\prime}(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi^{\prime}(x_{k})}^{x_{k}},u_{\rho}\models\exists^{\widetilde{o}}p_{c_{1}}^{x}\ldots\exists^{\widetilde{o}}p_{c_{l}}^{x}.\varphi_{\text{str}}(x)\wedge(\varphi^{\prime})_{s}^{\,f}.$ Finally, since $\chi^{\prime}(x_{i})=\chi(x_{i})$ for all $i$, we conclude that $t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},u_{\rho}\models(\langle\\!\langle x\rangle\\!\rangle^{o}\varphi^{\prime})_{s}^{\,f}.$ For the other direction, assume that $t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},u_{\rho}\models(\varphi)_{s}^{\,f},$ and recall that $(\varphi)_{s}^{\,f}=\exists^{\widetilde{o}}p_{c_{1}}^{x}\ldots\exists^{\widetilde{o}}p_{c_{l}}^{x}.\varphi_{\text{str}}(x)\wedge(\varphi^{\prime})_{s}^{\,f}$. Write $t=t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}}$. There exist $\widetilde{o}$-uniform $\ell_{p_{c}^{x}}$-labellings such that $t\otimes\ell_{p_{c_{1}}^{x}}\otimes\ldots\otimes\ell_{p_{c_{l}}^{x}}\models\varphi_{\text{str}}(x)\wedge(\varphi^{\prime})_{s}^{\,f}.$ By $\varphi_{\text{str}}(x)$, these labellings code for a strategy $\sigma$, and because they are $\widetilde{o}$-uniform, $\sigma$ is $o$-uniform. Let $\chi^{\prime}=\chi[x\mapsto\sigma]$. For all $1\leq i\leq k$, by assumption $x\neq x_{i}$, and thus $\chi^{\prime}(x_{i})=\chi(x_{i})$. The above can thus be rewritten $t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi^{\prime}(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi^{\prime}(x_{k})}^{x_{k}}\otimes\ell_{\chi^{\prime}(x)}^{x}\models\varphi_{\text{str}}(x)\wedge(\varphi^{\prime})_{s}^{\,f}.$ By induction hypothesis we have $\mathcal{G},\chi[x\mapsto\sigma],\rho\models\varphi^{\prime}$, hence $\mathcal{G},\chi,\rho\models\langle\\!\langle x\rangle\\!\rangle^{o}\varphi^{\prime}$. For $\varphi={\bf E}\psi$, assume first that $\mathcal{G},\chi,{\rho}\models{\bf E}\psi$. There exists a play $\pi\in\textnormal{Out}(\chi,\rho)$ such that $\mathcal{G},\chi,\pi,|\rho|-1\models\psi$. By induction hypothesis, $t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},\lambda_{\pi,|\rho|-1}\models(\psi)_{p}^{\,f}$. Since $\pi$ is an outcome of $\chi$, each agent $a\in\textit{dom}(\chi)\cap\textnormal{Ag}$ follows strategy $\chi(a)$ in $\pi$. Because $\textit{dom}(\chi)\cap\textnormal{Ag}=\textit{dom}(f)$ and for all $a\in\textit{dom}(f)$, $\chi(a)=\chi(f(a))$, each agent $a\in\textit{dom}(f)$ follows the strategy $\chi(f(a))$, which is coded by atoms $p_{c}^{f(a)}$ in the translation of $\Phi$. Therefore $\lambda_{\pi,|\rho|-1}$ also satisfies $\psi_{\text{out}}^{\,\chi}$, hence $t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},\lambda_{\pi,|\rho|-1}\models\psi_{\text{out}}^{\,\chi}\wedge(\psi)_{p}^{\,f}$, and we are done. For the other direction, assume that $t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}},u_{\rho}\models{\bf E}(\psi_{\text{out}}^{\,f}\wedge(\psi)_{p}^{\,f})$. There exists a path $\lambda$ in $t_{\mathcal{S}_{\mathcal{G}}}\otimes\ell_{\chi(x_{1})}^{x_{1}}\otimes\ldots\otimes\ell_{\chi(x_{k})}^{x_{k}}$ starting in node $u_{\rho}$ that satisfies both $\psi_{\text{out}}^{\,f}$ and $(\psi)_{p}^{\,f}$. By construction of $\mathcal{S}_{\mathcal{G}}$ there exists an infinite play $\pi$ such that $\pi_{\leq|\rho|-1}=\rho$ and $\lambda=\lambda_{\pi,|\rho|-1}$. By induction hypothesis, $\mathcal{G},\chi,\pi,|\rho|-1\models\psi$. Because $\lambda_{\pi,|\rho|-1}$ satisfies $\psi_{\text{out}}^{\,f}$, $\textit{dom}(\chi)\cap\textnormal{Ag}=\textit{dom}(f)$, and for all $a\in\textit{dom}(f)$, $\chi(a)=\chi(f(a))$, it is also the case that $\pi\in\textnormal{Out}(\chi,\rho)$, hence $\mathcal{G},\chi,\rho\models{\bf E}\psi$. The size of $\mathcal{S}_{\mathcal{G}}$, $(\varphi)_{s}^{\,f}$ and $(\psi)_{p}^{\,f}$ are easily verified. ∎ Applying Proposition 5.1 to the sentence $\Phi$, $\rho=v_{\iota}$, any assignment $\chi$, and the empty function $\emptyset$, we get: $\mathcal{G}\models\Phi\quad\mbox{if and only if}\quad t_{\mathcal{S}_{\mathcal{G}}}\models(\Phi)_{s}^{\,\emptyset}.$ Preserving hierarchy. To complete the proof of Theorem 2.9 it remains to check that $(\Phi)_{s}^{\,\emptyset}$ is a hierarchical $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula, which is the case because $\Phi$ is hierarchical in $\mathcal{G}$ and for every two observations $o_{i}$ and $o_{j}$ in Obs such that $\mathcal{O}(o_{i})\subseteq\mathcal{O}(o_{j})$, by definition of $\widetilde{o_{k}}$ we have that $\widetilde{o_{i}}\subseteq\widetilde{o_{j}}$. ### 5.2. Complexity We now establish the complexity of model checking hierarchical instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. As we did for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, we first define the simulation depth of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ state formulas. In the following inductive definition, $\mathcal{O}_{\varphi}$ denotes the intersection of all indistinguishability relations used in $\varphi$: $\mathcal{O}_{\varphi}:=\cap_{o\in\varphi}\mathcal{O}(o)$, with the empty intersection being defined as the identity relation (perfect information). Also, for a path formula $\psi$, $\max(\psi)$ is the set of maximal state subformulas in $\psi$. $\begin{array}[]{lcc}\mbox{sd}(p):=(0,\mbox{nd})&&\mbox{sd}(\neg\varphi):=(\mbox{sd}_{k}(\varphi),\mbox{alt})\\\\[7.0pt] \lx<EMAIL_ADDRESS>\lx@intercol\hfil\mbox{where }x=\begin{cases}\mbox{nd}&\mbox{if }\mbox{sd}_{x}(\varphi_{1})=\mbox{sd}_{x}(\varphi_{2})=\mbox{nd}\\\ <EMAIL_ADDRESS>\lx@intercol\mbox{sd}(\langle\\!\langle <EMAIL_ADDRESS>\lx@intercol\hfil\mbox{where }k=\begin{cases}\mbox{sd}_{k}(\varphi)&\mbox{if }\mbox{sd}_{x}(\varphi)=\mbox{nd}\mbox{ and }\mathcal{O}(o)=\mathcal{O}_{\varphi}\\\ <EMAIL_ADDRESS>\mbox{sd}((a,x)\varphi):=\mbox{sd}(\varphi)\\\\[7.0pt] \lx@intercol\mbox{sd}({\bf E}\psi):=\begin{cases}(0,\mbox{nd})&\mbox{if }\psi\in\textnormal{LTL}\\\ (\max_{\varphi\in\max(\psi)}\mbox{sd}_{k}(\varphi),\mbox{alt})&\mbox{otherwise}\end{cases}\hfil\lx@intercol\end{array}$ ###### Proposition 5.2. The model-checking problem for hierarchical instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ of simulation depth at most $k$ is $(k+1)$-Exptime -complete. ###### Proof. The upper bounds follow from the fact that the translated formulas in our reduction have essentially the same simulation depth as the original ones. However this is not quite right, because in the case where $\mbox{sd}_{x}(\varphi)=\mbox{nd}$ and $\mathcal{O}(o)=\mathcal{O}_{\varphi}$ we have $\mbox{sd}(\langle\\!\langle x\rangle\\!\rangle^{o}\varphi)=(\mbox{sd}_{k}(\varphi),\mbox{nd})$, while $\mbox{sd}((\langle\\!\langle x\rangle\\!\rangle^{o}\varphi)_{s}^{\,f})=(\mbox{sd}_{k}((\varphi)_{s}^{\,f})+1,\mbox{nd})$: indeed, while it is the case that $\mathcal{O}(o)=\mathcal{O}_{\varphi}$ implies that $\widetilde{o}=I_{(\varphi)_{s}^{\,f}}$, the translation introduces a conjunction with $\varphi_{\text{str}}(x)$, and even when $\mbox{sd}_{x}((\varphi)_{s}^{\,f})=\mbox{nd}$, we have $\mbox{sd}_{x}(\varphi_{\text{str}}(x)\wedge(\varphi)_{s}^{\,f})=\mbox{alt}$. According to Proposition 4.13, this should thus induce an additional exponential to check the translated formula. However, this can be avoided by noticing that the fixed formula $\varphi_{\text{str}}(x)={\bf A}{\bf G}\bigvee_{c\in\textnormal{Ac}}p_{c}^{x}$ can be checked by a simple _deterministic_ tree automaton with two states $q_{\text{check}}$ and $q_{\text{rej}}$: the automaton starts in state $q_{\text{check}}$, which is accepting (it has parity zero); when it visits a node $u$ in state $q_{\text{check}}$, if $\ell(u)$ satisfies $\bigvee_{c\in\textnormal{Ac}}p_{c}^{x}$, then the automaton sends state $q_{\text{check}}$ to all children of $u$, otherwise it sends the state $q_{\text{rej}}$ to all children. State $q_{\text{rej}}$ is rejecting (it has parity one) and is a sink: it sends itself to all children, independently on the label of the visited node. If we restrict $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ to deterministic strategies, the same observation can be made: the automaton that checks formula $\varphi_{\text{str}}^{\text{det}}(x)={\bf A}{\bf G}\bigvee_{c\in\textnormal{Ac}}(p_{c}^{x}\wedge\bigwedge_{c^{\prime}\neq c}\neg p_{c^{\prime}}^{x})$ is the same as the one described above, except that it checks whether $\bigvee_{c\in\textnormal{Ac}}(p_{c}^{x}\wedge\bigwedge_{c^{\prime}\neq c}\neg p_{c^{\prime}}^{x})$ is satisfied by the label of the current node. Given two tree automata $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, one deterministic and one nondeterministic, one can easily build a nondeterministic automaton $\mathcal{A}_{1}\cap\mathcal{A}_{2}$ of size $|\mathcal{A}_{1}|\times|\mathcal{A}_{2}|$ that accepts the intersection of their languages, so that in this case the conjunction does not introduce alternation, and thus we do not need an additional simulation before projecting to guess the strategy. We could refine the notion of simulation depth to reflect this, but we find that it would become very cumbersome for little added benefit, so we keep this observation in this proof. The lower bounds are inherited from $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ thanks to the polynomial reduction presented in Section 6.2.2, which preserves simulation depth. ∎ We point out that all instances of the model-checking problem for the perfect- information fragment are hierarchical, and thus this result provides improved upper-bounds for SL, which was only known to be in $k$-Exptime for formulas of length at most $k$ (Mogavero et al., 2014). Also the lower bounds for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ are inherited directly from the perfect-information fragment $\textnormal{{QCTL}}^{*}$, which reduces to the perfect-information fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ following the construction from Section 6.2.2. Therefore the lower bounds hold already for the perfect- information fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. Note however that this does not provide lower bounds for the usual, linear- time variant of Strategy Logic, where path quantifiers in $\textnormal{{QCTL}}^{*}$ formulas must be simulated with strategy quantifications which increase the simulation depth of the resulting Strategy Logic formulas. The exact complexity of the linear-time variant is not known, even in the perfect-information case. Simulation number. The intuition behind the alternation number as considered in (Mogavero et al., 2014) is to refine the classic alternation depth between existential and universal quantifiers by observing that subsentences of a sentence $\Phi$ to model-check can be treated independently thanks to a bottom-up labelling algorithm: innermost sentences are evaluated in all states of the model and replaced in $\Phi$ by atomic propositions that label the states where they hold. The alternation number of $\Phi$ is the maximum alternation depth of the successive subsentences that are treated by this bottom-up procedure, and it determines the complexity of the overall model- checking procedure. However, as discussed in Remark 1, the semantics of the outcome quantifier makes sentences sensitive to the assignment in which they are evaluated. As a result, to define the notion of alternation number in our setting, we introduce a notion of _independent subsentence_. Intuitively, a subsentence $\varphi$ of a sentence $\Phi$ is _independent_ if it redefines or unbinds the strategies of all players who are bound to a strategy when $\varphi$ is reached in the evaluation of $\Phi$. More precisely, we say that an agent $a$ is _bound_ in a syntactic subformula $\varphi$ of $\Phi$ if the path that leads to $\varphi$ in $\Phi$’s syntactic tree contains a binding operator $(a,x)$ for $a$ which is not followed by an unbinding $(a,\operatorname{?})$ for her. A subsentence $\varphi$ of $\Phi$ is _independent_ if all agents that are bound in $\varphi$ are either rebound by an operator $(a,x)$ or unbound by an operator $(a,\operatorname{?})$ before any outcome quantifier is met in $\varphi$. In an independent subsentence $\varphi$, the semantics of the outcome quantifier does not depend on strategies that are quantified outside $\varphi$, and in fact a subsentence $\varphi$ of $\Phi$ is independent if and only if the formula that corresponds to $\varphi$ in $(\Phi)_{s}^{\,\emptyset}$ is a subsentence of $(\Phi)_{s}^{\,\emptyset}$. Similarly to what we did for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ we now define the _simulation number_ $\mbox{sn}(\Phi)$ of an $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ sentence $\Phi$ as the maximum of the simulation depths for independent subsentences, where strict independent subsentences are counted as atoms. ###### Lemma 5.3. For every hierarchical instance $(\mathcal{G},\Phi)$ of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, $\mbox{sn}(\Phi)=\mbox{sn}((\Phi)_{s}^{\,\emptyset})$. The following then follows from Proposition 5.1, Lemma 5.3 and Proposition 4.14. ###### Proposition 5.4. The model-checking problem for hierarchical instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ of simulation number at most $k$ is $(k+1)$-Exptime -complete. We now compare the latter result with the complexity of model checking SL[NG], the nested goal fragment of Strategy Logic with perfect information (we refer the interested reader to (Mogavero et al., 2014) for a definition of this fragment). It is established in (Chatterjee et al., 2010b; Mogavero et al., 2014) that this problem is in $(k+1)$-Exptime for formulas of _alternation number_ $k$. We remark that the simulation number of an SL[NG] formula translated in our branching-time version of SL (this is done by adding outcome quantifiers between bindings and temporal operators) is equal to its alternation number plus one, and thus Proposition 5.4 gives a $(k+2)$-Exptime upper bound for SL[NG] formulas of alternation number $k$. In (Chatterjee et al., 2010b; Mogavero et al., 2014) the extra exponential is avoided by resorting to universal and nondeterministic tree automata, depending on whether the innermost strategy quantification is existential or universal, to deal with temporal formulas. Thus, the innermost strategy quantification can be dealt with without incurring an exponential blowup. The same thing cannot be done for $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, for two reasons. The first one is that in general the innermost strategy quantification may have imperfect information and thus require a narrowing of the automaton; this operation introduces alternation, which needs to be removed at the cost of one exponential before dealing with strategy quantification. The second reason is that even when the innermost strategy has perfect information, the outcome quantifier that we introduce in Strategy Logic allows the expression of $\textnormal{{CTL}}^{*}$ formulas which cannot be dealt with by nondeterministic and universal automata as is done in (Chatterjee et al., 2010b; Mogavero et al., 2014). ## 6\. Comparison with related logics In this section we first show that $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ subsumes SL and the main imperfect-information extensions of ATL. Then we show that model checking Coordination Logic (CL) reduces to model checking hierarchical instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ where the truth of all atomic propositions in the model is known by all agents (or more precisely, all observations in the concurrent game structures are fine enough to observe the truth value of all atomic propositions). ### 6.1. Comparison with ATL The main difference between SL and ATL-like strategic logics is that in the latter a strategy is always bound to some player, while in the former bindings and quantifications are separated. This separation adds expressive power, e.g., one can bind the same strategy to different players. Extending ATL with imperfect-information is done by giving each player an indistinguishability relation that its strategies must respect (Bulling and Jamroga, 2014). In $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ instead each strategy $x$ is assigned an indistinguishability relation $o$ when it is quantified. Associating observations to strategies rather than players allows us to obtain a logic $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ that is a clean generalisation of (perfect-information) SL, and subsumes imperfect-information extensions of $\textnormal{{ATL}}^{*}$ that associate observations to players. Concerning SL, it is rather easy to see that every sentence in SL has an equivalent in the fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ with deterministic strategies where all observation symbols are interpreted as perfect information. We now prove that $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ also subsumes $\textnormal{{ATL}}^{*}$ with imperfect information. ###### Proposition 6.1. For every $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$ formula555See (Bulling and Jamroga, 2014) for the definition of $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$, where subscript i refers to “imperfect information” and subscript R to “perfect recall”. Also, we consider the so-called _objective semantics_ for $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$. $\varphi$ there is an $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula $\varphi^{\prime}$ such that for every $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$ there is a $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}^{\prime}$ such that $\mathcal{G}\models\varphi$ if, and only if, $\mathcal{G}^{\prime}\models\varphi^{\prime}$. We recall that an $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$ formula $\langle A\rangle\psi$ reads as “there are strategies for players in $A$ such that $\psi$ holds whatever players in $\textnormal{Ag}\setminus A$ do”. Formula $\varphi^{\prime}$ is built from $\varphi$ by replacing each subformula of the form $\langle A\rangle\psi$, where $A=\\{a_{1},\ldots,a_{k}\\}\subset\textnormal{Ag}$ is a coalition of players and $\textnormal{Ag}\setminus A=\\{a_{k+1},\ldots,a_{n}\\}$ with formula $\langle\\!\langle x_{1}\rangle\\!\rangle^{o_{1}}\ldots\langle\\!\langle x_{k}\rangle\\!\rangle^{o_{k}}(a_{1},x_{1})\ldots(a_{k},x_{k})(a_{k+1},\operatorname{?})\ldots(a_{n},\operatorname{?}){\bf A}\,\psi^{\prime}$, where $\psi^{\prime}$ is the translation of $\psi$. Then $\mathcal{G}^{\prime}$ is obtained from $\mathcal{G}$ by interpreting each $o_{i}$ as the equivalence relation for player $i$ in $\mathcal{G}$, and interpreting $o_{p}$ as the identity relation. Third, $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ also subsumes the imperfect-information extension of $\textnormal{{ATL}}^{*}$ with strategy context (see (Laroussinie et al., 2015) for the definition of $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc}}$ with partial observation, which we refer to as $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc,i}}$). ###### Proposition 6.2. For every $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc,i}}$ formula $\varphi$ there is an $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula $\varphi^{\prime}$ such that for every $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$ there is a $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}^{\prime}$ such that $\mathcal{G}\models\varphi$ if, and only if, $\mathcal{G}^{\prime}\models\varphi^{\prime}$. The only difference between $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc,i}}$ and $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$ is the following: in $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize i,R}}$, when a subformula of the form $\langle A\rangle\psi$ is met, we quantify existentially on strategies for players in $A$ and quantify universally on possible outcomes obtained by letting other players behave however they want. Therefore, if any player in $\textnormal{Ag}\setminus A$ had previously been assigned a strategy, it is forgotten. In $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc,i}}$ on the other hand, these strategies are stored in a _strategy context_ , which is a _partial_ assignment $\chi$, defined for the subset of players currently bound to a strategy. A strategy context allows one to quantify universally only on strategies of players who are not in $A$ and who are not already bound to a strategy. It is then easy to adapt the translation presented for Proposition 6.1: it suffices not to unbind agents outside the coalition from their strategies. $\mathcal{G}^{\prime}$ is defined as for Proposition 6.1. ### 6.2. Comparison with Coordination Logic There is a natural and simple translation of instances of the model-checking problem of CL (Finkbeiner and Schewe, 2010) into the hierarchical instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. Moreover, the image of this translation consists of instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ with a very restricted form: atoms mentioned in the $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-formula are observable by all observations of the $\textrm{CGS}_{\textnormal{ii}}$ , i.e., for all $o\in\textnormal{Obs}$ and $p\in\textnormal{AP}$, $v\sim_{o}v^{\prime}$ implies that $p\in\ell(v)$ iff $p\in\ell(v^{\prime})$. ###### Proposition 6.3. There is an effective translation that, given a CL-instance $(\mathcal{S},\varphi)$ produces a hierarchical $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instance $(\mathcal{G},\Phi)$ such that 1. (1) $\mathcal{S}\models\varphi$ if, and only if, $\mathcal{G}\models\Phi$, 2. (2) For all atoms $p\in\textnormal{AP}$ and observations $o\in\textnormal{Obs}$, $v\sim_{o}v^{\prime}$ implies that $p\in\ell(v)$ iff $p\in\ell(v^{\prime})$. To do this, one first translates CL into (hierarchical) $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, the latter is defined in the next section. This step is a simple reflection of the semantics of CL in that of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$. Then one translates $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ into $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ by a simple adaptation of the translation of $\textnormal{{QCTL}}^{*}$ into $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc}}$ (Laroussinie and Markey, 2015). We briefly recall the syntax and semantics of CL, and refer to (Finkbeiner and Schewe, 2010) for further details. Notation for trees. Note that our definition for trees (see Section 3.2) differs slightly from the one in (Finkbeiner and Schewe, 2010), where the root is the empty word. Here we adopt this convention to stay closer to notations in (Finkbeiner and Schewe, 2010). Thus, $(Y,X)$-trees in CL are of the form $(\tau,l)$ where $\tau\subseteq X^{*}$ and $l:\tau\to 2^{Y}$. For two disjoint sets $X$ and $Y$, we identify $2^{X}\times 2^{Y}$ with $2^{X\cup Y}$. Let $X$ and $Y$ be two sets with $Z=X\cup Y$, and let $M$ and $N$ be two disjoint sets. Given an ${M}$-labelled $2^{Z}$-tree $t=(\tau,\ell_{M})$ and an ${N}$-labelled $2^{Z}$-tree $t^{\prime}=(\tau^{\prime},\ell_{N})$ with same domain $\tau=\tau^{\prime}$, we define $t\uplus t^{\prime}:=(\tau,\ell^{\prime})$, where for every $u\in\tau$, $\ell^{\prime}(u)=\ell_{M}(u)\cup\ell_{N}(u)$. Now, given a complete ${M}$-labelled $2^{X}$-tree $t=((2^{X})^{*},\ell_{M})$ and a complete ${N}$-labelled $2^{Y}$-tree $t^{\prime}=((2^{Y})^{*},\ell_{N})$, we define $t\oplus t^{\prime}:=t\\!\uparrow^{2^{Z\setminus X}}\uplus\,t^{\prime}\\!\uparrow^{2^{Z\setminus Y}}$. CL Syntax. Let $\mathcal{C}$ be a set of _coordination variables_ , and let $\mathcal{S}$ be a set of _strategy variables_ disjoint from $\mathcal{C}$. The syntax of CL is given by the following grammar: $\varphi::=x\mid\neg\varphi\mid\varphi\vee\varphi\mid{\bf X}\varphi\mid\varphi{\bf U}\varphi\mid\Finv C\exists s.\,\varphi$ where $x\in\mathcal{C}\cup\mathcal{S}$, $C\subseteq\mathcal{C}$ and $s\in\mathcal{S}$, and with the restriction that each coordination variable appears in at most one _subtree quantifier_ $\Finv C\exists s.\,$, and similarly for strategy variables. The notion of free and bound (coordination or strategy) variables is as usual. The set of free coordination variables in $\varphi$ is noted $\mathcal{F}_{\varphi}$. A bound coordination variable $c$ is _visible_ to a strategy variable $s$ if $s$ is in the scope of the quantifier that introduces $c$, and $\textit{Scope}_{\varphi}(s)$ is the union of the set of bound coordination variables visible to $s$ and the set of free coordination variables (note that this union is disjoint). We will see, in the semantics, that the meaning of a bound strategy variable $s$ is a strategy $f_{s}:(2^{\textit{Scope}_{\varphi}(s)})^{*}\to 2^{\\{s\\}}$. Free strategy variables are called _atomic propositions_ , and we denote the set of atomic propositions in $\varphi$ by $\textnormal{AP}_{\varphi}$. CL Semantics. A CL formula $\varphi$ is evaluated on a complete $\textnormal{AP}_{\varphi}$-labelled $2^{\mathcal{F}_{\varphi}}$-tree $t$. An $(\textnormal{AP}_{\varphi},2^{\mathcal{F}_{\varphi}})$-tree $t=(\tau,\ell)$ satisfies a CL formula $\varphi$ if for every path $\lambda$ that starts in the root we have $t,\lambda,0\models\varphi$, where the satisfaction of a formula at position $i\geq 0$ of a path $\lambda$ is defined inductively as follows: $\displaystyle t,\lambda,i\models$ $\displaystyle\,p$ if $\displaystyle\quad p\in\ell(\lambda_{i})$ $\displaystyle t,\lambda,i\models$ $\displaystyle\,\neg\varphi^{\prime}$ if $\displaystyle\quad t,\lambda,i\not\models\varphi^{\prime}$ $\displaystyle t,\lambda,i\models$ $\displaystyle\,\varphi_{1}\vee\varphi_{2}$ if $\displaystyle\quad t,\lambda,i\models\varphi_{1}\mbox{ or }t,\lambda,i\models\varphi_{2}$ $\displaystyle t,\lambda,i\models$ $\displaystyle\,{\bf X}\varphi^{\prime}$ if $\displaystyle\quad t,\lambda,i+1\models\varphi^{\prime}$ $\displaystyle t,\lambda,i\models$ $\displaystyle\,\varphi_{1}{\bf U}\varphi_{2}$ if $\displaystyle\quad\exists\,j\geq i\mbox{ s.t. }t,\lambda,j\models\varphi_{2}\text{ and }\forall k\text{ s.t. }i\leq k<j,\;t,\lambda,k\models\varphi_{1}$ $\displaystyle t,\lambda,i\models$ $\displaystyle\,\Finv C\exists s.\,\varphi^{\prime}\quad$ if $\displaystyle\quad\exists\,f:(2^{\textit{Scope}_{\varphi}(s)})^{*}\to 2^{\\{s\\}}\mbox{ s.t. }t_{\lambda_{i}}\oplus((2^{\textit{Scope}_{\varphi}(s)})^{*},f)\models\varphi^{\prime},$ where $t_{\lambda_{i}}$ is the subtree of $t$ rooted in $\lambda_{i}$. First, observe that in the last inductive case, $t_{\lambda_{i}}$ being a $2^{\mathcal{F}_{\varphi}}$-tree, $t_{\lambda_{i}}\oplus((2^{\textit{Scope}_{\varphi}(s)})^{*},f)$ is a $2^{\mathcal{F}_{\varphi}\cup\textit{Scope}_{\varphi}(s)}$-tree. By definition, $\textit{Scope}_{\varphi}(s)=\mathcal{F}_{\varphi}\cup C=\mathcal{F}_{\varphi^{\prime}}$. It follows that $\mathcal{F}_{\varphi}\cup\textit{Scope}_{\varphi}(s)=\textit{Scope}_{\varphi}(s)=\mathcal{F}_{\varphi^{\prime}}$, hence $\varphi^{\prime}$ is indeed evaluated on a $\mathcal{F}_{\varphi^{\prime}}$-tree. ###### Remark 5. Note that all strategies observe the value of all atomic propositions. Formally, for every CL-formula $\varphi$ of the form $\varphi=\Finv C_{1}\exists s_{1}.\,\ldots,\Finv C_{i}\exists s_{i}.\,\varphi^{\prime}$ evaluated on a $2^{\mathcal{F}_{\varphi}}$-tree $t=(\tau,\ell)$, $\varphi^{\prime}$ is evaluated on a $2^{\mathcal{F}_{\varphi}\cup C_{1}\cup\ldots\cup C_{i}}$-tree $t^{\prime}=(\tau^{\prime},\ell^{\prime})$ such that for every $p\in\textnormal{AP}_{\varphi}$, for every pair of nodes $u,u^{\prime}\in t^{\prime}$ such that $u\\!\downarrow_{2^{\mathcal{F}_{\varphi}}}=u^{\prime}\\!\downarrow_{2^{\mathcal{F}_{\varphi}}}$, it holds that $p\in\ell^{\prime}(u)$ iff $p\in\ell^{\prime}(u^{\prime})$. Thus, in CL one cannot directly capture strategic problems where atomic propositions are not observable to all players. The input to the model-checking problem for CL consists of a CL formula $\varphi$ and a finite representation of a $(\textnormal{AP}_{\varphi},2^{\mathcal{F}_{\varphi}})$-tree $t$. The standard assumption is to assume $t$ is a regular tree, i.e., is the unfolding of a finite structure. Precisely, a _finite representation_ of a $(\textnormal{AP}_{\varphi},2^{\mathcal{F}_{\varphi}})$-tree $t=(\tau,\ell^{\prime})$ is a structure $\mathcal{S}=(S,R,\ell,s_{\iota})$ such that * • $S=2^{\mathcal{F}_{\varphi}}$, * • $R=S\times S$, * • $\ell:S\to 2^{\textnormal{AP}_{\varphi}}$, * • $s_{\iota}\in S$, and $t=t_{\mathcal{S}}$ is the unfolding of $\mathcal{S}$. Thus, an _instance_ of the model-checking problem for CL is a pair $(\mathcal{S},\Phi)$ where $\mathcal{S}=(S,R,s_{{\iota}},\ell)$ is a finite representation of an $(\textnormal{AP}_{\varphi},2^{\mathcal{F}_{\varphi}})$-tree and $\Phi$ is a CL formula (over variables $\mathcal{S}\cup\mathcal{C}$). The _model-checking problem for CL_ is the following decision problem: given an instance $(\mathcal{S},\Phi)$, return ‘Yes’ if $t_{\mathcal{S}}\models\Phi$ and ‘No’ otherwise. We now describe a natural translation of CL-instances to $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instances. This translation: 1. (1) reduces the model-checking problem of CL to that of the hierarchical fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. 2. (2) shows that CL only produces instances in which all atoms are uniform with regard to all observations, i.e., instances $(\mathcal{G},\Phi)$ such that for every $p\in\textnormal{AP}$ and $o\in\textnormal{Obs}$, $v\sim_{o}v^{\prime}$ implies $p\in\ell(v)\leftrightarrow p\in\ell(v^{\prime})$. We will present the translation in two steps: first from CL-instances into $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$-instances, and then from $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$-instances to $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instances such that $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$-instances translate to hierarchical $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instances. #### 6.2.1. Translating CL to $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ Let $(\mathcal{S},\Phi)$ be an instance of the model-checking problem for CL, where $\mathcal{S}=(S,R,\ell,s_{{\iota}})$. We will construct a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$-instance $(\widetilde{\mathcal{S}},\widetilde{\varphi})$ such that $\mathcal{S}\models\Phi$ iff $\widetilde{\mathcal{S}}\models\widetilde{\Phi}$. Let $\widetilde{\textnormal{AP}}$ be the set of all strategy variables occurring in $\Phi$, let $\mathcal{C}(\Phi)$ be the set of coordination variables that appear in $\Phi$, and assume, w.l.o.g., that $\mathcal{C}(\varphi)=[n]$ for some $n\in\mathbb{N}$. Let $\textit{hidden}(\Phi):=\mathcal{C}(\Phi)\setminus\mathcal{F}_{\varphi}$. First, we define the CKS $\widetilde{\mathcal{S}}$ over $\widetilde{\textnormal{AP}}$: the idea is to add in the structure $\mathcal{S}$ the local states corresponding to coordination variables that are not seen by all the strategies. Formally, $\widetilde{\mathcal{S}}:=(\widetilde{S},\widetilde{R},\widetilde{s_{{\iota}}},\widetilde{\ell})$ where * • $\widetilde{S}=\prod_{c\in\mathcal{C}(\Phi)}L_{c}$ where $L_{c}=\\{c_{0},c_{1}\\}$, * • $\widetilde{R}=\widetilde{S}\times\widetilde{S}$, * • for every $s\in\widetilde{S}$, $\widetilde{\ell}(s)=\ell(s\\!\downarrow_{\mathcal{F}_{\varphi}})$, and * • $\widetilde{s_{{\iota}}}\in\widetilde{S}$ is any state $s$ such that $s\\!\downarrow_{\mathcal{F}_{\varphi}}=s_{{\iota}}$ Second, we define concrete observations corresponding to strategy variables in $\Phi$. As explained in (Finkbeiner and Schewe, 2010), and as reflected in the semantics of CL, the intuition is that a strategy variable $s$ in formula $\Phi$ observes coordination variables $\textit{Scope}_{\varphi}(s)$. Therefore, we simply define, for each strategy variable $s$ in $\Phi$, the concrete observation $\textnormal{{o}}_{s}:=\textit{Scope}_{\varphi}(s)$. Finally, we define the $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ formula $\widetilde{\Phi}$. This is done by induction on $\Phi$ as follows (recall that we take for atomic propositions in $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ the set of all strategy variables in $\Phi$): $\displaystyle\widetilde{x}$ $\displaystyle:=x$ $\displaystyle\widetilde{\neg\varphi}$ $\displaystyle:=\neg\widetilde{\varphi}$ $\displaystyle\widetilde{\varphi_{1}\vee\varphi_{2}}$ $\displaystyle:=\widetilde{\varphi_{1}}\vee\widetilde{\varphi_{2}}$ $\displaystyle\widetilde{{\bf X}\varphi}$ $\displaystyle:={\bf X}\,\widetilde{\varphi}$ $\displaystyle\widetilde{\varphi_{1}{\bf U}\varphi_{2}}$ $\displaystyle:=\widetilde{\varphi_{1}}\,{\bf U}\,\widetilde{\varphi_{2}}$ $\displaystyle\widetilde{\Finv C\exists s.\,\varphi}$ $\displaystyle:=\exists^{\textnormal{{o}}_{s}}s.\,{\bf A}\widetilde{\varphi}$ In the last case, note that $C\subseteq\textnormal{{o}}_{s}=\textit{Scope}_{\varphi}(s)$. Note that $\widetilde{\Phi}$ is a hierarchical $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$-formula. Also, one can easily check that the following holds: ###### Lemma 6.4. $t_{\mathcal{S}}\models\Phi\quad\mbox{iff}\quad t_{\widetilde{\mathcal{S}}}\models{\bf A}\widetilde{\Phi}$. Importantly, we notice that ${\bf A}\widetilde{\Phi}\in\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$, and that: ###### Lemma 6.5. For every $x\in\textnormal{AP}_{\varphi}$ and every $s$ quantified in $\Phi$, $t_{\widetilde{\mathcal{S}}}$ is $\textnormal{{o}}_{s}$-uniform in $x$. #### 6.2.2. Translation from $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ to $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ We now present a translation of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$-instances to $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-instances. It is a simple adaptation of the reduction from the model-checking problem for $\textnormal{{QCTL}}^{*}$ to the model-checking problem for $\textnormal{{ATL}}^{*}_{\textnormal{\scriptsize sc}}$ presented in (Laroussinie and Markey, 2015). Let $(\mathcal{S},\Phi)$ be an instance of the model-checking problem for $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, where $\mathcal{S}=(S,R,\ell,s_{{\iota}})$ and $S\subseteq\prod_{i\in[n]}L_{i}$. We assume, without loss of generality, that every atomic proposition is quantified at most once, and that if it appears quantified it does not appear free. Also, let ${\textnormal{AP}_{\exists}}(\Phi)=\\{p_{1},\ldots,p_{k}\\}$ be the set of atomic propositions quantified in $\Phi$, and for $i\in[k]$, let $\textnormal{{o}}_{i}$ be the concrete observation associated to the quantifier on $p_{i}$. We build the $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}^{\mathcal{S}}:=(\textnormal{Ac},V,E,\ell^{\prime},v_{\iota},\mathcal{O})$ over agents $\textnormal{Ag}:=\\{a_{0},a_{1},\ldots,a_{k}\\}$, observations $\textnormal{Obs}:=\\{o_{0},o_{1},\ldots,o_{k}\\}$ and atomic propositions $\textnormal{AP}:={\textnormal{AP}_{\exists}}(\Phi)\cup\\{p_{S}\\}$, where $p_{S}$ is a fresh atomic proposition. Intuitively, agent $a_{0}$ is in charge of choosing transitions in $\mathcal{S}$, while agent $a_{i}$ for $i\geq 1$ is in charge of choosing the valuation for $p_{i}\in{\textnormal{AP}_{\exists}}(\Phi)$. To this aim, we let $V:=\begin{array}[]{l}\\{v_{s}\mid s\in S\\}\;\cup\\\ \\{v_{s,i}\mid s\in S\mbox{ and }i\in[k]\\}\;\cup\\\ \\{v_{p_{i}}\mid 0\leq i\leq k\\}\;\cup\\\ \\{v_{\perp}\\}\end{array}$ and $\textnormal{Ac}:=\\{c^{s}\mid s\in S\\}\cup\\{c^{i}\mid 0\leq i\leq k\\}.$ In positions of the form $v_{s}$ with $s\in S$, transitions are determined by the action of agent $a_{0}$. First, she can choose to simulate a transition in $\mathcal{S}$: for every joint action $\bm{c}\in\textnormal{Ac}^{\textnormal{Ag}}$ such that $\bm{c}_{0}=c^{s^{\prime}}$, $E(v_{s},\bm{c}):=\begin{cases}v_{s^{\prime}}&\text{if }R(s,s^{\prime})\\\ v_{\perp}&\text{otherwise}.\end{cases}$ She can also choose to move to a position in which agent $a_{i}$ will choose the valuation for $p_{i}$ in the current node: for every joint action $\bm{c}\in\textnormal{Ac}^{\textnormal{Ag}}$ such that $\bm{c}_{0}=c^{i}$, $E(v_{s},\bm{c}):=\begin{cases}v_{s,i}&\text{if }i\neq 0\\\ v_{\perp}&\text{otherwise}.\end{cases}$ Next, in a position of the form $v_{s,i}$, agent $a_{i}$ determines the transition, which codes the labelling of $p_{i}$ in the current node: choosing $c^{i}$ means that $p_{i}$ holds in the current node, choosing any other action codes that $p_{i}$ does not hold. Formally, for every joint action $\bm{c}\in\textnormal{Ac}^{\textnormal{Ag}}$, $E(v_{s,i},\bm{c}):=\begin{cases}v_{p_{i}}&\text{if }\bm{c}_{i}=c^{i}\\\ v_{\perp}&\text{otherwise}.\end{cases}$ Positions of the form $v_{p_{i}}$ and $v_{\perp}$ are sink positions. The labelling function $\ell^{\prime}$ is defined as follows: $\ell^{\prime}(v):=\begin{cases}\ell(s)\cup\\{p_{S}\\}&\mbox{if }v=v_{s}\mbox{ for some }s\in S\\\ \emptyset&\mbox{if }v\in\\{v_{s,i}\mid s\in S,i\in[k]\\}\cup\,\\{v_{p_{0}},v_{\perp}\\}\\\ \\{p_{i}\\}&\mbox{if }v=v_{p_{i}}\text{ with }i\in[k]\end{cases}$ Finally we let $v_{{\iota}}:=v_{s_{{\iota}}}$ and we define the observation interpretation as follows: $\mathcal{O}(o_{0}):=\\{(v,v)\mid v\in V\\},$ meaning that agent $a_{0}$ has perfect information, and for $i\in[k]$, $\mathcal{O}(o_{i})$ is the smallest reflexive relation such that $\mathcal{O}(o_{i})\supseteq\bigcup_{s,s^{\prime}\in S}\\{(v_{s},v_{s^{\prime}}),(v_{s,i},v_{s^{\prime},i})\mid s\approx_{\textnormal{{o}}_{i}}s^{\prime}\\}.$ We explain the latter definition. First, observe that for every finite play $\rho$ in $\mathcal{G}^{\mathcal{S}}$ that stays in $V_{S}=\\{v_{s}\mid s\in S\\}$, writing $\rho=v_{s_{0}}\ldots v_{s_{n}}$, one can associate a finite path $\lambda_{\rho}=s_{0}\ldots s_{n}$ in $\mathcal{S}$. This mapping actually defines a bijection between the set of finite paths in $\mathcal{S}$ that start in $s_{{\iota}}$ and the set of finite plays in $\mathcal{G}^{\mathcal{S}}$ that remain in $V_{S}$. Now, according to the definition of the transition function, a strategy $\sigma_{i}$ for agent $i$ with $i\in[k]$ is only relevant on finite plays of the form $\rho=\rho^{\prime}\cdot v_{s,i}$, where $\rho^{\prime}\in V_{S}^{*}$, and $\sigma_{i}(\rho)$ is meant to determine whether $p_{i}$ holds in $\lambda_{\rho^{\prime}}$. If $\sigma_{i}$ is $o_{i}$-uniform, by definition of $\mathcal{O}(o_{i})$, it determines an $\textnormal{{o}}_{i}$-uniform labelling for $p_{i}$ in $t_{\mathcal{S}}$. Reciprocally, an $\textnormal{{o}}_{i}$-uniform labelling for $p_{i}$ in $t_{\mathcal{S}}$ induces an $\mathcal{O}(o_{i})$-strategy for agent $a_{i}$. It remains to transform $\Phi$ into an $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$-formula. We define the $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula $\widetilde{\Phi}$ by induction on $\Phi$ as follows: $\displaystyle\widetilde{p}$ $\displaystyle:=\begin{cases}{\bf E}{\bf X}{\bf X}p&\text{if }p=p_{i}\\\ p&\text{otherwise}\end{cases}$ $\displaystyle\widetilde{\neg\varphi}$ $\displaystyle:=\neg\widetilde{\varphi}$ $\displaystyle\widetilde{\varphi_{1}\vee\varphi_{2}}$ $\displaystyle:=\widetilde{\varphi_{1}}\vee\widetilde{\varphi_{2}}$ $\displaystyle\widetilde{{\bf E}\psi}$ $\displaystyle:={\bf E}({\bf G}p_{S}\wedge\widetilde{\psi})$ $\displaystyle\widetilde{\exists^{\textnormal{{o}}_{i}}p_{i}.\,\varphi}$ $\displaystyle:=\langle\\!\langle x_{i}\rangle\\!\rangle^{o_{i}}(a_{i},x_{i})\widetilde{\varphi}.$ The cases for path formulas are obtained by distributing over the operators. Observe that player 0 is never bound to a strategy. In the case for atomic propositions, the existential quantification on outcomes thus lets player 0 choose to move to a position where agent $i$ fixes the value for $p_{i}$ according to his strategy, fixed by the strategy quantifier in the translation for formulas of the form $\exists^{\textnormal{{o}}_{i}}p_{i}.\,\varphi$. In the translation of formulas of the form ${\bf E}\psi$, the existential quantification on outcomes lets player 0 choose a path in the original CKS $\mathcal{S}$. We have the following: ###### Lemma 6.6. $\mathcal{S}\models\Phi\quad\text{if and only if}\quad\mathcal{G}^{\mathcal{S}}\models\widetilde{\Phi}$. We observe that if $\Phi$ is hierarchical, then $(\widetilde{\Phi},\mathcal{G}^{\mathcal{S}})$ is a hierarchical instance, and: ###### Lemma 6.7. For every $p\in\textnormal{AP}_{f}(\Phi)$ and for every $i\in[k]$, if $t_{\mathcal{S}}$ is $\textnormal{{o}}_{i}$-uniform in $p$ then $v\sim_{o_{i}}v^{\prime}$ implies that $p\in\ell(v)$ iff $p\in\ell(v^{\prime})$. Combining Lemma 6.4 with Lemma 6.6 we get a reduction from the model-checking problem for CL to that for the hierarchical fragment of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, and Lemma 6.5 together with Lemma 6.7 show that in the models produced by this reduction, all atomic propositions are observable to all players. This implies that in CL one cannot reason about strategic problems with unobservable objectives. As a result it does not fully capture classic distributed synthesis (Pnueli and Rosner, 1990; Kupferman and Vardi, 2001), where the specification can talk about all variables, hidden and visible. It also shows that CL does not capture in a natural way ATL with imperfect information as defined in (Alur et al., 2002, Section 7.1), where imperfect information of agents is modelled by defining which atomic propositions they can observe. This, as well as unobservable objectives, can be naturally modelled in $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. ## 7\. Applications In this section we apply Theorem 2.9 to decide the existence of Nash Equilibria in hierarchical games of imperfect information. We then use a similar approach to obtain decidability results for the rational synthesis problem. In this section, for a tuple of agents $\bm{a}=(a_{i})_{i\in[m]}$ and tuple of strategy variables $\bm{x}=(x_{i})_{i\in[m]}$, we let $(\bm{a},\bm{x})$ be a macro for $(a_{1},x_{1})\ldots(a_{m},x_{m})$, and similarly for the unbinding operator $(\bm{a},\operatorname{?})$ which stands for $(a_{1},\operatorname{?})\ldots(a_{m},\operatorname{?})$. ### 7.1. Existence of Nash Equilibria in games with hierarchical observations A Nash equilibrium in a game is a tuple of strategies such that no player has an incentive to deviate. Let $\textnormal{Ag}=\\{a_{i}:i\in[n]\\}$. Assuming that agent $a_{i}$ has observation $o_{i}$ and LTL goal $\psi_{i}$, the following $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formula expresses the existence of a Nash equilibrium: $\displaystyle\Phi_{\textsc{NE}}:=$ $\displaystyle\langle\\!\langle x_{1}\rangle\\!\rangle^{o_{1}}\dots\langle\\!\langle x_{n}\rangle\\!\rangle^{o_{n}}(\bm{a},\bm{x})\bigwedge_{i\in[n]}\Big{[}\Big{(}\langle\\!\langle y_{i}\rangle\\!\rangle^{o_{i}}(a_{i},y_{i})\,{\bf A}\psi_{i}\Big{)}\to{\bf A}\psi_{i}\Big{]}$ where $\bm{a}=(a_{i})_{i\in[n]}$ and $\bm{x}=(x_{i})_{i\in[n]}$. Nash equilibria do not always exist when one restricts attention to pure strategies, as we do in this work. This is the case already in finite games, and by extension also in the infinite concurrent games played on graphs that we consider. This motivates the study of the Nash equilibria existence problem in such games. In the perfect information case, the problem has been solved for $\omega$-regular objectives, as well as more complex semi-quantitative objectives (Bouyer et al., 2015). When moving to imperfect information, for two players the problem is decidable for LTL objectives (Gutierrez et al., 2018) and parity objectives (Filiot et al., 2018). However, as for distributed synthesis, existence of Nash equilibria becomes undecidable for more than two players. This result is proved in (Bouyer, 2018) for constrained Nash equilibria (when one specifies for each player whether her objective is satisfied or not), and in (Gutierrez et al., 2018) for unconstrained equilibria. In both cases the proof proceeds by reduction from the distributed synthesis problem (Peterson et al., 2001; Pnueli and Rosner, 1990). The only known decidable cases for more than two players assume that all players receive the same information. In (Bouyer, 2018) the problem is solved on games where players observe the evolution of the game via _public signals_ and objectives are given by visible parity conditions or mean-payoff functions. In (Belardinelli et al., 2017a), an epistemic extension of strategy logic is used to solve the existence of Nash equilibria on games with _broadcast actions_ for objectives given as formulas from epistemic temporal logic. A stronger notion of Nash equilibria, called _locally consistent equilibria_ , is studied in (Ramanujam and Simon, 2010). In a locally consistent equilibrium, each player’s strategy has to be a best response not only to other players’ strategies in the equilibrium, but also to all strategies that are indistinguishable from those in the equilibrium. It is proved in (Ramanujam and Simon, 2010) that the existence of such equilibria is decidable on a model of games close in spirit to those with public signals studied in (Bouyer, 2018). Here we show that the existence of Nash equilibria is decidable for $n$ players when observations are hierarchical and objectives are given as LTL formulas. Note that this result is orthogonal to those described above, which all allow in a way or another some non-hierarchical information: in (Bouyer, 2018) players know their own actions in addition to the public signals, in (Ramanujam and Simon, 2010) they know their private local state, and in (Belardinelli et al., 2017a) they can have incomparable initial knowledge of the situation. ###### Definition 7.1. A $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$ presents _hierarchical observation_ (Berwanger et al., 2018) if the “finer-than” relation is a total ordering, i.e., if for all $o,o^{\prime}\in\textnormal{Obs}$, either $\mathcal{O}(o)\subseteq\mathcal{O}(o^{\prime})$ or $\mathcal{O}(o^{\prime})\subseteq\mathcal{O}(o)$. Let $\mathcal{G}$ be a $\textrm{CGS}_{\textnormal{ii}}$ with hierarchical observation, and since all agents have symmetric roles in the problem considered, assume without loss of generality that $\mathcal{O}(o_{n})\subseteq\ldots\subseteq\mathcal{O}(o_{1})$. Because of the nested strategy quantifiers $\langle\\!\langle y_{i}\rangle\\!\rangle^{o_{i}}$, the instance $(\mathcal{G},\Phi_{\textsc{NE}})$ is _not_ hierarchical even if $\mathcal{G}$ yields hierarchical observation (unless $\mathcal{O}(o_{i})=\mathcal{O}(o_{j})$ for all $i,j\in[n]$). However, considering the special observation symbol $o_{p}$ that is always interpreted as the identity relation (and thus represents perfect observation), and letting $\displaystyle\Phi^{\prime}:=$ $\displaystyle\langle\\!\langle x_{1}\rangle\\!\rangle^{o_{1}}\dots\langle\\!\langle x_{n}\rangle\\!\rangle^{o_{n}}(\bm{a},\bm{x})\bigwedge_{i\in[n]}\Big{[}\Big{(}\langle\\!\langle y_{i}\rangle\\!\rangle^{o_{p}}(a_{i},y_{i})\,{\bf E}\psi_{i}\Big{)}\to{\bf E}\psi_{i}\Big{]},$ we have that $\Phi^{\prime}$ forms a hierarchical instance with any $\textrm{CGS}_{\textnormal{ii}}$ that presents hierarchical observation. Besides, we can prove that for deterministic strategies, $\Phi^{\prime}$ is equivalent to $\Phi_{\textsc{NE}}$: ###### Lemma 7.2. If we consider deterministic strategies, then $\Phi_{\textsc{NE}}\equiv\Phi^{\prime}$. ###### Proof. Concerning the universal versus existential quantification on outcomes, it is enough to observe that assigning a deterministic strategy to each agent determines a unique outcome. Next, to change each inner $o_{i}$ for $o_{p}$, we exploit the fact that in a one-player game of partial observation (such a game occurs when all but one player have fixed their strategies), the player has a strategy enforcing some goal iff she has a uniform strategy enforcing that goal. To see this, it is enough to establish that for every $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$ and position $v$, $\mathcal{G},\chi,v\models\langle\\!\langle y_{i}\rangle\\!\rangle^{o_{p}}(a_{i},y_{i})\,{\bf E}\psi_{i}\leftrightarrow\langle\\!\langle y_{i}\rangle\\!\rangle^{o_{i}}(a_{i},y_{i})\,{\bf E}\psi_{i},$ for every $i\in[n]$ and every assignment $\chi$ such that $\chi(a_{j})$ is defined for all $j$. To this end, fix $i$ and $\chi$. The right-to-left implication is immediate (since $o_{p}$ is finer than $o_{i}$). For the converse, let $\sigma$ be an $o_{p}$-strategy (i.e., a perfect-information strategy) such that $\mathcal{G}^{\prime},\chi^{\prime},v_{{\iota}}\models\psi_{i}$, where $\chi^{\prime}=\chi[y_{i}\mapsto\sigma,a_{i}\mapsto\sigma]$. Because we consider deterministic strategies and $\chi^{\prime}$ assigns a strategy to each agent, it defines a unique outcome $\pi$ from the initial position, i.e., $\textnormal{Out}(\chi^{\prime},v_{\iota})=\\{\pi\\}$. We construct an $o_{i}$-strategy $\sigma^{\prime}$ such that if $a_{i}$ uses it instead of $\sigma$, we obtain the same outcome $\pi$, i.e., $\textnormal{Out}(\chi^{\prime\prime},v_{\iota})=\\{\pi\\}$, where $\chi^{\prime\prime}=\chi[y_{i}\mapsto\sigma^{\prime},a_{i}\mapsto\sigma^{\prime}]$. This can be done as follows: if $\rho\sim_{o_{i}}\pi_{\leq|\rho|-1}$ then define $\sigma^{\prime}(\rho):=\sigma(\pi_{\leq|\rho|-1})$, and otherwise let $\sigma^{\prime}(\rho):=c$ for some fixed action $c\in\textnormal{Ac}$. It is easy to see that $\sigma^{\prime}$ is an $o_{i}$-strategy and that $\chi^{\prime\prime}$ produces the same outcome as $\chi$ from $v_{\iota}$. ∎ ###### Corollary 7.3. If we consider deterministic strategies, then the existence of Nash Equilibria in games with hierarchical observation and $k$ different observations is in $(k+1)$-Exptime . ###### Proof. Deciding the existence of a Nash Equilibrium in a $\textrm{CGS}_{\textnormal{ii}}$ $\mathcal{G}$ amounts to model-checking formula $\Phi_{\textsc{NE}}$ in $\mathcal{G}$, which by Lemma 7.2 is equivalent to model-checking $\Phi^{\prime}$ in $\mathcal{G}$ if we restrict to deterministic strategies. Because $\Phi^{\prime}$ forms hierarchical instances with games that yield hierarchical observation, by Theorem 2.9 we can model check it on such games. Now because each $\psi_{i}$ is an LTL formula, we have that $\displaystyle\mbox{sd}\left(\langle\\!\langle y_{i}\rangle\\!\rangle^{o_{p}}(a_{i},y_{i})\,{\bf E}\psi_{i}\right)$ $\displaystyle=(0,\mbox{nd}),$ $\displaystyle\mbox{sd}\left(\bigwedge_{i\in[n]}\Big{[}\Big{(}\langle\\!\langle y_{i}\rangle\\!\rangle^{o_{p}}(a_{i},y_{i})\,{\bf E}\psi_{i}\Big{)}\to{\bf E}\psi_{i}\Big{]}\right)$ $\displaystyle=(0,\mbox{alt}),$ and finally we obtain that $\mbox{sd}(\Phi^{\prime})=(k,\mbox{nd})$, where $k$ is the number of different observations in $\mathcal{G}$, i.e., $k=|\\{\mathcal{O}(o_{1}),\ldots,\mathcal{O}(o_{n})\\}|$. By Proposition 5.2, we can model check $\Phi^{\prime}$ on $\mathcal{G}$ in time $(k+1)$-exponential, which concludes. ∎ We now show that, using the same trick, our main result can be applied to solve a more general problem called _rational synthesis_. ### 7.2. Rational distributed synthesis in games with hierarchical observations In classic synthesis, the environment is considered monolithic and “hostile”, in the sense that the system to be synthesised should be able to deal with all possible behaviours of the environment, even the most undesirable ones. This is a very strong requirement that can not always be met. When the environment can be considered rational, and its objective is known, it is reasonable to relax this requirement by asking that the system to synthesise behave well against the _rational_ behaviours of the environment. This problem is known as the _rational synthesis_ problem (Fisman et al., 2010; Kupferman et al., 2016; Condurache et al., 2016; Filiot et al., 2018). In the setting considered in the works above-mentioned, the system is seen as an agent $a$ and the environment is composed of several components, say $\\{e_{1},\ldots,e_{m}\\}$, that are assumed to be rational and follow individual objectives. While (Condurache et al., 2016) and (Filiot et al., 2018) consider various types of objectives such as reachability, safety or parity, here we consider LTL objectives as is done in (Fisman et al., 2010; Kupferman et al., 2016): the specification for the system is an LTL formula $\psi_{g}$, and the objective of each component $e_{i}$ of the environment is an LTL formula $\psi_{i}$. However note that the decidability results we establish would also hold for arbitrary omega-regular objectives. #### 7.2.1. Rational synthesis: state of the art Two variants of the rational synthesis problem have been considered: the _cooperative_ one, in which it is possible to tell the environment how to behave, as long as the suggested behaviour for each component forms an equilibrium, and the _non-cooperative_ one, in which the components of the environment may have any behaviour that forms an equilibrium. The existence of a solution to these problems can be expressed by the formulas $\Phi_{\text{c-RS}}$ and $\Phi_{\text{nc-RS}}$, respectively, defined as follows: $\displaystyle\Phi_{\text{c-RS}}$ $\displaystyle:=\langle\\!\langle x\rangle\\!\rangle^{o_{p}}\langle\\!\langle y_{1}\rangle\\!\rangle^{o_{p}}\ldots\langle\\!\langle y_{m}\rangle\\!\rangle^{o_{p}}(a,x)(\bm{e},\bm{y})\,\varphi_{\gamma}\wedge{\bf A}\psi_{g}$ $\displaystyle\Phi_{\text{nc-RS}}$ $\displaystyle:=\langle\\!\langle x\rangle\\!\rangle^{o_{p}}[\\![y_{1}]\\!]^{o_{p}}\ldots[\\![y_{m}]\\!]^{o_{p}}(a,x)(\bm{e},\bm{y})\,\varphi_{\gamma}\to{\bf A}\psi_{g}$ where $\bm{e}=(e_{i})_{i\in[m]}$, $\bm{y}=(y_{i})_{i\in[m]}$, and $\varphi_{\gamma}$ expresses that $\bm{y}$ forms an equilibrium for the environment. Also, as in the previous section, $o_{p}$ represents the perfect- information observation. Three different kinds of equilibria are considered in (Kupferman et al., 2016): profiles of dominant strategies, Nash equilibria, and subgame-perfect equilibria. Here we only consider Nash equilibria, because subgames of games with imperfect information should start in situations where all players have perfect information of the state, which we do not know how to express in $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$; and for dominant strategies, the natural formula to express them does not give rise to non- trivial decidable cases in the imperfect-information setting that we introduce later. The rational synthesis problem for Nash equilibria is obtained by replacing $\varphi_{\gamma}$ in the above formula with: $\displaystyle\varphi_{\text{NE}}$ $\displaystyle:=\bigwedge_{i\in[m]}\Big{[}\Big{(}\langle\\!\langle y^{\prime}_{i}\rangle\\!\rangle^{o_{p}}(e_{i},y^{\prime}_{i})\,{\bf A}\psi_{i}\Big{)}\to{\bf A}\psi_{i}\Big{]}$ It is proved in (Kupferman et al., 2016) that these problems are decidable for perfect information. Concerning imperfect information, because the existence of Nash equilibria is undecidable for three players, the problem is undecidable when the environment consists of at least three components (Filiot et al., 2018). Three decidable cases are known: when the environment consists of a single component (Filiot et al., 2018), when actions of all components are public (Belardinelli et al., 2017a), and when only the system has imperfect information while the (finitely many) components of the environment are perfectly informed (Filiot et al., 2018). We now extend the latter result by defining a generalisation of the rational synthesis problem that we call _rational distributed synthesis_ , and solving it in the case of hierarchical information. The case where the environment is perfectly informed and the system consists of a single component, solved in (Filiot et al., 2018), is a particular case of our Corollary 7.5 below666We only consider LTL objectives, but our automata construction can be adapted to handle all $\omega$-regular objectives.. However the other decidability result established in (Filiot et al., 2018) does not assume hierarchical information, and thus cannot be derived from the results we now present. #### 7.2.2. Rational distributed synthesis While for perfect information, distributed synthesis amounts to synthesis for a single meta-component which tells each component what to do, in the context of imperfect information it makes sense to consider that the system to be synthesised is composed of various components $\\{a_{1},\ldots,a_{n}\\}$ with different observation power, say $o_{i}$ for component $a_{i}$. We also let $o^{e}_{i}$ be the observation of the environment’s component $e_{i}$, for $i\in[m]$. We consider the imperfect-information variants of cooperative and non- cooperative rational synthesis defined by the following $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ formulas: $\displaystyle\Phi^{\textnormal{\scriptsize ii}}_{\text{c-RS}}$ $\displaystyle:=\langle\\!\langle x_{1}\rangle\\!\rangle^{o_{1}}\ldots\langle\\!\langle x_{n}\rangle\\!\rangle^{o_{n}}\langle\\!\langle y_{1}\rangle\\!\rangle^{o^{e}_{1}}\ldots\langle\\!\langle y_{m}\rangle\\!\rangle^{o^{e}_{m}}(\bm{a},\bm{x})(\bm{e},\bm{y})\,\varphi_{\gamma}\wedge{\bf A}\psi_{g}$ $\displaystyle\Phi^{\textnormal{\scriptsize ii}}_{\text{nc-RS}}$ $\displaystyle:=\langle\\!\langle x\rangle\\!\rangle^{o_{1}}\ldots\langle\\!\langle x_{n}\rangle\\!\rangle^{o_{n}}[\\![y_{1}]\\!]^{o^{e}_{1}}\ldots[\\![y_{m}]\\!]^{o^{e}_{m}}(\bm{a},\bm{x})(\bm{e},\bm{y})\,\varphi_{\gamma}\to{\bf A}\psi_{g}$ The formula for Nash equilibrium is adapted as follows: $\displaystyle\varphi^{\textnormal{\scriptsize ii}}_{\text{NE}}$ $\displaystyle:=\bigwedge_{i\in[m]}\Big{[}\Big{(}\langle\\!\langle y^{\prime}_{i}\rangle\\!\rangle^{o^{e}_{i}}(e_{i},y^{\prime}_{i})\,{\bf A}\psi_{i}\Big{)}\to{\bf A}\psi_{i}\Big{]}$ The only difference with the perfect-information case is that we use the observation of the different components of the environment instead of the perfect-information observation. We call the problems expressed by formulas $\Phi^{\textnormal{\scriptsize ii}}_{\text{c-RS}}$ and $\Phi^{\textnormal{\scriptsize ii}}_{\text{nc-RS}}$ _cooperative rational distributed synthesis_ and _non-cooperative rational distributed synthesis_ , respectively. As in the previous section on the existence of Nash equilibria, one can see that even if there is a total hierarchy on all observations, these formula do not yield hierarchical instances unless all observations are the same. However, the trick applied in the proof of Corollary 7.3 also applies here, both for Nash equilibria and subgame-perfect equilibria, i.e., we can replace each $o^{e}_{i}$ with $o_{p}$ in $\varphi^{\textnormal{\scriptsize ii}}_{\text{NE}}$ without affecting the semantics of formulas $\Phi^{\textnormal{\scriptsize ii}}_{\text{c-RS}}$ and $\Phi^{\textnormal{\scriptsize ii}}_{\text{nc-RS}}$. As a result, when there is a hierarchy on observations $o_{1},\ldots,o_{n},o^{e}_{1},\ldots,o^{e}_{m}$, the cooperative rational distributed synthesis is decidable. ###### Corollary 7.4. If we consider deterministic strategies and hierarchical observations, then cooperative rational distributed synthesis is decidable. For the non-cooperative variant, one cannot switch universal quantifications on strategies for the environments with existential quantifications for the system in order to obtain hierarchical instances, as the resulting formula would then capture a different problem. As a consequence, in addition to a hierarchy on observations $o_{1},\ldots,o_{n},o^{e}_{1},\ldots,o^{e}_{m}$, we need to have that the components of the environment observe better than the components of the system or, in other words, that the least informed component of the environment observes better than the best informed component of the system. When it is the case, we say that the environment is _more informed_ than the system. ###### Corollary 7.5. Non-cooperative rational distributed synthesis is decidable for deterministic strategies and hierarchical observations where the environment is more informed than the system. This result applies for instance when there is hierarchical information amongst the components of the system, and the environment has perfect information. Note that when the system consists of a single component, this corresponds to the second decidability result in (Filiot et al., 2018). As we mentioned in the introduction, considering that the opponent has perfect information is something classic in two-player games with imperfect information, as doing so ensures that the strategy one synthesises is winning no matter how much the opponent observes. In Reif’s words, this amounts to considering the possibility that the opponent may “cheat” and use information that it normally does not have access to (Reif, 1984). The non-cooperative rational synthesis problem is not precisely a two-player game, but it resembles one in the sense that the system as a whole (composed of its various components $a_{1},\ldots,a_{n}$) should win against any “rational” behaviour of the environment as a whole. In this view, considering that the components of the environment have perfect information thus yields a distributed system that is robust to possible leaks of hidden information to the environment. ###### Remark 6. When all components of the environment have perfect information, $\Phi^{\textnormal{\scriptsize ii}}_{\text{c-RS}}$ and $\Phi^{\textnormal{\scriptsize ii}}_{\text{nc-RS}}$ already form hierarchical instances with games where there is hierarchical observation amongst the system’s components, and one does not need to resort to the trick used in the proof of Corollary 7.3. A consequence is that in that case, corollaries 7.4 and 7.5 also hold for nondeterministic strategies. ## 8\. Conclusion We introduced $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, a logic for reasoning about strategic behaviour in multi-player games with imperfect information. The syntax specifies the observations with which strategies have to work, and thus allows one to reason about strategic problems in settings where agents can change observation power, for instance by being eventually granted access to previously hidden information. Moreover our logic contains an outcome quantifier and an unbinding operator which simplify the semantics, make it easier to express branching-time properties, allow us to naturally consider nondeterministic strategies, and make the correspondence with $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$ tighter, enabling us to derive precise complexity results for the model-checking of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$. We isolated the class of hierarchical formula/model pairs $(\Phi,\mathcal{G})$ and proved that for such instances one can decide whether $\mathcal{G}\models\Phi$. The proof reduces (hierarchical) instances of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ to (hierarchical) formulas of $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize ii}}$, a low-level logic that we introduced, and that serves as a natural bridge between $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ and automata constructions. We also studied in detail the complexity of the model-checking problems solved in this work. To do so we introduced a new measure on formulas called _simulation depth_. This measure, though being a purely syntactic notion, reflects the complexity of automata constructions required to treat a given formula. Since one can alternate quantifiers in $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, our decidability result goes beyond synthesis and can be used to easily obtain the decidability of many strategic problems. In this work we applied it to the problem of existence of Nash equilibria in games with hierarchical observation, and to the imperfect-information generalisations of rational synthesis that we called (cooperative and non-cooperative) _rational distributed synthesis_. Our result has also been used to prove that the existence of admissible strategies in games with hierarchical information is decidable (Brenguier et al., 2017). An interesting direction for future work would be to try and adapt the notion of hierarchical instances to allow for situations in which hierarchies can change along a play, as done in (Berwanger et al., 2018). We would also like to consider alternatives to the synchronous perfect recall setting considered here, such as the classic asynchronous perfect recall setting (Fagin et al., 1995; Puchala, 2010), or the more recent notion of causal knowledge (Genest et al., 2015). Finally, it is often interesting in presence of imperfect information to introduce epistemic operators to reason explicitely about what agents know. We already generalised the main result of this work to an extension of $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$ with such operators (Maubert and Murano, 2018); we would like to see if this can be used to reason about subgame-perfect equilibria in games with imperfect information, which do not seem to be easy to characterise in $\textnormal{{SL}}_{\textnormal{\scriptsize ii}}$, as mentioned in Section 7.2.1. Indeed, in games with imperfect information, the notion of subgame specifies that the initial situation should be known to all players (Selten, 1965), a property that epistemic logics are meant to be able to express. ###### Acknowledgements. We thank anonymous reviewers for their valuable comments on a previous version of this work. 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We let $K_{1},K_{2}\in\mathbb{N}$ be such that the number of states of $\mathcal{N}$ is bounded by $2^{K_{1}nl\log(nl)}$ and the number of colours by $K_{2}nl$. Proposition 4.12 follows directly from the following. ###### Proposition A.1. Let $\Phi$ be a $\textnormal{{QCTL}}^{*}_{\textnormal{\scriptsize i,$\tiny{\subseteq}$}}$ formula, $\mathcal{S}$ a CKS , and let ${\textnormal{AP}_{\exists}}={\textnormal{AP}_{\exists}}(\Phi)$. For every subformula $\varphi$ of $\Phi$ and state $s\in\mathcal{S}$, it holds that: * • if $\mbox{sd}_{k}(\varphi)=0$, $\mathcal{A}_{s}^{\varphi}$ has at most $f_{\mathcal{S}}^{\varphi}$ states and 2 colours, * • if $\mbox{sd}_{k}(\varphi)\geq 1$, $\mathcal{A}_{s}^{\varphi}$ has at most $\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}$ states, and its number of colours is at most $\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)-1\mid f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}$, with $f_{\mathcal{S}}^{\varphi}=(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\varphi||\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi)}2^{K_{\psi}|\varphi|{\bf E}\mathrm{d}(\varphi)}$. In addition, if $\mathcal{A}_{s}^{\varphi}$ has state set $Q$, for each $q\in Q$ and $a\in 2^{\textnormal{AP}_{\exists}}$, we have $|\delta(q,a)|\leq|\mathcal{S}||Q|^{|\mathcal{S}|}2^{H|\varphi|}$, where $H=1+{\bf E}\mathrm{d}(\varphi)$. ###### Proof. We prove the result by induction on $\varphi$. $\bm{\varphi=p:}$ in this case $\mbox{sd}_{k}(\varphi)=\exists\mathrm{d}(\varphi)={\bf E}\mathrm{d}(\varphi)=0$. By construction, $\mathcal{A}_{s}^{\varphi}$ has one state $q_{\iota}$ and two colours, so that the first part of the claim holds. In addition, each formula of its transition function is of size one, so that the second part of the claim also holds. $\bm{\varphi=\neg\varphi^{\prime}:}$ Complementing an ATA does not change the number of states, number of colours or size of formulas in the transition function, so that the result follows by induction hypothesis and the fact that $|\varphi^{\prime}|\leq|\varphi|$ and ${\bf E}\mathrm{d}(\varphi)={\bf E}\mathrm{d}(\varphi^{\prime})$. $\bm{\varphi=\varphi_{1}\vee\varphi_{2}:}$ To establish the claim about number of states and colours we split cases. First we consider the case where $\mbox{sd}_{k}(\varphi)=0$. In that case we also have $\mbox{sd}_{k}(\varphi_{1})=\mbox{sd}_{k}(\varphi_{2})=0$. By induction hypothesis, for $i\in\\{1,2\\}$, $\mathcal{A}_{s}^{\varphi_{i}}$ has at most $f_{\mathcal{S}}^{\varphi_{i}}$ states and $2$ colours. These automata are then narrowed down, but the narrowing operation leaves the size of formulas in the transition function unchanged (in fact they may become smaller, but not bigger, see (Kupferman and Vardi, 1999)). Therefore, by construction $\mathcal{A}_{s}^{\varphi}$ has at most $1+f_{\mathcal{S}}^{\varphi_{1}}+f_{\mathcal{S}}^{\varphi_{2}}$ states and two colours. Now we have that $\displaystyle 1+f_{\mathcal{S}}^{\varphi_{1}}+f_{\mathcal{S}}^{\varphi_{2}}$ $\displaystyle=1+\sum_{i\in\\{1,2\\}}(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi_{i})}|\varphi_{i}||\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi_{i})}2^{K_{\psi}|\varphi_{i}|{\bf E}\mathrm{d}(\varphi_{i})}$ $\displaystyle=1+(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\varphi||\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi)}\sum_{i\in\\{1,2\\}}2^{K_{\psi}|\varphi_{i}|{\bf E}\mathrm{d}(\varphi)}$ $\displaystyle\leq 1+(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\varphi||\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi)}2^{K_{\psi}(|\varphi_{1}|+|\varphi_{2}|){\bf E}\mathrm{d}(\varphi)}$ $\displaystyle 1+f_{\mathcal{S}}^{\varphi_{1}}+f_{\mathcal{S}}^{\varphi_{2}}$ $\displaystyle\leq(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\varphi||\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi)}2^{K_{\psi}(|\varphi_{1}|+|\varphi_{2}|+1){\bf E}\mathrm{d}(\varphi)}$ We get that (8) $1+f_{\mathcal{S}}^{\varphi_{1}}+f_{\mathcal{S}}^{\varphi_{2}}\leq f_{\mathcal{S}}^{\varphi}$ which concludes the claim about the number of states. Now for the case where $\mbox{sd}_{k}(\varphi)\geq 1$. By definition of nondeterminisation depth, for at least one $i\in\\{1,2\\}$ we have $\mbox{sd}_{k}(\varphi_{i})\geq 1$. Also, the number of colours used in $\mathcal{A}_{s}^{\varphi}$ is the maximum between the number of colours used in $\mathcal{A}_{s}^{\varphi_{1}}$ and those used in $\mathcal{A}_{s}^{\varphi_{2}}$. By induction hypothesis it is the case that $\mathcal{A}_{s}^{\varphi_{i}}$ has at most $\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi_{i})-1\mid f_{\mathcal{S}}^{\varphi_{i}}\log f_{\mathcal{S}}^{\varphi_{i}}\big{)}$ colours if $\mbox{sd}_{k}(\varphi_{i})\geq 1$, or 2 if $\mbox{sd}_{k}(\varphi_{i})=0$. Therefore, the number of colours in $\mathcal{A}_{s}^{\varphi}$ is at most $\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi_{i})-1\mid f_{\mathcal{S}}^{\varphi_{i}}\log f_{\mathcal{S}}^{\varphi_{i}}\big{)}$ for some $i$, which is less than $\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)-1\mid f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}$. For the number of states $|Q|$ in $\mathcal{A}_{s}^{\varphi}$, we have that $|Q|=1+|Q_{1}|+|Q_{2}|$, where $Q_{i}$ is the set of states of $\mathcal{A}_{s}^{\varphi_{i}}$. By induction hypothesis we get $\displaystyle|Q|$ $\displaystyle\leq 1+\sum_{i\in\\{1,2\\}}\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi_{i})\mid f_{\mathcal{S}}^{\varphi_{i}}\log f_{\mathcal{S}}^{\varphi_{i}}\big{)}$ $\displaystyle\leq 1+\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid\sum_{i\in\\{1,2\\}}f_{\mathcal{S}}^{\varphi_{i}}\log f_{\mathcal{S}}^{\varphi_{i}}\big{)}$ $\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid(\sum_{i\in\\{1,2\\}}f_{\mathcal{S}}^{\varphi_{i}}+1)\log f_{\mathcal{S}}^{\varphi}\big{)}$ $\displaystyle|Q|$ $\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}\mbox{\hskip 56.9055pt (using Equation~{}\eqref{eq-boum})}$ which concludes the claim about the number of states. Concerning the size of formulas in the transition function, for all states from $\mathcal{A}_{s}^{\varphi_{1}}$ and $\mathcal{A}_{s}^{\varphi_{2}}$ the transition function is unchanged and the result thus holds by induction hypothesis. For the remaining state $q_{\iota}$, we have by definition $\delta(q_{\iota},a)=\delta^{1}(q_{\iota}^{1},a)\vee\delta^{2}(q_{\iota}^{2},a)$ and thus $|\delta(q_{\iota},a)|=|\delta^{1}(q_{\iota}^{1},a)|+|\delta^{2}(q_{\iota}^{2},a)|+1$. By induction hypothesis we get that $\displaystyle|\delta(q_{\iota},a)|$ $\displaystyle\leq|\mathcal{S}||Q_{1}|^{|\mathcal{S}|}2^{H(\varphi_{1})|\varphi_{1}|}+|\mathcal{S}||Q_{2}|^{|\mathcal{S}|}2^{H(\varphi_{2})|\varphi_{2}|}+1$ $\displaystyle\leq|\mathcal{S}|2^{H(\varphi)(|\varphi_{1}|+|\varphi_{2}|)}(|Q_{1}|^{|\mathcal{S}|}+|Q_{2}|^{|\mathcal{S}|})$ $\displaystyle\leq|\mathcal{S}|2^{H(\varphi)|\varphi|}(|Q_{1}|+|Q_{2}|)^{|\mathcal{S}|}$ And thus $|\delta(q_{\iota},a)|\leq|\mathcal{S}|2^{H(\varphi)|\varphi|}|Q|^{|\mathcal{S}|}$ as required. $\bm{\varphi={\bf E}\psi:}$ The word automaton built for the LTL skeleton of $\psi$ is in fact a Büchi automaton, and thus uses only two colours. The number of colours used by $\mathcal{A}_{s}^{\varphi}$ is therefore the maximum number of colours used by the automata $\mathcal{A}_{s}^{\varphi_{i}}$ built for the maximal state subformulas $\varphi_{i}$ in $\psi$, and the result follows by induction hypothesis. Concerning the number of states, let $|Q_{\varphi}|$ (resp. $|Q_{i}|$, $|Q_{\psi}|$) be the number of states in $\mathcal{A}_{s}^{\varphi}$ (resp. $\mathcal{A}_{s}^{\varphi_{i}}$, $\mathcal{W}^{\psi}$). Note that the number of states in $\mathcal{A}_{s^{\prime}}^{\varphi_{i}}$ does not depend on $s^{\prime}$. Recall that $\max(\psi)=\\{\varphi_{1},\ldots,\varphi_{n}\\}$ is the set of maximal state subformulas of $\psi$, and let $\psi^{\prime}$ be the LTL skeleton of $\psi$, i.e., the LTL formula obtained from $\psi$ by replacing maximal state subformulas $\varphi_{i}$ with propositions $p_{\varphi_{i}}$. We thus have $\displaystyle|Q|$ $\displaystyle=|Q_{\psi}||\mathcal{S}|+2|\mathcal{S}|\sum_{i\in[n]}|Q_{i}|$ $\displaystyle\leq 2^{K_{\psi}|\psi^{\prime}|}|\mathcal{S}|+2|\mathcal{S}|\sum_{i\in[n]}\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi_{i})\mid f_{\mathcal{S}}^{\varphi_{i}}\log f_{\mathcal{S}}^{\varphi_{i}}\big{)}$ $\displaystyle|Q|$ $\displaystyle\leq 2^{K_{\psi}|\psi^{\prime}|}|\mathcal{S}|\left(1+\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid\sum_{i\in[n]}f_{\mathcal{S}}^{\varphi_{i}}\log f_{\mathcal{S}}^{\varphi_{i}}\big{)}\right)$ And thus (9) $|Q|\leq 2^{K_{\psi}|\psi^{\prime}|}|\mathcal{S}|\left(1+\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid\log f_{\mathcal{S}}^{\varphi}\sum_{i\in[n]}f_{\mathcal{S}}^{\varphi_{i}}\big{)}\right)$ Now observe that for each $i\in[n]$ we have that ${\bf E}\mathrm{d}(\varphi_{i})\leq{\bf E}\mathrm{d}(\varphi)-1$, and $\exists\mathrm{d}(\varphi_{i})=\exists\mathrm{d}(\varphi)$. Therefore, $\displaystyle\sum_{i\in[n]}f_{\mathcal{S}}^{\varphi_{i}}$ $\displaystyle=(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}\sum_{i\in[n]}|\varphi_{i}||\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi_{i})}2^{K_{\psi}|\varphi_{i}|{\bf E}\mathrm{d}(\varphi_{i})}$ $\displaystyle\leq(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi)-1}(\sum_{i\in[n]}|\varphi_{i}|)2^{K_{\psi}({\bf E}\mathrm{d}(\varphi)-1)\sum_{i\in[n]}|\varphi_{i}|}$ Using this in Equation (9) we get $\displaystyle|Q|$ $\displaystyle\leq 2^{K_{\psi}|\psi^{\prime}|}|\mathcal{S}|\left(1+\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi)-1}(\sum_{i\in[n]}|\varphi_{i}|)2^{K_{\psi}({\bf E}\mathrm{d}(\varphi)-1)\sum_{i\in[n]}|\varphi_{i}|}\log f_{\mathcal{S}}^{\varphi}\big{)}\right)$ $\displaystyle\leq 2^{K_{\psi}|\psi^{\prime}|}\left(1+\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi)}(\sum_{i\in[n]}|\varphi_{i}|)2^{K_{\psi}({\bf E}\mathrm{d}(\varphi)-1)\sum_{i\in[n]}|\varphi_{i}|}\log f_{\mathcal{S}}^{\varphi}\big{)}\right)$ $\displaystyle\leq 2^{K_{\psi}|\psi^{\prime}|}\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi)}(1+\sum_{i\in[n]}|\varphi_{i}|)2^{K_{\psi}({\bf E}\mathrm{d}(\varphi)-1)\sum_{i\in[n]}|\varphi_{i}|}\log f_{\mathcal{S}}^{\varphi}\big{)}$ $\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi)}|\varphi|2^{K_{\psi}B}\log f_{\mathcal{S}}^{\varphi}\big{)},$ where $B=({\bf E}\mathrm{d}(\varphi)-1)\sum_{i\in[n]}|\varphi_{i}|+|\psi^{\prime}|$. To conclude it only remains to show that $B\leq|\varphi|{\bf E}\mathrm{d}(\varphi)$. Because $\varphi={\bf E}\psi$, it holds that ${\bf E}\mathrm{d}(\varphi)\geq 1$. If ${\bf E}\mathrm{d}(\varphi)=1$, we have $B=|\psi^{\prime}|\leq|\varphi|{\bf E}\mathrm{d}(\varphi)$. Now if ${\bf E}\mathrm{d}(\varphi)\geq 2$, we have $B=({\bf E}\mathrm{d}(\varphi)-2)\sum_{i\in[n]}|\varphi_{i}|+|\psi^{\prime}|+\sum_{i\in[n]}|\varphi_{i}|$ Clearly, $\sum_{i\in[n]}|\varphi_{i}|\leq|\varphi|$, and $|\psi^{\prime}|+\sum_{i\in[n]}|\varphi_{i}|\leq 2|\varphi|$, and the result follows. Note that it could seem that $|\psi^{\prime}|+\sum_{i\in[n]}|\varphi_{i}|\leq|\varphi|$. It is true if one defines the size of a formula as the number of connectors, but not if one also counts atomic propositions, as we do here. However it is true that $|\psi^{\prime}|+\sum_{i\in[n]}|\varphi_{i}|\leq 2|\varphi|$, independently of the definition of formulas’ size. It remains to establish the claim about the size of transition formulas. By definition, for every state $q$ of $\mathcal{A}_{s}^{\varphi}$ that comes from some $\mathcal{A}^{i}_{s^{\prime}}$ or $\overline{\mathcal{A}^{i}_{s^{\prime}}}$, the transition function is unchanged and thus the result follows by induction hypothesis and the fact that narrowing and complementation do not increase the size of formulas in transition functions. Now for the remaining states, for each $(q^{\psi},s^{\prime})\in Q$ and every $a\in 2^{{\textnormal{AP}_{\exists}}(\Phi)}$, we have $\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$ $\displaystyle\leq\sum_{a^{\prime}\in 2^{\max(\psi)}}\left(|\delta_{\psi}((q^{\psi},s^{\prime}),a^{\prime})|+1+\sum_{\varphi_{i}\in a^{\prime}}(|\delta^{i}_{s^{\prime}}(q^{i}_{s^{\prime}},a)|+1)+\sum_{\varphi_{i}\notin a^{\prime}}(|\overline{\delta^{i}_{s^{\prime}}}(\overline{q^{i}_{s^{\prime}}},a)|+1)\right)$ Now by induction hypothesis, and because complementation does not increase the size of formulas, we get: (10) $|\delta((q^{\psi},s^{\prime}),a)|\leq\sum_{a^{\prime}\in 2^{\max(\psi)}}\left(|\delta_{\psi}((q^{\psi},s^{\prime}),a^{\prime})|+2\sum_{i\in[n]}|\mathcal{S}|2^{H(\varphi_{i})|\varphi_{i}|}|Q_{i}|^{|\mathcal{S}|}\right)+2^{|\max(\psi)|}+2|\max(\psi)|2^{|\max(\psi)|},$ where $|Q_{i}|$ is the number of states in automaton $\mathcal{A}_{s^{\prime}}^{\varphi_{i}}$. Now by definition, $\displaystyle|\delta_{\psi}((q^{\psi},s^{\prime}),a^{\prime})|$ $\displaystyle=\left(\sum_{q^{\prime}\in\Delta^{\psi}(q^{\psi},a^{\prime})}\sum_{s^{\prime\prime}\in R(s^{\prime})}1\right)+|\Delta^{\psi}(q^{\psi},a^{\prime})||R(s^{\prime})|-1$ $\displaystyle|\delta_{\psi}((q^{\psi},s^{\prime}),a^{\prime})|$ $\displaystyle\leq 2|\Delta^{\psi}(q^{\psi},a^{\prime})||R(s^{\prime})|-1$ We thus have (11) $|\delta_{\psi}((q^{\psi},s^{\prime}),a^{\prime})|\leq 2|Q_{\psi^{\prime}}||\mathcal{S}|-1$ where $Q_{\psi^{\prime}}$ is the set of states of the word automaton $\mathcal{W}^{\psi}$. Using this in Equation 10 we get: $\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$ $\displaystyle\leq 2^{|\max(\psi)|}\left(2|Q_{\psi^{\prime}}||\mathcal{S}|-1+2\sum_{i\in[n]}|\mathcal{S}|2^{H(\varphi_{i})|\varphi_{i}|}|Q_{i}|^{|\mathcal{S}|}\right)+2^{|\max(\psi)|}+2|\max(\psi)|2^{|\max(\psi)|}$ $\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$ $\displaystyle\leq 2^{|\max(\psi)|+1}|\mathcal{S}|\left(|Q_{\psi^{\prime}}|+\sum_{i\in[n]}2^{H(\varphi_{i})|\varphi_{i}|}|Q_{i}|^{|\mathcal{S}|}\right)+2|\max(\psi)|2^{|\max(\psi)|}$ But for natural numbers $\\{a_{i},b_{i}\\}_{i\in[n]}$, it holds that $\sum_{i\in[n]}2^{a_{i}}b_{i}=2^{\sum_{i\in[n]}a_{i}}\sum_{i\in[n]}b_{i}-\sum_{i\in[n]}2^{a_{i}}(2^{\sum_{j\neq i}a_{j}}-1)b_{i}$ Applying this to $a_{i}=H(\varphi_{i})|\varphi_{i}|$ and $b_{i}=|Q_{i}|^{|\mathcal{S}|}$ we obtain $\sum_{i\in[n]}2^{H(\varphi_{i})|\varphi_{i}|}|Q_{i}|^{|\mathcal{S}|}=2^{\sum_{i\in[n]}H(\varphi_{i})|\varphi_{i}|}\sum_{i\in[n]}|Q_{i}|^{|\mathcal{S}|}-\sum_{i\in[n]}2^{H(\varphi_{i})|\varphi_{i}|}(2^{\sum_{j\neq i}H(\varphi_{j})|\varphi_{j}|}-1)|Q_{i}|^{|\mathcal{S}|}$ We thus get that $|\delta((q^{\psi},s^{\prime}),a)|\leq 2^{|\max(\psi)|+1}|\mathcal{S}|\left(|Q_{\psi^{\prime}}|+2^{\sum_{i\in[n]}H(\varphi_{i})|\varphi_{i}|}\sum_{i\in[n]}|Q_{i}|^{|\mathcal{S}|}\right)+C,$ with $\displaystyle C$ $\displaystyle=2|\max(\psi)|2^{|\max(\psi)|}-2^{|\max(\psi)|+1}|\mathcal{S}|\sum_{i\in[n]}2^{H(\varphi_{i})|\varphi_{i}|}(2^{\sum_{j\neq i}H(\varphi_{j})|\varphi_{j}|}-1)|Q_{i}|^{|\mathcal{S}|}$ $\displaystyle=2^{|\max(\psi)|}\left(2|\max(\psi)|-2|\mathcal{S}|\sum_{i\in[n]}2^{H(\varphi_{i})|\varphi_{i}|}(2^{\sum_{j\neq i}H(\varphi_{j})|\varphi_{j}|}-1)|Q_{i}|^{|\mathcal{S}|}\right)$ If $n=|\max(\psi)|>1$, i.e., there are at least two maximal state subformulas, then $\sum_{j\neq i}H(\varphi_{j})|\varphi_{j}|>0$, hence $2|\mathcal{S}|\sum_{i\in[n]}2^{H(\varphi_{i})|\varphi_{i}|}(2^{\sum_{j\neq i}H(\varphi_{j})|\varphi_{j}|}-1)|Q_{i}|^{|\mathcal{S}|}\geq 4n=4|\max(\psi)|$, which implies that $C\leq 0$, and thus $\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$ $\displaystyle\leq 2^{|\max(\psi)|+1}|\mathcal{S}|\left(|Q_{\psi^{\prime}}|+2^{\sum_{i\in[n]}H(\varphi_{i})|\varphi_{i}|}\sum_{i\in[n]}|Q_{i}|^{|\mathcal{S}|}\right)$ $\displaystyle\leq 2^{|\max(\psi)|+1}|\mathcal{S}|2^{\sum_{i\in[n]}H(\varphi_{i})|\varphi_{i}|}\left(|Q_{\psi^{\prime}}|^{|\mathcal{S}|}+\sum_{i\in[n]}|Q_{i}|^{|\mathcal{S}|}\right)$ $\displaystyle\leq|\mathcal{S}|2^{|\max(\psi)|+1+(H(\varphi)-1)\sum_{i\in[n]}|\varphi_{i}|}\left(|Q_{\psi^{\prime}}|+\sum_{i\in[n]}|Q_{i}|\right)^{|\mathcal{S}|}$ $\displaystyle\leq|\mathcal{S}|2^{|\varphi|+(H(\varphi)-1)|\varphi|}|Q|^{|\mathcal{S}|}$ $\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$ $\displaystyle\leq|\mathcal{S}|2^{H(\varphi)|\varphi|}|Q|^{|\mathcal{S}|}$ It remains to consider the case where $\max(\psi)=\\{\varphi_{1}\\}$. In that case there are only two letters in the alphabet $2^{\max(\psi)}$, which are $\emptyset$ and $\\{\varphi_{1}\\}$. The transition formulas then simplify and one gets that $\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$ $\displaystyle\leq|\delta_{\psi}((q^{\psi},s^{\prime}),\emptyset)|+1+|\overline{\delta^{1}_{s^{\prime}}}(\overline{q^{1}_{s^{\prime}}},a)|+1+|\delta_{\psi}((q^{\psi},s^{\prime}),\\{\varphi_{1}\\})|+1+|\delta^{1}_{s^{\prime}}(q^{1}_{s^{\prime}},a)|$ Using Equation (11) and the induction hypothesis we get $\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$ $\displaystyle\leq 4|Q_{\psi^{\prime}}||\mathcal{S}|-2+2|\mathcal{S}|2^{H(\varphi_{1})|\varphi_{1}|}|Q_{1}|^{|\mathcal{S}|}+3$ $\displaystyle\leq 1+2|\mathcal{S}|(2|Q_{\psi^{\prime}}|+2^{H(\varphi_{1})|\varphi_{1}|}|Q_{1}|^{|\mathcal{S}|})$ $\displaystyle\leq 1+2|\mathcal{S}|2^{(H(\varphi)-1)|\varphi_{1}|}(|Q_{\psi^{\prime}}|^{|\mathcal{S}|}+|Q_{1}|^{|\mathcal{S}|})$ $\displaystyle\leq 1+|\mathcal{S}|2^{H(\varphi)|\varphi|}(|Q_{\psi^{\prime}}|^{|\mathcal{S}|}+|Q_{1}|^{|\mathcal{S}|})$ $\displaystyle|\delta((q^{\psi},s^{\prime}),a)|$ $\displaystyle\leq|\mathcal{S}|2^{H(\varphi)|\varphi|}|Q|^{|\mathcal{S}|}$ $\bm{\varphi=\exists}^{\bm{\textnormal{{o}}}}\bm{p.\,\varphi^{\prime}:}$ We first establish the claim for states and colours, and we start with the case $\mbox{sd}_{k}(\varphi)=\mbox{sd}_{k}(\varphi^{\prime})$. By definition we necessarily have that $\mbox{sd}_{x}(\varphi^{\prime})=\mbox{nd}$, i.e., $\mathcal{A}_{s}^{\varphi^{\prime}}$ is nondeterministic, and $\textnormal{{o}}=I_{\varphi^{\prime}}$, therefore there is no need to use narrowing or nondeterminisation here. $\mathcal{A}_{s}^{\varphi}$ is obtained by directly projecting $\mathcal{A}_{s}^{\varphi^{\prime}}$, an operation that does not change the number of states or colours, so that the claim for states and colours follows directly by induction hypothesis. Now we consider the case where $\mbox{sd}_{k}(\varphi)\neq\mbox{sd}_{k}(\varphi^{\prime})$, which implies that $\mbox{sd}_{k}(\varphi)\geq 1$. Let $n$ be the number of states and $l$ the number of colours in $\mathcal{A}_{s}^{\varphi^{\prime}}$. In this case $\mathcal{A}_{s}^{\varphi^{\prime}}$ is first narrowed down, which does not change number of states or colours. The resulting automaton is then nondeterminised, yielding an automaton with at most $2^{K_{1}nl\log nl}$ states and $K_{2}nl$ colours. Again, we split cases: if $\mbox{sd}_{k}(\varphi^{\prime})=0$, by induction hypothesis, $n\leq f_{\mathcal{S}}^{\varphi^{\prime}}$ and $l=2$. For the number of colours, observing that $\exists\mathrm{d}(\varphi)=\exists\mathrm{d}(\varphi^{\prime})+1$, we have $\displaystyle K_{2}nl\leq 2K_{2}f_{\mathcal{S}}^{\varphi^{\prime}}$ $\displaystyle=2K_{2}(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi^{\prime})}|\varphi^{\prime}||\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi^{\prime})}2^{K_{\psi}|\varphi^{\prime}|{\bf E}\mathrm{d}(\varphi^{\prime})}$ $\displaystyle\leq(4K_{1}+2K_{2})^{\exists\mathrm{d}(\varphi)}|\varphi||\mathcal{S}|^{{\bf E}\mathrm{d}(\varphi)}2^{K_{\psi}|\varphi|{\bf E}\mathrm{d}(\varphi)}$ $\displaystyle K_{2}nl$ $\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)-1\mid f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}$ For the number of states, we have that $\displaystyle 2^{K_{1}nl\log nl}$ $\displaystyle\leq 2^{2K_{1}f_{\mathcal{S}}^{\varphi^{\prime}}\log(2f_{\mathcal{S}}^{\varphi^{\prime}})}\leq\mathrm{exp}\big{(}\mbox{sd}_{k}{\varphi}\mid f_{\mathcal{S}}^{\varphi}\log(f_{\mathcal{S}}^{\varphi})\big{)}$ Now for the final case, if $\mbox{sd}_{k}(\varphi)=\mbox{sd}_{k}(\varphi^{\prime})+1$ and $\mbox{sd}_{k}(\varphi^{\prime})\geq 1$, by induction hypothesis $n\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})\mid f_{\mathcal{S}}^{\varphi^{\prime}}\log f_{\mathcal{S}}^{\varphi^{\prime}}\big{)}$ and $l\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})-1\mid f_{\mathcal{S}}^{\varphi^{\prime}}\log f_{\mathcal{S}}^{\varphi^{\prime}}\big{)}$. For the number of colours in $\mathcal{A}_{s}^{\varphi}$ we thus get $\displaystyle K_{2}nl$ $\displaystyle\leq K_{2}\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})-1\mid f_{\mathcal{S}}^{\varphi^{\prime}}\log f_{\mathcal{S}}^{\varphi^{\prime}}2^{f_{\mathcal{S}}^{\varphi^{\prime}}\log f_{\mathcal{S}}^{\varphi^{\prime}}}\big{)}$ $\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})\mid 2K_{2}f_{\mathcal{S}}^{\varphi^{\prime}}\log f_{\mathcal{S}}^{\varphi^{\prime}}\big{)}$ $\displaystyle K_{2}nl$ $\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)-1\mid f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}$ Concerning the number of states, we observe that $\displaystyle nl$ $\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})-1\mid f_{\mathcal{S}}^{\varphi^{\prime}}\log f_{\mathcal{S}}^{\varphi^{\prime}}2^{f_{\mathcal{S}}^{\varphi^{\prime}}\log f_{\mathcal{S}}^{\varphi^{\prime}}}\big{)}$ $\displaystyle nl$ $\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})\mid 2f_{\mathcal{S}}^{\varphi^{\prime}}\log f_{\mathcal{S}}^{\varphi^{\prime}}\big{)}$ $\displaystyle K_{1}nl\log nl$ $\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})-1\mid 2K_{1}f_{\mathcal{S}}^{\varphi^{\prime}}\log f_{\mathcal{S}}^{\varphi^{\prime}}2^{2f_{\mathcal{S}}^{\varphi^{\prime}}\log f_{\mathcal{S}}^{\varphi^{\prime}}}\big{)}$ $\displaystyle K_{1}nl\log nl$ $\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})\mid 4K_{1}f_{\mathcal{S}}^{\varphi^{\prime}}\log f_{\mathcal{S}}^{\varphi^{\prime}}\big{)}$ $\displaystyle K_{1}nl\log nl$ $\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi^{\prime})\mid f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}$ $\displaystyle 2^{K_{1}nl\log nl}$ $\displaystyle\leq\mathrm{exp}\big{(}\mbox{sd}_{k}(\varphi)\mid f_{\mathcal{S}}^{\varphi}\log f_{\mathcal{S}}^{\varphi}\big{)}$ It only remains to establish the claim for the size of transition formulas. Since $\mathcal{A}_{s}^{\varphi}$ is nondeterministic, formulas $\delta(q,a)$ are written in disjunctive normal form and for every direction $x\in S_{\varphi}$ each disjunct contains exactly one element of $\\{x\\}\times Q$, where $Q$ is the set of states in $\mathcal{A}_{s}^{\varphi}$. As a result, each formula $\delta(q,a)$ is of size $\displaystyle|\delta(q,a)|$ $\displaystyle\leq|Q|^{|S_{\varphi}|}(2|S_{\varphi}|-1)+|Q|^{|S_{\varphi}|}-1$ $\displaystyle\leq 2|S_{\varphi}||Q|^{|S_{\varphi}|}$ $\displaystyle|\delta(q,a)|$ $\displaystyle\leq 2^{H(\varphi)|\varphi|}|\mathcal{S}||Q|^{|\mathcal{S}|}$ ∎

2024-09-04T02:54:58.721052

2003.04738

arxiv-papers

# EKOR strata on Shimura varieties with parahoric reduction Jens Hesse Technische Universität Darmstadt<EMAIL_ADDRESS> ###### Abstract We investigate the geometry of the special fiber of the integral model of a Shimura variety with parahoric level at a given prime place. To be more precise, we deal with the EKOR stratification which interpolates between the Ekedahl-Oort and Kottwitz-Rapoport stratifications. In the Siegel case we give a geometric description by suitably generalizing the theory of $G$-zips of Moonen, Wedhorn, Pink and Ziegler to our context. ###### Contents 1. 1 Background 1. 1.1 Shimura data of Hodge type 2. 1.2 Bruhat-Tits buildings 3. 1.3 Bruhat-Tits group schemes 4. 1.4 Siegel integral models 5. 1.5 Local structure of the integral model 1. 1.5.1 Generizations and irreducible components 6. 1.6 The local model 1. 1.6.1 The Siegel case 2. 1.6.2 The relation between the integral and the local model 3. 1.6.3 The Pappas-Zhu construction 2. 2 EKOR strata and zips in the case of parahoric reduction 1. 2.1 The Ekedahl-Oort, Kottwitz-Rapoport and EKOR stratifications 1. 2.1.1 Iwahori-Weyl group and the admissible subset 2. 2.1.2 Kottwitz-Rapoport stratification 3. 2.1.3 Ekedahl-Oort stratification 4. 2.1.4 EKOR stratification 2. 2.2 $\overline{\mathcal{G}}_{K}$-zips in the Siegel case 1. 2.2.1 Preliminaries 2. 2.2.2 Lattice chains, zips, admissibility 3. 2.2.3 An explicit description of $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ 4. 2.2.4 $\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ in the Siegel case 5. 2.2.5 The example of $\operatorname{GSp}(4)$ Introduction Shimura varieties are objects of arithmetic geometry (namely varieties over number fields) that naturally arise in the search for generalized, non-abelian reciprocity laws (i.e., in the Langlands program) and as moduli spaces of abelian varieties (with certain extra structures on them). One way of approaching these objects is to try to understand their mod-$p$ reduction (which has to be carefully defined first). Insofar as a moduli interpretation in the above sense exists and continues to exist likewise for the mod-$p$ reduction111There need not be a _literal_ moduli interpretation, but in any event the stratifications in question derive from a close connection to moduli problems., it allows us to stratify the moduli space according to several invariants of the abelian varieties parametrized, e.g., the isomorphism classes of their $p$-torsion. (An important observation is that these stratifications genuinely live in the characteristic $p$ world, making use of Frobenius endomorphisms and so on.) This, very roughly, is the general theme everything in this article revolves around. More precisely, we will be dealing with Shimura varieties of Hodge type and parahoric level structure, at some fixed prime $v\mid p$ of the number field over which the Shimura variety is defined. Under some reasonably mild assumptions, cf. 1.17, Kisin and Pappas [KP15] constructed a canonical integral model for such a Shimura variety. We try to understand some aspects of the geometry of the special fiber of said integral model, namely the EKOR strata (an interpolation between the Ekedahl-Oort strata, which in the case of hyperspecial level are roughly the patches where the isomorphism class of the $p$-torsion associated with the abelian variety is constant, and the Kottwitz- Rapoport strata, which roughly are the patches where the Hodge filtration looks constant) and defining them in a geometrical way. Let us now go into more detail. On the integral model $\mathscr{S}_{K}$ ($K$ parahoric level) we have a “universal” abelian scheme (the quotation marks indicating that it is not really universal for some moduli problem on $\mathscr{S}_{K}$, but it comes from a universal abelian scheme via pullback) and we have various kinds of Hodge tensors. We also have a “universal” isogeny chain of abelian schemes tightly connected to the “universal” abelian scheme. The overarching goal (and what we meant above by “defining the EKOR strata in a geometrical way”) is to construct a “nice” algebraic stack $\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ and a “nice” morphism $\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ from the mod-$p$ reduction of the Shimura variety to it, such that the fibers are the EKOR strata. Shen, Yu and Zhang [SYZ19] solved this problem on individual Kottwitz-Rapoport strata and globally after perfection, but not in the form stated here (i.e., globally without passing to perfections). In the Siegel case we propose a solution which specializes to that of Shen, Yu and Zhang on Kottwitz-Rapoport strata, and should not be difficult to generalize to many (P)EL cases. We show that $\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ is surjective. However, we have to leave the question of whether $\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ is smooth (which would be part of “nice”) an open conjecture. For hyperspecial level, the EKOR stratification agrees with the Ekedahl-Oort stratification, and the goal just set out is achieved by the stack of $\overline{\mathcal{G}}_{K}$-zips, first defined in special cases by Moonen and Wedhorn in [MW04] and then generally by Pink, Wedhorn and Ziegler in [PWZ11, PWZ15]; the relation to Shimura varieties being established in increasing generality in [MW04], by Viehmann and Wedhorn in [VW13], and finally by Zhang in [Zha15]. One way of looking at the transition from hyperspecial level to general parahoric level (at the very least in nice enough (P)EL cases) is from the point of view of moduli problems of abelian varieties with extra structure, where in the hyperspecial case we are really dealing just with that and in the general case we are dealing with isogeny chains of abelian varieties with extra structure, indexed by lattice chains coming from the Bruhat-Tits building of the reductive $p$-adic Lie group in question. The basic idea in generalizing zips from the hyperspecial to the general parahoric case then is that one should be dealing with chains of zips in the old sense. The zip of an abelian variety encodes the following information: the Hodge filtration, the conjugate filtration, and the Cartier isomorphism relating the two. In the general case, every abelian variety in the isogeny chain has a Hodge filtration, a conjugate filtration and a Cartier isomorphism. Problems now arise because we are dealing with $p$-primary isogenies on $p$-torsion points, implying that the transition morphisms in these chains have non- vanishing kernels. This introduces additional difficulty compared to the hyperspecial case; there is a naive way of defining a zip stack, but eventually we need to consider a certain admissible locus in it, which so far suffers from the absence of a nice moduli description. Passing to perfections however simplifies things and allows us to prove that the admissible locus is closed. From here we arrive at the stack that we are really interested in by dividing out a certain group action involving the unipotent radical of the special fiber of the parahoric group scheme. A careful inspection shows that on Kottwitz-Rapoport strata we arrive at the same result as in [SYZ19]. To sum up the results, ###### Theorem A: In the Siegel case, there is an algebraic stack $\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ and a surjective morphism $\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$, whose fibers are the EKOR strata and such that on Kottwitz-Rapoport strata, one gets the stack and map constructed in [SYZ19]. For $\operatorname{GSp}(4)$ we do some calculations to illustrate the theory; section 2.2.5. ### Acknowledgements This article essentially is an extract of my doctoral thesis [Hes20] (another extract222In particular, there is a large overlap between the “Background” sections of the two articles., dealing with the foliation into central leaves, is [Hes20a]). I thank Torsten Wedhorn for suggesting the topic of the dissertation, his support and patiently answering my questions. Moreover I thank Eva Viehmann and Paul Hamacher for their hospitality and helpful discussions during a month-long stay in Munich at the TU München. I am also grateful to Timo Richarz and Timo Henkel for numerous helpful discussions. This research was supported by the Deutsche Forschungsgemeinschaft (DFG), project number WE 2380/5. ## 1 Background ### 1.1 Shimura data of Hodge type This article deals with aspects of the geometry of Shimura varieties (of Hodge type), which are the (systems of) varieties associated with Shimura data (of Hodge type). ###### Definition 1.1. A _Shimura datum of Hodge type_ is a pair $(G,X)$, where $G$ is a reductive algebraic group over $\mathbb{Q}$ and $X\subseteq\operatorname{Hom}_{\mathbb{R}\text{-grp}}(\mathbb{S},G_{\mathbb{R}})$ is a $G(\mathbb{R})$-conjugacy class ($\mathbb{S}:=\operatorname{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m,\mathbb{C}}$ being the Deligne torus) subject to the following conditions: 1. (1) For $h\in X$, the induced Hodge structure $\mathbb{S}\xrightarrow{h}G_{\mathbb{R}}\xrightarrow{\mathrm{Ad}}\operatorname{GL}(\operatorname{Lie}(G_{\mathbb{R}}))$ satisfies $\operatorname{Lie}(G_{\mathbb{C}})=\operatorname{Lie}(G_{\mathbb{C}})^{-1,1}\oplus\operatorname{Lie}(G_{\mathbb{C}})^{0,0}\oplus\operatorname{Lie}(G_{\mathbb{C}})^{1,-1}$. 2. (2) $\operatorname{int}(h(i))\colon G^{\mathrm{ad}}_{\mathbb{R}}\to G^{\mathrm{ad}}_{\mathbb{R}}$ is a Cartan involution, i.e., $\\{g\in G^{\mathrm{ad}}(\mathbb{C})\;|\;gh(i)=h(i)\overline{g}\\}$ is compact. Another way of phrasing this condition: Every finite-dimensional real representation $V$ of $G^{\mathrm{ad}}_{\mathbb{R}}$ carries a $G^{\mathrm{ad}}_{\mathbb{R}}$-invariant bilinear form $\varphi$ such that $(u,v)\mapsto\varphi(u,h(i)v)$ is symmetric and positive definite. It is enough to show that this holds for one _faithful_ finite-dimensional real representation $V$. 3. (3) $G^{\mathrm{ad}}$ _cannot_ be non-trivially written as $G^{\mathrm{ad}}\cong H\times I$ over $\mathbb{Q}$ with $\mathbb{S}\to G_{\mathbb{R}}\xrightarrow{\mathrm{proj}}H_{\mathbb{R}}$ trivial. 4. (4) There exists an embedding $(G,X)\hookrightarrow(\operatorname{GSp}(V),S^{\pm})$, where $(\operatorname{GSp}(V),S^{\pm})$ is the Shimura datum associated with a finite-dimensional symplectic $\mathbb{Q}$-vector space $V$ (see below). That is, we have an embedding $G\hookrightarrow\operatorname{GSp}(V)$ of $\mathbb{Q}$-group schemes such that the induced map $\operatorname{Hom}_{\mathbb{R}\text{-grp}}(\mathbb{S},G_{\mathbb{R}})\hookrightarrow\operatorname{Hom}_{\mathbb{R}\text{-grp}}(\mathbb{S},\operatorname{GSp}(V_{\mathbb{R}}))$ restricts to a map $X\hookrightarrow S^{\pm}$. ###### Example 1.2. Let $W$ be a finite-dimensional $\mathbb{R}$-vector space. $\mathbb{R}$-group homomorphisms $\mathbb{S}\to\operatorname{GL}(W)$ then correspond to Hodge decompositions of $W$, i.e., to decompositions $W_{\mathbb{C}}=\oplus_{(p,q)\in\mathbb{Z}^{2}}W_{\mathbb{C}}^{p,q}$, such that $W_{\mathbb{C}}^{p,q}$ is the complex conjugate of $W_{\mathbb{C}}^{q,p}$ for all $(p,q)\in\mathbb{Z}^{2}$. Under this correspondence, $h\colon\mathbb{S}\to\operatorname{GL}(W)$ corresponds to the Hodge decomposition $W_{\mathbb{C}}^{p,q}=\\{w\in W_{\mathbb{C}}\;|\;\forall z\in\mathbb{S}(\mathbb{R})=\mathbb{C}^{\times}\colon h(z)w=z^{-p}\bar{z}^{-q}w\\}$. Hodge decompositions of $W$ of type $(-1,0)+(0,-1)$ correspond to complex structures on $W$: If $h\colon\mathbb{S}\to\operatorname{GL}(W)$ yields such a Hodge decomposition, then $h(i)$ gives an $\mathbb{R}$-endomorphism $J$ of $W$ with $J\circ J=-\operatorname{id}_{W}$. Let $V=(V,\psi)$ be a finite-dimensional symplectic $\mathbb{Q}$-vector space. We say that a complex structure $J$ on $V_{\mathbb{R}}$ is positive (resp. negative) if $\psi_{J}:=\psi_{\mathbb{R}}(\\_,J\\_)$ is a positive definite (resp. negative definite) symmetric bilinear form on $V_{\mathbb{R}}$. Define $S^{+}$ (resp. $S^{-}$) to be the set of positive (resp. negative) complex structures on $(V_{\mathbb{R}},\psi_{\mathbb{R}})$ and $S^{\pm}:=S^{+}\sqcup S^{-}$. We can make this more concrete: A symplectic basis of $(V_{\mathbb{R}},\psi_{\mathbb{R}})$ is a basis $e_{1},\dotsc,\allowbreak e_{g},e_{-g},\dotsc,\allowbreak e_{-1}$, such that $\psi_{\mathbb{R}}$ is of the form $\begin{pmatrix}&\tilde{I}_{g}\\\ -\tilde{I}_{g}&\end{pmatrix}$ with respect to this basis, where $\tilde{I}_{g}=\begin{pmatrix}&&1\\\ &\iddots&\\\ 1&&\end{pmatrix}$ is the antidiagonal identity matrix.333Occasionally (in particular when doing concrete matrix calculations), it is more convenient to number the basis vectors $1,\dotsc,g,-1,\dotsc,-g$ instead of $1,\dotsc,g,-g,\dotsc,-1$. Then the standard symplectic form is given by $\left(\begin{smallmatrix}&I_{g}\\\ -I_{g}&\end{smallmatrix}\right)$, $I_{g}$ being the $g\times g$ identity matrix. Let $J$ be the endomorphism of $V_{\mathbb{R}}$ of the form $\begin{pmatrix}&-\tilde{I}_{g}\\\ \tilde{I}_{g}&\end{pmatrix}$ with respect to this basis. Then $J\in S^{+}$ and what we have described is a surjective map $\\{\text{symplectic bases of }(V_{\mathbb{R}},\psi_{\mathbb{R}})\\}\twoheadrightarrow S^{+}.$ In particular we see that $\operatorname{Sp}(V_{\mathbb{R}},\psi_{\mathbb{R}}):=\\{f\in\operatorname{GL}(V_{\mathbb{R}})\;|\;\psi_{\mathbb{R}}(f(\\_),f(\\_))=\psi_{\mathbb{R}}\\}$ (by virtue of acting simply transitively on the symplectic bases) acts transitively on $S^{+}\cong\operatorname{Sp}(V_{\mathbb{R}},\psi_{\mathbb{R}})/\operatorname{SpO}(V_{\mathbb{R}},\psi_{\mathbb{R}},J)$ (where we define $\operatorname{SpO}(V_{\mathbb{R}},\psi_{\mathbb{R}},J):=\operatorname{Sp}(V_{\mathbb{R}},\psi_{\mathbb{R}})\cap O(V_{\mathbb{R}},\psi_{J})=U((V_{\mathbb{R}},J),\psi_{J})$ for a fixed choice of $J\in S^{+}$) and therefore the general symplectic group $\operatorname{GSp}(V_{\mathbb{R}},\psi_{\mathbb{R}}):=\\{f\in\operatorname{GL}(V_{\mathbb{R}})\;|\;\psi_{\mathbb{R}}(f(\\_),f(\\_))=c\cdot\psi_{\mathbb{R}}\text{ for some }c\in\mathbb{R}^{\times}\\}$ acts transitively on $S^{\pm}$ (note that the element of the form $e_{\pm i}\mapsto e_{\mp i}$ of $\operatorname{GSp}(V_{\mathbb{R}},\psi_{\mathbb{R}})$ for any given choice of symplectic basis $\left(e_{i}\right)_{i}$ permutes $S^{+}$ and $S^{-}$). ###### Definition 1.3. Condition (1) of Definition 1.1 implies that the action of $\mathbb{G}_{m,\mathbb{R}}$ (embedded in $\mathbb{S}$ in the natural way) on $\operatorname{Lie}(G_{\mathbb{R}})$ is trivial, so that $h$ induces a homomorphism ${w\colon\mathbb{G}_{m,\mathbb{R}}\to\operatorname{Cent}(G_{\mathbb{R}})}$. This homomorphism is independent of the choice of $h\in X$ and is called the _weight homomorphism_ of $(G,X)$. Moreover, we denote by $\\{\mu\\}$ the the $G(\mathbb{C})$-conjugacy class of the cocharacter $\mu_{h}:=h\circ(\operatorname{id}_{\mathbb{G}_{m,\mathbb{C}}},1)\colon\mathbb{G}_{m,\mathbb{C}}\to\mathbb{G}_{m,\mathbb{C}}^{2}\cong\mathbb{S}_{\mathbb{C}}\to G_{\mathbb{C}}$, where $h$ is as above. Obviously, the conjugacy class $\\{\mu\\}$ is independent of the particular choice of $h\in X$. ###### Remark 1.4. Let $L/\mathbb{Q}$ be a field extension such that $G_{L}$ contains a split maximal torus $T$. Let $W:=\operatorname{Norm}_{G(L)}(T)/T$ be the Weyl group. Then the natural map $W\backslash\operatorname{Hom}_{L\text{-grp}}(\mathbb{G}_{m,L},T)\to G(L)\backslash\operatorname{Hom}_{L\text{-grp}}(\mathbb{G}_{m,L},G_{L})$ is bijective. Since the left hand side remains unchanged if we go from $L=\bar{\mathbb{Q}}$ (where as usual $\bar{\mathbb{Q}}$ denotes an algebraic closure of $\mathbb{Q}$) to $L=\mathbb{C}$, we see that $\\{\mu\\}$ contains a cocharacter defined over $\bar{\mathbb{Q}}$ and that we may then also consider $\\{\mu\\}$ as a $G(\bar{\mathbb{Q}})$-conjugacy class. ###### Definition 1.5. The _reflex field_ $\mathbf{E}=\mathbf{E}(G,X)$ of $(G,X)$ is the field of definition of $\\{\mu\\}$, i.e., the fixed field in $\bar{\mathbb{Q}}$ of $\\{\gamma\in\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\;|\;\gamma(\\{\mu\\})=\\{\mu\\}\\}$. ###### Example 1.6. The reflex field of the Shimura datum $(\operatorname{GSp}_{2g,\mathbb{Q}},S^{\pm})$ of Example 1.2 is $\mathbb{Q}$. To wit, one of the cocharacters in the conjugacy class $\\{\mu\\}$ is $\mu(z)=\left(\begin{smallmatrix}z&&&&&\\\ &\ddots&&&&\\\ &&z&&&\\\ &&&1&&\\\ &&&&\ddots&\\\ &&&&&1\end{smallmatrix}\right).$ ###### Notation 1.7. We denote the ring of (rational) adeles by $\mathbb{A}:=\mathbb{A}_{\mathbb{Q}}$, the subring of finite adeles by $\mathbb{A}_{f}:=\mathbb{A}_{\mathbb{Q},f}$ and the subring of finite adeles away from some fixed prime $p$ by $\mathbb{A}_{f}^{p}$. ###### Definition and Remark 1.8. Let $K\subseteq G(\mathbb{A}_{f})$ be a compact open subgroup. The _Shimura variety of level $K$ associated with $(G,X)$_ is the double coset space $\operatorname{Sh}_{K}(G,X):=G(\mathbb{Q})\backslash(X\times(G(\mathbb{A}_{f})/K)).$ A priori, this is just a set, but if $K$ is sufficiently small (i.e., “neat” in the sense of [Bor69, Pin90]), $\operatorname{Sh}_{K}(G,X)$ can be canonically written as a finite disjoint union of hermitian symmetric domains.444If $K$ fails to be sufficiently small, one might very reasonably argue that our definition of the Shimura variety of level $K$ really is the definition of the _coarse_ Shimura variety and that one should be working with stacks instead. Since we will only be interested in sufficiently small level, this is inconsequential for us. In particular, this gives $\operatorname{Sh}_{K}(G,X)$ the structure of a complex manifold. In fact, by the theorem of Baily-Borel, this complex manifold attains the structure of a quasi-projective complex variety in a canonical way. By work of Deligne, Milne and Borovoi, this variety is defined already (and again in a canonical way) over the reflex field $\mathbf{E}$. So in particular, it is defined over a number field independent of $K$. This is important when varying $K$ and it is the reason why we consider the whole Shimura variety instead of its connected components over $\mathbb{C}$ on their own. It is possible for the Shimura variety to have multiple connected components over $\mathbb{C}$ while being connected over $\mathbf{E}$. More detailed explanations may be found in [Mil05]. ### 1.2 Bruhat-Tits buildings Let $K$ be a complete discrete valuation field with ring of integers $\mathcal{O}$, uniformizer $\varpi$ and perfect residue field $\kappa:=\mathcal{O}/\varpi$. ###### Notation 1.9. For a (connected) reductive group $G$ over $K$, we denote by $\mathcal{B}(G,K)$ the extended (or enlarged) and by $\mathcal{B}^{\mathrm{red}}(G,K)$ the reduced (i.e., non-extended) Bruhat-Tits building of $G$ over $K$ [BT84]. Moreover, $\mathcal{B}^{\mathrm{abstract}}(G,K)$ denotes the underlying abstract simplicial complex. ###### Remark 1.10. Let $V$ be a finite-dimensional $K$-vector space. As described in [KP15, 1.1.9] (originally in [BT84a]), the points of $\mathcal{B}(\operatorname{GL}(V),K)$ correspond to graded periodic lattice chains $(\mathcal{L},c)$, i.e., * • $\emptyset\neq\mathcal{L}$ is a totally ordered set of full $\mathcal{O}$-lattices in $V$ stable under scalar multiplication (i.e., $\Lambda\in\mathcal{L}\iff\varpi\Lambda\in\mathcal{L}$), * • $c\colon\mathcal{L}\to\mathbb{R}$ is a strictly decreasing function such that $c(\varpi^{n}\Lambda)=c(\Lambda)+n$. ###### Remark 1.11. Fix such an $\mathcal{L}$ and let $\Lambda^{0}\in\mathcal{L}$. Then every homothety class of lattices has a unique representative $\Lambda$ such that $\Lambda\subseteq\Lambda^{0}$ and $\Lambda\not\subseteq\varpi\Lambda^{0}$. Consider such representatives $\Lambda^{i}$ for all of the distinct homothety classes of lattices that make up $\mathcal{L}$. Because $\mathcal{L}$ is totally ordered and $\Lambda^{i}\not\subseteq\varpi\Lambda^{0}$, it follows that $\Lambda^{i}\supseteq\varpi\Lambda^{0}$ for all $i$ and that $\left\\{\Lambda^{i}/\varpi\Lambda^{0}\right\\}_{i}$ is a flag of non-trivial linear subspaces of $\Lambda^{0}/\varpi\Lambda^{0}\cong\kappa^{n}$, where $n:=\dim V$. Consequently, the number $r$ of homothety classes is in $\\{1,\dotsc,n\\}$; it is called the _period length_ (or _rank_) of $\mathcal{L}$. Numbering the $\Lambda^{i}$ in descending order we hence obtain $r$ lattices $\Lambda^{0},\Lambda^{1},\dotsc,\Lambda^{r-1}$ such that $\Lambda^{0}\supsetneqq\Lambda^{1}\supsetneqq\dotsb\supsetneqq\Lambda^{r-1}\supsetneqq\varpi\Lambda^{0}$ (1.12) and $\mathcal{L}$ is given by the the strictly descending sequence of lattices $\Lambda^{qr+i}=\varpi^{q}\Lambda^{i},\quad q\in\mathbb{Z},\;0\leq i<r.$ ###### Remark 1.13. Let $V$ be a finite-dimensional symplectic $K$-vector space. $\mathcal{B}(\operatorname{GSp}(V),K)$ embeds into the subset of $\mathcal{B}(\operatorname{GL}(V),K)$ consisting of those $(\mathcal{L},c)$ such that $\Lambda\in\mathcal{L}\implies\Lambda^{\vee}\in\mathcal{L}$. Passing to the underlying abstract simplicial complexes means forgetting about the grading $c$ and $\mathcal{B}^{\mathrm{abstract}}(\operatorname{GSp}(V),K)=\\{\mathcal{L}\in\mathcal{B}^{\mathrm{abstract}}(\operatorname{GL}(V),K)\;|\;\Lambda\in\mathcal{L}\implies\Lambda^{\vee}\in\mathcal{L}\\}.$ If $\mathcal{L}\in\mathcal{B}^{\mathrm{abstract}}(\operatorname{GSp}(V),K)$ and $\\{\Lambda^{i}\\}_{i}$ is as in Remark 1.11, then there is an involution $t\colon\mathbb{Z}\to\mathbb{Z}$ with $\left(\Lambda^{i}\right)^{\vee}=\Lambda^{t(i)}$, $t(i+qr)=t(i)-qr$, and $i<j\implies t(i)>t(j)$. So $-a:=t(0)>t(1)>\dotsb>t(r)=-a-r$, which implies $t(i)=-i-a$. Thus $i_{0}-t(i_{0})=2i_{0}+a\in\\{0,1\\}$ for some unique $i_{0}\in\mathbb{Z}$. Hence, upon renumbering the $\Lambda^{i}$, we may assume that $a\in\\{0,1\\}$. We therefore have $\displaystyle\varpi\Lambda^{0}\subsetneqq\Lambda^{r-1}\subsetneqq\Lambda^{r-2}\subsetneqq\dotsb\subsetneqq\Lambda^{0}\subseteq\left(\Lambda^{0}\right)^{\vee}=\Lambda^{-a}\subsetneqq\left(\Lambda^{1}\right)^{\vee}=\Lambda^{-1-a}$ $\displaystyle\subsetneqq\dotsb\subsetneqq\left(\Lambda^{r-1}\right)^{\vee}=\Lambda^{-r+1-a}\subseteq\Lambda^{-r}=\varpi^{-1}\Lambda^{0}.$ ###### Example 1.14. See also section 2.2.5 for some elaborations on the building of $\operatorname{GSp}_{4}(\mathbb{Q}_{p})$. ### 1.3 Bruhat-Tits group schemes ###### Notation 1.15. Let $E$ be a finite field extension of $\mathbb{Q}_{p}$. Denote by $\breve{E}$ the completion of the maximal unramified extension of $E$ (hence $\breve{E}=E\cdot\breve{\mathbb{Q}}_{p}$). ###### Remark 1.16. If $E/\mathbb{Q}_{p}$ is unramified, then ${{\cal O}_{\breve{E}}}=W(\bar{\mathbb{F}}_{p})$, $\bar{\mathbb{F}}_{p}$ denoting an algebraic closure of $\mathbb{F}_{p}$ and $W\colon\mathrm{Ring}\to\mathrm{Ring}$ being the ($p$-adic) Witt vectors functor. This generalizes to the ramified case using _ramified Witt vectors_ instead, see e.g. [Haz78, Chap. IV, (18.6.13)] or [Ahs11, Chapter 1]. Let $(G,X)$ be a Shimura datum of Hodge type, let $(G,X)\hookrightarrow(\operatorname{GSp}(V),S^{\pm})$ be an embedding as in Definition 1.1 (4), and let $x\in\mathcal{B}(G,\mathbb{Q}_{p})$ be a point in the Bruhat-Tits building of $G$ over $\mathbb{Q}_{p}$. We consider the associated Bruhat-Tits scheme ${\cal G}_{x}$, i.e., the affine smooth model of $G_{\mathbb{Q}_{p}}$ over $\mathbb{Z}_{p}$ such that ${\cal G}_{x}(\breve{\mathbb{Z}}_{p})\subseteq G(\breve{\mathbb{Q}}_{p})$ is the stabilizer of the facet of $x$ in ${\cal B}(G,\breve{\mathbb{Q}}_{p})\overset{\text{\cite[cite]{[\@@bibref{}{landvogt}{}{}, Prop.\leavevmode\nobreak\ 2.1.3]}}}{=}\mathcal{B}(G,\mathbb{Q}^{\mathrm{ur}}_{p})$. Let $K_{p}:={\cal G}_{x}(\mathbb{Z}_{p})\subseteq G(\mathbb{Q}_{p})$ and let $K^{p}\subseteq G(\mathbb{A}_{f}^{p})$ be a sufficiently small open compact subgroup. Define $K:=K_{p}K^{p}\subseteq G(\mathbb{A}_{f})$. ###### Assumptions 1.17. From now on, we will always make the following assumptions: * • $\mathcal{G}_{x}=\mathcal{G}_{x}^{\circ}$ is connected. * • $G$ splits over a tamely ramified extension of $\mathbb{Q}_{p}$. * • $p\nmid\\#\pi_{1}(G^{\mathrm{der}})$. ###### Notation 1.18. In order not to make notation overly cumbersome, we usually denote the base change $G_{\mathbb{Q}_{p}}$ of $G$ to $\mathbb{Q}_{p}$ by $G$ again. (Later, we will almost exclusively be dealing with $G_{\mathbb{Q}_{p}}$.) ### 1.4 Siegel integral models With notation as above let $\displaystyle N_{p}$ $\displaystyle:=\operatorname{Stab}_{\operatorname{GSp}(V)(\mathbb{Q}_{p})}(\mathcal{L})\quad\text{(as before)},$ $\displaystyle J_{p}$ $\displaystyle:=\operatorname{Stab}_{\operatorname{GL}(V^{\S})(\mathbb{Q}_{p})}(\Lambda^{\S})\cap\operatorname{GSp}(V^{\S})(\mathbb{Q}_{p}).$ Let $N^{p}\subseteq\operatorname{GSp}(V)(\mathbb{A}_{f}^{p})$ and $J^{p}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{A}_{f}^{p})$ be sufficiently small open compact subgroups, and $N:=N_{p}N^{p}$, $J:=J_{p}J^{p}$. In this subsection, we are going to describe integral models of $\operatorname{Sh}_{N}(\operatorname{GSp}(V),S^{\pm})$ and of $\operatorname{Sh}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$ over $\mathbb{Z}_{(p)}$ and relate the two. ###### Remark 1.19. By [RZ96, Definition 6.9], the integral model $\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})$ is given by the moduli problem $(\mathbb{Z}_{(p)}\text{-scheme})\ni S\mapsto\left\\{(A,\bar{\lambda},\eta^{p})\right\\}/{\scriptstyle\cong}$, where: 1. (a) $A=\left(A_{\Lambda}\right)_{\Lambda\in\mathcal{L}}$ is an $\mathcal{L}$-set of abelian schemes, i.e., * • for every $\Lambda\in\mathcal{L}$, an abelian $S$-scheme up to $\mathbb{Z}_{(p)}$-isogeny $A_{\Lambda}$ (i.e., $A_{\Lambda}$ is an object of the category $(\text{abelian }S\text{-schemes})\otimes\mathbb{Z}_{(p)}$, where the category $\mathcal{A}\otimes R$ for $\mathcal{A}$ an preadditive category and $R$ a ring has the same objects as $\mathcal{A}$ and $\operatorname{Hom}_{\mathcal{A}\otimes R}(X,Y)=\operatorname{Hom}(X,Y)\otimes_{\mathbb{Z}}R$ for all objects $X,Y$), * • for every inclusion $\Lambda_{1}\subseteq\Lambda_{2}$ a $\mathbb{Z}_{(p)}$-isogeny $\rho_{\Lambda_{2},\Lambda_{1}}\colon A_{\Lambda_{1}}\to A_{\Lambda_{2}}$, * • $\rho_{\Lambda_{3},\Lambda_{1}}=\rho_{\Lambda_{3},\Lambda_{2}}\circ\rho_{\Lambda_{2},\Lambda_{1}}$ if $\Lambda_{1}\subseteq\Lambda_{2}\subseteq\Lambda_{3}$ in $\mathcal{L}$, * • the height of $\rho_{\Lambda_{2},\Lambda_{1}}$ is $\log_{p}|\Lambda_{2}/\Lambda_{1}|$. Here $\rho_{\Lambda_{2},\Lambda_{1}}$ gives rise to a well-defined homomorphism of $p$-divisible groups, and what we mean is that the kernel of this homomorphism (which is a finite locally free commutative group scheme, which we also refer to simply as the kernel of $\rho_{\Lambda_{2},\Lambda_{1}}$) is to have order $|\Lambda_{2}/\Lambda_{1}|$. * • For every $\Lambda\in\mathcal{L}$, there is an isomorphism (called _periodicity isomorphism_) $\theta_{\Lambda}\colon A_{\Lambda}\to A_{p\Lambda}$ such that $\rho_{\Lambda,p\Lambda}\circ\theta_{\Lambda}=[p]\colon A_{\Lambda}\to A_{\Lambda}$ is the multiplication-by-$p$ isogeny. 2. (b) $\bar{\lambda}\colon A\to\tilde{A}$ is a $\mathbb{Q}$-homogeneous principal polarization, i.e., a $\underline{\mathbb{Q}^{\times}}$-orbit of a principal polarization $\lambda\colon A\to\tilde{A}$. Here $\tilde{A}$ is the $\mathcal{L}$-set of abelian schemes over $S$ up to prime-to-$p$ isogeny given by $\tilde{A}_{\Lambda}:=(A_{\Lambda^{\vee}})^{\vee}$. And being a polarization $\lambda$ means being a quasi-isogeny of $\mathcal{L}$-sets $\lambda\colon A\to\tilde{A}$ such that $A_{\Lambda}\xrightarrow{\lambda_{\Lambda}}\tilde{A}_{\Lambda}=(A_{\Lambda^{\vee}})^{\vee}\xrightarrow{\varrho_{\Lambda^{\vee},\Lambda}^{\vee}}(A_{\Lambda})^{\vee}$ is a polarization of $A_{\Lambda}$ for all $\Lambda$. If $\lambda_{\Lambda}$ can be chosen to be an isomorphism up to prime-to-$p$ isogeny for all $\Lambda$, then we speak of a principal polarization. In that case, when referring to $\lambda_{\Lambda}$, we mean a $\lambda_{\Lambda}$ which is an isomorphism up to prime-to-$p$ isogeny. 3. (c) $\eta^{p}$ is a level-$N^{p}$-structure, i.e. (if $S$ is connected), it is a $\pi_{1}(S,s)$-invariant $N^{p}$-orbit of symplectic similitudes $\eta^{p}\colon V_{\mathbb{A}_{f}^{p}}\to H_{1}(A_{s},\mathbb{A}_{f}^{p})$ (where $s$ is some geometric basepoint and $H_{1}(A_{s},\mathbb{A}_{f}^{p})$ with its $\pi_{1}(S,s)$-action corresponds to the Tate $\mathbb{A}_{f}^{p}$-module of $A$ (cf. [RZ96, 6.8]), which is a smooth $\mathbb{A}_{f}^{p}$-sheaf). Note that this forces the abelian schemes $A_{\Lambda}$ to be $(\dim_{\mathbb{Q}}V)$-dimensional. ###### Definition 1.20. Set $\Lambda^{\S}_{\mathbb{Z}_{(p)}}:=\Lambda^{\S}_{\mathbb{Z}_{p}}\cap V^{\S}_{\mathbb{Q}}=\prod_{i=-(r-1)-a}^{r-1}\Lambda_{\mathbb{Z}_{(p)}}^{i}$. We choose a lattice $\Lambda^{\S}_{\mathbb{Z}}\subseteq V^{\S}$ such that $\Lambda^{\S}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}=\Lambda^{\S}_{\mathbb{Z}_{(p)}}$ and $\Lambda^{\S}_{\mathbb{Z}}\subseteq(\Lambda^{\S}_{\mathbb{Z}})^{\vee}$. ###### Remark 1.21. Set $d:=\bigl{|}\left(\Lambda_{\mathbb{Z}}^{\S}\right)^{\vee}/\Lambda_{\mathbb{Z}}^{\S}\bigr{|}$. By [Kis10, 2.3.3, 3.2.4], the integral model $\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$ is given by the moduli problem $(\mathbb{Z}_{(p)}\text{-schemes})\ni S\mapsto\left\\{(A^{\S},\lambda^{\S},\epsilon^{p})\right\\}/{\scriptstyle\cong}$, where 1. (a) $A^{\S}$ is an abelian scheme over $S$ up to $\mathbb{Z}_{(p)}$-isogeny, 2. (b) $\lambda^{\S}\colon A^{\S}\to\left(A^{\S}\right)^{\vee}$ is a polarization of degree $d$ (i.e., the polarization of the (well-defined) associated $p$-divisible group has degree $d$), 3. (c) $\epsilon^{p}$ is a level-$J^{p}$-structure, i.e. (if $S$ is connected), it is a $\pi_{1}(S,s)$-invariant $J^{p}$-orbit of symplectic similitudes $\epsilon^{p}\colon V^{\S}_{\mathbb{A}_{f}^{p}}\to H_{1}(A^{\S}_{s},\mathbb{A}_{f}^{p})$. Note that this forces the abelian schemes $A^{\S}$ to be $(\dim_{\mathbb{Q}}V^{\S})$-dimensional. This completes the descriptions of the moduli problems, and we turn to the question of the relationship between the two. Consider (for appropriate $N^{p},J^{p}$; see below) the morphism $\chi\colon\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})\to\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$ given on $S$-valued points by sending $(A,\bar{\lambda},\eta^{p})$ to $(A^{\S},\lambda^{\S},\epsilon^{p})$, where 1. (a) $\displaystyle A^{\S}:=\prod_{i=-(r-1)-a}^{r-1}A_{\Lambda^{i}}$, 2. (b) $\displaystyle\lambda^{\S}:=\prod_{i=-(r-1)-a}^{r-1}\left(\rho_{\left(\Lambda^{i}\right)^{\vee},\Lambda^{i}}^{\vee}\circ\lambda_{\Lambda^{i}}\right)$, 3. (c) $\epsilon^{p}$ is the product $\prod_{i=-(r-1)-a}^{r-1}\eta^{p}$, to be interpreted as the product over $\eta^{p}\colon V_{\mathbb{A}_{f}^{p}}\to H_{1}(A_{\Lambda^{i},s},\mathbb{A}_{f}^{p})\cong H_{1}(A_{s},\mathbb{A}_{f}^{p})$, where the isomorphism $H_{1}(A_{\Lambda^{i},s},\mathbb{A}_{f}^{p})\cong H_{1}(A_{s},\mathbb{A}_{f}^{p})$ is by definition the identity for some fixed $i=i_{0}$ and otherwise induced by the transition map $\rho_{\Lambda^{i},\Lambda^{i_{0}}}$. We need that $N^{p}$ is mapped into $J^{p}$ by $\operatorname{GSp}(V)\hookrightarrow\operatorname{GSp}(V^{\S})$ for this to make sense. ###### Lemma 1.22. Let $S$ be a scheme, $\ell\neq p$ prime numbers. If $\ell$ does not appear as a residue characteristic of $S$, then the Tate module functors $\displaystyle H_{1}(\\_,\mathbb{Z}_{\ell})$ $\displaystyle\colon(\text{abelian }S\text{-schemes})\to(\text{\~{A}©tale }\mathbb{Z}_{\ell}\text{-local systems on }S),$ $\displaystyle H_{1}(\\_,\mathbb{Q}_{\ell})$ $\displaystyle\colon(\text{abelian }S\text{-schemes})\to(\text{\~{A}©tale }\mathbb{Q}_{\ell}\text{-local systems on }S)$ (cf. [Gro74, III, 5.4 and 6.2] for precise definitions) are faithful. If only $p$ and $0$ appear as residue characteristics of $S$, then the Tate module functor $H_{1}(\\_,\mathbb{A}_{f}^{p})\colon(\text{abelian }S\text{-schemes})\to(\text{\~{A}©tale }\mathbb{A}_{f}^{p}\text{-local systems on }S)$ is faithful. ###### Proof: First note that the statements about $H_{1}(\\_,\mathbb{Q}_{\ell})$ and $H_{1}(\\_,\mathbb{A}_{f}^{p})$ follows from the statement about $H_{1}(\\_,\mathbb{Z}_{\ell})$, which is why it is enough to only look at $H_{1}(\\_,\mathbb{Z}_{\ell})$. A homomorphism of abelian $S$-schemes $f\colon A\to B$ vanishes if and only if it vanishes over every (geometric) fiber of $S$: Indeed, if it vanishes fiberwise, then it is flat by the fiber criterion for flatness. Applying that criterion again we see that the closed immersion and fiberwise isomorphism $\ker(f)\hookrightarrow A$ is flat, which means that is an isomorphism. This way we are reduced to the case where $R$ is an (algebraically closed) field of characteristic different from $\ell$. In this setting the faithfulness is well-known (the salient point being that the $\ell$-primary torsion is dense). □ ###### Lemma 1.23. Let $H$ be a totally disconnected locally compact555By (our) definition, locally compact implies Hausdorff. group (i.e., a locally profinite group) and let $N\subseteq H$ be a compact subgroup. Then $N=\bigcap_{\begin{subarray}{c}N\subseteq J\\\ J\subseteq H\text{ open compact subgrp.}\end{subarray}}J.$ Note that this is (a variant of) a well-known theorem by van Dantzig if $N=\\{1\\}$ [Dan36]. ###### Proof: We make use of the following fact [AT08, Prop. 3.1.7]: A Hausdorff space is locally compact and totally disconnected if and only if the open compact sets form a basis of the topology. (Van Dantzig’s theorem is the group version of this, which talks only about a neighborhood basis of the identity and open compact _subgroups_.) First we show that $N$ is contained in some open compact subset $K\subseteq H$. For every $x\in N$ choose a compact open neighborhood $x\in K_{x}\subseteq H$. This is possible by the fact cited above. Then there is a finite subset $I\subseteq N$ such that $N\subseteq\bigcup_{x\in I}K_{x}=:K$. Next, for every $x\in N$ choose an open neighborhood of the identity $U_{x}$ such that $xU_{x}K\subseteq K$. With $N\subseteq U:=\bigcup_{x\in N}xU_{x}$ we obtain $UK\subseteq K$. Replacing $U$ by $U\cap U^{-1}$, we may moreover assume it is symmetric. The subgroup generated by $U$ is open (hence closed) and contained in $K$, hence is an open compact subgroup. Thus $N$ even is contained in an open compact sub _group_ ; in other words, we may assume that $H$ is compact, i.e., is a profinite group. Then $H/N$ is compact666Hausdorff quotient spaces of compact spaces are compact again, but for “locally compact” the analogous statement is not true in general! and totally disconnected777Take $x,y\in H$ such that $xN\neq yN$. We show that any subspace $S\subseteq H/N$ containing both $xN$ and $yN$ is disconnected. Let $U\subseteq H/N$ be a neighborhood of $xN$ not containing $yN$. Let $x\in V\subseteq\pi^{-1}(U)$ be open and compact, where $\pi\colon H\to H/N$ is the projection. Then $yN\notin\pi(V)\subseteq H/N$ is open and compact (hence closed) and we have $S=(\pi(V)\cap S)\sqcup S\setminus\pi(V)$ where both $\pi(V)\cap S$ and $S\setminus\pi(V)$ are open in $S$. This shows that $S$ is disconnected. (i.e., is a Stone space). By the fact cited above, $H/N\supseteq\\{1\\}=\bigcap_{L\subseteq H/N\text{ open compact subset}}L.$ Observe that the quotient map $H\to H/N$ is proper to deduce $N=\bigcap_{\begin{subarray}{c}N\subseteq M\\\ M\subseteq H\text{ open compact subset}\end{subarray}}M.$ Say $M$ is an open and compact subset of $H$ containing $N$. As we have shown above, there is an open compact subgroup $J\subseteq H$ in between $N$ and $M$, and this is all we need to complete the proof. □ ###### Proposition 1.24. For every compact open subgroup $N^{p}\subseteq\operatorname{GSp}(V)(\mathbb{A}_{f}^{p})$ $\chi\colon\mathscr{S}_{N}(\operatorname{GSp}(V),S^{\pm})\to\mathscr{S}_{J}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$ is a well-defined morphism for all compact open subgroups $N^{p}\subseteq J^{p}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{A}_{f}^{p})$ and is a closed immersion for all sufficiently small compact open subgroups $N^{p}\subseteq J^{p}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{A}_{f}^{p})$. ###### Proof: The fact that it’s well-defined is clear from the construction. To show the second statement, as in [Del71, Prop. 1.15], it is enough to show that $\mathscr{S}_{N_{p}N^{p}}(\operatorname{GSp}(V),S^{\pm})\to\varprojlim_{J^{p}}\mathscr{S}_{J_{p}J^{p}}(\operatorname{GSp}(V^{\S}),S^{\S,\pm})$ is a closed immersion, i.e., a proper monomorphism. We begin by proving that it is a monomorphism, i.e., injective on $S$-valued points ($S$ arbitrary $\mathbb{Z}_{(p)}$-scheme). So, say $(A_{1},\lambda_{1},\eta_{1}^{p})$ and $(A_{2},\lambda_{2},\eta_{2}^{p})$ both map to $(A^{\S},\lambda^{\S},\epsilon_{J^{p}}^{p})$. That means precisely that there is an isomorphism of abelian $S$-schemes up to $\mathbb{Z}_{(p)}$-isogeny $\phi\colon\prod_{i=-(r-1)-a}^{r-1}A_{1,\Lambda^{i}}\xrightarrow{\cong}\prod_{i=-(r-1)-a}^{r-1}A_{2,\Lambda^{i}}$ such that $\phi^{\vee}\circ\prod_{i=-(r-1)-a}^{r-1}\left(\rho_{2,\left(\Lambda^{i}\right)^{\vee},\Lambda^{i}}^{\vee}\circ\lambda_{2,\Lambda^{i}}\right)\circ\phi=\prod_{i=-(r-1)-a}^{r-1}\left(\rho_{1,\left(\Lambda^{i}\right)^{\vee},\Lambda^{i}}^{\vee}\circ\lambda_{1,\Lambda^{i}}\right)$ and $H_{1}(\phi,\mathbb{A}_{f}^{p})\circ\epsilon_{1,J^{p}}^{p}=\epsilon_{2,J^{p}}^{p}\mod{J^{p}}.$ We claim that $\phi$ comes from isomorphisms $\phi_{i}\colon A_{1,\Lambda^{i}}\xrightarrow{\cong}A_{2,\Lambda^{i}}.$ Certainly there is but one candidate for $\phi_{i}$: define $\phi_{i}$ to be the composition $A_{1,\Lambda^{i}}\xrightarrow{\mathrm{incl}}\prod_{i=-(r-1)-a}^{r-1}A_{1,\Lambda^{i}}\xrightarrow{\phi}\prod_{i=-(r-1)-a}^{r-1}A_{2,\Lambda^{i}}\xrightarrow{\mathrm{proj}}A_{2,\Lambda^{i}}.$ Our claim then is that $\phi=\prod_{i=-(r-1)-a}^{r-1}\phi_{i}.$ Apply $H^{1}(\\_,\mathbb{A}_{f}^{p})$ on both sides. For the left hand side, we have $H_{1}(\phi,\mathbb{A}_{f}^{p})=\epsilon_{2,J^{p}}^{p}\circ\left(\epsilon_{1,J^{p}}^{p}\right)^{-1}\mod{J^{p}}.$ and the right hand side of this equation is block diagonal. So $H_{1}(\phi,\mathbb{A}_{f}^{p})=\prod_{i=-(r-1)-a}^{r-1}H_{1}(\phi_{i},\mathbb{A}_{f}^{p})\mod{J^{p}}.$ Since (by Lemma 1.23) $N^{p}=\bigcap_{\begin{subarray}{c}N_{\ell}\subseteq J_{\ell}\\\ J_{\ell}\subseteq\operatorname{GSp}(V^{\S})(\mathbb{Q}_{\ell})\text{ cpt. open subgrp.}\end{subarray}}J_{\ell},$ it follows that (with $\ell\neq p$) $H_{1}(\phi,\mathbb{Q}_{\ell})=\prod_{i=-(r-1)-a}^{r-1}H_{1}(\phi_{i},\mathbb{Q}_{\ell})\mod{N_{\ell}},$ hence (since $N_{\ell}$ acts block-diagonally) that $H_{1}(\phi,\mathbb{Q}_{\ell})=\prod_{i=-(r-1)-a}^{r-1}H_{1}(\phi_{i},\mathbb{Q}_{\ell})$. Since $H_{1}(\\_,\mathbb{Q}_{\ell})$ is faithful (Lemma 1.22), this implies $\phi=\prod_{i=-(r-1)-a}^{r-1}\phi_{i}$, as desired. Next, consider the extension by zero of $\left(H_{1}(\rho_{1/2,\Lambda^{j},\Lambda^{i}},\mathbb{A}_{f}^{p})\right)_{i,j}$ (where for “$1/2$” either “$1$” or “$2$” can be plugged in) to a map $H_{1}(A^{\S},\mathbb{A}_{f}^{p})\to H_{1}(A^{\S},\mathbb{A}_{f}^{p})$. Under the isomorphism given by the $J^{p}$-level structure this corresponds, up to the $J^{p}$-action, to the map $V^{\S}_{\mathbb{A}_{f}^{p}}\to V^{\S}_{\mathbb{A}_{f}^{p}}$ given by mapping the $i$’th copy of $V_{\mathbb{A}_{f}^{p}}$ identically to the $j$’th copy and the rest to zero. Thus $\rho_{1/2,i,j}$ yield the same up to $J^{p}$ after applying $H_{1}(\\_,\mathbb{A}_{f}^{p})$, hence they are equal in the $\mathbb{Z}_{(p)}$-isogeny category. Consequently, $\chi$ is a monomorphism. For properness, we will use the valuative criterion. Let $R$ be a discrete valuation ring with field of fractions $K$ and assume that a $K$-point $A^{\S}=\prod_{i=-(r-1)-a}^{r-1}A_{\Lambda^{i}}$ with its additional structures coming from $(A_{\Lambda^{i}})_{i}$ extends to an $R$-point $\mathcal{A}^{\S}$. Consider the map $A^{\S}\to A_{\Lambda^{i_{0}}}\to A^{\S}$ where the first map is a projection and the second an inclusion. By the Néron mapping property, this extends to a map $\mathcal{A}^{\S}\to\mathcal{A}^{\S}$. Define $\mathcal{A}_{\Lambda^{i_{0}}}$ to be the image of this map. The Néron mapping property also allows us to extend the transition isogenies $\rho_{\Lambda^{i_{0}},\Lambda^{j_{0}}}\colon\allowbreak{A_{\Lambda^{j_{0}}}\to A_{\Lambda^{i_{0}}}}$, $i_{0}\leq j_{0}$, the periodicity isomorphisms, and the polarization. Since $\pi_{1}(\operatorname{Spec}K)$ surjects onto $\pi_{1}(\operatorname{Spec}R)$ (see [Stacks, Tag 0BQM]), extending the level structure away from $p$ is trivial. □ ### 1.5 Local structure of the integral model #### 1.5.1 Generizations and irreducible components Let $\mathscr{X}\to\operatorname{Spec}\mathcal{O}_{\breve{E}}$ be a flat scheme locally of finite type; denote the special fiber by $X\to\operatorname{Spec}\bar{\mathbb{F}}_{p}$ and the generic fiber by $\mathcal{X}\to\operatorname{Spec}\breve{E}$. We assume that $\mathcal{X}$ is locally integral (e.g. smooth). For example, we can consider $(\mathscr{X},X,\mathcal{X})=(\mathscr{S}^{-}_{K}(G,X)_{{{\cal O}_{\breve{E}}}},\mathscr{S}^{-}_{K}(G,X)_{{{\cal O}_{\breve{E}}}}\otimes_{{{\cal O}_{\breve{E}}}}\bar{\mathbb{F}}_{p},\allowbreak{\operatorname{Sh}_{K}(G,X)\otimes_{E}\breve{E}})$. Let $\bar{x}\in X(\bar{\mathbb{F}}_{p})$. ###### Lemma 1.25. There is a generization $x$ of $\bar{x}$ which lies in the generic fiber $\mathcal{X}$, and is a closed point in there, i.e., $x\in\mathcal{X}(L)$ for a finite extension $L/\breve{E}$. ###### Definition 1.26. We shall call such a point $x$ a _closed point generization_ of $\bar{x}$ for short. ###### Proof: Due to flatness (going-down) there is _some_ generization in the generic fiber; call it $x_{0}$. By [Stacks, Tag 053U] the following set is dense (and in particular non-empty) in the closure of $\\{x_{0}\\}$ in $\mathcal{X}$: $\left\\{x\in\mathscr{X}\;|\;x\text{ is a specialization of }x_{0}\text{ and a closed point generization of }\bar{x}\right\\}.$ □ ###### Lemma 1.27. Notation as in the preceding lemma. The specialization $x\leadsto\bar{x}$ can be realized by an ${\cal O}_{L}$-valued point of $\mathscr{X}$. ###### Proof: First off, by [EGA2, 7.1.9], it can be realized by a morphism $\operatorname{Spec}R=\\{\eta,s\\}\to\mathscr{X}$ of ${{\cal O}_{\breve{E}}}$-schemes, where $R$ is a discrete valuation ring such that $L\cong\kappa(\eta)=\operatorname{Quot}(R)$ as field extensions of $\kappa(x)$. We hence get local homomorphisms of local rings ${{\cal O}_{\breve{E}}}\to{\cal O}_{\mathscr{X},\bar{x}}\to R$. Thus the discrete valuation on $L$ defined by $R$ extends the discrete valuation on $\breve{E}$. But there is but one such extension and its valuation ring is ${\cal O}_{L}$ (by definition). □ ###### Lemma 1.28. Mapping $x$ to the unique irreducible component of $\mathscr{X}$ that contains $x$ establishes a surjection from the set of closed point generizations $x$ of $\bar{x}$ to the set of irreducible components of $\mathscr{X}$ containing $\bar{x}$. ###### Proof: If $x_{0}\in\mathcal{X}$ is a generization of $\bar{x}$, then $x_{0}$ lies in a unique irreducible component of $\mathscr{X}$ because $\mathcal{X}$ is locally irreducible. Hence the map described above is well-defined. Now for surjectivity: Given an irreducible component $C$ of $\mathscr{X}$ containing $\bar{x}$, let $x_{0}\in C$ be the generic point. Then $x_{0}$ must be in the generic fiber (else we would be able to find a generization in the generic fiber by going-down). Now go through the proof of Lemma 1.25 with this particular choice of $x_{0}$. □ ### 1.6 The local model To give a very rough idea of what the _local model_ to be discussed in this section is supposed to accomplish: It should be an $\mathcal{O}_{E}$-scheme that is étale-locally isomorphic to $\mathscr{S}_{K}(G,X)$, but easier to understand by virtue of being of a more “linear-algebraic flavor”. In actuality however, the theory of local models quickly gets quite complicated once one departs from the simplest examples. #### 1.6.1 The Siegel case We do start with the simplest example. We consider the standard Iwahori subgroup $I_{p}\subseteq\operatorname{GSp}_{2g}(\mathbb{Z}_{p})$, defined as the preimage of the standard Borel subgroup of $\operatorname{GSp}_{2g}(\mathbb{F}_{p})$. In terms of the building (cf. Remark 1.13), it corresponds to the lattice chain $\mathcal{L}_{\mathrm{full}}$ given by $\displaystyle\Lambda^{0}=\mathbb{Z}_{p}^{2g}$ $\displaystyle\supsetneqq\Lambda^{1}=\mathbb{Z}_{p}^{2g-1}\oplus p\mathbb{Z}_{p}\supsetneqq\Lambda^{2}=\mathbb{Z}_{p}^{2g-2}\oplus p\mathbb{Z}_{p}^{2}$ (1.29) $\displaystyle\supsetneqq\dotsb\supsetneqq\Lambda^{2g-1}=\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{2g-1}\supsetneqq p\Lambda^{0}=p\mathbb{Z}_{p}^{2g}$ of period length $2g$. Consider a subset $J=\\{j_{0}>\dotsb>j_{m-1}\\}\subseteq\\{1,\dotsc,2g\\}$ such that for each $j\in J$ with $1\leq j\leq 2g-1$ also $2g-j\in J$, and let $K_{p}$ be the parahoric subgroup associated with the partial lattice chain $\mathcal{L}\subseteq\mathcal{L}_{\mathrm{full}}$ obtained from $\left\\{\Lambda^{j}\;|\;j\in J\right\\}$. Define a scheme $\tilde{\mathscr{S}}_{K}(G,X)$ over $\mathscr{S}_{K}(G,X)$ as follows: $\tilde{\mathscr{S}}_{K}(G,X)(S)=\left\\{(A,\bar{\lambda},\eta^{p},\tau)\;\middle|\;\begin{tabular}[]{@{}l@{}}$(A,\bar{\lambda},\eta^{p})\in\mathscr{S}_{K}(\operatorname{GSp}_{2g},S^{\pm})(S)$,\\\ $\tau\colon H_{\mathrm{dR}}^{1}(A)\xrightarrow{\sim}\mathcal{L}\otimes\mathcal{O}_{S}$ isomorphism of lattice chains\end{tabular}\right\\}$ for every $\mathbb{Z}_{p}$-scheme $S$. By [RZ96, Appendix to Chap. 3], $\tilde{\mathscr{S}}_{K}(G,X)\to\mathscr{S}_{K}(G,X)$ is a Zariski torsor under the automorphism group of $\mathcal{L}$, i.e., the Iwahori group scheme. This motivates the definition of the local model $M^{\mathrm{loc}}_{K_{p}}\to\operatorname{Spec}\mathbb{Z}_{p}$ as the “moduli space of Hodge filtrations”; more precisely: ###### Remark 1.30. (See [Gör03, 91].) $M^{\mathrm{loc}}_{K_{p}}(S)$ is the set of isomorphism classes of commutative diagrams $\Lambda^{j_{0}}_{S}$$\mathcal{F}^{j_{0}}$$\Lambda^{j_{1}}_{S}$$\mathcal{F}^{j_{1}}$$\dotsb$$\dotsb$$\Lambda^{j_{0}}_{S}$$\mathcal{F}^{j_{0}}$$\Lambda^{j_{m-1}}_{S}$$\mathcal{F}^{j_{m-1}}$$\cdot p$ with $\Lambda^{j}_{S}:=\Lambda^{j}\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{S}$, $\mathcal{F}^{j}\subseteq\Lambda^{j}_{S}$ locally direct summand of rank $g$, such that for all $j\in J$, $\mathcal{F}^{j}\to\Lambda^{j}_{S}\overset{\psi}{\cong}(\Lambda^{2g-j}_{S})^{*}\to(\mathcal{F}^{2g-j})^{*}$ vanishes, $\psi$ being the symplectic pairing. By Grothendieck-Messing theory, one obtains a diagram $\tilde{\mathscr{S}}_{K}(G,X)$$\mathscr{S}_{K}(G,X)$$M^{\mathrm{loc}}_{K}$smooth of rel. dim. $\dim\operatorname{Aut}(\mathcal{L})$$\operatorname{Aut}(\mathcal{L})$-torsor Since both morphisms in this diagram are smooth of the same dimension, it follows that for every finite field extension $\mathbb{F}_{q}/\mathbb{F}_{p}$ and every point $x\in\mathscr{S}_{K}(G,X)(\mathbb{F}_{q})$, there exists a point $y\in M^{\mathrm{loc}}_{K}(\mathbb{F}_{q})$ and an isomorphism $\mathcal{O}_{\mathscr{S}_{K}(G,X),x}^{h}\cong\mathcal{O}_{M^{\mathrm{loc}}_{K},y}^{h}$ of henselizations. In many (P)EL situations one has similar descriptions with the obvious extra structures. Sometimes however the so-called “naive” local models so obtained additionally need to be flattened, which leaves one without any self-evident moduli interpretation. #### 1.6.2 The relation between the integral and the local model Generalizing the Siegel example, we axiomatically characterize the relationship between the integral model of the Shimura variety and its local model: One wants a _local model diagram_ , i.e., a diagram of $\mathcal{O}_{E}$-schemes functorial in $K$ $\tilde{\mathscr{S}}_{K}(G,X)$$\mathscr{S}_{K}(G,X)$$M^{\mathrm{loc}}_{K}$equivariant and smooth of rel. dim. $\dim\mathcal{G}_{\mathcal{O}_{E}}$$\mathcal{G}_{\mathcal{O}_{E}}$-torsor (1.31) where $M^{\mathrm{loc}}_{K}$ is a projective flat $\mathcal{O}_{E}$-scheme with an action of $\mathcal{G}\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{E}$ and generic fiber the canonical model of $G_{\bar{\mathbb{Q}}_{p}}/P_{\mu^{-1}}$ over $E$. By Kisin-Pappas [KP15] we do actually have such a diagram in our situation. #### 1.6.3 The Pappas-Zhu construction In [PZ13], Pappas and Zhu give a construction of the local model in quite a general context, in particular with no assumptions going beyond our running assumptions 1.17. ###### Remark 1.32. To this end, they construct an affine smooth group scheme $\underline{\mathcal{G}}_{K}\to\mathbb{A}^{1}_{\mathbb{Z}_{p}}=\operatorname{Spec}\mathbb{Z}_{p}[t]$ with the following key properties: 1. (1) $\underline{\mathcal{G}}_{K}$ has connected fibers, 2. (2) $\underline{\mathcal{G}}_{K}$ is reductive over $\operatorname{Spec}\mathbb{Z}_{p}[t^{\pm 1}]$, 3. (3) $\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t],t\mapsto p}\mathbb{Z}_{p}\cong\mathcal{G}_{K}$, in particular * • $\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t],t\mapsto p}\mathbb{Q}_{p}\cong G_{\mathbb{Q}_{p}}$ and * • $\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t]}\mathbb{F}_{p}:=\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t],t\mapsto 0}\mathbb{F}_{p}\cong\mathcal{G}_{K}\otimes\mathbb{F}_{p}$, 4. (4) $\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t]}\mathbb{Q}_{p}[\mkern-2.0mu[t]\mkern-2.0mu]$ is parahoric for $\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t]}\mathbb{Q}_{p}(\mkern-4.0mu(t)\mkern-4.0mu)$, 5. (5) $\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t]}\mathbb{F}_{p}[\mkern-2.0mu[t]\mkern-2.0mu]$ is parahoric for $\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t]}\mathbb{F}_{p}(\mkern-4.0mu(t)\mkern-4.0mu)$. ###### Definition and Remark 1.33. Let $X_{\mu}$ be the canonical model of $G_{\bar{\mathbb{Q}}_{p}}/P_{\mu^{-1}}$ over $E$, where for a cocharacter $\nu$ one defines $P_{\nu}:=\\{g\in G\;|\;\lim_{t\to 0}\nu(t)g\nu(t)^{-1}\text{ exists}\\}$. Let $S_{\mu}$ be the closed subvariety of $\operatorname{Gr}_{G}\times_{\mathbb{Q}_{p}}E$ with $S_{\mu}(\bar{\mathbb{Q}}_{p})=G(\bar{\mathbb{Q}}_{p}[\mkern-2.0mu[t]\mkern-2.0mu])\mu(t)G(\bar{\mathbb{Q}}_{p}[\mkern-2.0mu[t]\mkern-2.0mu])/G(\bar{\mathbb{Q}}_{p}[\mkern-2.0mu[t]\mkern-2.0mu]).$ Then $S_{\mu}$ can be $G_{E}$-equivariantly identified with $X_{\mu}$. ###### Definition 1.34. The local model $M^{\mathrm{loc}}_{G,\mu,K}$ now is defined to be the Zariski closure of $X_{\mu}\subseteq\operatorname{Gr}_{G}\times_{\mathbb{Q}_{p}}E$ in $\operatorname{Gr}_{\underline{\mathcal{G}}_{K},\mathbb{Z}_{p}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{E}$, where $\operatorname{Gr}_{\underline{\mathcal{G}}_{K},\mathbb{Z}_{p}}:=\operatorname{Gr}_{\underline{\mathcal{G}}_{K},\mathbb{A}_{\mathbb{Z}_{p}}^{1}}\otimes_{\mathbb{A}^{1}_{\mathbb{Z}_{p}},\,u\mapsto p}\mathbb{Z}_{p}$ is a base change of the global affine Graßmannian as defined in [PZ13]. ## 2 EKOR strata and zips in the case of parahoric reduction ###### Notation 2.1. We still fix a Shimura datum $(G,X)$ of Hodge type, a parahoric subgroup $K_{p}\subseteq G(\mathbb{Q}_{p})$ (associated with a Bruhat-Tits group scheme $\mathcal{G}=\mathcal{G}_{K}=\mathcal{G}_{K_{p}}\to\operatorname{Spec}\mathbb{Z}_{p}$ associated with a facet $\mathfrak{f}$) and a sufficiently small open compact subgroup $K^{p}\subseteq G(\mathbb{A}_{f}^{p})$. Define $\overline{\mathcal{G}}_{K}:=\mathcal{G}_{K}\otimes_{\mathbb{Z}_{p}}\kappa$. We also keep up our standard assumptions 1.17. We now want to discuss the EKOR stratification on the special fiber of the integral model with parahoric level structure. The EKOR stratification interpolates between the Ekedahl-Oort (EO) and the Kottwitz-Rapoport (KR) stratification (see Remark 2.22 below for a precise formulation). We begin by explaining the basics about these stratifications and the combinatorics involved in the first section of this chapter. ### 2.1 The Ekedahl-Oort, Kottwitz-Rapoport and EKOR stratifications #### 2.1.1 Iwahori-Weyl group and the admissible subset ###### Notation 2.2. 1. (1) We fix an Iwahori subgroup $I_{p}\subseteq K_{p}$, i.e., $I_{p}$ is the group of $\mathbb{Z}_{p}$-points of the parahoric group scheme $\mathcal{I}$ associated with an alcove $\mathfrak{a}$ (facet of maximal dimension) such that $\mathfrak{f}\subseteq\overline{\mathfrak{a}}$. As usual, we also define $\breve{I}:=\mathcal{I}(\breve{\mathbb{Z}}_{p})\subseteq\breve{K}$. 2. (2) Let $T\subseteq G$ be a maximal torus such that $T_{\breve{\mathbb{Q}}_{p}}$ is contained in a Borel subgroup of $G_{\breve{\mathbb{Q}}_{p}}$888Note that by Steinberg’s theorem, $G_{\breve{\mathbb{Q}}_{p}}$ is quasi-split. [Ser97, Chap. III, § 2] and let $S$ be the maximal $\breve{\mathbb{Q}}_{p}$-split torus contained in $T_{\breve{\mathbb{Q}}_{p}}$. We can and do choose $T$ such that the alcove $\mathfrak{a}$ is contained in the apartment associated with $S$. By $N$ we denote the normalizer of $T$. 3. (3) Let $(V,R)$ be the relative root system of $(G_{\breve{\mathbb{Q}}_{p}},T_{\breve{\mathbb{Q}}_{p}})$, i.e., $V$ is the $\mathbb{R}$-vector space $X^{*}_{\breve{\mathbb{Q}}_{p}}(T_{\breve{\mathbb{Q}}_{p}})\otimes_{\mathbb{Z}}\mathbb{R}$ and $R\subseteq X^{*}_{\breve{\mathbb{Q}}_{p}}(T_{\breve{\mathbb{Q}}_{p}})$ is (as usual) such that we have a decomposition $\mathfrak{g}:=\operatorname{Lie}(G_{\bar{\mathbb{Q}}_{p}})=\operatorname{Lie}(T_{\bar{\mathbb{Q}}_{p}})\oplus\bigoplus_{\alpha\in R}\mathfrak{g}_{\alpha}.$ Contrary to the absolute situation, $\dim\mathfrak{g}_{\alpha}$ may be greater than $1$. ###### Definition 2.3. 1. (1) The _(finite relative) Weyl group_ of $G$ (over $\breve{\mathbb{Q}}_{p}$) is $W:=N(\breve{\mathbb{Q}}_{p})/T(\breve{\mathbb{Q}}_{p})$. It is the Weyl group of the root system $(V,R)$, i.e., the group generated by the orthogonal reflections through the hyperplanes defined by the elements of $R$. 2. (2) As described in [Lan00, 1.2.3], one defines a set of affine roots $R_{\mathrm{aff}}\supseteq R$ on $V$ using the valuation on $\breve{\mathbb{Q}}_{p}$. By $W_{a}\subseteq\mathrm{Aff}(V^{*})=\operatorname{GL}(V^{*})\ltimes V^{*}$ we denote the _affine Weyl group_ of the affine root system $(V,R_{\mathrm{aff}})$, i.e., the group generated by the orthogonal reflections through the affine hyperplanes defined by the elements of $R_{\mathrm{aff}}$. 3. (3) $\widetilde{W}:=N(\breve{\mathbb{Q}}_{p})/(T(\breve{\mathbb{Q}}_{p})\cap\breve{I})$ is the _Iwahori-Weyl group_. 4. (4) $W_{K}:=(N(\breve{\mathbb{Q}}_{p})\cap\breve{K})/(T(\breve{\mathbb{Q}}_{p})\cap\breve{I})\subseteq\widetilde{W}$. (Recall that $\breve{K}=\mathcal{G}(\breve{\mathbb{Z}}_{p})$.) ###### Remarks 2.4. 1. (1) We have $W\subseteq W_{a}$. With the systems of generators indicated above, $W$ and $W_{a}$ become (affine) Coxeter groups; in particular we can talk about reduced words and have length functions, cf. [BB05]. 2. (2) $W_{I}$ is the trivial group. ###### Proposition 2.5. [HR08, Prop. 8] The Bruhat-Tits decomposition $G(\breve{\mathbb{Q}}_{p})=\bigcup_{w\in\widetilde{W}}\breve{K}w\breve{K}$ identifies $\breve{K}\backslash G(\breve{\mathbb{Q}}_{p})/\breve{K}\cong W_{K}\backslash\widetilde{W}/W_{K}.$ ###### Proposition 2.6. [HR08, Prop. 13] Let $\breve{K}$ be the maximal parahoric subgroup of $G(\breve{\mathbb{Q}}_{p})$ associated with a special vertex in the apartment corresponding to $S$. Then $W_{K}\to W$ is an isomorphism and $\widetilde{W}\cong W\ltimes X_{*}(T)_{\operatorname{Gal}(\bar{\mathbb{Q}}_{p}/\breve{\mathbb{Q}}_{p})}$.999Notation: Let $\Gamma$ be a group and $M$ a $\mathbb{Z}[\Gamma]$-module. Then $M_{\Gamma}:=\mathbb{Z}\otimes_{\mathbb{Z}[\Gamma]}M=M/\langle\gamma m-m\;|\;\gamma\in\Gamma,\;m\in M\rangle$ is the module of $\Gamma$-coinvariants of $M$. ###### Notation 2.7. We denote the map $X_{*}(T)_{\operatorname{Gal}(\bar{\mathbb{Q}}_{p}/\breve{\mathbb{Q}}_{p})}\to\widetilde{W}$ of the proposition by $\nu\mapsto t_{\nu}$. ###### Proposition 2.8. [HR08, Lemma 14] Let $\Omega\subseteq\widetilde{W}$ be the subgroup consisting of those elements that preserve the base alcove $\mathfrak{a}$. There is an exact sequence $1\to W_{a}\to\widetilde{W}\to\Omega\to 1,$ with a canonical right splitting (namely the inclusion $\Omega\hookrightarrow\widetilde{W}$), i.e., $\widetilde{W}\cong W_{a}\rtimes\Omega$. ###### Definition 2.9. The semidirect product decomposition of the preceding proposition means that $\widetilde{W}$ is a “quasi-Coxeter” group. In practical terms, this means: 1. (1) We define a length function $\ell$ on $\widetilde{W}$ as follows: $\ell(w_{a},\omega):=\ell(w_{a})$ for all $w_{a}\in W_{a}$ and $\omega\in\Omega$, where on the right hand side we use the length function of the affine Coxeter group $W_{a}$. Note that $\Omega=\ell^{-1}(0)$. 2. (2) Likewise, we extend the Bruhat partial order from $W_{a}$ to $\widetilde{W}$ by defining $(w_{a,1},\omega_{1})\leq(w_{a,2},\omega_{2})\;:\Longleftrightarrow\;w_{a,1}\leq w_{a,2}\text{ and }\omega_{1}=\omega_{2}.$ Note that $w_{1}\leq w_{2}$ ($w_{1},w_{2}\in\widetilde{W}$) implies $\ell(w_{1})\leq\ell(w_{2})$. ###### Definition 2.10. 1. (1) Let $\\{\mu\\}$ be a $W_{\mathrm{abs}}$-conjugacy class of geometric cocharacters of $T$ (cf. Remark 1.4), $W_{\mathrm{abs}}:=N(\bar{\mathbb{Q}}_{p})/T(\bar{\mathbb{Q}}_{p})$ being the absolute Weyl group. Let $\bar{\mu}\in X_{*}(T)_{\operatorname{Gal}(\bar{\mathbb{Q}}_{p}/\breve{\mathbb{Q}}_{p})}$ be the image of a cocharacter in $\\{\mu\\}$ whose image in $X_{*}(T)\otimes_{\mathbb{Z}}\mathbb{R}$ is contained in the closed Weyl chamber corresponding to some Borel subgroup of $G$ containing $T$ and defined over $\breve{\mathbb{Q}}_{p}$. 2. (2) $\mathrm{Adm}(\mu):=\mathrm{Adm}(\\{\mu\\}):=\\{w\in\widetilde{W}\;|\;w\leq qt_{\bar{\mu}}q^{-1}=t_{q\bar{\mu}}\text{ for some }q\in W\\}$ is the _$\\{\mu\\}$ -admissible subset_ of $\widetilde{W}$. 3. (3) $\mathrm{Adm}(\\{\mu\\})^{K}:=W_{K}\mathrm{Adm}(\\{\mu\\})W_{K}\subseteq\widetilde{W}$. 4. (4) $\mathrm{Adm}(\\{\mu\\})_{K}:=\mathrm{KR}(K,\\{\mu\\}):=W_{K}\backslash\mathrm{Adm}(\\{\mu\\})^{K}/W_{K}\subseteq W_{K}\backslash\widetilde{W}/W_{K}$. 5. (5) Define ${}^{K}\widetilde{W}\subseteq\widetilde{W}$ to be the set of representatives of minimal length for the quotient $W_{K}\backslash\widetilde{W}$. 6. (6) ${}^{K}\mathrm{Adm}(\\{\mu\\}):=\mathrm{EKOR}(K,\\{\mu\\}):=\mathrm{Adm}(\\{\mu\\})^{K}\cap{}^{K}\widetilde{W}\subseteq{}^{K}\widetilde{W}$. ###### Lemma 2.11. (See [SYZ19, Thm. 1.2.2].) ${}^{K}\mathrm{Adm}(\\{\mu\\})=\mathrm{Adm}(\\{\mu\\})\cap{}^{K}\widetilde{W}$. #### 2.1.2 Kottwitz-Rapoport stratification Recall from Section 1.6.2 that we have an integral model and a local model diagram $\mathscr{S}_{K}\leftarrow\tilde{\mathscr{S}}_{K}\to M^{\mathrm{loc}}_{K}$ or, equivalently, a (smooth) morphism of stacks $\mathscr{S}_{K}\to[\mathcal{G}_{K}\backslash M^{\mathrm{loc}}_{K}]$ (over $\mathcal{O}_{E}$). As explained in Section 1.6.3, by the construction in [PZ13], the special fiber $M^{\mathrm{loc}}_{K}\otimes\kappa$ of $M^{\mathrm{loc}}_{K}$ is a closed subscheme of the affine flag variety $\operatorname{Gr}_{\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}}\kappa}=\mathcal{F}l_{\underline{\mathcal{G}}_{K}\otimes\kappa[\mkern-2.0mu[t]\mkern-2.0mu]}$, which is the ind-projective ind-scheme over $\kappa$ given as the fpqc sheafification (which exists in this case!) of the presheaf $R\mapsto\underline{\mathcal{G}}_{K}(R(\mkern-4.0mu(t)\mkern-4.0mu))/\underline{\mathcal{G}}_{K}(R[\mkern-2.0mu[t]\mkern-2.0mu])$. ###### Definition 2.12. Define $L^{+}(\underline{\mathcal{G}}_{K}\otimes\kappa[\mkern-2.0mu[t]\mkern-2.0mu])$ to be the $\kappa$-functor sending a $\kappa$-algebra $R$ to $\underline{\mathcal{G}}_{K}(R[\mkern-2.0mu[t]\mkern-2.0mu])$. We let $L^{+}(\underline{\mathcal{G}}_{K}\otimes\kappa[\mkern-2.0mu[t]\mkern-2.0mu])$ act on $\operatorname{Gr}_{\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}}\kappa}$ from the left and call this action $a$ (within this subsection). The orbits of this action on $\operatorname{Gr}_{\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}}\bar{\kappa}}$ are the _Schubert cells_. ###### Remarks 2.13. 1. (1) The Schubert cells can be indexed by $W_{K}\backslash\widetilde{W}/W_{K}$ by Proposition 2.5 with the following in mind: Strictly speaking, using the Bruhat-Tits decomposition here, we arrive at something involving the Iwahori- Weyl group of $\underline{\mathcal{G}}_{K}\otimes\bar{\kappa}(\mkern-4.0mu(t)\mkern-4.0mu)$. However, by [PZ13, 9.2.2], this is isomorphic to the Iwahori-Weyl group of $G_{\breve{\mathbb{Q}}_{p}}$. 2. (2) $M^{\mathrm{loc}}_{K}\otimes\bar{\kappa}$ is a union of Schubert cells, namely of those indexed by $\mathrm{KR}(K,\\{\mu\\}):=W_{K}\backslash(W_{K}\mathrm{Adm}(\\{\mu\\})W_{K})/W_{K}$, cf. [PZ13, Theorem 9.3]. ###### Remark 2.14. By construction, $M^{\mathrm{loc}}_{K}$ has an action $b$ of $\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}[t],t\mapsto p}\mathcal{O}_{E}\cong\mathcal{G}_{K}\otimes\mathcal{O}_{E}$. For $w\in\widetilde{W}$ choose a representative $\dot{w}\in L\underline{\mathcal{G}}_{K}(\bar{\kappa})$ (with Remark 2.13 (1) in mind) and let $e_{0}\in\operatorname{Gr}_{\underline{\mathcal{G}}_{K}\otimes_{\mathbb{Z}_{p}}\kappa}$ be the distinguished base point (associated with the identity). For $w\in W_{K}\mathrm{Adm}(\\{\mu\\})W_{K}$, the orbit map of $\dot{w}\cdot e_{0}$ for the action $a$ factors through the homomorphism $L^{+}(\underline{\mathcal{G}}_{K}\otimes\bar{\kappa}[\mkern-2.0mu[t]\mkern-2.0mu])\to\mathcal{G}_{K}\otimes\kappa\cong\underline{\mathcal{G}}_{K}\otimes\bar{\kappa}$. The orbits associated with the two $\mathcal{G}_{K}\otimes\kappa$-actions $a$ and $b$ on $M^{\mathrm{loc}}_{K}\otimes\kappa$ agree. The orbits of the $\mathcal{G}_{K}\otimes\kappa$-action on $M^{\mathrm{loc}}_{K}\otimes\kappa$ are indexed by $\mathrm{KR}(K,\\{\mu\\})$. ###### Definition 2.15. The stratifications thus obtained on $M^{\mathrm{loc}}_{K}\otimes\kappa$ and $\mathscr{S}_{K}\otimes\kappa$ are called _Kottwitz-Rapoport stratifications_. That is to say that Kottwitz-Rapoport strata on $\mathscr{S}_{K}\otimes\kappa$ are by definition pullbacks of Kottwitz-Rapoport strata on $M^{\mathrm{loc}}_{K}$, which in turn are $\mathcal{G}_{K}\otimes\kappa$-orbits. #### 2.1.3 Ekedahl-Oort stratification The Ekedahl-Oort stratification is only defined in the case of good reduction, i.e., if $K_{p}$ is hyperspecial or, equivalently, if $\mathcal{G}_{K}$ is a _reductive_ model of $G_{\mathbb{Q}_{p}}$. Then $G_{\mathbb{Q}_{p}}$ splits over $\breve{\mathbb{Q}}_{p}$ (by definition of “hyperspecial”, cf. [Tit79, 1.10.2]). We therefore put ourselves in the situation of good reduction for this subsection. ###### Remark 2.16. Then $W$ as defined in Definition 2.3 (1) agrees with the absolute Weyl group of $G_{\mathbb{Q}_{p}}=\mathcal{G}_{K}\otimes\mathbb{Q}_{p}$, which in turn agrees with the absolute Weyl group of $\overline{\mathcal{G}}_{K}:=\mathcal{G}_{K}\otimes\kappa$, cf. [VW13, App. A.5]. ###### Definition 2.17. Define $I$ to be the type (interpreted as a subset of simple reflections) of the parabolic subgroup of $G_{\mathbb{Q}_{p}}$ defined by $\mu^{-1}$ (cf. Remark 1.33), and ${}^{I}W\subseteq W$ to be the system of representatives of the quotient group $W_{I}\backslash W$ containing the element of least length of every coset. ###### Theorem 2.18. [MW04, PWZ15, PWZ11, Zha18] There is a smooth algebraic stack $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}_{\kappa}:=\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\mu}_{\kappa}$ over $\kappa$ with underlying topological space ${}^{I}W$ together with a smooth morphism $\mathscr{S}_{K}\otimes\kappa\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\mu}_{\kappa}.$ The stratification of $\mathscr{S}_{K}\otimes\kappa$ thus obtained is the _Ekedahl-Oort stratification_. #### 2.1.4 EKOR stratification ###### Definition 2.19. Let $L$ be a valued field extension of $\mathbb{Q}_{p}$ with ring of integers $\mathcal{O}$, maximal ideal $\mathfrak{m}$ and residue field $\lambda$. The _pro-unipotent radical_ of $\mathcal{G}_{K}(\mathcal{O})$ is $\mathcal{G}_{K}(\mathcal{O})_{1}:=\\{g\in\mathcal{G}_{K}(\mathcal{O})\;|\;(g\mod\mathfrak{m})\in\bar{R}_{K}(\lambda)\\},$ where $\bar{R}_{K}$ is the unipotent radical of $\mathcal{G}_{K}\otimes_{\mathbb{Z}_{p}}\lambda$. In particular, if $K$ is hyperspecial, then $\mathcal{G}_{K}(\mathcal{O})_{1}=\ker(\mathcal{G}_{K}(\mathcal{O})\to\mathcal{G}_{K}(\lambda))$. Also, $\overline{\breve{K}}:=\breve{K}/\breve{K}_{1}\cong\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}(\bar{\mathbb{F}}_{p})$, where $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ is the maximal reductive quotient of $\overline{\mathcal{G}}_{K}:=\mathcal{G}_{K}\otimes\kappa$. ###### Remark 2.20. [HR17, after Cor. 6.2] We have a commutative diagram $G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1})$${}^{K}\widetilde{W}$$W_{K}\backslash\widetilde{W}/W_{K}.$$\breve{K}\backslash G(\breve{\mathbb{Q}}_{p})/\breve{K}$ Consider the map $v_{K}\colon\mathscr{S}_{K}\otimes\kappa\to G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1}),$ which is the composition of the central leaves map $\Upsilon_{K}\colon\mathscr{S}_{K}\otimes\kappa\to G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}$ (see [Hes20a]) with the projection $G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}\to G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1})$. The Kottwitz-Rapoport map $\lambda_{K}\colon{\mathscr{S}_{K}\otimes\kappa}\to\breve{K}\backslash G(\breve{\mathbb{Q}}_{p})/\breve{K}$ factors through this map. The fibers of $v_{K}$ are called _EKOR strata_. By [HR17, Thm. 6.15], they are locally closed subsets of $\mathscr{S}_{K}\otimes\kappa$. ###### Remarks 2.21. 1. (1) One can explicitly express the image of a EKOR stratum under a change-of- parahoric map as a union of EKOR strata on the target [HR17, Prop. 6.11]. 2. (2) The closure of an EKOR stratum is a union of EKOR strata and one can explicitly describe the associated order relation [HR17, Thm. 6.15]. ###### Remark 2.22. In the hyperspecial case, the EKOR stratification agrees with the Ekedahl-Oort stratification. In the Iwahori case, it agrees with the Kottwitz-Rapoport stratification (${}^{K}\widetilde{W}=\widetilde{W}=W_{K}\backslash\widetilde{W}/W_{K}$ in that case). By definition, the EKOR stratification always is a refinement of the Kottwitz- Rapoport stratification. So one way of approaching the EKOR stratification is by looking at a fixed Kottwitz-Rapoport stratum and trying to understand how it is subdivided into EKOR strata. To get this started, let us recall some calculations from the proof of [HR17, Thm. 6.1]. Fixing a Kottwitz-Rapoport stratum means restricting our view to $\breve{K}w\breve{K}/\breve{K}_{\sigma}$ rather than the whole of $G(\breve{\mathbb{Q}}_{p})/\breve{K}_{\sigma}$, for some fixed $w\in\mathrm{KR}(K,\\{\mu\\})$. The EKOR strata in the Kottwitz-Rapoport stratum associated with $w$ are therefore indexed by $\breve{K}w\breve{K}/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1})$. Define $\sigma^{\prime}:=\sigma\circ\operatorname{Ad}(w)$ and consider the bijection $\displaystyle\breve{K}/(\breve{K}\cap w^{-1}\breve{K}w)_{\sigma^{\prime}}$ $\displaystyle\xrightarrow{\sim}\breve{K}w\breve{K}/\breve{K}_{\sigma},$ $\displaystyle k$ $\displaystyle\mapsto wk,$ $\displaystyle k_{2}\sigma(k_{1})$ $\displaystyle\mapsfrom k_{1}wk_{2}.$ Let $J$ be the set of simple affine reflections in $W_{K}$, let $\bar{B}$ be the image of $\breve{I}$ in $\overline{\breve{K}}$ and $\bar{T}\subseteq\bar{B}$ the maximal torus. Set $J_{1}:=J\cap w^{-1}Jw$. ###### Proposition 2.23. (See [Mor93, Lemma 3.19].) The image of $\breve{K}\cap w^{-1}\breve{K}w$ in $\overline{\breve{K}}$ is $\bar{P}_{J_{1}}$, i.e., the standard parabolic subgroup of $\overline{\breve{K}}$ associated with $J_{1}$. ###### Remark 2.24. He and Rapoport invoke Carter’s book [Car93] at this point, which primarily pertains to the case of (usual) BN-pairs attached to reductive groups. Morris [Mor93] shows that the relevant results carry over likewise to the case of generalized (or affine) BN-pairs. Then we get a map $\displaystyle\breve{K}w\breve{K}/\breve{K}_{\sigma}\to\breve{K}/(\breve{K}\cap w^{-1}\breve{K}w)_{\sigma^{\prime}}$ $\displaystyle\to\overline{\breve{K}}/(\bar{P}_{J_{1}})_{\sigma^{\prime}}$ $\displaystyle\to\overline{\breve{K}}/(\bar{L}_{J_{1}})_{\sigma^{\prime}}(\bar{U}_{J_{1}})_{\sigma^{\prime}}\to\overline{\breve{K}}/(\bar{L}_{J_{1}})_{\sigma^{\prime}}(\bar{U}_{J_{1}}\times\bar{U}_{\sigma^{\prime}(J_{1})}),$ which factors through a bijection $\breve{K}w\breve{K}/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1})\xrightarrow{\sim}\overline{\breve{K}}/(\bar{L}_{J_{1}})_{\sigma^{\prime}}(\bar{U}_{J_{1}}\times\bar{U}_{\sigma^{\prime}(J_{1})})\cong\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\mathcal{Z}_{w}}(\bar{\mathbb{F}}_{p})/{{\scriptstyle\cong}}.$ Here, $\mathcal{Z}_{w}$ is the (connected) algebraic zip datum $\mathcal{Z}_{w}=(\overline{\mathcal{G}}^{\mathrm{rdt}},\bar{P}_{J_{1}},\bar{P}_{\sigma^{\prime}(J_{1})},\sigma^{\prime})$, as described in [SYZ19]. In [SYZ19], Shen, Yu and Zhang show that this observation “globalizes” (with the drawback that “global” here still just refers to the Kottwitz-Rapoport stratum101010They also give another “globalization”; the drawback there being that it only works after perfection.) in a pleasant way. To wit, one gets a smooth morphism [SYZ19, Theorem A] $\zeta_{w}\colon\overline{\mathscr{S}}_{K}^{w}\to\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\mathcal{Z}_{w}}_{\kappa}$ (the source being a Kottwitz-Rapoport stratum). ### 2.2 $\overline{\mathcal{G}}_{K}$-zips in the Siegel case Here we work with the Siegel Shimura datum, cf. Example 1.2. #### 2.2.1 Preliminaries ###### Notation 2.25. Fix $p\neq 2$,111111As in [RZ96], the principal reason for this restriction is our use of the equivalence between alternating and skew-symmetric. See Definition 2.30 (e). $g\in\mathbb{Z}_{\geq 1}$ and a subset $J\subseteq\mathbb{Z}$ with $J=-J$ and $J+2g\mathbb{Z}=J$. Associated with $J$ is the partial lattice chain $\left\\{\Lambda^{j}\;|\;j\in J\right\\}$, where $\Lambda^{j}$ are defined as in equation (1.29). Let $K_{p}$ be the corresponding parahoric subgroup of $\operatorname{GSp}_{2g}(\mathbb{Q}_{p})$, i.e., the stabilizer of said lattice chain. It contains the Iwahori subgroup $I_{p}$ associated with the full lattice chain (1.29). For the maximal torus $T$ we take the usual diagonal (split) torus. ###### Remark 2.26. The Weyl group is $\displaystyle W$ $\displaystyle=\\{\pi\in S_{2g}=\operatorname{Aut}(\\{\pm 1,\pm 2,\dotsc,\pm g\\})\;|\;\pi(-n)=-\pi(n)\text{ for }n=1,2,\dotsc,g\\}$ $\displaystyle\cong S_{g}\ltimes\\{\pm 1\\}^{g}.$ Here the transposition $(n\quad m)$ of $S_{g}=\operatorname{Aut}(\\{1,2,\dotsc,g\\})$ corresponds to the element ${(n\quad m)(-n\quad{-m})}$ of $\operatorname{Aut}(\\{\pm 1,\pm 2,\dotsc,\pm g\\})$ and the element of $\\{\pm 1\\}^{g}$ which has a $-1$ in position $i$ and $1$ everywhere else corresponds to $(i\quad{-i})$. The affine Weyl group is $W_{a}=W\ltimes Y_{0}$ and the Iwahori-Weyl group $\widetilde{W}=W\ltimes Y$ with $\displaystyle\mathbb{Z}^{g+1}$ $\displaystyle\cong Y=\\{(\nu_{1},\dotsc,\nu_{g},\nu_{-g},\dotsc,\nu_{-1})\in\mathbb{Z}^{2g}:\nu_{1}+\nu_{-1}=\dotsb=\nu_{g}+\nu_{-g}\\}$ $\displaystyle\supseteq Y_{0}=\\{(\nu_{1},\dotsc,\nu_{g},\nu_{-g},\dotsc,\nu_{-1})\in\mathbb{Z}^{2g}:0=\nu_{1}+\nu_{-1}=\dotsb=\nu_{g}+\nu_{-g}\\}\cong\mathbb{Z}^{g}.$ The simple affine roots (whose walls bound the base alcove $\mathfrak{a}$) are $\displaystyle 1-2e_{-1}+e_{0}=1+2e_{1}-e_{0},$ $\displaystyle e_{-1}-e_{-2}=e_{2}-e_{1},e_{-2}-e_{-3},\dotsc,e_{-(g-1)}-e_{-g},$ $\displaystyle 2e_{-g}-e_{0}=e_{0}-2e_{g},$ where $e_{1},\dotsc,e_{g},e_{-g},\dotsc,e_{-1}\colon T\to\mathbb{G}_{m}$ are the obvious cocharacters and $e_{0}=e_{1}+e_{-1}=\dotsb=e_{g}+e_{-g}$. The reflections corresponding to the simple affine roots are $((1\quad{-1}),\left(\begin{smallmatrix}-1\\\ 0\\\ \vdots\\\ 0\\\ 1\end{smallmatrix}\right)),(-1\quad{-2})(1\quad 2),\dotsc,(-g\quad{-(g-1)})(g\quad{g-1}),(g\quad{-g}).$ The length zero subgroup $\Omega\subseteq\widetilde{W}$ is generated by $((w_{0},\epsilon),y)\in(S_{g}\ltimes\\{\pm 1\\}^{g})\ltimes Y$, where $w_{0}\in S_{g}$ is the longest element, $\epsilon=(-1,-1,\dotsc,-1)$ and $y=(0^{g},1^{g})$. ###### Remark 2.27. One also can choose $\dotsb\subseteq p\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}^{2g-1}\subseteq\mathbb{Z}_{p}^{2g}\subseteq\dotsb$ instead of $\dotsb\subseteq\mathbb{Z}_{p}^{2g-1}\oplus p\mathbb{Z}_{p}\subseteq\mathbb{Z}_{p}^{2g}\subseteq\dotsb$ as the standard lattice chain. Then the simple affine roots would be $1-2e_{1}+e_{0},e_{1}-e_{2}=e_{2}-e_{1},e_{2}-e_{3},\dotsc,e_{g-1}-e_{g},2e_{g}-e_{0}.$ ###### Remark 2.28. $\widetilde{W}=W\ltimes Y=N(\mathbb{Q}_{p})/T(\mathbb{Z}_{p})$ and $N(\mathbb{Q}_{p})\to W\ltimes Y$ has a section $W\ltimes Y\to N(\mathbb{Q}_{p})$, which sends $(\pi,\underline{\nu})\in W\ltimes Y$ to $T_{\underline{\nu}}P_{w}$, where $T_{\underline{\nu}}=\left(\begin{smallmatrix}p^{\nu_{1}}&&&&\\\ &p^{\nu_{2}}&&&\\\ &&\ddots&&\\\ &&&p^{\nu_{-2}}&\\\ &&&&p^{\nu_{-1}}\end{smallmatrix}\right)$ and $P_{w}$ is the permutation matrix with $P_{w}(e_{i})=e_{w(i)}$. ###### Remark 2.29. Using the results of [KR00] we also easily can compute $\mathrm{Adm}(\\{\mu\\})$. One potential source of confusion at this point is that, due to our choice of the base alcove (cf. Remark 2.27), in our setup we need to use $\omega_{i}:=(0^{2g-i},1^{i})$ instead of $\omega_{i}:=(1^{i},0^{2g-i})$ (notation of [KR00]), cf. [Yu08, 1268]. With that convention in place, we have that $x\in\widetilde{W}$ is $\\{\mu\\}$-admissible ($\mu=(1^{g},0^{g})$) if and only if $(0,\dotsc,0)\leq x(\omega_{i})-\omega_{i}\leq(1,\dotsc,1)\quad\text{for all }0\leq i<2g$ (component-wise comparison). #### 2.2.2 Lattice chains, zips, admissibility ###### Definition 2.30. Let $S$ be a $\mathbb{Z}_{p}$-scheme. A _Siegel lattice chain in the weak sense on $S$ of type $J$_ is a tuple $(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet})$, where 1. (a) for all $j\in J$, $\mathcal{V}^{j}$ is a vector bundle on $S$ of rank $2g$, 2. (b) $\mathcal{L}$ is a line bundle on $S$, 3. (c) for all $i,j\in J$ with $j>i$, $\alpha_{j,i}\colon\mathcal{V}^{j}\to\mathcal{V}^{i}$ is a vector bundle homomorphism, such that the $\bigl{(}\alpha_{j,i}\bigr{)}$ satisfy the obvious cocycle condition (and we also define $\alpha_{i,i}:=\operatorname{id}$), 4. (d) for all $j\in J$, $\theta_{j}\colon\mathcal{V}^{j}\xrightarrow{\sim}\mathcal{V}^{j-2g}$ is a vector bundle isomorphism such that the $\bigl{(}\theta_{j}\bigr{)}$ are compatible with the $\bigl{(}\alpha_{j,i}\bigr{)}$ in that $\theta_{i}\circ\alpha_{j,i}=\alpha_{j-2g,i-2g}\circ\theta_{j}$ and $\alpha_{j,j-2g}=p\cdot\theta_{j}$, 5. (e) for all $j\in J$ a vector bundle isomorphism $\psi_{j}\colon\mathcal{V}^{j}\xrightarrow{\sim}(\mathcal{V}^{-j})^{*}\otimes\mathcal{L}$ compatible with $\bigl{(}\theta_{j}\bigr{)}$ and $\bigl{(}\alpha_{j,i}\bigr{)}$, such that $-\psi_{j}(x,y)=\psi_{-j}(y,x)$ for all $(x,y)\in\mathcal{V}^{j}\times\mathcal{V}^{-j}$.121212By “$(x,y)\in\mathcal{V}^{j}\times\mathcal{V}^{-j}$” we of course mean that there is an open subset $U\subseteq S$ such that $(x,y)\in(\mathcal{V}^{j}\times\mathcal{V}^{-j})(U)$. We also have a _standard_ Siegel lattice chain in the weak sense on $\operatorname{Spec}\mathbb{Z}_{p}$ (and hence by base change on every $\mathbb{Z}_{p}$-scheme $S$) of type $J$, namely the one given by the lattice chain $\left\\{\Lambda^{j}\;|\;j\in J\right\\}$. We can think of the standard Siegel lattice chain as either having varying $\mathcal{V}^{j}$ with the $\alpha_{j,i}$ being the obvious inclusion maps (e.g. (if $\\{0,1\\}\subseteq J$), $\mathcal{V}^{1}={\mathbb{Z}_{p}^{2g-1}\oplus p\mathbb{Z}_{p}}\xrightarrow{\alpha_{1,0}=\mathrm{inclusion}}\mathbb{Z}_{p}^{2g}=\mathcal{V}^{0}$) or as having constant $\mathcal{V}^{j}=\mathbb{Z}_{p}^{2g}$ with the $\alpha_{j,i}$ being diagonal matrices with all entries either $p$ or $1$ (e.g., $\mathcal{V}^{1}=\mathbb{Z}_{p}^{2g}\xrightarrow{\alpha_{1,0}=\operatorname{diag}(1,1,\dotsc,1,p)}\mathbb{Z}_{p}^{2g}=\mathcal{V}^{0}$). Usually the latter point of view is more convenient. A _Siegel lattice chain on $S$ of type $J$_ (or _Siegel lattice chain in the strong sense on $S$ of type $J$_) then is a Siegel lattice chain in the weak sense on $S$ of type $J$ that Zariski-locally on $S$ is isomorphic to the standard chain. ###### Remarks 2.31. 1. (1) Let $(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet})$ be a Siegel lattice chain in the weak sense on $S$ of type $J$. Then $\tilde{\psi_{j}}:=(\tilde{\alpha}_{j,-j}^{*}\otimes\operatorname{id}_{\mathcal{L}})\circ\psi_{j}\colon\mathcal{V}^{j}\otimes\mathcal{V}^{j}\to\mathcal{L}$ is alternating. Here $\tilde{\alpha}_{j,-j}$ is defined as follows: Let $n\in\mathbb{Z}$ be maximal with $j-2gn\geq-j$. Then $\tilde{\alpha}_{j,-j}:=\alpha_{j-2gn,-j}\circ\theta_{j-2g(n-1)}\circ\dotsb\circ\theta_{j}$. 2. (2) Note that this means that $\tilde{\psi}_{j}$ is (twisted) symplectic if $-j\in j+2g\mathbb{Z}$, i.e., if $j\in g\mathbb{Z}$. ###### Proof: (of (1)) Let $x,y\in\mathcal{V}^{j}$. Then $\displaystyle\tilde{\psi}_{j}(x,y)$ $\displaystyle=\psi_{j}(x,\tilde{\alpha}_{j,-j}(y))$ $\displaystyle=\psi_{j}(x,(\alpha_{j-2gn,-j}\circ\theta_{j-2g(n-1)}\circ\dotsb\circ\theta_{j})(y))$ $\displaystyle=\psi_{2gn-j}(\alpha_{j,2gn-j}(x),(\theta_{j-2g(n-1)}\circ\dotsb\circ\theta_{j})(y))$ $\displaystyle=\psi_{2g(n-1)-j}((\theta_{2gn-j}\circ\alpha_{j,2gn-j})(x),(\theta_{j-2g(n-2)}\circ\dotsb\circ\theta_{j})(y))$ $\displaystyle=\dotsb$ $\displaystyle=\psi_{-j}((\theta_{-j+2g}\circ\dotsb\circ\theta_{2gn-j}\circ\alpha_{j,2gn-j})(x),y)$ $\displaystyle=-\psi_{j}(y,(\theta_{-j+2g}\circ\dotsb\circ\theta_{2gn-j}\circ\alpha_{j,2gn-j})(x))$ $\displaystyle=-\tilde{\psi}_{j}(y,x).$ □ ###### Reminder 2.32. $\mathcal{G}_{K}$ is the automorphism group of the standard Siegel lattice chain. The following definition is a generalization of [VW13, Definition 3.1] in the Siegel case. ###### Definition 2.33. Let $S$ be an $\mathbb{F}_{p}$-scheme. A $\overline{\mathcal{G}}_{K}$-zip over $S$ is a tuple $(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet},\mathcal{C}^{\bullet},\mathcal{D}^{\bullet},\varphi_{0}^{\bullet},\varphi_{1}^{\bullet},\varphi_{\mathcal{L}})$, where 1. (a) $(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet})$ is a Siegel lattice chain on $S$ of type $J$, 2. (b) for all $j\in J$, $\mathcal{C}^{j}\subseteq\mathcal{V}^{j}$ are locally direct summands of rank $g$ compatible with $\alpha_{\bullet\bullet},\theta_{\bullet}$, such that $\mathcal{C}^{j}\hookrightarrow\mathcal{V}^{j}\overset{\psi_{j}}{\cong}(\mathcal{V}^{-j})^{*}\otimes\mathcal{L}\to(\mathcal{C}^{-j})^{*}\otimes\mathcal{L}$ vanishes. (cf. Remark 1.30 for the origins of this condition.) 3. (c) $\mathcal{D}^{\bullet}\subseteq\mathcal{V}^{\bullet}$ satisfies the same conditions as $\mathcal{C}^{\bullet}\subseteq\mathcal{V}^{\bullet}$, 4. (d) $\varphi_{0}^{j}\colon(\mathcal{C}^{j})^{(p)}\xrightarrow{\sim}\mathcal{V}^{j}/\mathcal{D}^{j}$ and $\varphi_{1}^{j}\colon(\mathcal{V}^{j}/\mathcal{C}^{j})^{(p)}\xrightarrow{\sim}\mathcal{D}^{j}$ are isomorphisms of vector bundles compatible with $\alpha_{\bullet\bullet}$ and $\theta_{\bullet}$ and $\varphi_{\mathcal{L}}\colon\mathcal{L}^{(p)}\xrightarrow{\sim}\mathcal{L}$ is an isomorphism of line bundles, such that $(\mathcal{C}^{j})^{(p)}$$(\mathcal{V}^{-j}/\mathcal{C}^{-j})^{*,(p)}\otimes\mathcal{L}^{(p)}$$(\mathcal{D}^{-j})^{*}\otimes\mathcal{L}$$\mathcal{V}^{j}/\mathcal{D}^{j}$$\varphi_{0}^{j}$$(\varphi_{1}^{-j})^{*}\otimes\varphi_{\mathcal{L}}^{-1}$$\psi_{j}^{(p)}$$\psi_{j}$ commutes, i.e., ${\psi_{j}(\varphi_{0}^{j}(\\_),\varphi_{1}^{-j}(\\_))=\varphi_{\mathcal{L}}\circ\psi_{j}^{(p)}(\\_,\\_)\colon}{(\mathcal{C}^{j})^{(p)}\times(\mathcal{V}^{-j}/\mathcal{C}^{-j})^{(p)}\to\mathcal{L}^{(p)}\to\mathcal{L}}.$ Since $\varphi_{\mathcal{L}}$ evidently is uniquely determined by the other data, we sometimes leave it out. We obtain a fibered category $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}\to\mathrm{Sch}_{\mathbb{F}_{p}}$. ###### Remark 2.34. $\psi_{j}$ gives rise to isomorphisms $\displaystyle\mathcal{C}^{j}$ $\displaystyle\xrightarrow{\sim}(\mathcal{V}^{-j}/\mathcal{C}^{-j})^{*}\otimes\mathcal{L},$ $\displaystyle\mathcal{V}^{j}/\mathcal{C}^{j}$ $\displaystyle\xrightarrow{\sim}(\mathcal{C}^{-j})^{*}\otimes\mathcal{L},$ $\displaystyle\mathcal{D}^{j}$ $\displaystyle\xrightarrow{\sim}(\mathcal{V}^{-j}/\mathcal{D}^{-j})^{*}\otimes\mathcal{L},$ $\displaystyle\mathcal{V}^{j}/\mathcal{D}^{j}$ $\displaystyle\xrightarrow{\sim}(\mathcal{D}^{-j})^{*}\otimes\mathcal{L}.$ This way $\mathcal{V}^{\bullet}/\mathcal{C}^{\bullet}\oplus\mathcal{C}^{\bullet}$ and $\mathcal{D}^{\bullet}\oplus\mathcal{V}^{\bullet}/\mathcal{D}^{\bullet}$ become Siegel lattice chains in the weak(!) sense of type $J$. The Cartier isomorphism then is an isomorphism in the category of Siegel lattice chains in the weak sense of type $J$. Over an algebraically closed field, we call the isomorphism type of the Siegel lattice chain in the weak sense $\mathcal{V}^{\bullet}/\mathcal{C}^{\bullet}\oplus\mathcal{C}^{\bullet}$ the _Kottwitz-Rapoport type of the $\overline{\mathcal{G}}_{K}$-zip_. We also define a linearly rigidified version of $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$ as follows. ###### Definition 2.35. We define the fibered category $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}\to\mathrm{Sch}_{\mathbb{F}_{p}}$ just like $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$ but with the extra condition that $(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet})$ be the standard Siegel lattice chain (rather than just locally isomorphic to it). ###### Lemma 2.36. We always have a closed embedding of $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$ into a product of (classical) $\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$’s, and therefore $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$ is a scheme. ###### Proof: Set $J^{\prime}:=J\cap\\{0,\dotsc,2g-1\\}$. Let $\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}=E_{(1^{g},0^{g})}\backslash(\operatorname{GL}_{2g}\times\operatorname{GL}_{2g})$ be the $\mathbb{F}_{p}$-scheme of trivialized $\operatorname{GL}_{2g}$-zips (so that $[\operatorname{GL}_{2g}\backslash\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}]=\operatorname{GL}_{2g}\text{-}\mathrm{Zip}$) with respect to the the cocharacter $(1^{g},0^{g})$, and $\prod_{j\in J^{\prime}}\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ the product of $\\#J^{\prime}$ copies of this scheme. On $J^{\prime}$ we define $-j:=2g-j$ for $1\leq j\leq 2g-1$ and $-0:=0$. Then we get a monomorphism $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}\hookrightarrow\prod_{j\in J^{\prime}}\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ (2.37) by sending $(\mathcal{C}^{\bullet},\mathcal{D}^{\bullet},\varphi_{0}^{\bullet},\varphi_{1}^{\bullet})$ to $\left(\mathcal{C}^{j},\mathcal{D}^{j},\varphi_{0}^{j},\varphi_{1}^{j}\right)_{j\in J^{\prime}}$. The extra conditions for an element of $\prod_{j\in J^{\prime}}\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ to be in $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$ are as in Definition 2.33: 1. (1) $\mathcal{C}^{\bullet},\mathcal{D}^{\bullet},\varphi_{0}^{\bullet},\varphi_{1}^{\bullet}$ are compatible with the transition maps (or, to put it differently, ${(\mathcal{C}^{j}\oplus\mathcal{V}^{j}/\mathcal{C}^{j})^{(p)}}\xrightarrow[\cong]{\varphi_{0}^{j}\oplus\varphi_{1}^{j}}{\mathcal{V}^{j}/\mathcal{D}^{j}\oplus\mathcal{D}^{j}}$ is compatible with the transition maps), 2. (2) $\mathcal{C}^{j}\hookrightarrow\mathcal{V}^{j}\overset{\psi_{j}}{\cong}(\mathcal{V}^{-j})^{*}\to(\mathcal{C}^{-j})^{*}$ vanishes. 3. (3) $\mathcal{D}^{j}\hookrightarrow\mathcal{V}^{j}\overset{\psi_{j}}{\cong}(\mathcal{D}^{-j})^{*}\to(\mathcal{D}^{-j})^{*}$ vanishes. 4. (4) There is a (necessarily unique) isomorphism $\varphi_{\mathcal{L}}\colon\mathcal{L}^{(p)}\xrightarrow{\sim}\mathcal{L}=\mathcal{O}_{S}$ of line bundles, such that $(\mathcal{C}^{j})^{(p)}$$(\mathcal{V}^{-j}/\mathcal{C}^{-j})^{*,(p)}$$(\mathcal{D}^{-j})^{*}$$\mathcal{V}^{j}/\mathcal{D}^{j}$$\varphi_{0}^{j}$$(\varphi_{1}^{-j})^{*}\otimes\varphi_{\mathcal{L}}^{-1}$$\psi_{j}^{(p)}$$\psi_{j}$ commutes. We claim that the conditions are closed on $\prod_{j\in J^{\prime}}\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ (hence the monomorphism is a closed immersion). To see this, we recall the construction of the scheme $\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ as executed in [MW04, (3.10), (3.11), (4.3)]. Recall our notational convention regarding the parabolic subgroup associated with a cocharacter $\chi$ from Definition 1.33. As in [MW04], we denote by $\mathrm{Par}_{\chi}$ the scheme of parabolic subgroups of type $\chi$. There is a group scheme $H$ defined by the cartesian diagram $H$$\mathcal{P}_{((-1)^{g},0^{g})}/\mathcal{U}_{((-1)^{g},0^{g})}$$\square$$\mathrm{Par}_{((-1)^{g},0^{g})}\times\mathrm{Par}_{(1^{g},0^{g})}$$\mathrm{Par}_{((-1)^{g},0^{g})}$$(\;)^{(p)}\circ\operatorname{pr}_{1}$ where $\mathcal{P}_{((-1)^{g},0^{g})}\to\mathrm{Par}_{((-1)^{g},0^{g})}$ is the universal parabolic group scheme and $\mathcal{U}_{((-1)^{g},0^{g})}$ its unipotent radical, such that $\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ is an $H$-Zariski torsor over $\mathrm{Par}_{((-1)^{g},0^{g})}\times\mathrm{Par}_{(1^{g},0^{g})}$, where $\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}\to\mathrm{Par}_{((-1)^{g},0^{g})}\times\mathrm{Par}_{(1^{g},0^{g})}$ is given by $(C,D,\varphi_{0},\varphi_{1})\mapsto(C,D)$. Clearly, compatibility of $\mathcal{C}^{\bullet},\mathcal{D}^{\bullet}$ with the transition maps is a closed condition on $\prod_{j\in J^{\prime}}\mathrm{Par}_{((-1)^{g},0^{g})}\times\mathrm{Par}_{(1^{g},0^{g})}$ and then also on $\prod_{j\in J^{\prime}}\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$. Similar for the conditions (2) and (3). Locally, we can choose complements (not necessarily compatible with the transition maps) and then $\varphi_{\bullet}^{j}$ yield sections $g^{j}$ of $\operatorname{GL}_{2g}$ as in [MW04, definition of $g\in G(S)$ in the proof of (4.3)]. The $g^{j}$ are well-defined up to $\mathcal{U}_{((-1)^{g},0^{g})}^{(p)}\times\mathcal{U}_{(1^{g},0^{g})}$, and we want them to be compatible with the transition maps coming from the Siegel lattice chains in the weak sense $\mathcal{C}^{j}\oplus\mathcal{V}^{j}/\mathcal{C}^{j}$ and $\mathcal{V}^{j}/\mathcal{D}^{j}\oplus\mathcal{D}^{j}$, respectively. With our complements in place, these transition maps correspond to maps $\mathcal{V}^{j}\to\mathcal{V}^{j-n}$. The question of whether $g^{j}$ is compatible with these maps is independent of the choice of complements (basically because the transition maps $\mathcal{V}^{j}\to\mathcal{V}^{j-n}$ depend on the choice of complements similar to how $g^{j}$ depends on that choice). So in effect we can view the conditions on $\varphi_{0}^{\bullet},\varphi_{1}^{\bullet}$ of (1) as closed conditions on $\prod_{j\in J^{\prime}}\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim\sim}$, where $\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim\sim}\to\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim}$ (an fpqc quotient map) additionally comes with complementary spaces of $C,D$ ($\operatorname{GL}_{2g}\text{-}\mathrm{Zip}^{\sim\sim}=\tilde{X}_{\tau}$ in the notation of [MW04, proof of (4.3)]). We also can reformulate condition (4) in those terms. □ ###### Corollary 2.38. $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$ is the algebraic quotient stack $[\overline{\mathcal{G}}_{K}\backslash\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}]$. Here by definition an element $\phi\in\overline{\mathcal{G}}_{K}$ acts on $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$ by replacing $(\mathcal{C}^{\bullet},\mathcal{D}^{\bullet},\varphi_{0}^{\bullet},\varphi_{1}^{\bullet},\varphi_{\mathcal{L}})$ by $(\phi\mathcal{C}^{\bullet},\phi\mathcal{D}^{\bullet},\phi\varphi_{0}^{\bullet}\phi^{-(p)},\phi\varphi_{1}^{\bullet}\phi^{-(p)},\varphi_{\mathcal{L}})$. ###### Definition 2.39. We let an element $(X,Y)\in\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$ act on $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$ by replacing $(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet},\mathcal{C}^{\bullet},\mathcal{D}^{\bullet},\varphi_{0}^{\bullet},\varphi_{1}^{\bullet})$ by $(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet},X\mathcal{C}^{\bullet},Y\mathcal{D}^{\bullet},Y\varphi_{0}^{\bullet}X^{-(p)},Y\varphi_{1}^{\bullet}X^{-(p)}).$ ###### Notation 2.40. Let $\mathscr{S}_{K}\to\operatorname{Spec}\mathbb{Z}_{p}$ be the integral model of the Siegel Shimura variety of level $K$ (where $K=K_{p}K^{p}$ with $K^{p}$ sufficiently small), and recall $\tilde{\mathscr{S}}_{K}$ from Section 1.6.1. Moreover, define $\overline{\mathscr{S}}_{K}:=\mathscr{S}_{K}\otimes_{\mathbb{Z}_{p}}\mathbb{F}_{p}$ and $\tilde{\overline{\mathscr{S}}}_{K}:=\tilde{\mathscr{S}}_{K}\otimes_{\mathbb{Z}_{p}}\mathbb{F}_{p}$. So $\tilde{\overline{\mathscr{S}}}_{K}\to\overline{\mathscr{S}}_{K}$ is a $\overline{\mathcal{G}}_{K}$-torsor. ###### Remark 2.41. We have morphisms $\tilde{\overline{\mathscr{S}}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$ (take first de Rham cohomology with Frobenius and Verschiebung) and $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}\to\overline{M}^{\mathrm{loc}}_{K}$ (take the $\mathcal{C}^{\bullet}$-filtration) and therefore $\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}\to[\overline{\mathcal{G}}_{K}\backslash\overline{M}^{\mathrm{loc}}_{K}].$ ###### Remark 2.42. In particular, $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$ has a Kottwitz-Rapoport stratification, which agrees with the notion of Kottwitz- Rapoport type as defined in Remark 2.34. For $w\in\mathrm{KR}(K,\\{\mu\\})$ denote the associated Kottwitz-Rapoport stratum by $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}_{w}$, i.e., we interpret $w$ as a $\bar{\mathbb{F}}_{p}$-valued point of $[\overline{\mathcal{G}}_{K}\backslash\overline{M}^{\mathrm{loc}}_{K}]$ and form $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}_{w}$ as a fiber product. ###### Construction 2.43. Fix $w\in\mathrm{Adm}(\\{\mu\\})^{K}\subseteq\widetilde{W}$ (so that $W_{K}wW_{K}\in\mathrm{KR}(K,\\{\mu\\})$). We define a standard $\overline{\mathcal{G}}_{K}$-zip of KR type $W_{K}wW_{K}$. Using Remark 2.28, we interpret $w$ as an element of $N(\mathbb{Q}_{p})\subseteq G(\mathbb{Q}_{p})$. The admissibility condition implies that we can interpret it as an endomorphism $w^{\bullet}$ of the standard lattice chain $\mathcal{V}^{\bullet}$ over $\mathbb{Z}_{p}$.131313Take up the second point of view described in Definition 2.30 regarding $\mathcal{V}^{\bullet}$. Define $\underline{\nu}^{(0)}:=\underline{\nu}$, $\underline{\nu}^{(1)}:=\underline{\nu}+\left(\begin{smallmatrix}0\\\ \vdots\\\ 0\\\ 0\\\ -1\end{smallmatrix}\right)+w\left(\begin{smallmatrix}0\\\ \vdots\\\ 0\\\ 0\\\ 1\end{smallmatrix}\right)$, $\underline{\nu}^{(2)}:=\underline{\nu}+\left(\begin{smallmatrix}0\\\ \vdots\\\ 0\\\ -1\\\ -1\end{smallmatrix}\right)+w\left(\begin{smallmatrix}0\\\ \vdots\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)$, and so on. Then $w^{j}=T_{\underline{\nu}^{(j)}}P_{w}$ for $0\leq j<2g$. From the formulation of the admissibility condition as in Remark 2.29, we see that $w\in\mathrm{Adm}(\\{\mu\\})^{K}$ is equivalent to the condition that $\underline{\nu}^{(j)}$ be a permutation of $(1^{g},0^{g})$ for all relevant $j$. We denote the standard Siegel lattice chain over $\mathbb{Z}_{p}$ by $\mathscr{V}^{\bullet}$ and its base change to $\mathbb{F}_{p}$ by $\mathcal{V}^{\bullet}$. Define $\mathscr{C}_{w}^{\bullet}:=pw^{\bullet,-1}\mathscr{V}^{\bullet}$ and $\mathscr{D}_{w}^{\bullet}:=\sigma(w^{\bullet})\mathscr{V}^{\bullet}$. Then $\mathcal{C}_{w}^{\bullet}:=\mathscr{C}_{w}^{\bullet}\otimes\mathbb{F}_{p}=\ker(w^{\bullet}\colon\mathcal{V}^{\bullet}\to\mathcal{V}^{\bullet})$, so $(\mathcal{V}^{\bullet}/\mathcal{C}_{w}^{\bullet})^{(p)}\xrightarrow{\sim}\mathcal{D}_{w}^{\bullet}:=\mathscr{D}_{w}^{\bullet}\otimes\mathbb{F}_{p}$ via $\sigma(w^{\bullet})$ and $(\mathcal{C}_{w}^{\bullet})^{(p)}\xrightarrow{\sim}\mathcal{V}^{\bullet}/\mathcal{D}_{w}^{\bullet}$ via $p^{-1}\sigma(w^{\bullet})$. This defines a standard element $\widetilde{\mathrm{Std}}(w)$ of $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}(\mathbb{F}_{p})$ and a standard element $\mathrm{Std}(w)$ of $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}_{w}(\mathbb{F}_{p})$. ###### Definition and Remark 2.44. (See also [SYZ19, Lemma 3.3.2].) $\mathcal{G}_{w}:=\operatorname{Aut}(\mathscr{C}_{w}^{\bullet}\subseteq\mathscr{V}^{\bullet})$ is a Bruhat-Tits group scheme with generic fiber $G_{\mathbb{Q}_{p}}$ and $\breve{\mathbb{Z}}_{p}$-points $\breve{K}\cap w^{-1}\breve{K}w$; and similarly for $\mathcal{G}_{\sigma(w)^{-1}}:=\operatorname{Aut}(\mathscr{D}_{w}^{\bullet}\subseteq\mathscr{V}^{\bullet})$ with $\breve{K}\cap\sigma(w)\breve{K}\sigma(w)^{-1}$. ###### Definition 2.45. We keep $w\in\mathrm{Adm}(\\{\mu\\})^{K}\subseteq\widetilde{W}$ fixed and define $\widetilde{E}_{w}\subseteq\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$ to be the stabilizer of $\widetilde{\mathrm{Std}}(w)$. So $\widetilde{E}_{w}$ consists of those $(X^{\bullet},Y^{\bullet})\in\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$ such that $X^{\bullet}\mathcal{C}_{w}^{\bullet}=\mathcal{C}_{w}^{\bullet}$, $Y^{\bullet}\mathcal{D}_{w}^{\bullet}=\mathcal{D}_{w}^{\bullet}$, and $Y^{\bullet}\circ\varphi_{j}^{\bullet}\circ X^{\bullet,-(p)}=\varphi_{j}^{\bullet}$ for $j=0,1$. In the notation of [SYZ19, Lemma 3.3.2] we have $\widetilde{E}_{w}=\overline{\mathcal{G}}_{w}\times_{\overline{\mathcal{G}}_{w}^{L,(p)}}\overline{\mathcal{G}}_{\sigma(w)^{-1}}.$ (2.46) Here $\overline{\mathcal{G}}_{w}^{L}$ is the image of $\overline{\mathcal{G}}_{w}$ in $\mathrm{DiagAut}(\mathcal{C}_{w}^{\bullet}\oplus\mathcal{V}^{\bullet}/\mathcal{C}_{w}^{\bullet})$ (the automorphisms of $\mathcal{C}_{w}^{\bullet}\oplus\mathcal{V}^{\bullet}/\mathcal{C}_{w}^{\bullet}$ respecting both $\mathcal{C}_{w}^{\bullet}$ and $\mathcal{V}^{\bullet}/\mathcal{C}_{w}^{\bullet}$). The orbit of $\widetilde{\mathrm{Std}}(w)$ in $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$ is the fppf quotient $(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})/\widetilde{E}_{w}$, cf. [DG80, II, § 5, no. 3]. ###### Lemma 2.47. We have commutative diagrams $\overline{\mathcal{G}}_{w}$$\overline{\mathcal{G}}$$\overline{\mathcal{G}}^{\mathrm{rdt}}$$\bar{P}_{J_{1}}$ and $\overline{\mathcal{G}}_{\sigma(w)^{-1}}$$\overline{\mathcal{G}}$$\overline{\mathcal{G}}^{\mathrm{rdt}}$$\bar{P}_{\sigma^{\prime}(J_{1})}$ and $\overline{\mathcal{G}}_{w}^{L}$$\overline{\mathcal{G}}$$\overline{\mathcal{G}}^{\mathrm{rdt}}$$\bar{L}_{J_{1}}$. ###### Proof: This follows from Proposition 2.23. □ ###### Lemma 2.48. The image of $\widetilde{E}_{w}$ under $\overline{\mathcal{G}}\times\overline{\mathcal{G}}\to\overline{\mathcal{G}}^{\mathrm{rdt}}\times\overline{\mathcal{G}}^{\mathrm{rdt}}$ is $E_{\mathcal{Z}_{w}}$. ###### Proof: This follows from Lemma 2.47. □ ###### Lemma 2.49. Assume $0\in J$. The $\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$-orbit of $\widetilde{\mathrm{Std}}(w)$ for $w\in\mathrm{Adm}(\\{\mu\\})^{K}$ depends only on $W_{K}wW_{K}$. ###### Proof: Let $x,y\in W_{K}\subseteq W$. As above we get endomorphisms $x^{\bullet},y^{\bullet}$ of $\mathcal{V}^{\bullet}$, which in this case are in fact automorphisms. Now $\widetilde{\mathrm{Std}}(w)=((y^{\bullet})^{-1},\sigma(x^{\bullet}))\cdot\widetilde{\mathrm{Std}}(w)$. □ ###### Definition 2.50. Define $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}$ to be the union of the $\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$-orbits of the standard zips $\widetilde{\mathrm{Std}}(w)$ for $w\in\mathrm{Adm}(\\{\mu\\})^{K}$. Here an orbit by definition is the image of the orbit map endowed with the reduced subscheme structure, and—as we prove just below—the union of orbits just referred to is a closed subset, which we again endow with the reduced subscheme structure. Define $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}:=[\overline{\mathcal{G}}_{K}\backslash\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}]\subseteq[\overline{\mathcal{G}}_{K}\backslash\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}]=\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$. ###### Lemma 2.51. $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}$ is a closed subset of $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$. ###### Proof: This being a purely topological question, we may freely pass to perfections, which will be convenient since Dieudonné theory is simpler over perfect rings. By “perfection” we mean the inverse perfection in the terminology of [BG18, Section 5]. Consider therefore $(\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim})^{\mathrm{perf}}$ as a sheaf on $\mathrm{Perf}_{\mathbb{F}_{p}}$, the fpqc site of affine perfect $\mathbb{F}_{p}$-schemes. Again denoting the standard Siegel lattice chain over $\mathbb{Z}_{p}$ by $\mathscr{V}^{\bullet}$ and its base change to $\mathbb{F}_{p}$ by $\mathcal{V}^{\bullet}$, we can describe the elements of $(\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim})^{\mathrm{perf}}(R)=\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}(R)$, where $R$ is a perfect $\mathbb{F}_{p}$-algebra as being given by > homomorphisms > $\mathcal{V}_{R}^{\bullet,(p)}\xrightarrow{F^{\bullet}}\mathcal{V}_{R}^{\bullet}\xrightarrow{V^{\bullet}}\mathcal{V}_{R}^{\bullet,(p)}$ > such that > $\ker(F^{\bullet})=:\mathcal{C}^{\bullet,(p)}=\operatorname{im}(V^{\bullet})$ > and > $\operatorname{im}(F^{\bullet})=:\mathcal{D}^{\bullet}=\ker(V^{\bullet})$ > and $\psi_{j}(F^{j}\\_,\\_)=u\sigma(\psi_{j}(\\_,V^{-j}\\_))$ for some $u\in > R^{\times}$ and $\mathcal{C}^{\bullet,(p)},\mathcal{D}^{\bullet}$ have the > same rank (namely $g$). To see that $\mathcal{C}^{\bullet,(p)},\mathcal{D}^{\bullet}$ are direct summands of $\mathcal{V}_{R}^{\bullet,(p)},\mathcal{V}_{R}^{\bullet}$ (which makes the last part of the characterization given above meaningful), one argues as in [Lau14, Lemma 2.4] (since both are finitely presented, it is enough to show flatness and to that end, one looks at the fiber dimensions). Define a presheaf $\mathcal{X}$ on $\mathrm{Sch}_{\mathbb{Z}_{p}}$ in the same way but for the following changes: $\mathcal{V}^{\bullet}$ is replaced by $\mathscr{V}^{\bullet}$, and we impose the condition that both compositions $F^{\bullet}\circ V^{\bullet}$ and $V^{\bullet}\circ F^{\bullet}$ are multiplication by $p$, and the $\ker=\operatorname{im}$-conditions are only required to hold modulo $p$. We also slightly reformulate these $\ker=\operatorname{im}$-conditions: We impose the condition that the reductions $\bar{F}^{\bullet},\bar{V}^{\bullet}$ be fiberwise of rank $g$ over $R/p$. (Note that the argument that $\mathcal{C}^{\bullet,(p)},\mathcal{D}^{\bullet}$ are direct summands only works over reduced rings.) Then $\mathcal{X}$ is a separated $\mathbb{Z}_{p}$-scheme. To see this, we build it up from scratch as follows. $\operatorname{End}(\mathscr{V}^{j})$ obviously is a $\mathbb{Z}_{p}$-scheme (an affine space), hence so is $\operatorname{Hom}(\mathscr{V}^{j,(p)},\mathscr{V}^{j})$ since $\mathscr{V}_{j}^{(p)}\cong\mathscr{V}_{j}$. $\operatorname{Hom}(\mathscr{V}^{\bullet,(p)},\mathscr{V}^{\bullet})$ is a locally closed subscheme of a finite product of such schemes. Homomorphisms $\mathscr{V}^{\bullet,(p)}\xrightarrow{F^{\bullet}}\mathscr{V}^{\bullet}\xrightarrow{V^{\bullet}}\mathscr{V}^{\bullet,(p)}$ such that both compositions are multiplication by $p$ form a closed subscheme $\mathcal{X}^{\prime}$ of $\operatorname{Hom}(\mathscr{V}^{\bullet,(p)},\mathscr{V}^{\bullet})\times\operatorname{Hom}(\mathscr{V}^{\bullet},\mathscr{V}^{\bullet,(p)})$. In the special fiber $\mathcal{X}^{\prime}_{\mathbb{F}_{p}}$ we now consider the $\ker=\operatorname{im}$-conditions and show that they define an open subscheme $\bar{\mathcal{X}}^{\prime\prime}$. Then $\mathcal{X}=\mathcal{X}^{\prime}\times_{\mathcal{X}^{\prime}_{\mathbb{F}_{p}}}\bar{\mathcal{X}}^{\prime\prime}$. Indeed, the extra conditions are that all $F^{\bullet},V^{\bullet}$ have some non-vanishing $g$-minor—evidently open conditions. The upshot is that we defined a $\mathbb{Z}_{p}$-scheme $\mathcal{X}$ such that $(\mathcal{X}\times_{\mathbb{Z}_{p}}\mathbb{F}_{p})^{\mathrm{perf}}=(\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim})^{\mathrm{perf}}$ and such that we have an obvious morphism $\tilde{\mathscr{S}}_{K}\to\mathcal{X}$, which takes a principally polarized isogeny chain of abelian schemes to the evaluation of the Dieudonné crystal on the trivial thickening.141414This makes use of the crystalline-de Rham comparison to make a trivialization of the de Rham cohomology into a trivialization of the crystalline cohomology. Observe that $\mathcal{X}$ also has a natural $\mathcal{G}_{K}\times\mathcal{G}_{K}$-action: We interpret $\mathcal{G}_{K}$ as $\operatorname{Aut}(\mathscr{V}^{\bullet})$ and the action of $(X^{\bullet},Y^{\bullet})$ transforms $(F^{\bullet},V^{\bullet})$ into $(Y^{\bullet}\circ F^{\bullet}\circ X^{\bullet,-(p)},X^{(p)}\circ V^{\bullet}\circ Y^{\bullet,-1})$. The identity $(\mathcal{X}\times_{\mathbb{Z}_{p}}\mathbb{F}_{p})^{\mathrm{perf}}=(\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim})^{\mathrm{perf}}$ is an identity of $\overline{\mathcal{G}}_{K}^{\mathrm{perf}}\times\overline{\mathcal{G}}_{K}^{\mathrm{perf}}$-varieties. Now we claim that $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}=\left(\mathcal{X}_{\mathbb{F}_{p}}\times_{\mathcal{X}}\overline{\mathcal{X}_{\mathbb{Q}_{p}}}\right)^{\mathrm{perf}}$ topologically, where $\overline{\mathcal{X}_{\mathbb{Q}_{p}}}$ is the flat closure of the generic fiber in $\mathcal{X}$. This of course implies $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}\subseteq\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}$ being closed. Both sets are constructible, so it suffices to check it on a very dense subset, say the $\bar{\mathbb{F}}_{p}$-valued points. Using Lemmas 1.25 and 1.27, we see that $(\mathcal{X}_{\mathbb{F}_{p}}\times_{\mathcal{X}}\overline{\mathcal{X}_{\mathbb{Q}_{p}}})(\bar{\mathbb{F}}_{p})$ consists precisely of those elements $\bar{x}\in\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}^{\sim}(\bar{\mathbb{F}}_{p})$ such that there exists a finite field extension $L/\breve{\mathbb{Q}}_{p}$ and a point $x\in\mathcal{X}(\mathcal{O}_{L})$ lifting $\bar{x}$. (We’ll also say that $\bar{x}$ is _liftable_ in this situation.) Since $\mathcal{G}_{K}$ is flat over $\mathbb{Z}_{p}$, this liftability condition for $\mathcal{G}_{K}$ (in lieu of $\mathcal{X}$) is always satisfied. Consequently, $(\mathcal{X}_{\mathbb{F}_{p}}\times_{\mathcal{X}}\overline{\mathcal{X}_{\mathbb{Q}_{p}}})(\bar{\mathbb{F}}_{p})$ is stable under the $\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$-action. Also, the standard zips clearly are liftable. Thus, $(\mathcal{X}_{\mathbb{F}_{p}}\times_{\mathcal{X}}\overline{\mathcal{X}_{\mathbb{Q}_{p}}})(\bar{\mathbb{F}}_{p})\supseteq\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}(\bar{\mathbb{F}}_{p})$. For the converse inclusion, there are injective maps from $\mathcal{X}(\mathcal{O}_{L})$ to $\mathcal{X}(L)$ to $\mathcal{G}_{K}(L)$ such that the corresponding Schubert cell (in the local model) is indexed by the image mod $\mathcal{G}_{K}(\mathcal{O}_{L})\times\mathcal{G}_{K}(\mathcal{O}_{L})^{\mathrm{op}}$, cf. Proposition 2.5.151515Note that $\mathcal{G}_{K}(\mathcal{O}_{L})\backslash\mathcal{G}_{K}(L)/\mathcal{G}_{K}(\mathcal{O}_{L})\cong W_{K}\backslash\widetilde{W}/W_{K}$ for every strictly henselian discretely valued field $L$ by [HR08, Prop. 8]. (And also in the construction of $\widetilde{W}$ and $W_{K}$ any such field, not just $L=\breve{\mathbb{Q}}_{p}$, can be used.) This proves it since we know which Schubert cells belong to the local model. □ ###### Remark 2.52. Regarding the orbit closure relations for $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}$, let us point out that $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}\to\bar{M}^{\mathrm{loc}}_{K}$ is $\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$-equivariant, where the action of $\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$ on $M^{\mathrm{loc}}_{K}$ factors through the first projection map, and this map is a bijection on orbits. Writing $w^{\prime}\preceq w$ if $(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})\cdot\mathrm{Std}(w^{\prime})\subseteq\overline{(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})\cdot\mathrm{Std}(w)}$, it follows from these observations that $w^{\prime}\leq w$ implies $w^{\prime}\preceq w$. Here $\leq$ is the Bruhat order on $W_{K}\backslash\widetilde{W}/W_{K}$ as explained in [PRS13, section 4.2]. It appears reasonable to suspect that $\preceq$ and $\leq$ in fact agree. ###### Conjecture 2.53. The closure of ${(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})\cdot\widetilde{\mathrm{Std}}(w)}$ is given by the disjoint union of ${(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})\cdot\widetilde{\mathrm{Std}}(w^{\prime})}$ for $w^{\prime}\leq w$. ###### Lemma 2.54. The map $\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}$ factors through $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}$. ###### Proof: It is sufficient to check this on $k=\bar{\mathbb{F}}_{p}$-valued points. The map $\overline{\mathscr{S}}_{K}(k)\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}(k)$ factors through $\Upsilon_{K}\colon\overline{\mathscr{S}}_{K}(k)\to\bigcup_{w\in\mathrm{KR}(K,\\{\mu\\})}\breve{K}w\breve{K}/\breve{K}_{\sigma}$ with $\breve{K}w\breve{K}/\breve{K}_{\sigma}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}(k)$ given by sending $xwy$ to $(\bar{y}^{-1},\sigma(\bar{x}))\cdot\mathrm{Std}(w)$ (similar to Lemma 2.49). □ #### 2.2.3 An explicit description of $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ In order to get a better feeling for the passage from $\overline{\mathcal{G}}_{K}$ to the maximal reductive quotient $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}=\overline{\mathcal{G}}_{K}/R_{u}\overline{\mathcal{G}}_{K}$ (with $R_{u}\overline{\mathcal{G}}_{K}$ being the unipotent radical of $\overline{\mathcal{G}}_{K}$), which is key in the definition of the EKOR stratification, we describe $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ in explicit, linear-algebraic terms in the Siegel case. Let $(\mathcal{V}^{\bullet},\mathcal{L},\alpha_{\bullet\bullet},\theta_{\bullet},\psi_{\bullet})$ be the standard Siegel lattice chain on $S$ of type $J$. Assume $0\in J$. In what follows, we sometimes use $j$ as a shorthand for $\mathcal{V}^{j}$. By a _symmetric transition map_ , we mean a transition map from $j^{\prime}$ to $j^{\prime\prime}$, where $n\in\mathbb{Z}$, $j^{\prime},j^{\prime\prime}\in J$, $ng\geq j^{\prime}\geq j^{\prime\prime}>(n-2)g$, and $j^{\prime}+j^{\prime\prime}\in 2g\mathbb{Z}$. We will also call this the symmetric transition map of $(j^{\prime},n)$ (or of $j^{\prime}$ if $n$ doesn’t matter). By a _one-sided transition map_ , we mean a transition map from $j^{\prime}$ to $j^{\prime\prime}$, where $n\in\mathbb{Z}$, $j^{\prime},j^{\prime\prime}\in J$, $ng\geq j^{\prime}\geq j^{\prime\prime}\geq(n-1)g$. Call it right-anchored if $j^{\prime}=ng$ and left-anchored if $j^{\prime\prime}=(n-1)g$. We then also speak of the right-anchored transition map of $j^{\prime\prime}$ and the left-anchored transition map of $j^{\prime}$, respectively. The kernels of the symmetric transition maps are symplectic subbundles of $\mathcal{O}_{S}^{2g}$ (even of the form $\mathcal{O}_{S}^{I}$, where $I\subseteq\\{\pm 1,\dotsc,\pm g\\}$ is symmetric (i.e., $-I=I$)), and the kernels of the one-sided transition maps are totally isotropic subbundles (even of the form $\mathcal{O}_{S}^{I}$, where $I\subseteq\\{1,\dotsc,g\\}$ or $I\subseteq\\{-1,\dotsc,-g\\}$). Let $\mathcal{O}_{S}^{I_{j}}$ be the kernel of the symmetric transition map of $j$. Then $I_{j}\sqcup I_{-j}=\\{\pm 1,\dotsc,\pm g\\}$. Every kernel of a one-sided transition map is a subbundle of a kernel of an anchored transition map inside of which it is complemented by the kernel of another one-sided transition map. The kernel of the left-anchored transition map of $j$ is a subbundle of the kernel of the symmetric transition map of $-j$ inside of which it is complemented by the kernel of the right-anchored transition map of $-j$. Likewise, the kernel of the right-anchored transition map of $j$ is a subbundle of the kernel of the symmetric transition map of $j$ inside of which it is complemented by the kernel of the left-anchored transition map of $-j$. Now consider the standard symplectic bundle $\mathcal{O}_{S}^{2g}$ together with the kernels of all the symmetric transition maps and all the one-sided transition maps. So we have a symplectic bundle with a bunch of symplectic subbundles coming in complementary pairs, some of which come with a further decomposition into complementary Lagrangians, some of which come with further decompositions into complementary subbundles (of course still totally isotropic). We will also call these kernels _distinguished subspaces_. Below we prove that $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ is the automorphism group scheme $\mathcal{A}$ of these data. Clearly, $\mathcal{A}$ is reductive; in fact it is a Levi subgroup of a parabolic of $\operatorname{GSp}_{2g}$. We have a map $\overline{\mathcal{G}}_{K}\to\mathcal{A}$; the image of an $S$-point $f^{\bullet}$ under $\overline{\mathcal{G}}_{K}\to\mathcal{A}$ on the kernel of a transition map starting at $j$ is given by $f^{j}$. Note that $f^{j}=\tau\circ f^{j}$ on $\ker(\tau)$ for every transition map $\tau$ starting at $j$. $\overline{\mathcal{G}}_{K}\to\mathcal{A}$ has a natural section $\mathcal{A}\to\overline{\mathcal{G}}_{K}$, where in the image all the $f^{j}$ are the same as automorphisms of $\mathcal{O}_{S}^{2g}$. (This is well- defined!) ###### Proposition 2.55. $\mathcal{A}=\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$. ###### Proof: Let us show that $\mathcal{K}:=\ker(\overline{\mathcal{G}}_{K}\to\mathcal{A})$ is unipotent. Consider $\overline{\mathcal{G}}_{K}$ as a subgroup of $\prod_{j\in J/2g\mathbb{Z}}\operatorname{GL}_{2g}\subseteq\operatorname{GL}_{N}$. We claim that said kernel is contained in $\prod_{j\in J/2g\mathbb{Z}}U^{(j)}$, $U^{(j)}$ being a conjugate of the standard unipotent subgroup $\left(\begin{smallmatrix}1&\ast&\ast&\dotsb&\ast\\\ &1&\ast&\dotsb&\ast\\\ &&\ddots&\dotsb&\vdots\end{smallmatrix}\right)$ of $\operatorname{GL}_{2g}$. Indeed, say $f^{\bullet}$ is in the kernel. Then $f^{j}$ acts as the identity on the kernel of the symmetric transition map of $j$ and $f^{-j}$ acts as the identity on the kernel of the symmetric transition map of $-j$. On the image of the symmetric transition map $\tau_{j}$ of $j$, $f^{-j}$ agrees with $\tau_{j}\circ f^{j}$. Note that $\operatorname{im}(\tau_{j})=\ker(\tau_{-j})$. So $\tau_{j}\circ f^{j}$ is the identity on $\ker(\tau_{-j})$. Hence, if $x\in\ker(\tau_{-j})$, then $x=\tau_{j}(x)$ and $f^{j}(x)\equiv x\mod\ker(\tau_{j})$. Thus with respect to the decomposition $\ker(\tau_{j})\oplus\ker(\tau_{-j})$, $f^{j}$ is of the form $\begin{pmatrix}1&\ast\\\ &1\end{pmatrix}$. Now we have $\overline{\mathcal{G}}_{K}=\mathcal{A}\ltimes\mathcal{K}$, in particular $\overline{\mathcal{G}}_{K}\cong\mathcal{A}\times_{\mathbb{F}_{p}}\mathcal{K}$ as schemes. Since both $\overline{\mathcal{G}}_{K}$ and $\mathcal{A}$ are reduced and connected, so is $\mathcal{K}$. All in all, we see that $\mathcal{A}$ is indeed $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ and $\mathcal{K}=R_{u}\overline{\mathcal{G}}_{K}$ is the unipotent radical of $\overline{\mathcal{G}}_{K}$. □ ###### Example 2.56. * • If $J=\mathbb{Z}$, then $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}=\mathbb{G}_{m}^{g+1}$ is the standard maximal torus of $\operatorname{GSp}_{2g}$. * • If $g=2$ and $J=2\mathbb{Z}$, then $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ is the automorphism group of the standard twisted symplectic space $\mathbb{F}_{p}^{4}$ with its standard Lagrangian decomposition, i.e., $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\cong\operatorname{GL}_{2}\times\mathbb{G}_{m}$. * • If $g=2$ and $J/2g\mathbb{Z}=\\{-1,0,1\\}$, then $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ is the automorphism group of the standard twisted symplectic space $\mathbb{F}_{p}^{4}$ with its standard decomposition in twisted symplectic subspaces and the totally isotropic rank-$1$ subbundles generated by $e_{\pm 1}$, i.e., $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\cong\operatorname{GL}_{2}\times\mathbb{G}_{m}$. * • Let $g=8$. We have the local Dynkin diagram where we labelled the simple affine roots as follows: $1-2e_{-1}+e_{0}$ is labelled 0, $e_{-i}-e_{-(i+1)}$ is labelled $i$ for $1\leq i\leq 7$, and $2e_{-8}-e_{0}$ is labelled 8. Consider $J/2g\mathbb{Z}=\\{0,\pm 3,\pm 5\\}$. Then the Dynkin diagram of $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ should (according to [Tit79, 3.5.1]) be the one we get by removing $0,3,5$ and the adjacent edges. So we expect something along the lines161616i.e., having the same Dynkin diagram as of $\operatorname{GSp}(6)\times\operatorname{GL}(2)\times\operatorname{GL}(3)$. We have the following (bases of) kernels of symmetric transition maps: $\\{\pm 1,\pm 2,\pm 3\\},\\{\pm 4,\pm 5,\pm 6,\pm 7,\pm 8\\},\\{\pm 1,\pm 2,\pm 3,\pm 4,\pm 5\\},\\{\pm 6,\pm 7,\pm 8\\},$ and the following kernels of one-sided transition maps: $\displaystyle\\{-3,-2,-1\\},\\{-5,-4\\},\\{-5,-4,-3,-2,-1\\},$ $\displaystyle\\{4,5\\},\\{1,2,3,4,5\\},\\{1,2,3\\}.$ So an element $A$ of $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ is given by specifying linear automorphisms $A_{123}$ of $\langle 1,2,3\rangle$ and $A_{45}$ of $\langle 4,5\rangle$ and a symplectic similitude $A_{\pm 6,\pm 7,\pm 8}$ of $\langle\pm 6,\pm 7,\pm 8\rangle$, such that $\left.A\right|_{\langle 1,2,3\rangle}=A_{123}$, $\left.A\right|_{\langle 4,5\rangle}=A_{45}$, $\left.A\right|_{\langle\pm 6,\pm 7,\pm 8\rangle}=A_{\pm 6,\pm 7,\pm 8}$, where $\left.A\right|_{\langle-1,-2,-3\rangle}$ is uniquely determined by $A_{123}$, $c(A_{\pm 6,\pm 7,\pm 8})$ ($c$ being the multiplier character) and the imposition that $A$ be a symplectic similitude, and similarly for $\left.A\right|_{\langle-4,-5\rangle}$. If for example we consider $J/2g\mathbb{Z}=\\{0,\pm 2,\pm 3,\pm 5\\}$ instead, we expect something along the lines of $\operatorname{GSp}(6)\times\operatorname{GL}(2)\times\operatorname{GL}(2)$ and indeed we additionally get the subbundles $\displaystyle\\{-2,-1\\},\\{1,2\\},\\{-3\\},\\{-5,-4,-3\\},$ $\displaystyle\\{3,4,5\\},\\{3\\},\\{3,4,5,6,7,8,-8,-7,-6,-5,-4,-3\\},\\{1,2,-2,-1\\}.$ So an element $A$ of $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}$ is given by specifying linear automorphisms $A_{12}$ of $\langle 1,2\rangle$ and $A_{45}$ of $\langle 4,5\rangle$ and a symplectic similitude $A_{\pm 6,\pm 7,\pm 8}$ of $\langle\pm 6,\pm 7,\pm 8\rangle$ in a similar way to above. #### 2.2.4 $\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ in the Siegel case Recall that we denote the unipotent radical of $\overline{\mathcal{G}}_{K}$ by $R_{u}\overline{\mathcal{G}}_{K}$. We divide out of $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}$ the action of the smooth normal subgroup $R_{u}\overline{\mathcal{G}}_{K}\times R_{u}\overline{\mathcal{G}}_{K}\subseteq\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$ and observe that $\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$ still acts on $[R_{u}\overline{\mathcal{G}}_{K}\times R_{u}\overline{\mathcal{G}}_{K}\backslash\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}]=:\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}^{\sim}$ (not a scheme). We also define $\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}:=[(\Delta({\overline{\mathcal{G}}_{K}})\cdot(R_{u}\overline{\mathcal{G}}_{K}\times R_{u}\overline{\mathcal{G}}_{K}))\backslash\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}]$. ###### Proposition 2.57. We have well-defined morphisms $\displaystyle(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})/\widetilde{E}_{w}$ $\displaystyle\to(\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\times\overline{\mathcal{G}}_{K}^{\mathrm{rdt}})/E_{\mathcal{Z}_{w}},$ $\displaystyle\quad(X,Y)$ $\displaystyle\mapsto(X^{\mathrm{rdt}},Y^{\mathrm{rdt}}),$ $\displaystyle\overline{\mathcal{G}}_{K}/\widetilde{E}_{w}$ $\displaystyle\to\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}/E_{\mathcal{Z}_{w}},$ $\displaystyle\quad X$ $\displaystyle\mapsto X^{\mathrm{rdt}},$ and a bijectiion $(\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K})/(\widetilde{E}_{w}\cdot(R_{u}\overline{\mathcal{G}}_{K}\times R_{u}\overline{\mathcal{G}}_{K}))\to(\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\times\overline{\mathcal{G}}_{K}^{\mathrm{rdt}})/E_{\mathcal{Z}_{w}}.$ ###### Proof: The first assertion follows from the definition of $E_{\mathcal{Z}_{w}}$ and equation (2.46). The second then follows from Lemma 2.48. □ ###### Lemma 2.58. Assume $0\in J$. The underlying topological spaces of the stacks in consideration are as follows: 1. (1) $|[\overline{\mathcal{G}}_{K}\backslash\overline{M}^{\mathrm{loc}}]|=\mathrm{KR}(K,\\{\mu\\})\overset{\text{def.}}{=}W_{K}\backslash(W_{K}\mathrm{Adm}(\\{\mu\\})W_{K})/W_{K}$. 2. (2) $|\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}|=\mathrm{EKOR}(K,\\{\mu\\})=\mathrm{Adm}(\\{\mu\\})^{K}\cap{}^{K}\widetilde{W}$ $\overset{\text{\ref{iw- diagr}}}{\cong}\bigcup_{w\in\mathrm{KR}(K,\\{\mu\\})}\breve{K}w\breve{K}/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1})$. ###### Proof: (1) is well-known as explained in Section 2.1.2. (2): By Lemma 2.49, the $\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$-orbits in $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}$ are indexed by $\mathrm{Adm}(\\{\mu\\})_{K}=\mathrm{KR}(K,\\{\mu\\})$. Let us further investigate the $\overline{\mathcal{G}}_{K}\times\overline{\mathcal{G}}_{K}$-orbit of $\widetilde{\mathrm{Std}}(w)$ in $\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}^{\sim}$ for some fixed $w\in\mathrm{Adm}(\\{\mu\\})^{K}$. By Proposition 2.57, its underlying topological space agrees with that of $\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\sim,\mathcal{Z}_{w}}$. By [SYZ19] we know that $|\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\mathcal{Z}_{w}}|\cong\breve{K}w\breve{K}/\breve{K}_{\sigma}(\breve{K}_{1}\times\breve{K}_{1})$, whence the lemma. □ ###### Corollary 2.59. We have a morphism $\left(\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}_{w}\right)_{\mathrm{red}}=\text{orbit of }\mathrm{Std}(w)\to\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\mathcal{Z}_{w}}.$ This defines the EKOR stratification on $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}_{w}$. All in all, we get an EKOR stratification on $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}$. The morphism factors through $(\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}_{w})_{\mathrm{red}}$, and $(\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}_{w})_{\mathrm{red}}\to\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\mathcal{Z}_{w}}$ is an isomorphism. ###### Corollary 2.60. For every point of $[\overline{\mathcal{G}}_{K}\backslash\overline{M}_{K}]$, $\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ is smooth as a map between the associated reduced fiber of $\overline{\mathscr{S}}_{K}\to[\overline{\mathcal{G}}_{K}\backslash\overline{M}_{K}]$ and the associated reduced fiber of $\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}\to[\overline{\mathcal{G}}_{K}\backslash\overline{M}_{K}]$. ###### Proof: This follows from the preceding corollary by [SYZ19, Theorem A] (which says that the map $\overline{\mathscr{S}}_{K}^{w}\to\overline{\mathcal{G}}_{K}^{\mathrm{rdt}}\text{-}\mathrm{Zip}^{\mathcal{Z}_{w}}$ is smooth, cf. subsection 2.1.4). □ The key obstacle in going forward toward proving smoothness of $\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ now is that we do not know whether the fibers of $\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}\to[\overline{\mathcal{G}}_{K}\backslash\overline{M}_{K}]$ are reduced. ###### Conjecture 2.61. We conjecture that the answer is affirmative. In fact, we conjecture that $\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}\to[\overline{\mathcal{G}}_{K}\backslash\overline{M}_{K}]$ is smooth. ###### Corollary 2.62. $\overline{\mathscr{S}}_{K}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$ is surjective. ###### Proof: This follows from the description of the topological space and what is already known from [HR17, first paragraph of section 6.3]. □ We get a commutative diagram $\tilde{\overline{\mathscr{S}}}_{K}$$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}$$\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}^{\sim}$$\overline{\mathscr{S}}_{K}$$\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}$$\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}$$[\overline{\mathcal{G}}_{K}\backslash\overline{M}^{\mathrm{loc}}_{K}]$ ###### Remark 2.63. Since $R_{u}\overline{\mathcal{G}}_{K}$ is smooth, $\overline{\mathcal{G}}_{K}\text{-}\mathrm{AdmZip}^{\sim}\to\overline{\mathcal{G}}_{K}\text{-}\mathrm{EKORZip}^{\sim}$ is smooth. ###### Remark 2.64. Another open question at this point is: what is the relationship between $\overline{\mathcal{G}}_{K}\text{-EKORZip}^{\mathrm{perf}}$ and the shtuka approach of [SYZ19, Section 4]? ###### Remark 2.65. It should be straightforward to generalize (taking into account the extra structure) our constructions to those (P)EL cases where the local model is the “naive” local model of Rapoport-Zink [RZ96]. #### 2.2.5 The example of $\operatorname{GSp}(4)$ To illustrate some aspects, we look at the example $2g=4$. ##### The apartment. We describe the (extended) apartment. We follow the general outline of [Lan00], in particular as far as notation is concerned. The roots are $\pm(2e_{1}-e_{0}),\pm(2e_{2}-e_{0}),\pm(e_{1}-e_{2}),\pm(e_{1}+e_{2}-e_{0})$. The simple affine roots and the (various variants of the) Weyl group are as described in Remark 2.26. The root one-parameter subgroups171717The parameter being additive here; i.e., we’re talking about homomorphisms $\mathbb{G}_{a}\to G$. are given as follows: $\displaystyle u_{e_{1}-e_{2}}(x)$ $\displaystyle=\begin{pmatrix}1&x&&\\\ &1&&\\\ &&1&-x\\\ &&&1\end{pmatrix},$ $\displaystyle u_{e_{2}-e_{1}}(x)$ $\displaystyle=\begin{pmatrix}1&&&\\\ x&1&&\\\ &&1&\\\ &&-x&1\end{pmatrix},$ $\displaystyle u_{2e_{1}-e_{0}}(x)$ $\displaystyle=\begin{pmatrix}1&&&x\\\ &1&&\\\ &&1&\\\ &&&1\end{pmatrix},$ $\displaystyle u_{e_{0}-2e_{1}}(x)$ $\displaystyle=\begin{pmatrix}1&&&\\\ &1&&\\\ &&1&\\\ x&&&1\end{pmatrix},$ $\displaystyle u_{2e_{2}-e_{0}}(x)$ $\displaystyle=\begin{pmatrix}1&&&\\\ &1&x&\\\ &&1&\\\ &&&1\end{pmatrix},$ $\displaystyle u_{e_{0}-2e_{2}}(x)$ $\displaystyle=\begin{pmatrix}1&&&\\\ &1&&\\\ &x&1&\\\ &&&1\end{pmatrix},$ $\displaystyle u_{e_{1}+e_{2}-e_{0}}(x)$ $\displaystyle=\begin{pmatrix}1&&x&\\\ &1&&x\\\ &&1&\\\ &&&1\end{pmatrix},$ $\displaystyle u_{e_{0}-e_{1}-e_{2}}(x)$ $\displaystyle=\begin{pmatrix}1&&&\\\ &1&&\\\ x&&1&\\\ &x&&1\end{pmatrix}$ For $a\in R$ define $w_{a}(x):=u_{a}(x)u_{-a}(-x^{-1})u_{a}(x)$. ###### Remark 2.66. $N(\mathbb{Q}_{p})$ is generated by $T(\mathbb{Q}_{p})$ and all $w_{a}(x)$ as above. ###### Remark 2.67. $w_{a}(x)=m(u_{-a}(-x^{-1}))$ in Landvogt’s notation [Lan00]. We have $V_{1}:=X_{*}(T)\otimes\mathbb{R}=\\{(x_{1},x_{2},x_{-2},x_{-1})\in\mathbb{R}^{4}\;|\;x_{1}+x_{-1}=x_{2}+x_{-2}\\}$ and $\nu_{1}\colon T(\mathbb{Q}_{p})\to V_{1},\;\begin{pmatrix}d_{1}&&&\\\ &d_{2}&&\\\ &&cd_{2}^{-1}&\\\ &&&cd_{1}^{-1}\end{pmatrix}\mapsto\begin{pmatrix}-v_{p}(d_{1})\\\ -v_{p}(d_{2})\\\ -v_{p}(cd_{2}^{-1})\\\ -v_{p}(cd_{1}^{-1})\end{pmatrix}.$ Also, $V_{0}=\\{v\in V_{1}\;|\;a(v)=0\;\forall a\in\Phi\\}=\mathbb{R}(1,1,1,1)$, $V:=V_{1}/V_{0}$. The extended apartment $A=A^{\mathrm{ext}}$ now is an affine $V_{1}$-space together with the map $\nu_{1}\colon N(\mathbb{Q}_{p})\to\operatorname{Aff}(A)=\operatorname{GL}(V_{1})\ltimes V_{1}$, whose restriction to $T(\mathbb{Q}_{p})$ is given as above and (cf. Remark 2.66) $\displaystyle\nu_{1}(w_{2e_{1}-e_{0}}(x))$ $\displaystyle=(\left(\begin{smallmatrix}&&&1\\\ &1&&\\\ &&1&\\\ 1&&&\end{smallmatrix}\right),\left(\begin{smallmatrix}-v_{p}(x)\\\ 0\\\ 0\\\ v_{p}(x)\end{smallmatrix}\right)),$ $\displaystyle\nu_{1}(w_{2e_{2}-e_{0}}(x))$ $\displaystyle=(\left(\begin{smallmatrix}1&&&\\\ &&1&\\\ &1&&\\\ &&&1\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\ -v_{p}(x)\\\ v_{p}(x)\\\ 0\end{smallmatrix}\right)),$ $\displaystyle\nu_{1}(w_{e_{1}-e_{2}}(x))$ $\displaystyle=(\left(\begin{smallmatrix}&1&&\\\ 1&&&\\\ &&&1\\\ &&1&\end{smallmatrix}\right),\left(\begin{smallmatrix}-v_{p}(x)\\\ v_{p}(x)\\\ -v_{p}(x)\\\ v_{p}(x)\end{smallmatrix}\right)),$ $\displaystyle\nu_{1}(w_{e_{1}+e_{2}-e_{0}}(x))$ $\displaystyle=(\left(\begin{smallmatrix}&&1&\\\ &&&1\\\ 1&&&\\\ &1&&\end{smallmatrix}\right),\left(\begin{smallmatrix}-v_{p}(x)\\\ -v_{p}(x)\\\ v_{p}(x)\\\ v_{p}(x)\end{smallmatrix}\right)),$ etc. (Recipe: Write $w_{a}(x)$ as a product of a diagonal matrix $\operatorname{diag}(d_{1},d_{2},d_{-2},d_{-1})$ and a permutation matrix $P$ (this need not be a factorization in $\operatorname{GSp}(4)$); then $\nu_{1}(w_{a}(x))=(P,\left(\begin{smallmatrix}-v_{p}(d_{1})\\\ -v_{p}(d_{2})\\\ -v_{p}(d_{-2})\\\ -v_{p}(d_{-1})\end{smallmatrix}\right)).)$ The reduced apartment $A^{\mathrm{red}}$ is the affine $V$-space together with $\nu\colon N(\mathbb{Q}_{p})\to\operatorname{Aff}(A^{\mathrm{red}})=\operatorname{GL}(V)\ltimes V$ given by the same formulas. The walls (or rather, wall conditions) are given as follows ($n\in\mathbb{Z}$): $\displaystyle 2e_{1}-e_{0}$ $\displaystyle:n=x_{0}-2x_{1},$ $\displaystyle 2e_{2}-e_{0}$ $\displaystyle:n=x_{0}-2x_{2},$ $\displaystyle e_{1}-e_{2}$ $\displaystyle:n=x_{2}-x_{1},$ $\displaystyle e_{1}+e_{2}-e_{0}$ $\displaystyle:n=x_{0}-x_{1}-x_{2}.$ Figure 1: The reduced apartment with the base alcove highlighted. ##### Lattice chains and parahoric subgroups. By [BT84a], the extended building $\mathcal{B}(\operatorname{GL}(X),\mathbb{Q}_{p})$ is in bijection with norms181818Defining conditions for a norm: $\alpha(tx)=\alpha(x)+\operatorname{ord}_{p}(t)$, $\alpha(x+y)\geq\min(\alpha(x),\alpha(y))$, $\alpha(x)=\infty\iff x=0$ $\alpha\colon X\to\mathbb{R}\cup\\{\infty\\}$. Norms in turn are in bijection with graded lattice chains (cf. Remark 1.10). Indeed, if $\alpha$ is a norm, define $\Delta_{\alpha}$ to be the set of its balls centered around zero and $c_{\alpha}(\Lambda):=\inf_{\lambda\in\Lambda}\alpha(\lambda)$. Conversely, given a graded lattice chain $(\Delta,c)$, define a norm $\alpha$ by $\alpha(x):=c(\Lambda)$ for the smallest $\Lambda\in\Delta$ with $x\in\Lambda$. To go from the extended apartment of $\operatorname{GL}(X)$, an affine $\mathbb{R}^{n}$-space, where $n=\dim X$, to norms, fix a basis $e_{1},\dotsc,e_{n}$ of $X$. Then $v\in\mathbb{R}^{n}$ corresponds to the norm $\alpha_{v}$ with $\alpha_{v}(\sum t_{i}e_{i})=\min_{i}(\operatorname{ord}_{p}(t_{i})-v_{i}).$ There are seven types of points in the extended apartment (in each case we choose one in the base alcove to represent all of its type) corresponding to the vertices, edges and interior of the base alcove: * • standard hyperspecial: $x_{\mathrm{hs}}=(0,0,0,0)$ * • paramodular: $x_{\mathrm{paramod}}=(-1/2,0,0,1/2)$ * • Klingen: $x_{\mathrm{Klingen}}=(-1/4,0,0,1/4)$ * • Siegel: $x_{\mathrm{Siegel}}=(-1/4,-1/4,1/4,1/4)$ * • Iwahori: $x_{\mathrm{Iwahori}}=(-1/4,-1/8,1/8,1/4)$ * • another hyperspecial: $x=(-1/2,-1/2,1/2,1/2)$ * • another parahoric: $x=(-1/2,-1/4,1/4,1/2)$ The last two are conjugates (by the Atkin-Lehner element) of the standard hyperspecial and the Klingen parahoric, respectively (see e.g. [Rös18, 151]); therefore we will neglect them in the sequel. For a set of lattices $S$ denote by $\langle S\rangle$ the closure under homotheties, i.e., $\langle S\rangle:=\\{p^{n}s\;|\;n\in\mathbb{Z},\;s\in S\\}$. Then: * • $\Delta_{\mathrm{hs}}=\langle\mathbb{Z}_{p}^{4}\rangle$ and $c_{\mathrm{hs}}(\mathbb{Z}_{p}^{4})=0$. * • $\Delta_{\mathrm{paramod}}=\langle\mathbb{Z}_{p}^{3}\oplus p\mathbb{Z}_{p},\;\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3}\rangle$ and $c_{\mathrm{paramod}}(\mathbb{Z}_{p}^{3}\oplus p\mathbb{Z}_{p})=-\frac{1}{2}$, $c_{\mathrm{paramod}}(\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3})=0$. * • $\Delta_{\mathrm{Klingen}}=\langle\mathbb{Z}_{p}^{4},\mathbb{Z}_{p}^{3}\oplus p\mathbb{Z}_{p},\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3}\rangle$ and $c_{\mathrm{Klingen}}(\mathbb{Z}_{p}^{4})=-1/4$, $c_{\mathrm{Klingen}}(\mathbb{Z}_{p}^{3}\oplus p\mathbb{Z}_{p})=0$, $c_{\mathrm{Klingen}}(\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3})=1/4$. * • $\Delta_{\mathrm{Siegel}}=\langle\mathbb{Z}_{p}^{4},\mathbb{Z}_{p}^{2}\oplus p\mathbb{Z}_{p}^{2}\rangle$ and $c_{\mathrm{Siegel}}(\mathbb{Z}_{p}^{4})=-1/4$, $c_{\mathrm{Siegel}}(\mathbb{Z}_{p}^{2}\oplus p\mathbb{Z}_{p}^{2})=1/4$. * • $\Delta_{\mathrm{Iwahori}}=\langle\mathbb{Z}_{p}^{4},\mathbb{Z}_{p}^{3}\oplus p\mathbb{Z}_{p},\mathbb{Z}_{p}^{2}\oplus p\mathbb{Z}_{p}^{2},\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3}\rangle$ and $c_{\mathrm{Iwahori}}(\mathbb{Z}_{p}^{4})=-1/4$, $c_{\mathrm{Iwahori}}(\mathbb{Z}_{p}^{3}\oplus p\mathbb{Z}_{p})=-1/8$, $c_{\mathrm{Iwahori}}(\mathbb{Z}_{p}^{2}\oplus p\mathbb{Z}_{p}^{2})=1/8$, $c_{\mathrm{Iwahori}}(\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3})=1/4$. The associated parahoric subgroups are * • hyperspecial: $\operatorname{GSp}_{4}(\mathbb{Z}_{p})$ * • paramodular: $\operatorname{GSp}_{4}(\mathbb{Q}_{p})\cap\begin{pmatrix}\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&p^{-1}\mathbb{Z}_{p}\\\ p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\ p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\ p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&\mathbb{Z}_{p}\end{pmatrix}$ * • Klingen: $\operatorname{GSp}_{4}(\mathbb{Z}_{p})\cap\begin{pmatrix}\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\ p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\ p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\ p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&\mathbb{Z}_{p}\end{pmatrix}$ * • Siegel: $\operatorname{GSp}_{4}(\mathbb{Z}_{p})\cap\begin{pmatrix}\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\ \mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\ p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\ p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\end{pmatrix}$ * • Iwahori: $\operatorname{GSp}_{4}(\mathbb{Z}_{p})\cap\begin{pmatrix}\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\ p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\ p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\ p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&p\mathbb{Z}_{p}&\mathbb{Z}_{p}\end{pmatrix}$ ###### Remark 2.68. Dualizing with respect to the symplectic form, we have $\displaystyle(\mathbb{Z}_{p}^{4})^{\vee}$ $\displaystyle=\mathbb{Z}_{p}^{4},$ $\displaystyle(\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3})^{\vee}$ $\displaystyle=p^{-1}(\mathbb{Z}_{p}^{3}\oplus p\mathbb{Z}_{p}),$ $\displaystyle(\mathbb{Z}_{p}^{3}\oplus p\mathbb{Z}_{p})^{\vee}$ $\displaystyle=p^{-1}(\mathbb{Z}_{p}\oplus p\mathbb{Z}_{p}^{3}),$ $\displaystyle(\mathbb{Z}_{p}^{2}\oplus p\mathbb{Z}_{p}^{2})^{\vee}$ $\displaystyle=p^{-1}(\mathbb{Z}_{p}^{2}\oplus p\mathbb{Z}_{p}^{2}).$ ##### Admissible set. We compute the admissible set in the way outlined in Remark 2.29. The cocharacter $\mu$ is $(1,1,0,0)$. We obtain $\displaystyle\mathrm{Adm}(\\{\mu\\})=\bigl{\\{}$ $\displaystyle\Bigl{(}\operatorname{id},\left(\begin{smallmatrix}1\\\ 1\\\ 0\\\ 0\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}(2\quad{-2}),\left(\begin{smallmatrix}1\\\ 0\\\ 1\\\ 0\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}\operatorname{id},\left(\begin{smallmatrix}1\\\ 0\\\ 1\\\ 0\end{smallmatrix}\right)\Bigr{)},$ $\displaystyle\Bigl{(}(1\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}(1\quad 2\quad{-1}\quad{-2}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right)\Bigr{)},$ $\displaystyle\Bigl{(}(1\quad 2)({-2}\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}\operatorname{id},\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right)\Bigr{)},$ $\displaystyle\Bigl{(}(1\quad{-1})(2\quad{-2}),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}(1\quad{-1}),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)\Bigr{)},$ $\displaystyle\Bigl{(}(1\quad{-2})(2\quad{-1}),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}(1\quad{-2}\quad{-1}\quad 2),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)\Bigr{)},$ $\displaystyle\Bigl{(}(2\quad{-2}),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)\Bigr{)},\quad\Bigl{(}\operatorname{id},\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)\Bigr{)}\bigr{\\}},$ or, in terms of Frobenii (cf. Construction 2.43) $\displaystyle\Bigl{\\{}$ $\displaystyle\left(\begin{smallmatrix}p&0&0&0\\\ 0&p&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{smallmatrix}\right),\left(\begin{smallmatrix}p&0&0&0\\\ 0&0&1&0\\\ 0&p&0&0\\\ 0&0&0&1\end{smallmatrix}\right),\left(\begin{smallmatrix}p&0&0&0\\\ 0&1&0&0\\\ 0&0&p&0\\\ 0&0&0&1\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&0&1\\\ 0&p&0&0\\\ 0&0&1&0\\\ p&0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&1&0\\\ p&0&0&0\\\ 0&0&0&1\\\ 0&p&0&0\end{smallmatrix}\right),$ $\displaystyle\left(\begin{smallmatrix}0&1&0&0\\\ p&0&0&0\\\ 0&0&0&1\\\ 0&0&p&0\end{smallmatrix}\right),\left(\begin{smallmatrix}1&0&0&0\\\ 0&p&0&0\\\ 0&0&1&0\\\ 0&0&0&p\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&0&1\\\ 0&0&1&0\\\ 0&p&0&0\\\ p&0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&0&1\\\ 0&1&0&0\\\ 0&0&p&0\\\ p&0&0&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&0&1&0\\\ 0&0&0&1\\\ p&0&0&0\\\ 0&p&0&0\end{smallmatrix}\right),$ $\displaystyle\left(\begin{smallmatrix}0&1&0&0\\\ 0&0&0&1\\\ p&0&0&0\\\ 0&0&p&0\end{smallmatrix}\right),\left(\begin{smallmatrix}1&0&0&0\\\ 0&0&1&0\\\ 0&p&0&0\\\ 0&0&0&p\end{smallmatrix}\right),\left(\begin{smallmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&p&0\\\ 0&0&0&p\end{smallmatrix}\right)\Bigr{\\}}.$ ##### Siegel level. From now on, we consider the Siegel level structure. Denote the Siegel parahoric by $K$ and the standard hyperspecial subgroup by $H$. Here $W_{K}$ is generated by $({-1}\quad{-2})(1\quad 2)$, while $W_{H}$ is generated by $W_{K}$ and $(2\quad{-2})$. Recalling Remark 2.31 (2), we note that one has a natural morphism $\overline{\mathcal{G}}_{K}\text{-}\mathrm{Zip}\to\overline{\mathcal{G}}_{H}\text{-}\mathrm{Zip}\times\overline{\mathcal{G}}_{H}\text{-}\mathrm{Zip}$. We have $\displaystyle\mathrm{KR}(K,\\{\mu\\})$ $\displaystyle=\Bigl{\\{}(\operatorname{id},\left(\begin{smallmatrix}1\\\ 1\\\ 0\\\ 0\end{smallmatrix}\right)),((1\quad{-2})(2\quad{-1}),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)),((1\quad 2\quad{-1}\quad{-2}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right)),$ $\displaystyle(\operatorname{id},\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)),((1\quad{-2}\quad{-1}\quad 2),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)),((1\quad 2)({-2}\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))\Bigr{\\}},$ $\displaystyle\mathrm{EKOR}(K,\\{\mu\\})$ $\displaystyle=\mathrm{KR}(K,\\{\mu\\})\cup\Bigl{\\{}(\operatorname{id},\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right)),\quad((2\quad{-2}),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right)),\quad((1\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))\Bigr{\\}}.$ In the following table, $w^{j}$ is the isomorphism type of the $\overline{\mathcal{G}}_{H}$-zip at position $j$. For $\mathcal{C}^{\bullet},\mathcal{D}^{\bullet}$ we give (indices of) basis vectors. “$\leftarrow$” means “same as in the column adjacent to the left”. $\alpha_{0}\colon\bar{\mathbb{F}}_{p}^{4}\to\bar{\mathbb{F}}_{p}^{4}$ is the projection onto the plane spanned by the $1,2$-coordinates, $\alpha_{2}$ the projection onto the plane spanned by the $-2,-1$-coordinates. By $\alpha_{j,\mathcal{C}^{\bullet}/\mathcal{D}^{\bullet}}$ we denote the induced maps on $\mathcal{V}^{\bullet}/\mathcal{C}^{\bullet}\oplus\mathcal{C}^{\bullet}$ and $\mathcal{D}^{\bullet}\oplus\mathcal{V}^{\bullet}/\mathcal{D}^{\bullet}$, respectively. Each $\mathcal{C}^{j}\subseteq\mathcal{V}^{j}$ has a canonical complement in terms of standard basis vectors. Importantly however, we will not always have a complementary _chain_ of linear subspaces. In any event, below we say what the $\alpha_{j,\mathcal{C}^{\bullet}/\mathcal{D}^{\bullet}}$ are the projection onto if interpreted as described. For instance, the projection onto $\emptyset$ is the zero map. So in that case $\mathcal{V}^{\bullet}/\mathcal{C}^{\bullet}\oplus\mathcal{C}^{\bullet}$ (or $\mathcal{D}^{\bullet}\oplus\mathcal{V}^{\bullet}/\mathcal{D}^{\bullet}$) is a chain of vector spaces with zero transition maps. $w$ | KR-type | $\mathcal{C}^{0}$ | $\mathcal{D}^{0}$ | $\mathcal{C}^{2}$ | $\mathcal{D}^{2}$ | $w^{0}$ | $w^{2}$ | $\alpha_{2,\mathcal{C}^{\bullet}}$ | $\alpha_{0,\mathcal{C}^{\bullet}}$ | $\alpha_{2,\mathcal{D}^{\bullet}}$ | $\alpha_{0,\mathcal{D}^{\bullet}}$ ---|---|---|---|---|---|---|---|---|---|---|--- $(\operatorname{id},\left(\begin{smallmatrix}1\\\ 1\\\ 0\\\ 0\end{smallmatrix}\right))$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\\{-2,-1\\}$ | $(-2\quad 1)({-1}\quad 2)$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ $((1\quad{-2})(2\quad{-1}),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right))$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\operatorname{id}$ | $\leftarrow$ | $\emptyset$ | $\emptyset$ | $\emptyset$ | $\emptyset$ $((1\quad 2\quad{-1}\quad{-2}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{-2,1\\}$ | $\\{-2,1\\}$ | $\\{-2,-1\\}$ | $(-2\quad 2)$ | $\leftarrow$ | $\\{1\\}$ | $\\{-1\\}$ | $\\{2\\}$ | $\\{-2\\}$ $(\operatorname{id},\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right))$ | $\leftarrow$ | $\\{-2,-1\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{1,2\\}$ | $(-2\quad 1)({-1}\quad 2)$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ $((1\quad{-2}\quad{-1}\quad 2),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right))$ | $\leftarrow$ | $\\{-2,1\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,1\\}$ | $(-2\quad 2)$ | $\leftarrow$ | $\\{2\\}$ | $\\{-2\\}$ | $\\{1\\}$ | $\\{-1\\}$ $((1\quad 2)({-2}\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))$ | $\leftarrow$ | $\\{-2,1\\}$ | $\\{-2,1\\}$ | $\\{-2,1\\}$ | $\\{-2,1\\}$ | $\operatorname{id}$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ $(\operatorname{id},\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))$ | $((1\quad 2)({-2}\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))$ | $\\{-1,2\\}$ | $\\{-1,2\\}$ | $\\{-2,1\\}$ | $\\{-2,1\\}$ | $(-2\quad 1)({-1}\quad 2)$ | $\leftarrow$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,-1\\}$ $((2\quad{-2}),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right))$ | $((1\quad{-2}\quad{-1}\quad 2),\left(\begin{smallmatrix}0\\\ 0\\\ 1\\\ 1\end{smallmatrix}\right))$ | $\\{-1,2\\}$ | $\\{-2,-1\\}$ | $\\{1,2\\}$ | $\\{-2,1\\}$ | $(-2\quad{-1}\quad 2\quad 1)$ | $\leftarrow$ | $\\{1\\}$ | $\\{-1\\}$ | $\\{1\\}$ | $\\{-1\\}$ $((1\quad{-1}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))$ | $((1\quad 2\quad{-1}\quad{-2}),\left(\begin{smallmatrix}0\\\ 1\\\ 0\\\ 1\end{smallmatrix}\right))$ | $\\{1,2\\}$ | $\\{-1,2\\}$ | $\\{-2,1\\}$ | $\\{-2,-1\\}$ | $(-2\quad{-1}\quad 2\quad 1)$ | $\leftarrow$ | $\\{2\\}$ | $\\{-2\\}$ | $\\{2\\}$ | $\\{-2\\}$ ###### Observations 2.69. * • We always have $w^{0}=w^{2}$. 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2003.04739

arxiv-papers

# On the Co-Design of AV-Enabled Mobility Systems Gioele Zardini1,3, Nicolas Lanzetti2,3, Mauro Salazar3,4, Andrea Censi1, Emilio Frazzoli1, and Marco Pavone3 1Institute for Dynamic Systems and Control, ETH Zürich<EMAIL_ADDRESS>Control Laboratory, ETH Zürich<EMAIL_ADDRESS>of Aeronautics and Astronautics, Stanford University<EMAIL_ADDRESS>Systems Technology Group, Eindhoven University of Technology<EMAIL_ADDRESS>preliminary version of this paper was presented at the 99th Annual Meeting of the Transportation Research Board [1].This research was supported by the National Science Foundation under CAREER Award CMMI-1454737, the Toyota Research Institute (TRI), and ETH Zürich. This article solely reflects the opinions and conclusions of its authors and not NSF, TRI, or any other entity. ###### Abstract The design of autonomous vehicles (AVs) and the design of AV-enabled mobility systems are closely coupled. Indeed, knowledge about the intended service of AVs would impact their design and deployment process, whilst insights about their technological development could significantly affect transportation management decisions. This calls for tools to study such a coupling and co- design AVs and AV-enabled mobility systems in terms of different objectives. In this paper, we instantiate a framework to address such co-design problems. In particular, we leverage the recently developed theory of co-design to frame and solve the problem of designing and deploying an intermodal Autonomous Mobility-on-Demand system, whereby AVs service travel demands jointly with public transit, in terms of fleet sizing, vehicle autonomy, and public transit service frequency. Our framework is modular and compositional, allowing one to describe the design problem as the interconnection of its individual components and to tackle it from a system-level perspective. To showcase our methodology, we present a real-world case study for Washington D.C., USA. Our work suggests that it is possible to create user-friendly optimization tools to systematically assess costs and benefits of interventions, and that such analytical techniques might gain a momentous role in policy-making in the future. ## I Introduction Arguably, the current design process for AVs largely suffers from the lack of clear, specific requirements in terms of the service such vehicles will be providing. Yet, knowledge about their intended service (e.g., last-mile versus point-to-point travel) might dramatically impact how the AVs are designed, and, critically, significantly ease their development process. For example, if for a given city we knew that for an effective on-demand mobility system autonomous cars only need to drive up to 25 mph and only on relatively easy roads, their design would be greatly simplified and their deployment could certainly be accelerated. At the same time, from the system-level perspective of transportation management, knowledge about the trajectory of technology development for AVs would certainly impact decisions on infrastructure investments and provision of service. In other words, the design of the AVs and the design of a mobility system leveraging AVs are intimately coupled. This calls for methods to reason about such a coupling, and in particular to _co-design_ the AVs and the associated AV-enabled mobility system. A key requirement in this context is the ability to account for a range of heterogeneous objectives that are often not directly comparable (consider, for instance, travel time and emissions). Accordingly, the goal of this paper is to lay the foundations for a framework through which one can co-design future AV-enabled mobility systems. Specifically, we show how one can leverage the recently developed mathematical theory of co-design [2, 3, 4], which provides a general methodology to co- design complex systems in a modular and compositional fashion. This tool delivers the set of rational design solutions lying on the Pareto front, allowing one to reason about costs and benefits of the individual design options. The framework is instantiated in the setting of co-designing intermodal AMoD systems [5], whereby fleets of self-driving vehicles provide on-demand mobility jointly with public transit. Aspects subject to co-design include fleet size, AV-specific characteristics, and public transit service frequency. ### I-A Literature Review Our work lies at the interface of the design of urban public transportation services and the design of AMoD systems. The first research stream is reviewed in [6, 7], and comprises _strategic_ long-term infrastructure modifications and _operational_ short-term scheduling. The joint design of traffic network topology and control infrastructure has been presented in [8]. Public transportation scheduling has been solved jointly with the design of the transit network in a passengers’ and operators’ cost-optimal fashion in [9], using demand-driven approaches in [10], and in an energy-efficient way in [11]. However, these works only focus on the public transit system and do not consider its joint design with an AMoD system. The research on the design of AMoD systems is reviewed in [12] and mainly pertains their fleet sizing. In this regard, studies range from simulation-based approaches [13, 14, 15, 16] to analytical methods [17]. In [18], the authors jointly design the fleet size and the charging infrastructure, and formulate the arising design problem as a mixed integer linear program. The authors of [19] solve the fleet sizing problem together with the vehicle allocation problem. Finally, [20] co-designs the AMoD fleet size and its composition. More recently, the joint design of multimodal transit networks and AMoD systems was formulated in [21] as a bilevel optimization problem and solved with heuristics. Overall, the problem- specific structure of existing design methods for AMoD systems is not amenable to a modular and compositional problem formulation. Moreover, previous work does not capture important aspects of AV-enabled mobility systems, such as other transportation modes and AV-specific design parameters (e.g., the level of autonomy). ### I-B Statement of Contribution In this paper we lay the foundations for the systematic study of the design of AV-enabled mobility systems. Specifically, we leverage the mathematical theory of co-design [2] to devise a framework to study the design of intermodal AMoD (I-AMoD) systems in terms of fleet characteristics and public transit service, enabling the computation of the _rational_ solutions lying on the Pareto front of minimal travel time, transportation costs, and emissions. Our framework allows one to structure the design problem in a modular way, in which each different transportation option can be “plugged in” in a larger model. Each model has minimal assumptions: Rather than properties such as linearity and convexity, we ask for very general monotonicity assumptions. For example, we assume that the cost of automation increases monotonically with the speed achievable by the AV. We are able to obtain the full Pareto front of _rational_ solutions, or, given policies, to weigh incomparable costs (such as travel time and emissions) and to present actionable information to the stakeholders of the mobility ecosystem. We showcase our methodology through a real-world case study of Washington D.C., USA. We show how, given the model, we can easily formulate and answer several questions regarding the introduction of new technologies and investigate possible infrastructure interventions. ### I-C Organization The remainder of this paper is structured as follows: Section II reviews the mathematical theory of co-design. Section III presents the co-design problem for AV-enabled mobility systems. We showcase our approach with real-world case studies for Washington D.C., USA, in Section IV. Section V concludes the paper with a discussion and an overview on future research directions. ## II Background This paper builds on the mathematical theory of co-design, presented in [2]. In this section, we present a review of the main contents needed for this work. ### II-A Orders We will use basic facts from order theory, which we review in the following. ###### Definition II.1 (Poset). A partially ordered set (poset) is a tuple $\langle\mathcal{P},\preceq_{\mathcal{P}}\rangle$, where $\mathcal{P}$ is a set and $\preceq_{\mathcal{P}}$ is a partial order, defined as a reflexive, transitive, and antisymmetric relation. Given a poset, we can formalize the idea of “Pareto front” through antichains. ###### Definition II.2 (Antichains). A subset $S\subseteq\mathcal{P}$ is an antichain iff no elements are comparable: For $x,y\in S$, $x\preceq y$ implies $x=y$. We denote by $\textsf{A}\mathcal{P}$ the set of all antichains in $\mathcal{P}$. ###### Definition II.3 (Directed set). A subset $S\subseteq\mathcal{P}$ is directed if each pair of elements in $S$ has an upper bound: For all $a,b\in S$, there exists a $c\in S$ such that $a\preceq c$ and $b\preceq c$. ###### Definition II.4 (Completeness). A poset is a complete partial order (CPO) if each of its directed subsets has a supremum and a least element. For instance, the poset $\langle\mathbb{R}_{+},\leq\rangle$, with $\mathbb{R}_{+}\coloneqq\\{x\in\mathbb{R}\,|\,x\geq 0\\}$, is not complete, as its directed subset $\mathbb{R}_{+}\subseteq\mathbb{R}_{+}$ does not have an upper bound (and therefore a supremum). Nonetheless, we can make it complete by artificially adding a top element $\top$, i.e., by defining $\langle\overline{\mathbb{R}}_{+},\leq\rangle$ with $\overline{\mathbb{R}}_{+}\coloneqq\mathbb{R}_{+}\cup\\{\top\\}$ and $a\leq\top$ for all $a\in\mathbb{R}_{+}$. Similarly, we can complete $\mathbb{N}$ to $\overline{\mathbb{N}}$. In this setting, Scott-continuous maps will play a key role. Intuitively, Scott-continuity can be understood as a stronger notion of monotonicity. ###### Definition II.5 (Scott continuity). A map $f:\mathcal{P}\rightarrow\mathcal{Q}$ between two posets $\langle\mathcal{P},\preceq_{\mathcal{P}}\rangle$ and $\langle\mathcal{Q},\preceq_{\mathcal{Q}}\rangle$ is Scott-continuous iff for each directed set $D\subseteq\mathcal{P}$ the image $f(D)$ is directed and $\sup f(D)=f(\sup D)$. ### II-B Mathematical Theory of Co-Design We start by presenting design problems with implementation (DPIs), which can then be composed and interconnected to form a co-design problem with implementation (CDPI). ###### Definition II.6 (DPI). A DPI is a tuple $\langle\mathcal{F},\mathcal{R},\mathcal{I},\textsf{{exe}},\textsf{{eva}}\rangle$: * • $\mathcal{F}$ is a poset, called functionality space; * • $\mathcal{R}$ is a poset, called resource space; * • $\mathcal{I}$ is a set, called implementation space; * • the map $\textsf{{exe}}:\mathcal{I}\to\mathcal{F}$ maps an implementation to the functionality it provides; * • the map $\textsf{{eva}}:\mathcal{I}\to\mathcal{R}$, maps an implementation to the resources it requires. Given a DPI we can define a map which, given a functionality $\textsf{{f}}\in\mathcal{F}$, returns all the non-comparable resources (i.e., the antichain) which provide f. ###### Definition II.7 (Functionality to resources map). Given a DPI $\langle\mathcal{F},\mathcal{R},\mathcal{I},\textsf{{exe}},\textsf{{eva}}\rangle$ define the map $h:\mathcal{F}\to\textsf{{A}}\mathcal{R}$ as $\displaystyle h:$ $\displaystyle\mathcal{F}$ $\displaystyle\to$ $\displaystyle\textsf{{A}}\mathcal{R}$ (1) f $\displaystyle\mapsto$ $\displaystyle\min_{\preceq_{\mathcal{R}}}\\{\textsf{{eva}}(\textsf{{i}})\,|\,\textsf{{i}}\in\mathcal{I}\wedge\textsf{{f}}\preceq\textsf{{exe}}(\textsf{{i}})\\}.$ In particular, if a functionality is infeasible, then $h(\textsf{{f}})=\emptyset$. We now turn our attention to “monotone” DPIs. ###### Definition II.8 (Monotone DPI). We say a DPI $\langle\mathcal{F},\mathcal{R},\mathcal{I},\textsf{{exe}},\textsf{{eva}}\rangle$ is monotone if: 1. 1. The posets $\mathcal{F}$ and $\mathcal{R}$ are CPOs. 2. 2. The map $h$ (see Definition II.7) is Scott-continuous. Individual DPIs can be composed in series (i.e., the functionality of a DPI is the resource of a second DPI) and in parallel (i.e., two DPIs share the same resource or functionality) to obtain a CDPI. Notably, such compositions preserve monotonicity and, thus, all related algorithmic properties. For further details we refer to [2]. ## III Co-Design of AV-enabled Mobility Systems ### III-A Intermodal AMoD Framework #### III-A1 Multi-Commodity Flow Model The transportation system and its different modes are modeled using the digraph $\mathcal{G}=\left(\mathcal{V},\mathcal{A}\right)$, shown in Footnote 2. Figure 1: The I-AMoD network consists of a road, a walking, and a public transportation digraph. The coloured circles represent stops or intersections and the black arrows denote road links, pedestrian pathways, or public transit arcs. Dashed lines are nodes which are close geographically, while grey arrows denote the mode-switching arcs connecting them222We thank Ms. Sonia Monti for the illustration.. It is described through a set of nodes $\mathcal{V}$ and a set of arcs $\mathcal{A}\subseteq\mathcal{V}\times\mathcal{V}$. Specifically, it contains a road network layer $\mathcal{G}_{\mathrm{R}}=\left(\mathcal{V}_{\mathrm{R}},\mathcal{A}_{\mathrm{R}}\right)$, a public transportation layer $\mathcal{G}_{\mathrm{P}}=\left(\mathcal{V}_{\mathrm{P}},\mathcal{A}_{\mathrm{P}}\right)$, and a walking layer $\mathcal{G}_{\mathrm{W}}=\left(\mathcal{V}_{\mathrm{W}},\mathcal{A}_{\mathrm{W}}\right)$. The road network is characterized through intersections $i\in\mathcal{V}_{\mathrm{R}}$ and road segments $(i,j)\in\mathcal{A}_{\mathrm{R}}$. Similarly, public transportation lines are modeled through station nodes $i\in\mathcal{V}_{\mathrm{P}}$ and line segments $(i,j)\in\mathcal{A}_{\mathrm{P}}$. The walking network contains walkable streets $(i,j)\in\mathcal{A}_{\mathrm{W}}$, connecting intersections $i\in\mathcal{V}_{\mathrm{W}}$. Our model allows mode-switching arcs $\mathcal{A}_{\mathrm{C}}\subseteq\mathcal{V}_{\mathrm{R}}\times\mathcal{V}_{\mathrm{W}}\cup\mathcal{V}_{\mathrm{W}}\times\mathcal{V}_{\mathrm{R}}\cup\mathcal{V}_{\mathrm{P}}\times\mathcal{V}_{\mathrm{W}}\cup\mathcal{V}_{\mathrm{W}}\times\mathcal{V}_{\mathrm{P}}$, connecting the road and the public transportation layers to the walking layer. Consequently, $\mathcal{V}=\mathcal{V}_{\mathrm{W}}\cup\mathcal{V}_{\mathrm{R}}\cup\mathcal{V}_{\mathrm{P}}$ and $\mathcal{A}=\mathcal{A}_{\mathrm{W}}\cup\mathcal{A}_{\mathrm{R}}\cup\mathcal{A}_{\mathrm{P}}\cup\mathcal{A}_{\mathrm{C}}$. Consistently with the structural properties of road and walking networks in urban environments, we assume the graph $\mathcal{G}$ to be strongly connected. We represent customer movements by means of travel requests. A travel request refers to a customer flow starting its trip at a node $o\in\mathcal{V}$ and ending it at a node $d\in\mathcal{V}$. ###### Definition III.1 (Travel request). A travel request $\rho$ is a triple $(o,d,\alpha)\in\mathcal{V}\times\mathcal{V}\times\mathbb{R}_{+}$, described by an origin node $o\in\mathcal{V}$, a destination node $d\in\mathcal{V}$, and the request rate $\alpha>0$, namely, the number of customers who want to travel from $o$ to $d$ per unit time. To ensure that a customer is not forced to use a given transportation mode, we assume all requests to lie on the walking digraph, i.e., $o_{m},d_{m}\in\mathcal{V}_{\mathrm{W}}$ for all $m\in\mathcal{M}\coloneqq\\{1,\ldots,M\\}$. The flow $f_{m}\left(i,j\right)\geq 0$ represents the number of customers per unit time traversing arc $(i,j)\in\mathcal{A}$ and satisfying a travel request $m$. Furthermore, $f_{0}\left(i,j\right)\geq 0$ denotes the flow of empty AVs on road arcs $(i,j)\in\mathcal{A}_{\mathrm{R}}$. This accounts for rebalancing flows of AVs between a customer’s drop-off and the next customer’s pick-up. Assuming AVs to carry one customer at a time, the flows satisfy $\displaystyle\sum_{i:(i,j)\in\mathcal{A}}f_{m}\left(i,j\right)+\mathds{1}_{j=o_{m}}\cdot\alpha_{m}=\sum_{k:(j,k)\in\mathcal{A}}f_{m}\left(j,k\right)+\mathds{1}_{j=d_{m}}\cdot\alpha_{m}$ $\displaystyle\hskip 136.5733pt\forall m\in\mathcal{M},\,j\in\mathcal{V}$ (2a) $\displaystyle\sum_{i:(i,j)\in\mathcal{A}_{\mathrm{R}}}f_{\mathrm{tot}}\left(i,j\right)=\sum_{k:(j,k)\in\mathcal{A}_{\mathrm{R}}}f_{\mathrm{tot}}\left(j,k\right)\quad\forall j\in\mathcal{V}_{\mathrm{R}},$ (2b) where $\mathbb{1}_{j=x}$ denotes the boolean indicator function and $f_{\mathrm{tot}}\left(i,j\right)\coloneqq f_{0}\left(i,j\right)+\sum_{m\in\mathcal{M}}f_{m}\left(i,j\right)$. Specifically, (2a) guarantees flows conservation for every transportation demand, and (2b) preserves flow conservation for AVs on every road node. Combining conservation of customers (2a) with the conservation of AVs (2b) guarantees rebalancing AVs to match the demand. ### III-B Travel Time and Travel Speed The variable $t_{ij}$ denotes the time needed to traverse an arc $(i,j)$ of length $s_{ij}$. We assume a constant walking speed on walking arcs and infer travel times on public transportation arcs from the public transit schedules. Assuming that the public transportation system at node $j$ operates with frequency $\varphi_{j}$, switching from a pedestrian vertex $i\in\mathcal{V}_{\mathrm{W}}$ to a public transit station $j\in\mathcal{V}_{\mathrm{P}}$ takes, on average, $t_{ij}=t_{\mathrm{WS}}+0.5\cdot 1/\varphi_{j}\quad\forall(i,j)\in\mathcal{A}_{\mathrm{W}},$ (3) where $t_{\mathrm{WS}}$ is a constant sidewalk-to-station travel time. We assume that the average waiting time for AMoD vehicles is $t_{\mathrm{WR}}$, and that switching from the road graph and the public transit graph to the walking graph takes the transfer times $t_{\mathrm{RW}}$ and $t_{\mathrm{SW}}$, respectively. While each road arc $(i,j)\in\mathcal{A}_{\mathrm{R}}$ is characterized by a speed limit $v_{\mathrm{L,V},ij}$, AVs safety protocols impose a maximum achievable velocity $v_{\mathrm{V,a}}$. In order to prevent too slow and therefore dangerous driving behaviours, we only consider road arcs through which the AVs can drive at least at a fraction $\beta$ of the speed limit: Arc $(i,j)\in\mathcal{A}_{\mathrm{R}}$ is kept in the road network iff $v_{\mathrm{V,a}}\geq\beta\cdot v_{\mathrm{L,V},ij},$ (4) where $\beta\in(0,1]$. We set the velocity of all arcs fulfilling condition (4) to $v_{\mathrm{V},ij}=\min\\{v_{\mathrm{V,a}},v_{\mathrm{L,V},ij}\\}$ and compute the travel time to traverse them as $t_{ij}=s_{ij}/v_{\mathrm{V},ij}\quad\forall(i,j)\in\mathcal{A}_{\mathrm{R}}.$ (5) ### III-C Road Congestion We capture congestion effects with a threshold model. The total flow on each road arc $(i,j)\in\mathcal{A}_{\mathrm{R}}$, given by the sum of the AVs flow $f_{\mathrm{tot}}\left(i,j\right)$ and the baseline usage $u_{ij}$ (e.g., private vehicles), must remain below the nominal capacity $c_{ij}$ of the arc: $f_{\mathrm{tot}}\left(i,j\right)+u_{ij}\leq c_{ij}\quad\forall(i,j)\in\mathcal{A}_{\mathrm{R}}.$ (6) ### III-D Energy Consumption We compute the energy consumption of AVs for each road link considering an urban driving cycle, scaled so that the average speed $v_{\mathrm{avg,cycle}}$ matches the free-flow speed on the link. The energy consumption is then scaled as $e_{ij}=e_{\mathrm{cycle}}\cdot s_{ij}/s_{\mathrm{cycle}}\quad\forall(i,j)\in\mathcal{A}_{\mathrm{R}}.$ (7) For the public transportation system, we assume a constant energy consumption per unit time. This approximation is reasonable in urban environments, as the operation of the public transportation system is independent from the number of customers serviced, and its energy consumption is therefore customer- invariant. ### III-E Fleet Size We consider a fleet of $n_{\mathrm{V,max}}$ AVs. In a time-invariant setting, the number of vehicles on arc $(i,j)\in\mathcal{A}_{\mathrm{R}}$ is expressed as the product of the total vehicles flow on the arc and its travel time. Therefore, we constrain the number of used AVs as $n_{\mathrm{V,u}}=\sum_{(i,j)\in\mathcal{A}_{\mathrm{R}}}f_{\mathrm{tot}}\left(i,j\right)\cdot t_{ij}\leq n_{\mathrm{V,max}}.$ (8) ### III-F Discussion A few comments are in order. First, we assume the demand to be time-invariant and allow flows to have fractional values. This assumption is in line with the mesoscopic and system-level planning perspective of our study. Second, we model congestion effects using a threshold model. This approach can be interpreted as a municipality preventing AVs to exceed the critical flow density on road arcs. AVs can be therefore assumed to travel at free-flow speed [22]. This assumption is realistic for an initial low penetration of AMoD systems in the transportation market, especially when the AV fleet is of limited size. Finally, we allow AVs to transport one customer at the time [23]. ### III-G Co-Design Framework We integrate the I-AMoD framework presented in Section III-A in the co-design formalism, allowing one to decompose the CDPI of a complex system in the DPIs of its individual components in a modular, compositional, and systematic fashion. We aim at computing the antichain of resources, quantified in terms of costs, average travel time per trip, and emissions required to provide the mobility service to a set of customers. In order to achieve this, we decompose the CDPI in the DPIs of the individual AVs (Section III-G1), of the AV fleet (Section III-G3), and of the public transportation system (Section III-G2). The interconnection of the presented DPIs is presented in Section III-G4. #### III-G1 The Autonomous Vehicle Design Problem The AV DPI consists of selecting the maximal speed of the AVs. Under the rationale that driving safely at higher speed requires more advanced sensing and algorithmic capabilities, we model the achievable speed of the AVs $v_{\mathrm{V,a}}$ as a monotone function of the vehicle fixed costs $C_{\mathrm{V,f}}$ (resulting from the cost of the vehicle $C_{\mathrm{V,v}}$ and the cost of its automation $C_{\mathrm{V,a}}$) and of the mileage- dependent operational costs $C_{\mathrm{V,o}}$ (accounting for maintenance, cleaning, energy consumption, depreciation, and opportunity costs [24]). In this setting, the AV DPI provides the functionality $v_{\mathrm{V,a}}$ and requires the resources $C_{\mathrm{V,f}}$ and $C_{\mathrm{V,o}}$. Consequently, the functionality space is $\mathcal{F}_{\mathrm{V}}=\overline{\mathbb{R}}_{+}$, and the resources space is $\mathcal{R}_{\mathrm{V}}=\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}$. #### III-G2 The Subway Design Problem We design the public transit infrastructure by means of the service frequency introduced in Section III-B. Specifically, we assume that the service frequency $\varphi_{j}$ scales linearly with the size of the train fleet $n_{\mathrm{S}}$ as $\varphi_{j}/\varphi_{j,\mathrm{base}}=n_{\mathrm{S}}/n_{\mathrm{S,base}}.$ (9) We relate a train fleet of size $n_{\mathrm{S}}$ to the fixed costs $C_{\mathrm{S,f}}$ (accounting for train and infrastructural costs) and to the operational costs $C_{\mathrm{S,o}}$ (accounting for energy consumption, vehicles depreciation, and train operators’ wages). Given the passenger- independent public transit operation in today’s cities, we reasonably assume the operational costs $C_{\mathrm{S,o}}$ to be mileage independent and to only vary with the size of the fleet. Formally, the number of acquired trains $n_{\mathrm{S,a}}=n_{\mathrm{S}}-n_{\mathrm{S,base}}$ is a functionality, whereas $C_{\mathrm{S,f}}$ and $C_{\mathrm{S,o}}$ are resources. The functionality space is $\mathcal{F}_{\mathrm{S}}=\overline{\mathbb{N}}$ and the resources space is $\mathcal{R}_{\mathrm{S}}=\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}$. #### III-G3 The I-AMoD Framework Design Problem The I-AMoD DPI considers demand satisfaction as a functionality. Formally, $\mathcal{F}_{\mathrm{O}}=2^{\mathcal{V}\times\mathcal{V}\times\overline{\mathbb{R}}_{+}}$ with the partial order $\preceq_{\mathcal{F}_{\mathrm{O}}}$ defined by $\mathcal{D}_{1}\coloneqq\\{(o^{1}_{i},d^{1}_{i},\alpha^{1}_{i})\\}_{i=1}^{M_{1}}\preceq_{\mathcal{F}_{\mathrm{O}}}\\{(o^{2}_{i},d^{2}_{i},\alpha^{2}_{i})\\}_{i=1}^{M_{2}}\eqqcolon\mathcal{D}_{2}$ iff for all $(o^{1},d^{1},\alpha^{1})\in\mathcal{D}_{1}$ there is some $(o^{2},d^{2},\alpha^{2})\in\mathcal{D}_{2}$ with $o^{1}=o^{2}$, $d^{1}=d^{2}$, and $\alpha^{2}_{i}\geq\alpha^{1}_{i}$. In other words, $\mathcal{D}_{1}\preceq_{\mathcal{F}_{\mathrm{O}}}\mathcal{D}_{2}$ if every travel request in $\mathcal{D}_{1}$ is in $\mathcal{D}_{2}$ too. To successfully satisfy a given set of travel requests, we require the following resources: (i) the achievable speed of the AVs $v_{\mathrm{V,a}}$, (ii) the number of available AVs per fleet $n_{\mathrm{V,max}}$, (iii) the number of trains $n_{\mathrm{S,a}}$ acquired by the public transportation system, and (iv) the average travel time of a trip $t_{\mathrm{avg}}\coloneqq\frac{1}{\alpha_{\mathrm{tot}}}\cdot\sum_{m\in\mathcal{M},(i,j)\in\mathcal{A}}t_{ij}\cdot f_{m}\left(i,j\right),$ (10) with $\alpha_{\mathrm{tot}}\coloneqq\sum_{m\in\mathcal{M}}\alpha_{m}$, (v) the total distance driven by the AVs per unit time $s_{\mathrm{V,tot}}\coloneqq\sum_{(i,j)\in\mathcal{A}_{\mathrm{R}}}s_{ij}\cdot f_{\mathrm{tot}}\left(i,j\right),$ (11) (vi) the total AVs CO2 emissions per unit time $m_{\mathrm{CO_{2},V,tot}}\coloneqq\gamma\cdot\sum_{(i,j)\in\mathcal{A}_{\mathrm{R}}}e_{ij}\cdot f_{\mathrm{tot}}\left(i,j\right),$ (12) where $\gamma$ relates the energy consumption and the CO2 emissions. We assume that customers’ trips and AMoD rebalancing strategies are chosen to maximize customers’ welfare, defined through the average travel time $t_{\mathrm{avg}}$. Hence, we link the functionality and resources of the I-AMoD DPI through the following optimization problem: $\begin{split}\min_{\begin{subarray}{c}f_{m}\left(\cdot,\cdot\right)\geq 0,\\\ f_{0}\left(\cdot,\cdot\right)\geq 0\end{subarray}}t_{\mathrm{avg}}&=\frac{1}{\alpha_{\mathrm{tot}}}\sum_{m\in\mathcal{M},(i,j)\in\mathcal{A}}t_{ij}\cdot f_{m}\left(i,j\right)\\\ &\mathrm{s.t.\ Eq.}\eqref{eq:flowconstotal},\ \mathrm{Eq.}\eqref{eq:capacity},\ \mathrm{Eq.}\eqref{eq:fleetsizecar}.\end{split}$ (13) Formally, $\mathcal{F}_{\mathrm{O}}=\overline{\mathbb{R}}_{+}$, and $\mathcal{R}_{\mathrm{O}}=\overline{\mathbb{R}}_{+}\times\overline{\mathbb{N}}\times\overline{\mathbb{N}}\times\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}$. ###### Remark. In general, the optimization problem (13) might possess multiple optimal solutions, making the relation between resources and functionality ill-posed. To overcome this subtlety, if two solutions share the same average travel time, we select the one incurring in the lowest mileage. #### III-G4 The Monotone Co-Design Problem The functionality of the system is to provide mobility service to the customers. Formally, the functionality provided by the CDPI is the set of travel requests. To provide the mobility service, the following three resources are required. First, on the customers’ side, we require an average travel time, defined in (10). Second, on the municipality side, the resource is the total transportation cost of the intermodal mobility system. Assuming an average vehicles’ life $l_{\mathrm{V}}$, an average trains’ life $l_{\mathrm{S}}$, and a baseline subway fleet of $n_{\mathrm{S,base}}$ trains, we express the total costs as $C_{\mathrm{tot}}=C_{\mathrm{V}}+C_{\mathrm{S}},$ (14) where $C_{\mathrm{V}}$ is the AV-related cost $C_{\mathrm{V}}=\frac{C_{\mathrm{V,f}}}{l_{\mathrm{V}}}\cdot n_{\mathrm{V,max}}+C_{\mathrm{V,o}}\cdot s_{\mathrm{V,tot}},$ (15) and $C_{\mathrm{S}}$ is the public transit-related cost $C_{\mathrm{S}}=\frac{C_{\mathrm{S,f}}}{l_{\mathrm{S}}}\cdot n_{\mathrm{S,a}}+C_{\mathrm{S,o}}.$ (16) Third, on the environmental side, the resources are the total CO2 emissions $m_{\mathrm{CO_{2},tot}}=m_{\mathrm{CO_{2},V,tot}}+m_{\mathrm{CO_{2},S}}\cdot n_{\mathrm{S}},$ (17) where $m_{\mathrm{CO_{2},S}}$ represents the CO2 emissions of a single train. Formally, the set of travel requests $\\{\rho_{m}\\}_{m\in\mathcal{M}}$ is the CDPI functionality, whereas $t_{\mathrm{avg}}$, $C_{\mathrm{tot}}$, and $m_{\mathrm{CO_{2},tot}}$ are its resources. Consistently, the functionality space is $\mathcal{F}=\overline{\mathbb{R}}_{+}$ and the resources space is $\mathcal{R}=\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}$. Note that the resulting CDPI (Figure 2) is indeed monotone, since it consists of the interconnection of monotone DPIs [2]. #### III-G5 Discussion A few comments are in order. First, we lump the autonomy functionalities in its achievable velocity. We leave to future research more elaborated AV models, accounting for instance for accidents rates [25] and for safety levels. Second, we assume the service frequency of the subway system to scale linearly with the number of trains. We inherently rely on the assumption that the existing infrastructure can homogeneously accommodate the acquired train cars. To justify the assumption, we include an upper bound on the number of potentially acquirable trains in our case study design in Section IV. Third, we highlight that the I-AMoD framework is only one of the many feasible ways to map total demand to travel time, costs, and emissions. Specifically, practitioners can easily replace the corresponding DPI with more sophisticated models (e.g., simulation-based frameworks like AMoDeus [26]), as long as the monotonicity of the DPI is preserved. In our setting, we conjecture the customers’ and vehicles’ routes to be centrally controlled by the municipality in a socially-optimal fashion. Implicitly, we rely on the existence of effective incentives aligning private and societal interests. The study of such incentives represents an avenue for future research. Fourth, we assume a homogenous fleet of AVs. Nevertheless, our model is readily extendable to capture heterogeneous fleets. Finally, we consider a fixed travel demand, and compute the antichain of resources providing it. Nonetheless, our formalization can be readily extended to arbitrary demand models preserving the monotonicity of the CDPI (accounting for instance for elastic effects). We leave this topic to future research. I-AMoDVehicleSubway$\preceq$$\preceq$$\preceq$$\times$$\preceq$$\preceq$$\times$$\preceq$$\preceq$$\times$$\preceq$$\preceq$$\preceq$$+$$\preceq$$+$$\preceq$$\times$$\preceq$$\preceq$$+$$+$$\preceq$$+$$\preceq$$v_{\mathrm{V,a}}$$n_{\mathrm{S,a}}$$s_{\mathrm{V,tot}}$$C_{\mathrm{V,o}}$$C_{\mathrm{V,f}}$co- design constraint$C_{\mathrm{S,o}}$$C_{\mathrm{S,f}}$$n_{\mathrm{V,max}}$$C_{\mathrm{tot}}$$\alpha_{\mathrm{tot}}$$m_{\mathrm{CO_{2},V,tot}}$$n_{\mathrm{S}}$$m_{\mathrm{CO_{2},S}}$$m_{\mathrm{CO_{2},tot}}$$t_{\mathrm{avg}}$$l_{\mathrm{V}}$$l_{\mathrm{S}}$total costaverage travel timetotal emissionstotal request rate Figure 2: Schematic representation of the CDPI. In solid green the provided functionalities and in dashed red the required resources. The edges represent co-design constraints: The resources required by a first design problem are the lower bound for the functionalities provided by the second one. ## IV Results In this section, we leverage the framework presented in Section III to perform a real-world case study of Washington D.C., USA. Section IV-A details the case study. We then present numerical results in Sections IV-B and IV-C. ### IV-A Case Study We base our studies on a real-world case of the urban area of Washington D.C., USA. We import the road network and its features from OpenStreetMap [27]. The public transit network and its schedules are extracted from the GTFS data [28]. Demand data is obtained by merging the origin-destination pairs of the morning peak of May 31st 2017 provided by taxi companies [29] and the Washington Metropolitan Area Transit Authority (WMATA) [23]. Given the lack of reliable demand data for the MetroBus system, we focus our studies on the MetroRail system and its design, inherently assuming MetroBus commuters to be unaffected by our design methodology. To conform with the large presence of ride-hailing companies, we scale the taxi demand rate by factor of 5 [30]. Overall, the demand dataset includes 15,872 travel requests, corresponding to a demand rate of 24.22 $\nicefrac{{requests}}{{s}}$. To account for congestion effects, we compute the nominal road capacity as in [31] and assume an average baseline road usage of 93%, in line with [32]. We summarize the main parameters together with their bibliographic sources in Table I. In the remainder of this section, we tailor and solve the co-design problem presented in Section III through the PyMCDP solver [33], and investigate the influence of different AVs costs on the design objectives and strategies. | Parameter | Name | Value | Units | Source ---|---|---|---|---|--- | Road usage | | $u_{ij}$ | 93 | % | [32] Vehicle | | | | C1 | C2.1 | C2.2 | | Operational cost | $C_{\mathrm{V,o}}$ | 0.084 | 0.084 | 0.062 | $\nicefrac{{USD}}{{mile}}$ | [34, 35] Cost | $C_{\mathrm{V}}$ | 32,000 | 32,000 | 26,000 | $\nicefrac{{USD}}{{car}}$ | [34] Automation cost | 20 mph | $C_{\mathrm{V,a}}$ | 15,000 | 20,000 | 3,700 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39] 25 mph | 15,000 | 30,000 | 4,400 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39] 30 mph | 15,000 | 55,000 | 6,200 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39] 35 mph | 15,000 | 90,000 | 8,700 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39] 40 mph | 15,000 | 115,000 | 9,800 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39] 45 mph | 15,000 | 130,000 | 12,000 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39] 50 mph | 15,000 | 150,000 | 13,000 | $\nicefrac{{USD}}{{car}}$ | [35, 36, 37, 38, 39] Vehicle life | $l_{\mathrm{V}}$ | 5 | 5 | 5 | years | [34] CO2 per Joule | $\gamma$ | 0.14 | 0.14 | 0.14 | $\nicefrac{{g}}{{kJ}}$ | [40] Time $\mathcal{G}_{\mathrm{W}}$ to $\mathcal{G}_{\mathrm{R}}$ | $t_{\mathrm{WR}}$ | 300 | 300 | 300 | s | - Time $\mathcal{G}_{\mathrm{R}}$ to $\mathcal{G}_{\mathrm{W}}$ | $t_{\mathrm{RW}}$ | 60 | 60 | 60 | s | - | Speed fraction | $\beta$ | $\frac{1}{1.3}$ | $\frac{1}{1.3}$ | $\frac{1}{1.3}$ | - | - Public transit | Operational cost | 100 % | $C_{\mathrm{S,o}}$ | 148,000,000 | $\nicefrac{{USD}}{{year}}$ | [41] 133 % | 197,000,000 | $\nicefrac{{USD}}{{year}}$ | [41] 200 % | 295,000,000 | $\nicefrac{{USD}}{{year}}$ | [41] Fixed cost | $C_{\mathrm{S,f}}$ | 14,500,000 | $\nicefrac{{USD}}{{train}}$ | [42] Train life | $l_{\mathrm{S}}$ | 30 | years | [42] Emissions/train | $m_{\mathrm{CO_{2},S}}$ | 140,000 | $\nicefrac{{kg}}{{year}}$ | [43] Fleet baseline | $n_{\mathrm{S,base}}$ | 112 | trains | [42] Service frequency | $\varphi_{j,\mathrm{base}}$ | $1/6$ | $\nicefrac{{1}}{{min}}$ | [44] Time $\mathcal{G}_{\mathrm{W}}$ to $\mathcal{G}_{\mathrm{P}}$ | $t_{\mathrm{WS}}$ | $60$ | s | - Time $\mathcal{G}_{\mathrm{P}}$ to $\mathcal{G}_{\mathrm{W}}$ | $t_{\mathrm{SW}}$ | $60$ | s | - TABLE I: Parameters, variables, numbers, and units for the case studies. ### IV-B Case 1 - Constant Cost of Automation In line with [35, 36, 37, 38, 39], we first assume an average achievable- velocity-independent cost of automation. As discussed in Section III, we design the system by means of subway service frequency, AV fleet size, and achievable free-flow speed. Specifically, we allow the municipality to (i) increase the subway service frequency $\varphi_{j}$ by a factor of 0%, 33%, or 100%, (ii) deploy an AVs fleet of size $n_{\mathrm{V,max}}\in\\{0,500,1000,\ldots,6000\\}$ vehicles, and (iii) design the single AV achievable velocity $v_{\mathrm{V,a}}\in\\{20\,\mathrm{mph},25\,\mathrm{mph},\ldots,50\,\mathrm{mph}\\}$. We assume the AVs fleet to be composed of battery electric BEV-250 mile AVs [34]. In Figure 3(a), we show the solution of the co-design problem by reporting the antichain consisting of the total transportation cost, average travel time, and total CO2 emissions. These solutions are _rational_ (and not comparable) in the sense that there exists no instance which simultaneously yields lower cost, average travel time, and emissions. (a) Left: Three-dimensional representation of antichain elements and their projection in the cost-time space. Right: Two-dimensional projections. (b) Results for constant automation costs. On the left, the two-dimensional representation of the antichain elements: In red are the unfeasible strategies, in orange the feasible but irrational solutions, and in green the Pareto front. On the right, the implementations corresponding to the highlighted antichain elements, quantified in terms of achievable vehicle speed, AV fleet size, and train fleet size. Figure 3: Solution of the CDPI: state-of-the art case. For the sake of clarity, we opt for a two-dimensional antichain representation, by translating and including the emissions in the total cost. To do so, we consider the conversion factor 40 $\nicefrac{{USD}}{{kg}}$ [45]. Note that since this transformation preserves the monotonicity of the CDPI it smoothly integrates in our framework. Doing so, we can conveniently depict the co-design strategies through the two-dimensional antichain (Figure 3(b), left) and the corresponding municipality actions (Figure 3(b), right). Generally, as the municipality budget increases, the average travel time per trip required to satisfy the given demand decreases, reaching a minimum of about 17.1 min with an expense of around 43 $\nicefrac{{MilUSD}}{{month}}$. This configuration corresponds to a fleet of 5,500 AVs able to drive at 50 mph and to the doubling of the current MetroRail train fleet. On the other hand, the smallest rational investment of 12.9 $\nicefrac{{MilUSD}}{{month}}$ leads to a 42 % higher average travel time, corresponding to a inexistent autonomous fleet and an unchanged subway infrastructure. Notably, an expense of 23 $\nicefrac{{MilUSD}}{{month}}$ (48 % lower than the highest rational investment) only increases the minimal required travel time by 9 %, requiring a fleet of 4,000 vehicles able to drive at 35 mph and no acquisition of trains. Conversely, an investment of 15.6 $\nicefrac{{MilUSD}}{{month}}$ (just 2 $\nicefrac{{MilUSD}}{{month}}$ more than the minimal rational investment) provides a 3 min shorter travel time. Remarkably, the design of AVs able to exceed 40 mph only improves the average travel time by 6 %, and it is rational just starting from an expense of 22.8 $\nicefrac{{MilUSD}}{{month}}$. This suggests that the design of faster vehicles mainly results in higher emission rates and costs, without substantially contributing to a more time-efficient demand satisfaction. Finally, it is rational to improve the subway system only starting from a budget of 28.5 $\nicefrac{{MilUSD}}{{month}}$, leading to a travel time improvement of just 4 %. This trend can be explained with the high train acquisition and increased operation costs, related to the subway reinforcement. We expect this phenomenon to be more marked for other cities, considering the moderate operation costs of the MetroRail subway system due to its automation [44] and related benefits [46]. ### IV-C Case 2 - Speed-Dependent Automation Costs To relax the potentially unrealistic assumption of a velocity-independent automation cost, we consider a performance-dependent cost structure. The large variance in sensing technologies and their reported performances [47] suggests that this rationale is reasonable. Indeed, the technology required today to safely operate an autonomous vehicle at 50 mph is substantially more sophisticated, and therefore more expensive, than the one needed at 20 mph. To this end, we adopt the cost structure reported in Table I. Furthermore, the frenetic evolution of automation techniques intricates their monetary quantification. Therefore, we perform our studies with current (2020) costs as well as with their projections for the upcoming decade (2025) [48, 34]. #### IV-C1 Case 2.1 - 2020 We study the hypothetical case of an immediate AV fleet deployment. We introduce the aforementioned velocity-dependent automation cost structure and obtain the results reported in Figure 4(a). Comparing these results with the state-of-the-art parameters presented in Figure 3 confirms the previously observed trend concerning high vehicle speeds. Indeed, spending 24.9 $\nicefrac{{MilUSD}}{{month}}$ (55 % lower than the highest rational expense) only increases the average travel time by 10 %, requiring a fleet of 3,000 AVs at 40 mph and no subway interventions. Nevertheless, the comparison shows two substantial differences. First, the budget required to reach the minimum travel time of 17.1 min is 28 % higher compared to the previous case, and consists of the same strategy for the municipality, i.e., doubling the train fleet and having a fleet of 5,500 AVs at 50 mph. Second, the higher vehicle costs result in an average AV fleet growth of 5 %, an average velocity reduction of 9 %, and an average train fleet growth of 7 %. This trend suggests that, compared to Case 1, rational design strategies foster larger fleets and less performing AVs. (a) Results for speed-dependent automation costs in 2020. (b) Results for speed-dependent automation costs in 2025. Figure 4: Results for the speed-dependent automation costs. On the left, the two-dimensional representation of the antichain elements: In red are the unfeasible strategies, in orange the feasible but irrational solutions, and in green the Pareto front. On the right, the implementations corresponding to the highlighted antichain elements. #### IV-C2 Case 2.2 - 2025 Experts forecast a substantial decrease of automation costs (up to 90 %) in the next decade, mainly due to mass-production of the AVs sensing technology [48, 49]. In line with this prediction, we inspect the futuristic scenario by solving the CDPI for the adapted automation costs, and report the results in Figure 4(b). Two comments are in order. First, the maximal rational budget is 25 % lower than in the immediate adoption case. Second, the reduction in autonomy costs clearly eases the acquisition of more performant AVs, increasing the average vehicle speed by 10 %. As a direct consequence, the AV and train fleets are reduced in size by 5 % and 10 %, respectively. ### IV-D Discussion We conclude the analysis of our case study with two final comments. First, the presented case studies illustrate the ability of our framework to extract the set of rational design strategies for an AV-enabled mobility system. This way, stakeholders such as AVs companies, transportation authorities, and policy makers can get transparent and interpretable insights on the impact of future interventions. Second, we perform a sensitivity analysis through the variation of the autonomy cost structures. On the one hand, this reveals a clear transition from small fleets of fast AVs (in the case of low autonomy costs) to slow fleets of numerous AVs (in the case of high autonomy costs). On the other hand, our studies highlight that investments in the public transit infrastructure are rational only when large budgets are available. Indeed, the onerous train acquisition and operation costs lead to a comparative advantage of AV-based mobility. ## V Conclusion In this paper, we leveraged the mathematical theory of co-design to propose a design framework for AV-enabled mobility systems. Specifically, the nature of our framework allows both for the modular and compositional interconnection of the DPIs of different mobility options and for multiple objectives. Starting from the multi-commodity flow model of an I-AMoD system, we optimize the design of AVs and public transit both from a vehicle-centric and fleet-level perspective. In particular, we studied the problem of deploying a fleet of AVs providing on-demand mobility in cooperation with public transit, optimizing the speed achievable by the vehicles, the fleet size, and the service frequency of the subway lines. Our framework allows the stakeholders involved in the mobility ecosystem, from vehicle developers all the way to mobility-as- a-service companies and governmental authorities, to characterize rational trajectories for technology and investment development. We showcased our methodology on a real-world case study of Washington D.C., USA. Notably, our problem formulation allows for a systematic analysis of incomparable objectives, providing stakeholders with analytical insights for the socio- technical design of AV-enabled mobility systems. This work opens the field for the following future research streams: _Modeling:_ First, we would like to extend the presented framework to capture additional modes of transportation, such as micromobility, and heterogeneous fleets with different self-driving infrastructures, propulsion systems, and passenger capacity. Second, we would like to investigate variable demand models. Finally, we would like to analyze the interactions between multiple stakeholders, characterizing the equilibrium arising from their conflicting interests. _Algorithms:_ It is of interest to tailor co-design algorithmic frameworks to the particular case of transportation DPIs, possibly leveraging their specific structure. _Application:_ Finally, we would like to devise a user-friendly web interface which supports mobility stakeholders to reason about strategic interventions in urban areas. ## References * [1] G. Zardini, N. Lanzetti, M. Salazar, A. Censi, E. Frazzoli, and M. Pavone, “Towards a co-design framework for future mobility systems,” in _Annual Meeting of the Transportation Research Board_ , Washington D.C., United States, 2020. * [2] A. 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2024-09-04T02:54:58.766292

2003.04805

arxiv-papers

*mymilestone mymilestone/.style= shape=isosceles triangle, inner sep=0pt, draw=cyan, top color=white, bottom color=cyan!50 , mymilestone incomplete/.style= /pgfgantt/mymilestone, draw=yellow, bottom color=yellow!50 , mymilestone label font=, mymilestone left shift=0pt, mymilestone right shift=0pt stardate1cm, inner sep=0]

2024-09-04T02:54:58.774191

2003.04819

arxiv-papers

Karate Club: An API Oriented Open-Source Python Framework for Unsupervised Learning on Graphs Benedek Rozemberczki The University of Edinburgh United Kingdom Oliver Kiss Central European University Rik Sarkar The University of Edinburgh United Kingdom Graphs encode important structural properties of complex systems. Machine learning on graphs has therefore emerged as an important technique in research and applications. We present Karate Club – a Python framework combining more than 30 state-of-the-art graph mining algorithms. These unsupervised techniques make it easy to identify and represent common graph features. The primary goal of the package is to make community detection, node and whole graph embedding available to a wide audience of machine learning researchers and practitioners. Karate Club is designed with an emphasis on a consistent application interface, scalability, ease of use, sensible out of the box model behaviour, standardized dataset ingestion, and output generation. This paper discusses the design principles behind the framework with practical examples. We show Karate Club's efficiency in learning performance on a wide range of real world clustering problems and classification tasks along with supporting evidence of its competitive speed. § ACKNOWLEDGEMENTS Benedek Rozemberczki was supported by the Centre for Doctoral Training in Data Science, funded by EPSRC (grant EP/L016427/1). #1 #1#1#1 #1 #1 #1 #1#1 #1#1 [Abadi, Barham, Chen, Chen, Davis, Dean, Devin, Ghemawat, Irving, Isard, et alAbadi et al2016] authorpersonMartín Abadi, personPaul Barham, personJianmin Chen, personZhifeng Chen, personAndy Davis, personJeffrey Dean, personMatthieu Devin, personSanjay Ghemawat, personGeoffrey Irving, personMichael Isard, et al year2016. 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2024-09-04T02:54:58.783178

2003.04827

arxiv-papers

0pt0pt *34pc* **1 1in1in* 2 *1.5* subsubsection subsection # Dirichlet Polynomials David I. Spivak David Jaz Myers # Dirichlet Polynomials form a Topos David I. Spivak David Jaz Myers ###### Abstract One can think of power series or polynomials in one variable, such as $P(\mathcal{y})=2\mathcal{y}^{3}+\mathcal{y}+5$, as functors from the category $\mathsf{Set}$ of sets to itself; these are known as polynomial functors. Denote by $\mathsf{Poly}_{\mathsf{Set}}$ the category of polynomial functors on $\mathsf{Set}$ and natural transformations between them. The constants $0,1$ and operations $+,\times$ that occur in $P(\mathcal{y})$ are actually the initial and terminal objects and the coproduct and product in $\mathsf{Poly}_{\mathsf{Set}}$. Just as the polynomial functors on $\mathsf{Set}$ are the copresheaves that can be written as sums of representables, one can express any Dirichlet series, e.g. $\sum_{n=0}^{\infty}n^{\mathcal{y}}$, as a coproduct of representable presheaves. A Dirichlet polynomial is a finite Dirichlet series, that is, a finite sum of representables $n^{\mathcal{y}}$. We discuss how both polynomial functors and their Dirichlet analogues can be understood in terms of bundles, and go on to prove that the category of Dirichlet polynomials is an elementary topos. ## Chapter 0 Introduction Polynomials $P(\mathcal{y})$ and finite Dirichlet series $D(\mathcal{y})$ in one variable $\mathcal{y}$, with natural number coefficients $a_{i}\in\mathbb{N}$, are respectively functions of the form $\displaystyle P(\mathcal{y})$ $\displaystyle=a_{n}\mathcal{y}^{n}+\cdots+a_{2}\mathcal{y}^{2}+a_{1}\mathcal{y}^{1}+a_{0}\mathcal{y}^{0},$ (1) $\displaystyle D(\mathcal{y})$ $\displaystyle=a_{n}n^{\mathcal{y}}+\cdots+a_{2}2^{\mathcal{y}}+a_{1}1^{\mathcal{y}}+a_{0}0^{\mathcal{y}}.$ The first thing we should emphasize is that the algebraic expressions in (1) can in fact be regarded as _objects in a category_ , in fact two categories: $\mathsf{Poly}$ and $\mathsf{Dir}$. We will explain the morphisms later, but for now we note that in $\mathsf{Poly}$, $\mathcal{y}^{2}=\mathcal{y}\times\mathcal{y}$ is a product and $2\mathcal{y}=\mathcal{y}+\mathcal{y}$ is a coproduct, and similarly for $\mathsf{Dir}$. The operators—in both the polynomial and the Dirichlet case—are not just algebraic, they are category-theoretic. Moreover, these categories have a rich structure. The category $\mathsf{Poly}$ is well studied (see [GK12]). In particular, the following are equivalent: ###### Theorem 1. [GK12] For a functor $P\colon\mathsf{Fin}\to\mathsf{Fin}$, the following are equivalent: 1. 1. $P$ is polynomial. 2. 2. $P$ is a sum of representables. 3. 3. $P$ preserves connected limits – or equivalently, wide pullbacks. In Theorem 8 we prove an analogous result characterizing Dirichlet polynomials: ###### Theorem 2. For a functor $D\colon\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Fin}$, the following are equivalent: 1. 1. $D$ is a Dirichlet polynomial. 2. 2. $D$ is a sum of representables. 3. 3. $D$ sends connected colimits to limits – or equivalently, $D$ preserves wide pushouts. We will also show that $\mathsf{Dir}$ is equivalent to the arrow category of finite sets, $\mathsf{Dir}\simeq\mathsf{Fin}^{\to},$ and in particular that $\mathsf{Dir}$ is an elementary topos. If one allows _arbitary_ sums of functors represented by finite sets, one gets _analytic_ functors in the covariant case—first defined by Joyal in his seminal paper on combinatorial species [Joy81]—and _Dirichlet_ functors in the contravariant case—first defined by Baez and Dolan and appearing in Baez’s _This Week’s Finds_ blog [BD]. Baez and Dolan also drop the traditional negative sign in the exponent (that is, they use $n^{s}$ where $n^{-s}$ usually appears), but also find a nice way to bring it back by moving to groupoids. Here, we drop the negative sign and work with finite sets to keep things as simple as possible. Similar considerations hold with little extra work for infinite Dirichlet series or power series, and even more generally, by replacing $\mathsf{Fin}$ with $\mathsf{Set}$. ## Chapter 1 Polynomial and Dirichlet functors Recall that a _co-representable functor_ $\mathsf{Fin}\to\mathsf{Fin}$ is one of the form $\mathsf{Fin}(k,-)$ for a finite set $k=\\{`1\text{'},`2\text{'},\ldots,`k\text{'}\\}.$ We denote this functor by $\mathcal{y}^{k}$ and say it is _represented by_ $k\in\mathsf{Fin}$. Similarly, a _(contra-) representable functor_ $\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Fin}$ is contravariant functor of the form $\mathsf{Fin}(-,k)$; we denote this functor by $k^{\mathcal{y}}$. The functors $\mathcal{y}^{-}$ and $-^{\mathcal{y}}$ are the contravariant and covariant Yoneda embeddings, $\mathcal{y}^{k}\coloneqq\mathsf{Fin}(k,-)\qquad\text{and}\qquad k^{\mathcal{y}}\coloneqq\mathsf{Fin}(-,k).$ For example $\mathcal{y}^{3}(2)\cong 8$ and $3^{\mathcal{y}}(2)\cong 9$. Note that the functor $0^{\mathcal{y}}\not\cong 0$ is not the initial object in $\mathsf{Dir}$; it is given by $0^{\mathcal{y}}(s)=\begin{cases}1&\textnormal{ if }s=0\\\ 0&\textnormal{ if }s\geq 1.\end{cases}$ The coefficient $a_{0}$ of $1=\mathcal{y}^{0}$ in a polynomial $P$ is called its _constant_ term. We refer to the coefficient $D_{\text{zc}}\coloneqq a_{0}$ of $0^{\mathcal{y}}$ in a Dirichlet series $D$ as its _zero-content_ term. Rather than having no content, the content of the functor $D_{\text{zc}}{\cdot}0^{\mathcal{y}}$ becomes significant exactly when it is applied to zero. ###### Example 1. The reader can determine which Dirichlet polynomial $D(\mathcal{y})\in\mathsf{Dir}$ as in Eq. 1 has the following terms $\begin{array}[]{c|ccccccc}\mathcal{y}&\cdots&5&4&3&2&1&0\\\ \hline\cr D(\mathcal{y})&\cdots&96&48&24&12&6&7\end{array}$ Hint: its zero-content term is $D_{\text{zc}}=4$. The set $P(1)$ (resp. the set $D(0)$) has particular importance; it is the set of pure-power terms $\mathcal{y}^{k}$ in $P$ (resp. the pure-exponential terms $k^{\mathcal{y}}$ in $D$). For example if $P=\mathcal{y}^{2}+4\mathcal{y}+4$ and $D=2^{\mathcal{y}}+4+4{\cdot}0^{\mathcal{y}}$ then $P(1)=D(0)=9$. ###### Definition 2. A _polynomial functor_ is a functor $P\colon\mathsf{Fin}\to\mathsf{Fin}$ that can be expressed as a sum of co-representable functors. Similarly, we define a _Dirichlet functor_ to be a functor $D\colon\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Fin}$ that can be expressed as a sum of representable presheaves (contravariant functors): $P=\sum_{i=1}^{P(1)}\mathcal{y}^{p_{i}}\qquad\text{and}\qquad D=\sum_{i=1}^{D(0)}(d_{i})^{\mathcal{y}}.$ (1) That is, $P(X)=\sum_{i=1}^{P(1)}\mathsf{Fin}(p_{i},X)$ and $D(X)=\sum_{i=1}^{D(0)}\mathsf{Fin}(X,d_{i})$ as functors applied to $X\in\mathsf{Fin}$. See Theorem 1 above for well-known equivalent conditions in $\mathsf{Poly}$ and Theorem 8 below for a Dirichlet analogue. ## Chapter 2 The categories $\mathsf{Poly}$ and $\mathsf{Dir}$ For any small category $C$, let $\mathsf{Fin}^{C}$ denote the category whose objects are the functors $C\to\mathsf{Fin}$ and whose morphisms are the natural transformations between them. ###### Definition 1. The _category of polynomial functors_ , denoted $\mathsf{Poly}$, is the (skeleton of the) full subcategory of $\mathsf{Fin}^{\mathsf{Fin}}$ spanned by sums $P$ of representable functors. The _category of Dirichlet functors_ , denoted $\mathsf{Dir}$, is the (skeleton of the) full subcategory of $\mathsf{Fin}^{(\mathsf{Fin}^{\textnormal{op}})}$ spanned by the sums $D$ of representable presheaves. While we will not pursue it here, one can take $\mathsf{Poly}_{\mathsf{Set}}$ to be the full subcategory of functors $\mathsf{Set}\to\mathsf{Set}$ spanned by small coproducts of representables, and similarly for $\mathsf{Dir}_{\mathsf{Set}}$. ###### Lemma 2. The set of polynomial maps $P\to Q$ and Dirichlet maps $D\to E$ are given by the following formulas: $\mathsf{Poly}(P,Q)\coloneqq\prod_{i\in P(1)}Q(p_{i})\qquad\text{and}\qquad\mathsf{Dir}(D,E)\coloneqq\prod_{i\in D(0)}E(d_{i}).$ ###### Example 3. Let $P=2\mathcal{y}^{2}$, $Q=\mathcal{y}+1$, and let $D=2\cdot 2^{\mathcal{y}}$ and $E=1+0^{\mathcal{y}}$. Then there are nine ($9$) polynomial morphisms $P\to Q$, zero ($0$) polynomial morphisms $Q\to P$, one ($1$) Dirichlet morphism $D\to E$, and eight ($8$) Dirichlet morphisms $E\to D$. ###### Remark 4. Sums and products of polynomials in the usual algebraic sense agree exactly with sums and products in the categorical sense: if $P$ and $Q$ are polynomials, i.e. objects in $\mathsf{Poly}$, then their coproduct is the usual algebraic sum $P+Q$ of polynomials, and similarly their product is the usual algebraic product $PQ$ of polynomials. The same is true for $\mathsf{Dir}$: sums and products of Dirichlet polynomials in the usual algebraic sense agree exactly with sums and products in the categorical sense. #### Formal structures. We review some formal structures of the categories $\mathsf{Poly}$ and $\mathsf{Dir}$; all are straightforward to prove. There is an adjoint quadruple and adjoint 5-tuple as follows, labeled by where they send objects $n\in\mathsf{Fin}$, $P\in\mathsf{Poly}$, $D\in\mathsf{Dir}$: ${\mathsf{Fin}}$${\mathsf{Poly}}$$\scriptstyle{n}$$\scriptstyle{n\mathcal{y}}$$\scriptstyle{P(0)}$$\scriptstyle{P(1)}$${\scriptstyle\top}$${\scriptstyle\top}$${\scriptstyle\top}$ ${\mathsf{Fin}}$${\mathsf{Dir}}$$\scriptstyle{n\cdot 0^{\mathcal{y}}}$$\scriptstyle{n}$$\scriptstyle{n^{\mathcal{y}}}$$\scriptstyle{D(0)}$$\scriptstyle{D(1)}$${\scriptstyle\bot}$${\scriptstyle\bot}$${\scriptstyle\bot}$${\scriptstyle\bot}$ (1) All five of the displayed functors out of $\mathsf{Fin}$ are fully faithful. For each $k:\mathsf{Fin}$ the functors $P\mapsto P(k)$ and $D\mapsto D(k)$ have left adjoints, namely $n\mapsto n\mathcal{y}^{k}$ and $n\mapsto n{\cdot}k^{\mathcal{y}}$ respectively. These are functorial in $k$ and in fact extend to two-variable adjunctions $\mathsf{Fin}\times\mathsf{Poly}\to\mathsf{Poly}$ and $\mathsf{Fin}\times\mathsf{Dir}\to\mathsf{Dir}$. Indeed, for $n\in\mathsf{Fin}$ and $P,Q\in\mathsf{Poly}$ (respectively $D,E\in\mathsf{Dir}$), we have $\displaystyle\mathsf{Poly}(nP,Q)\cong\mathsf{Poly}(P,Q^{n})\cong\mathsf{Fin}(n,\mathsf{Poly}(P,Q)),$ $\displaystyle\mathsf{Dir}(nD,E)\cong\mathsf{Dir}(D,E^{n})\cong\mathsf{Fin}(n,\mathsf{Dir}(D,E)),$ where $nP$ and $nD$ denote $n$-fold coproducts and $P^{n}$ and $D^{n}$ denote $n$-fold products. Consider the unique function $0\to 1$. The natural transformation induced by it, denoted $\pi_{D}\colon D(1)\to D(0)$, is equivalent to two natural transformations on $\mathsf{Dir}$ via the adjunctions in Eq. 1: $n{\cdot}0^{\mathcal{y}}\to n,\qquad D(1)\xrightarrow{\pi_{D}}D(0),\qquad n\to n^{\mathcal{y}}.$ (2) The one labeled $\pi_{D}$ is also $D(0!)$, where $0!\colon 0\to 1$ is the unique function of that type. The composite of two polynomial functors $\mathsf{Fin}\to\mathsf{Fin}$ is again polynomial, $(P\circ Q)(n)\coloneqq P(Q(n))$; this gives a nonsymmetric monoidal structure on $\mathsf{Poly}$. The monoidal unit is $\mathcal{y}$. Day convolution for the cartesian product monoidal structure provides symmetric monoidal structure $\otimes\colon\mathsf{Poly}\times\mathsf{Poly}\to\mathsf{Poly}$, for which the monoidal unit is $\mathcal{y}$. This monoidal structure—like the Cartesian monoidal structure—distributes over $+$ We can write an explicit formula for $P\otimes Q$, with $P,Q$ as in Eq. 1: $P\otimes Q=\sum_{i=1}^{P(1)}\sum_{j=1}^{Q(1)}\mathcal{y}^{p_{i}q_{j}}$ (3) We call this the _Dirichlet product_ of polynomials, for reasons we will see in Remark 1. The Dirichlet monoidal structure is closed; that is, for any $A,Q:\mathsf{Poly}$ we define $[A,Q]\coloneqq\prod_{i:A(1)}Q\circ(a_{i}\mathcal{y}),$ (4) for example $[n\mathcal{y},\mathcal{y}]\cong\mathcal{y}^{n}$ and $[\mathcal{y}^{n},\mathcal{y}]\cong n\mathcal{y}$. For any polynomial $A$ there is an $(-\otimes A)\dashv[A,-]$ adjunction $\mathsf{Poly}(P\otimes A,Q)\cong\mathsf{Poly}(P,[A,Q]).$ (5) In particular we recover Lemma 2 using Eqs. 4 and 1. The cartesian monoidal structure on $\mathsf{Poly}$ is also closed, $\mathsf{Poly}(P\times A,Q)\cong\mathsf{Poly}(P,Q^{A})$, and the formula for $Q^{A}$ is similar to Eq. 4: $Q^{A}\coloneqq\prod_{i:A(1)}Q\circ(a_{i}+\mathcal{y}).$ If we define the _global sections_ functor $\Gamma\colon\mathsf{Poly}\to\mathsf{Fin}^{\textnormal{op}}$ by $\Gamma P\coloneqq\mathsf{Poly}(P,\mathcal{y})$, or explicitly $\Gamma(P)=[P,\mathcal{y}](1)=\prod_{i}p_{i}$, we find that it is left adjoint to the Yoneda embedding $\leavevmode\hbox to107.58pt{\vbox to31.03pt{\pgfpicture\makeatletter\hbox{\hskip 53.78891pt\lower-21.44022pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ 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}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.23367pt}{-19.28745pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\Gamma P\mapsfrom P}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.23886pt}{-8.5819pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\scriptstyle\top}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys<EMAIL_ADDRESS> Each of the categories $\mathsf{Poly}$ and $\mathsf{Dir}$ has pullbacks, which we denote using “fiber product notation” $A\times_{C}B$. We can use pullbacks in combination with monad units $\eta_{P}\colon P\to P(1)$ and $\eta_{D}\colon D\to D(0)$ arising from Eq. 1 to recover Eq. 1: $P=\sum_{i=1}^{P(1)}P\times_{P(1)}`i\text{'}\qquad\text{and}\qquad D=\sum_{i=1}^{D(0)}D\times_{D(0)}`i\text{'}.$ ###### Remark 5. By a result of Rosebrugh and Wood [RW94], the category of finite sets is characterized amongst locally finite categories by the existence of the five left adjoints to its Yoneda embedding $k\mapsto y^{k}\colon\mathsf{Fin}\to\mathsf{Fin}^{\mathsf{Fin}^{\textnormal{op}}}$. The adjoint sextuple displayed in (1) is just the observation that five of these six functors restrict to the subcategory $\mathsf{Dir}$. ## Chapter 3 $\mathsf{Poly}$ and $\mathsf{Dir}$ in terms of bundles There is a bijection between the respective object-sets of these two categories $\displaystyle\operatorname{Ob}(\mathsf{Poly})$ $\displaystyle\xrightarrow{\cong}\operatorname{Ob}(\mathsf{Dir})$ $\displaystyle\sum_{i=1}^{n}\mathcal{y}^{k_{i}}$ $\displaystyle\mapsto\sum_{i=1}^{n}(k_{i})^{\mathcal{y}}.$ (1) We call this mapping the _Dirichlet transform_ and denote it using an over- line $P\mapsto\overline{P}$. We will see in Theorem 6 that this bijection extends to an equivalence $\mathsf{Poly}_{\textnormal{cart}}\cong\mathsf{Dir}_{\textnormal{cart}}$ between the subcategories of cartesian maps. ###### Remark 1. With the Dirichlet transform in hand, we see why $P\otimes Q$ can be called the Dirichlet product, e.g. in Eq. 3. Namely, the Dirichlet transform is strong monoidal with respect to $\otimes$ and the cartesian monoidal structure $\times$ in $\mathsf{Dir}$: $\overline{P\otimes Q}=\overline{P}\times\overline{Q}.$ ###### Proposition 2. There is a one-to-one correspondence between the set of polynomials in one variable, the set of Dirichlet polynomials, and the set of (isomorphism classes of) functions $\pi\colon s\to t$ between finite sets. ###### Proof. We already established a bijection $P\mapsto\overline{P}$ between polynomials and finite Dirichlet series in Eq. 1. Given a finite Dirichlet series $D$, we have a function $\pi_{D}\colon D(1)\to D(0)$ as in Eq. 2. And given a function $\pi\colon s\to t$, define $D_{\pi}\coloneqq\sum_{i=1}^{t}(d_{i})^{\mathcal{y}}$, where $d_{i}\coloneqq\pi^{-1}(i)$ for each $1\leq i\leq t$. (N.B. Rather than constructing $D_{\pi}$ from $\pi$ by hand, one could instead use a certain orthogonal factorization system on $\mathsf{Dir}$.) It is easy to see that the roundtrip on Dirichlet series is identity, and that the round-trip for functions is a natural isomorphism. ∎ We will upgrade Proposition 2 to an equivalence $\mathsf{Poly}_{\textnormal{cart}}\simeq\mathsf{Dir}_{\textnormal{cart}}$ between certain subcategories of $\mathsf{Poly}$ and $\mathsf{Dir}$ in Theorem 6. ###### Example 3. Under the identification from Proposition 2, both the polynomial $2\mathcal{y}^{3}+\mathcal{y}^{2}+3$ and the Dirichlet series $2{\cdot}3^{\mathcal{y}}+1{\cdot}2^{\mathcal{y}}+3{\cdot}0^{\mathcal{y}}$ correspond to the function $\bullet$$1$$\bullet$$2$$\bullet$$3$$\bullet$$4$$\bullet$$5$$\bullet$$6$$6\cong D(0)\cong$$\bullet$$(1,1)$$\bullet$$(1,2)$$\bullet$$(1,3)$$\bullet$$(2,1)$$\bullet$$(2,2)$$\bullet$$(2,3)$$\bullet$$(3,1)$$\bullet$$(3,2)$$8\cong D(1)\cong$$\pi$ (2) We can think of a function $\pi\colon s\to t$, e.g. that shown in (2), as a _bundle_ of fibers $\pi^{-1}(`i\text{'})$, one for each element $`i\text{'}\in t$. In Definition 4 we define two different notions of morphism between bundles. We will see in Theorem 6 that they correspond to morphisms in the categories $\mathsf{Poly}$ and $\mathsf{Dir}$. For any function $\pi^{\prime}\colon s^{\prime}\to t^{\prime}$ and function $f\colon t\to t^{\prime}$, denote by $f^{*}(\pi^{\prime})$ the pullback function as shown ${s\times_{t^{\prime}}s^{\prime}}$${s^{\prime}}$${t}$${t^{\prime}}$$\scriptstyle{f^{*}(\pi^{\prime})}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{f}$${\lrcorner}$ ###### Definition 4. Let $\pi\colon s\to t$ and $\pi^{\prime}\colon s^{\prime}\to t^{\prime}$ be functions between finite sets. * • a _bundle morphism_ consists of a pair $(f,f_{\sharp})$ where $f\colon t\to t^{\prime}$ is a function and $f_{\sharp}\colon\pi\to f^{*}(\pi^{\prime})$ is a morphism in the slice category over $t$; * • a _container morphism_ consists of a pair $(f,f^{\sharp})$ where $f\colon t\to t^{\prime}$ is a function and $f^{\sharp}\colon f^{*}(\pi^{\prime})\to\pi$ is a morphism in the slice category over $t$. We say a bundle morphism $(f,f_{\sharp})$ (resp. a container morphism $(f,f^{\sharp})$) is _cartesian_ if $f_{\sharp}$ (resp. $f^{\sharp})$ is an isomorphism. ${s}$${t\times_{t^{\prime}}s^{\prime}}$${s^{\prime}}$${t}$${t^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{f^{*}\pi^{\prime}}$$\scriptstyle{f_{\sharp}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{f}$${\lrcorner}$ ${s}$${t\times_{t^{\prime}}s^{\prime}}$${s^{\prime}}$${t}$${t^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{f^{*}\pi^{\prime}}$$\scriptstyle{f^{\sharp}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{f}$${\lrcorner}$ Figure 1: The categories $\mathsf{Bun}$ and $\mathsf{Cont}$ have the same objects, functions $\pi\colon s\to t$. Here a morphism $(f,f_{\sharp})\colon\pi\to\pi^{\prime}$ in $\mathsf{Bun}$ and a morphism $(f,f^{\sharp})\colon\pi\to\pi^{\prime}$ in $\mathsf{Cont}$ are shown. Define $\mathsf{Bun}$ (resp. $\mathsf{Cont}$) to be the category for which an object is a function between finite sets and a morphism is a bundle morphism (resp. container morphism); see Fig. 1. Denote by $\mathsf{Bun}_{\textnormal{cart}}$ (resp. $\mathsf{Cont}_{\textnormal{cart}}$) the subcategory of cartesian bundle morphisms. One may note that $\mathsf{Bun}$ is the Grothendieck construction of the self- indexing $\mathsf{Fin}_{/(-)}\colon\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Cat}$, while $\mathsf{Cont}$ is the Grothendieck construction of its point-wise opposite $(\mathsf{Fin}_{/(-)})^{\textnormal{op}}\colon\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Cat}$. The name _container_ comes from the work of Abbot, Altenkirch, and Ghani abbott2003categoriesabbott2005containersabbot2003categoriesthesis (see Remark 2.18 in [GK12] for a discussion of the precise relationship between the notion of container and the notion of polynomial and polynomial functor). ###### Remark 5. By the universal property of pullbacks, $\mathsf{Bun}\simeq\mathsf{Fin}^{\to}$ is equivalent (in fact isomorphic) to the category of morphisms and commuting squares in $\mathsf{Fin}$. Furthermore, $\mathsf{Bun}_{\textnormal{cart}}$ is equivalent to the category of morphisms and pullback squares in $\mathsf{Fin}$, and $\mathsf{Bun}_{\textnormal{cart}}\simeq\mathsf{Cont}_{\textnormal{cart}}$ (as in both cases a cartesian morphism $(f,f_{\sharp})$ or $(f,f^{\sharp})$ is determined by $f$ alone). Next we show that $\mathsf{Bun}\simeq\mathsf{Dir}$ is also equivalent to the category of Dirichlet functors, from Definition 1. Recall that a natural transformation is called _cartesian_ if its naturality squares are pullbacks. ###### Theorem 6. We have equivalences of categories $\mathsf{Poly}\simeq\mathsf{Cont}\qquad\text{and}\qquad\mathsf{Dir}\simeq\mathsf{Bun}.$ In particular, this gives an equivalence $\mathsf{Poly}_{\textnormal{cart}}\simeq\mathsf{Dir}_{\textnormal{cart}}$ between the category of polynomial functors and cartesian natural transformations and the category of Dirichlet functors and cartesian natural transformations. ###### Proof. The functors $P_{-}\colon\mathsf{Cont}\to\mathsf{Poly}$ and $D_{-}\colon\mathsf{Bun}\to\mathsf{Dir}$ are defined on each object, i.e. function $\pi\colon s\to t$, by the formula $\pi\mapsto P_{\pi}$ and $\pi\mapsto D_{\pi}\coloneqq\overline{P_{\pi}}$ as in Proposition 2. For each $1\leq i\leq t$, denote the fiber of $\pi$ over $i$ by $k_{i}\coloneqq\pi^{-1}(i)$. For any finite set $X$, consider the unique map $X!\colon X\to 1$. Applying $P_{-}$ and $D_{-}$ to it, we obtain the corresponding representable: $P_{X!}\cong\mathcal{y}^{X}$ and $D_{X!}\cong X^{\mathcal{y}}$. We next check that there are natural isomorphisms $\displaystyle\mathsf{Poly}(P_{X!},P_{\pi})\cong P_{\pi}(X)=\sum X^{k_{i}}\cong\mathsf{Cont}(X!,\pi),$ $\displaystyle\mathsf{Dir}(D_{X!},D_{\pi})\cong D_{\pi}(X)=\sum_{i=1}^{t}(k_{i})^{X}\cong\mathsf{Bun}(X!,\pi).$ (3) In both lines, the first isomorphism is the Yoneda lemma and the second is a computation using Definition 4 (see Fig. 1). Thus we define $P_{-}$ on morphisms by sending $f\colon\pi\to\pi^{\prime}$ in $\mathsf{Cont}$ to the “compose-with-$f$” natural transformation, i.e. having $X$-component $\mathsf{Cont}(X!,f)\colon\mathsf{Cont}(X!,\pi)\to\mathsf{Cont}(X!,\pi^{\prime})$, which is clearly natural in $X$. We define $D_{-}$ on morphisms similarly: for $f$ in $\mathsf{Bun}$, use the natural transformation $\mathsf{Bun}(-!,f)$. By definition, every object in $\mathsf{Poly}$ and $\mathsf{Dir}$ is a coproduct of representables, so to prove that we have the desired equivalences, one first checks that coproducts in $\mathsf{Cont}$ and $\mathsf{Bun}$ are taken pointwise: $(\pi\colon s\to t)+(\pi^{\prime}\colon s^{\prime}\to t^{\prime})\cong(\pi+\pi^{\prime})\colon(s+s^{\prime})\to(t+t^{\prime}),$ and then that $P_{\pi+\pi^{\prime}}=P_{\pi}+P_{\pi^{\prime}}$ and $D_{\pi+\pi^{\prime}}=D_{\pi}+D_{\pi^{\prime}}$; see Remark 4. By Remark 5, we know that $\mathsf{Bun}_{\textnormal{cart}}\simeq\mathsf{Cont}_{\textnormal{cart}}$, and we have just established the equivalences $\mathsf{Poly}\simeq\mathsf{Cont}$ and $\mathsf{Dir}\simeq\mathsf{Bun}$. It thus remains to check that the latter equivalences identify cartesian natural transformations in $\mathsf{Poly}$ with cartesian morphisms in $\mathsf{Cont}$, and similarly for $\mathsf{Dir}$ and $\mathsf{Bun}$. For polynomial functors, we may refer to [GK12, Section 2]. Turning to Dirichlet functors, we want to show that for any $f\colon D\to D^{\prime}$ the square ${D(1)}$${D^{\prime}(1)}$${D(0)}$${D^{\prime}(0)}$$\scriptstyle{f_{1}}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{f_{0}}$ (4) is a pullback in $\mathsf{Set}$ iff for all functions $g\colon X\to X^{\prime}$, the naturality square ${D(X^{\prime})}$${D^{\prime}(X^{\prime})}$${D(X)}$${D^{\prime}(X)}$$\scriptstyle{f_{X^{\prime}}}$$\scriptstyle{D(g)}$$\scriptstyle{D^{\prime}(g)}$$\scriptstyle{f_{X}}$ (5) is a pullback in $\mathsf{Set}$; we will freely use the natural isomorphism $D_{\pi}(X)\cong\mathsf{Bun}(X!,\pi)$ from Eq. 3. The square in Eq. 4 is a special case of that in Eq. 5, namely for $g\coloneqq 0!$ the unique function $0\to 1$; this establishes the only-if direction. To complete the proof, suppose that Eq. 4 is a pullback, take an arbitrary $g\colon X\to X^{\prime}$, and suppose given a commutative solid-arrow diagram as shown: ${X}$${X^{\prime}}$${D(1)}$${D^{\prime}(1)}$${1}$${1}$${D(0)}$${D^{\prime}(0)}$$\scriptstyle{g}$ We can interpret the statement that Eq. 5 is a pullback as saying that there are unique dotted arrows making the diagram commute, since $DX\cong\mathsf{Bun}(X!,D0!)$ and similarly for the other corners of the square in Eq. 5. So, we need to show that if the front face is a pullback, then there are unique diagonal dotted arrows as shown, making the diagram commute. This follows quickly from the universal property of the pullback. ∎ ###### Corollary 7. $\mathsf{Dir}$ is an elementary topos. ###### Proof. For any finite category $C$, the functor category $\mathsf{Fin}^{C}$ is an elementary topos. The result now follows from Remarks 5 and 6, noting that $\mathsf{Dir}\simeq\mathsf{Fin}^{\to}$. ∎ As we mentioned in the introduction, this all goes through smoothly when one drops all finiteness conditions. The general topos of Dirichlet functors is the category of (arbitrary) sums of representables $\mathsf{Set}^{\textnormal{op}}\to\mathsf{Set}$, and this is equivalent to the arrow category $\mathsf{Set}^{\to}$ and so is itself a topos. We conclude with the equivalence promised in Dirichlet Polynomials. ###### Theorem 8. A functor $D\colon\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Fin}$ is a Dirichlet polynomial if and only if it preserves connected limits, or equivalently wide pullbacks. ###### Proof. Let $D(\mathcal{y})=\sum_{i:D(0)}(d_{i})^{\mathcal{y}}$, and suppose that $J$ is any connected category. Then for any diagram $X\colon J\to\mathsf{Fin}$, we have $\displaystyle D(\operatorname*{colim}X_{j})$ $\displaystyle=\sum_{i:D(0)}(d_{i})^{\operatorname*{colim}X_{j}}$ $\displaystyle\cong\sum_{i:D(0)}\lim(d_{i})^{X_{j}}$ $\displaystyle\cong\lim\sum_{i:D(0)}(d_{i})^{X_{j}}$ $\displaystyle=\lim D(X_{j})$ since connected limits commute with sums in any topos (in particular $\mathsf{Set}$). Now suppose $D\colon\mathsf{Fin}^{\textnormal{op}}\to\mathsf{Fin}$ is any functor that preserves connected limits; in particular, it sends wide pushouts to wide pullbacks. Every finite set $X$ can be expressed as the wide pushout ${X}$${1}$${1}$${\cdots}$${1}$${1}$${0}$ of its elements. Therefore, we have the following limit diagram: ${D(X)}$${D(1)}$${D(1)}$${\cdots}$${D(1)}$${D(1)}$${D(0)}$ That is, an element of $D(X)$ is a family of elements $a_{x}\in D(1)$, one for each $x\in X$, such that the $D(0!)(a_{x})$ are all equal in $D(0)$. But this is just a bundle map, i.e. $D(X)\cong\mathsf{Bun}(X!,D(0!))$ where $X!\colon X\to 1$ and $D(0!)\colon D(1)\to D(0)$. Thus by Theorem 6, the functor $D$ is the Dirichlet polynomial associated to the bundle $D(0!)$. ∎ ### Acknowledgments The authors thank Joachim Kock, André Joyal, and Brendan Fong for helpful comments that improved the quality of this note. Spivak also appreciates support by Honeywell Inc. as well as AFOSR grants FA9550-17-1-0058 and FA9550-19-1-0113. Jaz Myers appreciates support by his advisor Emily Riehl and the National Science Foundation grant DMS-1652600. ## References * [AAG03] Michael Gordon Abbott, Thorsten Altenkirch and Neil Ghani “Categories of Containers” In _FoSSaCS_ , 2003 * [AAG05] Michael Abbott, Thorsten Altenkirch and Neil Ghani “Containers: Constructing strictly positive types” Applied Semantics: Selected Topics In _Theoretical Computer Science_ 342.1, 2005, pp. 3–27 * [Abb03] Michael Gordon Abbott “Categories of Containers”, 2003 * [BD] John Baez and James Dolan “This Week’s Finds 300” Accessed: 2020-02-16 URL: http://math.ucr.edu/home/baez/week300.html * [GK12] Nicola Gambino and Joachim Kock “Polynomial functors and polynomial monads” In _Mathematical Proceedings of the Cambridge Philosophical Society_ 154.1 Cambridge University Press (CUP), 2012, pp. 153–192 * [Joy81] André Joyal “Une théorie combinatoire des séries formelles” In _Advances in Mathematics_ 42.1, 1981, pp. 1–82 * [RW94] Robert Rosebrugh and R.. Wood “An Adjoint Characterization of the Category of Sets” In _Proc. Amer. Math. Soc_ 122, 1994, pp. 409–413

2024-09-04T02:54:58.798419

2003.04857

arxiv-papers

# Image Restoration for Under-Display Camera Yuqian Zhou1, David Ren2, Neil Emerton3, Sehoon Lim3, Timothy Large3 1IFP, UIUC, 2CIL, UC Berkeley, 3Microsoft ###### Abstract The new trend of full-screen devices encourages us to position a camera behind a screen. Removing the bezel and centralizing the camera under the screen brings larger display-to-body ratio and enhances eye contact in video chat, but also causes image degradation. In this paper, we focus on a newly-defined Under-Display Camera (UDC), as a novel real-world single image restoration problem. First, we take a 4k Transparent OLED (T-OLED) and a phone Pentile OLED (P-OLED) and analyze their optical systems to understand the degradation. Second, we design a Monitor-Camera Imaging System (MCIS) for easier real pair data acquisition, and a model-based data synthesizing pipeline to generate Point Spread Function (PSF) and UDC data only from display pattern and camera measurements. Finally, we resolve the complicated degradation using deconvolution-based pipeline and learning-based methods. Our model demonstrates a real-time high-quality restoration. The presented methods and results reveal the promising research values and directions of UDC. ## 1 Introduction Under-display Camera (UDC) is a new imaging system that mounts display screen on top of a traditional digital camera lens, as shown in Figure 1. Such a system has mainly two advantages. First, it brings a new product trend of full-screen devices [11] with larger screen-to-body ratio, which can provide better user perceptive and intelligent experience [12]. Without seeing the bezel and extra buttons, users can easily access more functions by directly touching the screen. Second, it provides better human computer interaction. By putting the camera in the center of the display, it enhances teleconferencing experiences with perfect gaze tracking, and it is increasingly relevant for larger display devices such as laptops and TVs. Unlike pressure or fingerprint sensors that can be easily integrated into a display, it is relatively difficult to retain full functionality of an imaging sensor after mounting it behind a display. The imaging quality of a camera will be severely degraded due to lower light transmission rate and diffraction effects. As a result, images captured will be noisy and blurry. Therefore, while bringing better user experience and interaction, UDC may sacrifice the quality of photography, face processing [35] and other downstream vision tasks. Restoring and enhancing the images captured by UDC system will be desired. Figure 1: The newly proposed imaging system named Under-Display Camera (UDC). We mount display screen on top of a traditional digital camera lens. The design brings new trend of full-screen devices. Traditional image restoration approaches form the task as an inverse problem or an optimization problem like Maximum-a-Posterior (MAP). For the UDC problem, for practical purposes, the proposed image restoration algorithm and system are expected to work in real-time. Therefore, deconvolutional-based methods like Wiener Filter [14] should be preferred. Deconvolution is the inverse process of convolution and recovers the original signal from the point-spread-function (PSF)-convolved image. The fidelity of the deconvolution process is dependent on the space-invariance of the PSF over the image field of-view (FOV) and on a low condition number for the inverse of the PSF [19]. For strongly non-delta-function-like PSFs such as those encountered when imaging through a display, the value of condition number can be large. For such PSFs an additional denoising step may be essential. Another option is the emerging discriminative learning-based image restoration model. Data-driven discriminative learning-based image restoration models usually outperform traditional methods in specific tasks like image de-noising [42, 48, 47, 26, 43, 3], de-bluring [21, 28],de-raining [40, 39], de-hazing [13, 33], super-resolution [22, 37], and light-enhancement [9]. However, working on synthesis data with single degradation type, existing models can be hardly utilized to enhance real-world low-quality images with complicated or combined degradation types. To address complicated real degradation like the UDC problem, directly collecting real paired data or synthesizing near- realistic data after fully understanding the degradation model is necessary. In this paper, we present the first study to define and analyze the novel Under-Display Camera (UDC) system from both optics and image restoration viewpoints. For optics, we parse the optical system of the UDC pipeline and analyze the characteristics of light transmission. Then we relate the obtained intuitions and measurements to an image restoration pipeline, and propose two ways of resolving the single-image restoration: A deconvolution-based Wiener Filter [29] pipeline (DeP) and a data-driven learning-based approach. Specifically, we regard UDC restoration as a combination of tasks such as low- light enhancement, de-blurring, and de-noising. Without loss of generality, our analysis focuses on two types of displays, a 4K Transparent Organic Light-Emitting Diode (T-OLED) and a phone Pentile OLED (P-OLED), and a single camera type, a 2K FLIR RGB Point Grey research camera. To obtain the real imaging data and measure the optical factors of the system, we also propose a data acquisition system using the above optical elements. In summary, the main contributions of our paper are: (1) A brand new imaging system named Under-Display Camera (UDC) is defined, measured and analyzed. Extensive experiments reveal the image degradation process of the system, inspiring better approaches for restoring the captured images. (2) As a baseline, two practical and potential solutions are proposed, including conventional Wiener Filter and the recent learning-based method. (3) Adopting the newly-assembled image acquisition system, we collect the first Under- Display Camera (UDC) dataset which will be released and evaluated by the public. ## 2 Related Work #### Real-world Image Reconstruction and Restoration Image restoration for UDC [46, 24, 23, 49] can be categorized into the problem of Real-world restoration[3, 45]. It is becoming a new concept in low-level vision. In the past decades, low-level vision works on synthetic data (denoising on AWGN and SR on Bicubic), but the models are not efficient for images with real degradation such as real noises or blur kernels. Making models perform better on real-world inputs usually requires new problem analysis and a more challenging data collection. Recently, researchers also worked on challenging cases like lensless imaging problems [30, 27, 20], or integrating optics theory with High Dynamic Range imaging [34]. Previously, there has been two common ways to prepare adaptive training data for real- world problems: real data collection and near-realistic data synthesis. Recently, more real noise datasets such as DND [31], SIDD [2, 28], and RENOIR [5], have been introduced to address practical denoising problems. Abdelrahman et al. [3] proposed to estimate ground truth from captured smartphone noise images, and utilized the paired data to train and evaluate the real denoising algorithms. In addition to noise, Chen et al. first introduced the SID dataset [9] to resolve extreme low-light imaging. In the area of Single Image Super Resolution (SISR), researchers considered collecting optical zoom data [45, 10] to learn better computational zoom. Other restoration problems including reflection removal [36, 32] also follow the trend of real data acquisition. Collecting real data suffers from limitation of scene variety since most previous models acquire images of postcards, static objects or color boards. In this paper, we propose a novel monitor-camera imaging system, to add real degradation to the existing natural image datasets like DIV2K [4]. A realistic dataset can be synthesized if the degradation model is fully understood and resolved. One good practice of data synthesis is generating real noises on raw sensors or RGB images. CBDNet [17] and Tim et al. [8] synthesized realistic noise by unfolding the in-camera pipeline, and Abdelhamed et al. [1] better fitted the real noise distribution with flow- based generative models. Zhou et al. [48] adapted the AWGN-RVIN noise into real RGB noise by analyzing the demosacing process. Other physics-based synthesis was also explored in blur[7] or hazing[6]. For the UDC problem in this paper, we either collected real paired data, or synthesized near- realistic data from model simulation. In particular, we applied the theory of Fourier optics to simulate the diffraction effects, and further adjusted the data with other camera measurements. Our data synthesizing pipeline demonstrates a promising performance for addressing real complicated degradation problem. (a) (b) Figure 2: Image formation pipeline of under-display camera (UDC) problem. (a) Image Formation Pipeline. (b)Optics characteristics of UDC. The structure of the 4K T-OLED has a grating-like pixel layout. P-OLED differs from T-OLED in sub-pixel design. From left to right: Micrography of display patterns, PSFs (red light only) and MTFs (red, green, and blue). ## 3 Formulation In this section, we discuss the optical system and image formation process of the proposed UDC imaging system. We analyze the degradation type, light transmission rate and visualize the Point Spread Function (PSF). Moreover, we formulate the image formation pipeline to compute simulated PSF from measurements. ### 3.1 Optical System Analysis Optical Elements. In our experiments, we focus on the Organic Light-Emitting Diode (OLED) displays [38] as they have superior optical properties compared to the traditional LCDs (Liquid Crystal Display). Due to confidentiality reasons it is often difficult to obtain the sample materials used for demos from commercial companies. In this case, we select the displays with different transparencies to improve the generalization. Note that all the displays are non-active in our experiments, since in real scenario, the display can be turned off locally by setting black pixels on local regions of the OLED display when the camera is in operation to 1) reduce unnecessary difficulty from display contents while not affecting user experience and 2) provide users with the status of the device and thus ensure privacy. Owing to transparent materials being used in OLED display panels, visible lights can be better transmitted through the OLEDs than LCDs. In the meantime, pixels are also arranged such that open area is maximized. In particular, we focus on 4k Transparent OLED (T-OLED) and a phone Pentile OLED (P-OLED). Figure 2 is a micrograph illustration of the pixel layout in the two types of OLED displays. The structure of the 4K T-OLED has a grating-like pixel layout. P-OLED differs from T-OLED in sub-pixel design. It follows the basic structure of RGBG matrix. Table 1: Comparison of two displays in terms of light transmission rate and physical pixel layout and open areas. Metrics | T-OLED | P-OLED ---|---|--- Pixel Layout Type | Stripe | Pentile Open Area | 21$\%$ | 23$\%$ Transmission Rate | 20$\%$ | 2.9$\%$ Major Degradation | Blur, Noise | Low-light, Color Shift, Noise Light Transmission Rate. We measure the transmission efficiency of the OLEDs by using a spectrophotometer and white light source. Table 1 compares the light transmission rate of the two displays. For T-OLED, the open area occupies about 21$\%$, and the light transmission rate is around 20$\%$. For P-OLED, although the open area can be as large as 23$\%$, the light transmission rate is only 2.9$\%$. The loss of photons can be attributed mainly to the structure of P-OLED. First, the P-OLED has a finer pixel pitch, so photos are scattered to higher angles comparing to the T-OLED. As a result, high angle photons are not collected by the lens. Second, P-OLED is a flexible/bendable display, which has a poly-amide substrate on which the OLED is formed. Such a substrate has relatively low transmission efficiency, causing photons to be absorbed. The absorbed light with certain wavelengths may make the images captured through a polyamide-containing display panel by a UDC appear yellow. As a result, imaging through a P-OLED results in lower signal-to-noise ratio (SNR) comparing to using a T-OLED, and has a color shift issue. One real imaging example is shown in Figure 4. Diffraction Pattern and Point Spread Function (PSF). Light diffracts as it propagates through obstacles with sizes that are similar to its wavelength. Unfortunately, the size of the openings in the pixel layout is on the order of wavelength of visible light, and images formed will be degraded due to diffraction. Here we characterize our system by measuring the point spread function (PSF). We do so by pointing a collimated red laser beam ($\lambda=$ 650nm) at the display panel and recording the image formed on the sensor, as demonstrated in Figure 1 and 2. An ideal PSF shall resemble a delta function, which then forms a perfect image of the scene. However, light greatly spreads out in UDC. For T-OLED, light spreads mostly across the horizontal direction due to its nearly one dimensional structure in the pixel layout, while for P-OLED, light is more equally distributed as the pixel layout is complex. Therefore, images captured by UDC are either blurry (T-OLED) or hazy (P-OLED). Modulation Transfer Function (MTF) Modulation Transfer Function (MTF) is another important metric for an imaging system, as it considers the effect of finite lens aperture, lens performance, finite pixel size, noise, non- linearities, quantization (spatial and bit depth), and diffraction in our systems. We characterize the MTF of our systems by recording sinusoidal patterns with increasing frequency in both lateral dimensions, and we report them in Figure 2. For T-OLED, contrasts along the horizontal direction are mostly lost in the mid-band frequency due to diffraction. This phenomenon is due to the nearly one-dimensional pixel layout of the T-OLED. Figure 4 shows severe smearing horizontally when putting T-OLED in front of the camera. While for P-OLED, the MTF is almost identical to that of display-free camera, except with severe contrast loss. Fortunately, however, nulls have not been observed in any particular frequencies. ### 3.2 Image Formation Pipeline In this section, we derive the image formation process of UDC based on the analysis in the previous sections. Given a calibrated pixel layout and measurements using a specific camera, degraded images can be simulated from a scene. From the forward model, we can compute the ideal PSF and consequently synthesize datasets from ground truth images. Given an object in the scene $\mathbf{x}$, the degraded observation $\mathbf{y}$ can be modeled by a convolution process, $\mathbf{y}=(\gamma\mathbf{x})\otimes\mathbf{k}+\mathbf{n},$ (1) where $\gamma$ is the intensity scaling factor under the current gain setting and display type, $\mathbf{k}$ is the PSF, and $\mathbf{n}$ is the zero-mean signal-dependent noise. Notice that this is a simple noise model that approximately resembles the combination of shot noise and readout noise of the camera sensor, and it will be discussed in a later section. Intensity Scaling Factor ($\gamma$) The intensity scaling factor measures the changing ratio of the average pixel values after covering the camera with a display. It simultaneously relates to the physical light transmission rate of the display, as well as the digital gain $\delta$ setting of the camera. $\gamma$ can be computed from the ratio of $\delta$-gain amplified average intensity values $I_{d}(\delta,s)$ at position $s$ captured by UDC, to the 0-gain average intensity values $I_{nd}(0,s)$ by naked camera within an enclosed region $S$. It is represented by, $\gamma=\frac{\int_{S}I_{d}(\delta,s)ds}{\int_{S}I_{nd}(0,s)ds}$ (2) Diffraction Model We approximate the blur kernel $\mathbf{k}$, which is the Point Spread Function (PSF) of the UDC. As shown in Figure 1, in our model, we assume the display panel is at the principle plane of the lens. We also assume the input light is monochromatic plane wave with wavelength $\lambda$ (i.e. perfectly coherent), or equivalently light from a distance object with unit amplitude. Let the display pattern represented by transparency with complex amplitude transmittance be $g(m,n)$ at the Cartesian co-ordinate $(m,n)$, and let the camera aperture/pupil function $p(m,n)$ be 1 if $(m,n)$ lies inside the lens aperture region and 0 otherwise, then the display pattern inside the aperture range $g_{p}(m,n)$ becomes, $g_{p}(m,n)=g(m,n)p(m,n).$ (3) At the focal plane of the lens (i.e. 1 focal length away from the principle plane), the image measured is the intensity distribution of the complex field, which is proportional to the Fourier transform of the electric field at the principle plane [16]: $I(u,v)\propto\left|{\iint}^{\infty}_{-\infty}g_{p}(m,n)\exp\left[-j\frac{2\pi}{\lambda f}(mu+nv)\right]\text{d}m\text{d}n\right|^{2}.$ (4) Suppose $G_{p}(v_{m},v_{n})=\mathscr{F}(g_{p}(m,n))$, where $\mathscr{F}(\cdot)$ is the Fourier transform operator, then $I(u,v)\propto\left|G_{p}(v_{m},v_{n})\right|^{2}=\left|G_{p}(\frac{u}{\lambda f},\frac{v}{\lambda f})\right|^{2},$ (5) which performs proper scaling on the Fourier transform of the display pattern on the focal plane. Therefore, to compute the PSF $\mathbf{k}$ for image $\mathbf{x}$, we start from computing Discrete Fourier Transform (DFT) with squared magnitude $M(a,b)=|\hat{G_{p}}(a,b)|^{2}$ of the $N\times N$ microscope transmission images $\hat{g_{p}}$ of the display pattern and re-scaling it. Then, the spatial down-sampling factor $r$ (denoted by $\downarrow r$) becomes, $r=\frac{1}{\lambda f}\cdot{\delta_{N}N}\cdot{\rho},$ (6) where $\delta_{N}$ is the pixel size of the $\hat{g_{p}}$ images, and $\rho$ is the pixel size of the sensor. Finally, $\mathbf{k}$ can be represented as $k(i,j)=\frac{M_{\downarrow r}(i,j)}{\sum_{(\hat{i},\hat{j})}M_{\downarrow r}(\hat{i},\hat{j})}.$ (7) $k$ is a normalized form since we want to guarantee that it represents the density distribution of the intensity with diffraction effect. Note that only PSF for a single wavelength is computed for simplicity. However, scenes in the real-world are by no means monochromatic. Therefore, in order to calculate an accurate color image from such UDC systems, PSF for multiple wavelengths shall be computed. More details will be shown in Section 4.2. Adding Noises We follow the commonly used shot-read noise model [8, 18, 25] to represent the real noise on the imaging sensor. Given the dark and blur signal $w=(\gamma\mathbf{x})\otimes\mathbf{k}$, the shot and readout noise can be modeled by a heteroscedastic Gaussian, $\mathbf{n}\sim\mathcal{N}(\mu=0,\sigma^{2}=\lambda_{read}+\lambda_{shot}w),$ (8) where the variance $\sigma$ is signal-dependent, and $\lambda_{read}$ , $\lambda_{shot}$ are determined by camera sensor and gain values. ## 4 Data Acquisition and Synthesis We propose an image acquisition system called Monitor-Camera Imaging System (MCIS). In particular, we display natural images with rich textures on high- resolution monitor and capture them with a static camera. The method is more controllable, efficient, and automatic to capture a variety of scene contents than using mobile set-ups to capture limited static objects or real scenes. ### 4.1 Monitor-Camera Imaging System Figure 3: Monitor-Camera Imaging System (MCIS). MCIS consists of a 4K LCD monitor, the 2K FLIR RGB Point-Grey research camera, and a panel that is either T-OLED, P-OLED or Glass(i.e. no display). The camera is mounted on the center line of the 4K monitor, and adjusted to cover the full monitor range. (a) Display-free (b) TOLED (c) POLED Figure 4: Real samples collected by the proposed MCIS. Images captured by T-OLED are blur and noisy, while those captured by P-OLED are low-light, color-shifted and hazy. The system architecture is shown in Figure 3. MCIS consists of a 4K LCD monitor, the 2K FLIR RGB Point-Grey research camera, and a panel that is either T-OLED, P-OLED or Glass(i.e. no display). The camera is mounted on the center line of the 4K monitor, and adjusted to cover the full monitor range. We calibrate the camera gain by measuring a $256\times 256$ white square shown on the monitor and matching the RGB histogram. For fair comparison and simplicity, we adjust the focus and fix the aperture to f/1.8. It guarantees a reasonable pixel intensity range avoiding saturation while collecting data with no gain. Suppose we develop a real-time video system, the frame rate has to be higher than 8 fps. So the lowest shutter speed is 125 ms for the better image quality and the higher Signal-to-Noise Ratio (SNR). Table 2: Camera Settings for different set of collected data Parameteres | No-Display | T-OLED | P-OLED ---|---|---|--- Aperture | f/1.8 FPS/Shutter | 8/125ms Brightness | 0 Gamma | 1 Gain | 1 | 6 | 25(Full) White-balance | Yes | None | None We split 300 images from DIV2K dataset [4], and take turns displaying them on a 4K LCD in full screen mode. We either rotate or resize the images to maintain the Aspect Ratio. For training purposes, we capture two sets of images, which are the degraded images $\\{y_{i}\\}$, and the degradation-free set $\\{x_{i}\\}$. To capture $\\{x_{i}\\}$, we first cover the camera with a thin glass panel which has the same thickness as a display panel. This process allows us to avoid the pixel misalignment issues caused by light refraction inside the panel. To eliminate the image noises in $\\{x_{i}\\}$, we average the 16 repeated captured frames. Then we replace the glass with a display panel (T-OLED or P-OLED), calibrate the specific gain value avoiding saturation, and capture $\\{y_{i}\\}$. For each set, we record both the 16-bit 1-channel linear RAW CMOS sensor data as well as the 8-bit 3-channel linear RGB data after in-camera pipeline that includes demosaicing. The collected pairs are naturally well spatially-aligned in pixel-level. They can be directly used for deep model training without further transformations. Due to the yellow substrate inside the P-OLED, certain light colors, especially blue, are filtered out and changes the white balance significantly. We therefore did not further alter the white balance. The light transmission ratio of P-OLED is extremely low, so we set up the gain value to be the maximum (25) for higher signal values. All the detailed camera settings for the two display types are shown in Table 2. One real data sample is shown in Figure 4. As discussed and analyzed in Section 3.1, images captured by T-OLED are blur and noisy, while those captured by P-OLED are low-light, color- shifted and hazy. Table 3: Measured parameters for data synthesis Parameteres | T-OLED | P-OLED ---|---|--- | R | G | B | R | G | B $\gamma$ | 0.97 | 0.97 | 0.97 | 0.34 | 0.34 | 0.20 $\lambda$ (nm) | 640 | 520 | 450 | 640 | 520 | 450 r | 2.41 | 2.98 | 3.44 | 2.41 | 2.98 | 3.44 ### 4.2 Realistic Data Synthesis Pipeline We follow the image formation pipeline to simulate the degradation on the collected $\\{x_{i}\\}$. A model-based data synthesis method will benefit concept understanding and further generalization. Note that all the camera settings are the same as the one while collecting real data. We first transform the 16-bit raw sensor data $\\{x_{i}\\}$ into four bayer channels $x_{r}$, $x_{gr}$, $x_{gl}$, and $x_{b}$. Then, we multiply the measured intensity scaling factor $\gamma$, compute the normalized and scaled PSF $k$, and add noises to the synthesize degraded data. Measuring $\gamma$: To measure $\gamma$ for each channel using the MCIS, we select the region of interest $S$ to be a square region of size $256\times 256$, and display the intensity value input from 0 to 255 with stride 10 on the monitor. We then record the average intensity both with and without the display for each discrete intensity value, and plot the relationship between display-covered values and no-display-covered ones. Using linear regression, we obtain the ratios of lines for different RGGB channel. For T-OLED, the measured $\gamma$ is 0.97, same for all the channels. For P-OLED, $\gamma=0.20$ for the blue channel, and $\gamma=0.34$ for the other three channels. Computing PSF: Following equation 3, we acquire the transmission microscope images of the display pattern and crop them with the approximated circular aperture shape with diameter $3333\mu m$, the size of the camera aperture. In equation 6, the $\delta_{N}N$ is $3333\mu m$. $\rho$ equals to $1.55\mu m/pixel$ in Sony sensor. However, after re-arranging the raw image into four RGGB channels, $\rho$ becomes 3.1 for each channel. The focal length is $6000\mu m$. $\lambda=(640,520,450)$ for R, G, B channel, which are the approximated center peaks of the R, G, B filters respectively on the sensor. It yields the down-sampling ratio $r=(2.41,2.98,3.44)$ for the R, G, and B channels. Adding Noises: We measure $\lambda_{read}$ and $\lambda_{shot}$ to estimate the noise statistics. We display random patterns within the $256\times 256$ window on the monitor, collect the paired noisy and noise-free RAW data, and compute their differences. For each of the RGGB channel, we linearly regress the function of noise variance to the intensity value, and obtain the ratio as the shot noise variance, and the y-intersection as the readout noise variance. We then repeat the process 100 times and collect pairs of data points. Finally, we estimate the distribution and randomly sample $\lambda_{read}$ and $\lambda_{shot}$. All the measurements are listed in Table 3. Figure 5: Network structure of the proposed UNet. It takes a 4-channel RAW sensor data observation $y$, and outputs the restored 3-channel RGB image $x$. ## 5 Image Restoration Baselines We use the collected real paired data, synthetic paired data, simulated PSF, and all the necessary measurements to perform image restoration. We split the 300 pairs of images in the UDC dataset into 200 for training, 40 for validation and 60 images in the testing partition. All the images have a resolution of $1024\times 2048$. ### 5.1 Deconvolution Pipeline (DeP) The DeP is a general-purpose conventional pipeline concatenating denoising and deconvolution (Wiener Filter), which is an inverse process of the analyzed image formation. To better utilize the unsupervised Wiener Filter (WF) [29], we first apply the BM3D denoiser to each RAW channel separately, afterwards we linearly divide the measured $\gamma$ with the outputs for intensity scaling. After that, WF is applied to each channel given the pre-computed PSF $\mathbf{k}$. Finally, RAW images with bayer pattern are demosaiced by linear interpolation. The restored results are evaluated on the testing partition of the UDC dataset. ### 5.2 Learning-based Methods UNet. We propose a learning-based restoration network baseline as shown in Figure 5. The proposed model takes a 4-channel RAW sensor data observation $y$, and outputs the restored 3-channel RGB image $x$. The model conducts denoising, debluring, white-balancing, intensity scaling, and demosaicing in a single network, whose structure is basically a UNet. We split the encoder into two sub-encoders, one of which is for computing residual details to add, and the other one learns content encoding from degraded images. By splitting the encoder, compared with doubling the width of each layer, we will have fewer parameters, and make the inference and learning more efficient. To train the model from paired images, we apply the $L_{1}$ loss, which will at large guarantee the temporal stability compared with adversarial loss [15]. Besides, we also apply $SSIM$ and Perception Loss (VGG Loss) for ablation study. We crop patches of $256\times 256$, and augment the training data using the raw image augmentation [26] while preserving the RGGB bayer pattern. We train the model for 400 epochs using Adam optimizer ($\beta_{1}=0.9$, $\beta_{2}=0.999$ and $\epsilon=10^{-8}$) with learning rate $10^{-4}$ and decay factor 0.5 after 200 epoches. We also train the same structure using the synthetic data (denoted as UNet(Syn)) generated by the pipeline proposed in section 4.2. ResNet. Additionally, a data-driven ResNet trained with the same data is utilized for evaluation. To our knowledge, UNet and ResNet-based structures are two widely-used deep models for image restoration. We use 16 residual blocks with a feature width of 64 for our ResNet architecture, just as Lim et al. do for EDSR [22]. The model also takes 4-channel RAW data, and outputs 3-channel RGB images. The data-driven models cannot be directly adaptive to UDC inputs if only trained with bi-cubic degradation. We did not compare with their model structures because model novelty is not our main claim, and the presented two methods are the most general ways which can achieve real-time inference as the baselines. Other model variants can be further explored in future work. (a) T-OLED (b) DeP (c) UNet(Syn) (d) UNet (e) GT Figure 6: Restoration Results Comparison for T-OLED. GT: Ground Truth. (a) P-OLED (b) DeP (c) UNet(Syn) (d) UNet (e) GT Figure 7: Restoration Results Comparison for P-OLED. GT: Ground Truth. Table 4: Pipeline Comparison . 4K T-OLED P-OLED Pipeline Structure $\\#$P $\downarrow$ GFLOPs $\downarrow$ T $\downarrow$ PSNR/SSIM $\uparrow$ LPIPS $\downarrow$ PSNR/SSIM $\uparrow$ LPIPS $\downarrow$ DeP - - - 28.50/0.9117 0.4219 16.97/0.7084 0.6306 ResNet 1.37M 721.76 92.92 36.26/0.9703 0.1214 27.42/0.9176 0.2500 UNet(Syn) 8.93M 124.36 21.37 32.42/0.9343 0.1739 25.88/0.9006 0.3089 UNet 8.93M 124.36 21.37 36.71/0.9713 0.1209 30.45/0.9427 0.2219 Table 5: Ablation Study on UNet alternatives. Alternatives | | | | 4K T-OLED | P-OLED ---|---|---|---|---|--- | $\\#$P $\downarrow$ | GFLOPs $\downarrow$ | T $\downarrow$ | PSNR/SSIM $\uparrow$ | LPIPS $\downarrow$ | PSNR/SSIM $\uparrow$ | LPIPS $\downarrow$ UNet Basseline | 8.93M | 124.36 | 21.37 | 36.71/0.9713 | 0.1209 | 30.45/0.9427 | 0.2219 Double Width | 31.03M | 386.37 | 40.42 | 37.00/0.9730 | 0.1171 | 30.37/0.9425 | 0.2044 Single Encoder | 7.76M | 97.09 | 15.85 | 36.47/0.9704 | 0.1288 | 30.26/0.9387 | 0.2318 $L_{1}\rightarrow L_{1}+SSIM$ | 8.93M | 124.36 | 21.37 | 36.69/0.9714 | 0.1246 | 30.37/0.9403 | 0.2131 $L_{1}\rightarrow L_{1}+VGG$ | 8.93M | 124.36 | 21.37 | 36.31/0.9711 | 0.1130 | 30.37/0.9403 | 0.2130 ## 6 Experimental Results ### 6.1 Qualitative and Quantitative Comparisons The qualitative restoration results are shown in Figure 6 and 7. As shown, image Deconvolution Pipeline (DeP) successfully recovers image details but still introduces some artifacts, and suffers from the inaccuracy of the computed ideal PSF. The UNet-based model achieves better visual quality and denoising performance. The results of UNet trained with the synthetic data are visually better than DeP. The quantitative results are listed in Table 4. We report the performance in PSNR, SSIM, a perceptual metric LPIPS [44], inference time T (ms/MPixels) and GFLOPs. The inference time is tested with one single Titan X, and the GFLOPs is computed by input size of $512\times 1024\times 4$. ResNet achieves a comparable performance to UNet, but it requires more computation operations and longer inference time. The proposed UNet-based structure is efficient and effective, which can therefore be deployed for real-time inference for high- resolution inputs with a single GPU. In Table 4, we demonstrate that synthetic data still has gaps with the real data, though it has already greatly out- performed the DeP for the two display types. The domain gap mainly comes from the following aspects. First, due to the existing distances between display and lens, in real data there appears visible patterns of the display on the image plane. We recall in the assumption of the diffraction model, the display panel is exactly at the principle plane of the lens system. The cause of the visible bands are illustrated in the supplementary material. Second, the approximated light transmission rate may not be accurate, the measured values may be influenced by other environment light sources. Third, impulse noise caused by dead pixels or over-exposure in the camera sensors widely exist in the real dataset. Those factors provide more improvement space for this work. Figure 8: Face detection performance before and after applying restoration. Without display, the original face recall rate is 60$\%$. Covering the camera with T-OLED or P-OLED will decrease the recall rate to 8$\%$ and 0$\%$. After image restoration, the recall rates recovered back to 56$\%$ and 39$\%$. ### 6.2 Ablation Study For the best-performed UNet structure, we compare different UNet alternatives in Table 5. We increase the parameter size by splitting the original encoders into two sub-encoders, so the performance is also increased. The increment parameter size and inference time is far less than doubling the width of each layer of UNet, but the performance improvement is comparable (T-OLED), even better (P-OLED). We claim that the proposed UNet structure will both maintain a small number of parameters and operations, and achieve a real-time high- quality inference. To try alternative loss functions, we add $SSIM$ or $VGG$ loss in additional to $L_{1}$ loss with 1:1 ratio. However, the performance gains on either $SSIM$ or perceptual metric LPIPS are not significant enough, and are not visually distinctive. Adversarial loss is not implemented due to its temporal instability of GAN-based training. ### 6.3 Downstream Applications The proposed image restoration also enhances the performance of downstream applications including face detection. Figure 8 shows an example of detecting faces using MTCNN [41]. Without display, the original face recall rate is 60$\%$. Covering the camera with T-OLED or P-OLED will decrease the recall rate to 8$\%$ and 0$\%$. After image restoration, the recall rates are recovered to 56$\%$ and 39$\%$. ## 7 Conclusion and Limitations This paper defined and presented a novel imaging system named Under-Display- Camera (UDC). Deploying UDC to full-screen devices improves the user interaction as well as teleconferencing experience, but does harm to imaging quality and other downstream vision applications. We systematically analyzed the optical systems and modelled the image formation pipeline of UDC, and both collected real data using a novel acquisition system and synthesized realistic data and the PSF of the system using optical model. We then proposed to address the image restoration of UDC using a Deconvolution-based Pipeline (DeP) and data-driven learning-based methods. Our experiments showed that the former achieved basic restoration and the latter demonstrated an efficient high-quality restoration. The model trained with synthetic data also achieved a remarkable performance indicating the potential generalization ability. UDC problem has its promising research values in complicated degradation analysis. In real-world applications, other factors like an active display, reflection, lens flare etc. are still very challenging and complicated. Future work can be exploring UDC-specific restoration models and working with aperture and display researchers to analyze the influential factors of image degradation. It will make the restoration model generalized better for mass production, or helpful for down-stream tasks, as an ultimate goal. ## Appendix A Appendices (a) Display-free (b) T-OLED (c) P-OLED Figure A.1: More real data samples acquried by our MCIS set-up. (a) The image captured with camera covered by thin glass, (b)T-OLED, and (c) P-OLED. ### A.1 Real Data More examples in 8-bit RGB version in the UDC real dataset are shown in Fig. A.1. Each image has a high resolution of $1024\times 2048\times 3$. Images captured by T-OLED demonstrate a blur effect along the horizontal direction. Some spatial frequencies (i.e. vertical bands) are missing due to diffraction effects. Images captured by P-OLED are yellow-shifted, dark, and noisy. We also stored the 16-bit raw sensor data, which is mainly used for training and testing in the paper. ### A.2 Synthetic Data Figure A.2: Real and the computed point spread function (kernel). We follow the image formation pipeline to synthesize the near-realistic data. Given only the display pattern, and some specific measurements of the cameras, we could generate the blur kernels as shown in Fig. A.2 along with the degraded images for training. Fig. A.3 compares the synthetic data with the real data. Perceptually, two sets of data samples have similar visual characteristics. ### A.3 Visible Bands for T-OLED (a) Real data samples. (b) Synthetic data samples. Figure A.3: Comparison of real data and synthetic data. First row: T-OLED. Second row: P-OLED. (a) Synthetic data. (b) Real data with bands. Figure A.4: Visible bands in real data. In addition to the degradation formulated in the paper, there is another minor image artifact caused by the periodic grating-like pixel structure (i.e. T-OLED): superposition of periodic bands over the image at low to moderate visibility levels. As shown in the Fig. A.4, periodic bands are visible in the real data, but not in the synthetic data. We regard it as the main gap of data synthesis. Those bands are caused by the imperfect adhesion of the display to the camera lens. In the degradation model, we assume the display pattern or objects are exactly placed against the lens, while in practical set-up of our experiments, there is still a small distance between them. We can consider the grating as being imaged very out-of-focus on the sensor plane. There will be an image on the image sensors consisting of the grating convoluted with the very-out-of-focus point spread function – a circle. 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arxiv-papers

12020 # Multi-SimLex: A Large-Scale Evaluation of Multilingual and Cross-Lingual Lexical Semantic Similarity https://multisimlex.com/ Ivan Vulić ♠ ♠Equal contribution; English Faculty Building, 9 West Road Cambridge CB3 9DA, United Kingdom. E-mail: <EMAIL_ADDRESS>LTL, University of Cambridge Simon Baker ∗♠ LTL, University of Cambridge Edoardo Maria Ponti ∗♠ LTL, University of Cambridge Ulla Petti ∗ LTL, University of Cambridge Ira Leviant Technion City, Haifa 3200003, Israel. E-mail: <EMAIL_ADDRESS><EMAIL_ADDRESS>Faculty of Industrial Engineering and Management, Technion, IIT Kelly Wing ∗ LTL, University of Cambridge Olga Majewska ∗ LTL, University of Cambridge Eden Bar ∗∗ Faculty of Industrial Engineering and Management, Technion, IIT Matt Malone ∗ LTL, University of Cambridge Thierry Poibeau Rue Maurice Arnoux, 92120 Montrouge, France. E-mail<EMAIL_ADDRESS>LATTICE Lab, CNRS and ENS/PSL and Univ. Sorbonne nouvelle/USPC Roi Reichart ∗∗ Faculty of Industrial Engineering and Management, Technion, IIT Anna Korhonen ∗ LTL, University of Cambridge ###### Abstract We introduce Multi-SimLex, a large-scale lexical resource and evaluation benchmark covering datasets for 12 typologically diverse languages, including major languages (e.g., Mandarin Chinese, Spanish, Russian) as well as less- resourced ones (e.g., Welsh, Kiswahili). Each language dataset is annotated for the lexical relation of semantic similarity and contains 1,888 semantically aligned concept pairs, providing a representative coverage of word classes (nouns, verbs, adjectives, adverbs), frequency ranks, similarity intervals, lexical fields, and concreteness levels. Additionally, owing to the alignment of concepts across languages, we provide a suite of 66 cross-lingual semantic similarity datasets. Due to its extensive size and language coverage, Multi-SimLex provides entirely novel opportunities for experimental evaluation and analysis. On its monolingual and cross-lingual benchmarks, we evaluate and analyze a wide array of recent state-of-the-art monolingual and cross-lingual representation models, including static and contextualized word embeddings (such as fastText, M-BERT and XLM), externally informed lexical representations, as well as fully unsupervised and (weakly) supervised cross- lingual word embeddings. We also present a step-by-step dataset creation protocol for creating consistent, Multi-Simlex -style resources for additional languages. We make these contributions - the public release of Multi-SimLex datasets, their creation protocol, strong baseline results, and in-depth analyses which can be be helpful in guiding future developments in multilingual lexical semantics and representation learning - available via a website which will encourage community effort in further expansion of Multi- Simlex to many more languages. Such a large-scale semantic resource could inspire significant further advances in NLP across languages. ††issue: 1 ## 1 Introduction The lack of annotated training and evaluation data for many tasks and domains hinders the development of computational models for the majority of the world’s languages Snyder and Barzilay (2010); Adams et al. (2017); Ponti et al. (2019a). The necessity to guide and advance multilingual and cross-lingual NLP through annotation efforts that follow cross-lingually consistent guidelines has been recently recognized by collaborative initiatives such as the Universal Dependency (UD) project Nivre et al. (2019). The latest version of UD (as of March 2020) covers more than 70 languages. Crucially, this resource continues to steadily grow and evolve through the contributions of annotators from across the world, extending the UD’s reach to a wide array of typologically diverse languages. Besides steering research in multilingual parsing Zeman et al. (2018); Kondratyuk and Straka (2019); Doitch et al. (2019) and cross-lingual parser transfer Rasooli and Collins (2017); Lin et al. (2019); Rotman and Reichart (2019), the consistent annotations and guidelines have also enabled a range of insightful comparative studies focused on the languages’ syntactic (dis)similarities Bjerva and Augenstein (2018); Ponti et al. (2018a); Pires, Schlinger, and Garrette (2019). Inspired by the UD work and its substantial impact on research in (multilingual) syntax, in this article we introduce Multi-SimLex, a suite of manually and consistently annotated semantic datasets for 12 different languages, focused on the fundamental lexical relation of semantic similarity Budanitsky and Hirst (2006); Hill, Reichart, and Korhonen (2015). For any pair of words, this relation measures whether their referents share the same (functional) features, as opposed to general cognitive association captured by co-occurrence patterns in texts (i.e., the distributional information). Datasets that quantify the strength of true semantic similarity between concept pairs such as SimLex-999 Hill, Reichart, and Korhonen (2015) or SimVerb-3500 Gerz et al. (2016) have been instrumental in improving models for distributional semantics and representation learning. Discerning between semantic similarity and relatedness/association is not only crucial for theoretical studies on lexical semantics (see §2), but has also been shown to benefit a range of language understanding tasks in NLP. Examples include dialog state tracking Mrkšić et al. (2017); Ren et al. (2018), spoken language understanding Kim et al. (2016); Kim, de Marneffe, and Fosler-Lussier (2016), text simplification Glavaš and Vulić (2018); Ponti et al. (2018b); Lauscher et al. (2019), dictionary and thesaurus construction Cimiano, Hotho, and Staab (2005); Hill et al. (2016). Despite the proven usefulness of semantic similarity datasets, they are available only for a small and typologically narrow sample of resource-rich languages such as German, Italian, and Russian Leviant and Reichart (2015), whereas some language types and low-resource languages typically lack similar evaluation data. Even if some resources do exist, they are limited in their size (e.g., 500 pairs in Turkish Ercan and Yıldız (2018), 500 in Farsi Camacho-Collados et al. (2017), or 300 in Finnish Venekoski and Vankka (2017)) and coverage (e.g., all datasets which originated from the original English SimLex-999 contain only high-frequent concepts, and are dominated by nouns). This is why, as our departure point, we introduce a larger and more comprehensive English word similarity dataset spanning 1,888 concept pairs (see §4). Most importantly, semantic similarity datasets in different languages have been created using heterogeneous construction procedures with different guidelines for translation and annotation, as well as different rating scales. For instance, some datasets were obtained by directly translating the English SimLex-999 in its entirety Leviant and Reichart (2015); Mrkšić et al. (2017) or in part Venekoski and Vankka (2017). Other datasets were created from scratch Ercan and Yıldız (2018) and yet others sampled English concept pairs differently from SimLex-999 and then translated and reannotated them in target languages Camacho-Collados et al. (2017). This heterogeneity makes these datasets incomparable and precludes systematic cross-linguistic analyses. In this article, consolidating the lessons learned from previous dataset construction paradigms, we propose a carefully designed translation and annotation protocol for developing monolingual Multi-SimLex datasets with aligned concept pairs for typologically diverse languages. We apply this protocol to a set of 12 languages, including a mixture of major languages (e.g., Mandarin, Russian, and French) as well as several low-resource ones (e.g., Kiswahili, Welsh, and Yue Chinese). We demonstrate that our proposed dataset creation procedure yields data with high inter-annotator agreement rates (e.g., the average mean inter-annotator agreement for Welsh is 0.742). The unified construction protocol and alignment between concept pairs enables a series of quantitative analyses. Preliminary studies on the influence that polysemy and cross-lingual variation in lexical categories (see §2.3) have on similarity judgments are provided in §5. Data created according to Multi- SimLex protocol also allow for probing into whether similarity judgments are universal across languages, or rather depend on linguistic affinity (in terms of linguistic features, phylogeny, and geographical location). We investigate this question in §5.4. Naturally, Multi-SimLex datasets can be used as an intrinsic evaluation benchmark to assess the quality of lexical representations based on monolingual, joint multilingual, and transfer learning paradigms. We conduct a systematic evaluation of several state-of- the-art representation models in §7, showing that there are large gaps between human and system performance in all languages. The proposed construction paradigm also supports the automatic creation of 66 cross-lingual Multi-SimLex datasets by interleaving the monolingual ones. We outline the construction of the cross-lingual datasets in §6, and then present a quantitative evaluation of a series of cutting-edge cross-lingual representation models on this benchmark in §8. Contributions. We now summarize the main contributions of this work: 1) Building on lessons learned from prior work, we create a more comprehensive lexical semantic similarity dataset for the English language spanning a total of 1,888 concept pairs balanced with respect to similarity, frequency, and concreteness, and covering four word classes: nouns, verbs, adjectives and, for the first time, adverbs. This dataset serves as the main source for the creation of equivalent datasets in several other languages. 2) We present a carefully designed and rigorous language-agnostic translation and annotation protocol. These well-defined guidelines will facilitate the development of future Multi-SimLex datasets for other languages. The proposed protocol eliminates some crucial issues with prior efforts focused on the creation of multi-lingual semantic resources, namely: i) limited coverage; ii) heterogeneous annotation guidelines; and iii) concept pairs which are semantically incomparable across different languages. 3) We offer to the community manually annotated evaluation sets of 1,888 concept pairs across 12 typologically diverse languages, and 66 large cross- lingual evaluation sets. To the best of our knowledge, Multi-SimLex is the most comprehensive evaluation resource to date focused on the relation of semantic similarity. 4) We benchmark a wide array of recent state-of-the-art monolingual and cross- lingual word representation models across our sample of languages. The results can serve as strong baselines that lay the foundation for future improvements. 5) We present a first large-scale evaluation study on the ability of encoders pretrained on language modeling (such as bert Devlin et al. (2019) and xlm Conneau and Lample (2019)) to reason over word-level semantic similarity in different languages. To our own surprise, the results show that monolingual pretrained encoders, even when presented with word types out of context, are sometimes competitive with static word embedding models such as fastText Bojanowski et al. (2017) or word2vec Mikolov et al. (2013). The results also reveal a huge gap in performance between massively multilingual pretrained encoders and language-specific encoders in favor of the latter: our findings support other recent empirical evidence related to the “curse of multilinguality” Conneau et al. (2019); Bapna and Firat (2019) in representation learning. 6) We make all of these resources available on a website which facilitates easy creation, submission and sharing of Multi-Simlex-style datasets for a larger number of languages. We hope that this will yield an even larger repository of semantic resources that inspire future advances in NLP within and across languages. In light of the success of Universal Dependencies Nivre et al. (2019), we hope that our initiative will instigate a collaborative public effort with established and clear-cut guidelines that will result in additional Multi- SimLex datasets in a large number of languages in the near future. Moreover, we hope that it will provide means to advance our understanding of distributional and lexical semantics across a large number of languages. All monolingual and cross-lingual Multi-SimLex datasets–along with detailed translation and annotation guidelines–are available online at: https://multisimlex.com/. ## 2 Lexical Semantic Similarity ### 2.1 Similarity and Association The focus of the Multi-SimLex initiative is on the lexical relation of pure semantic similarity. For any pair of words, this relation measures whether their referents share the same features. For instance, graffiti and frescos are similar to the extent that they are both forms of painting and appear on walls. This relation can be contrasted with the cognitive association between two words, which often depends on how much their referents interact in the real world, or are found in the same situations. For instance, a painter is easily associated with frescos, although they lack any physical commonalities. Association is also known in the literature under other names: relatedness Budanitsky and Hirst (2006), topical similarity (McKeown et al., 2002), and domain similarity (Turney, 2012). Semantic similarity and association overlap to some degree, but do not coincide Kiela, Hill, and Clark (2015); Vulić, Kiela, and Korhonen (2017). In fact, there exist plenty of pairs that are intuitively associated but not similar. Pairs where the converse is true can also be encountered, although more rarely. An example are synonyms where a word is common and the other infrequent, such as to seize and to commandeer. Hill, Reichart, and Korhonen (2015) revealed that while similarity measures based on the WordNet graph (Wu and Palmer, 1994) and human judgments of association in the University of South Florida Free Association Database (Nelson, McEvoy, and Schreiber, 2004) do correlate, a number of pairs follow opposite trends. Several studies on human cognition also point in the same direction. For instance, semantic priming can be triggered by similar words without association (Lucas, 2000). On the other hand, a connection with cue words is established more quickly for topically related words rather than for similar words in free association tasks De Deyne and Storms (2008). A key property of semantic similarity is its gradience: pairs of words can be similar to a different degree. On the other hand, the relation of synonymy is binary: pairs of words are synonyms if they can be substituted in all contexts (or most contexts, in a looser sense), otherwise they are not. While synonyms can be conceived as lying on one extreme of the semantic similarity continuum, it is crucial to note that their definition is stated in purely relational terms, rather than invoking their referential properties (Lyons, 1977; Cruse, 1986; Coseriu, 1967). This makes behavioral studies on semantic similarity fundamentally different from lexical resources like WordNet Miller (1995), which include paradigmatic relations (such as synonymy). ### 2.2 Similarity for NLP: Intrinsic Evaluation and Semantic Specialization The ramifications of the distinction between similarity and association are profound for distributional semantics. This paradigm of lexical semantics is grounded in the distributional hypothesis, formulated by Firth (1957) and Harris (1951). According to this hypothesis, the meaning of a word can be recovered empirically from the contexts in which it occurs within a collection of texts. Since both pairs of topically related words and pairs of purely similar words tend to appear in the same contexts, their associated meaning confounds the two distinct relations Hill, Reichart, and Korhonen (2015); Schwartz, Reichart, and Rappoport (2015); Vulić et al. (2017b). As a result, distributional methods obscure a crucial facet of lexical meaning. This limitation also reflects onto word embeddings (WEs), representations of words as low-dimensional vectors that have become indispensable for a wide range of NLP applications (Collobert et al., 2011; Chen and Manning, 2014; Melamud et al., 2016, inter alia). In particular, it involves both static WEs learned from co-occurrence patterns Mikolov et al. (2013); Levy and Goldberg (2014); Bojanowski et al. (2017) and contextualized WEs learned from modeling word sequences (Peters et al., 2018; Devlin et al., 2019, inter alia). As a result, in the induced representations, geometrical closeness (measured e.g. through cosine distance) conflates genuine similarity with broad relatedness. For instance, the vectors for antonyms such as sober and drunk, by definition dissimilar, might be neighbors in the semantic space under the distributional hypothesis. Turney (2012), Kiela and Clark (2014), and Melamud et al. (2016) demonstrated that different choices of hyper-parameters in WE algorithms (such as context window) emphasize different relations in the resulting representations. Likewise, Agirre et al. (2009) and Levy and Goldberg (2014) discovered that WEs learned from texts annotated with syntactic information mirror similarity better than simple local bag-of-words neighborhoods. The failure of WEs to capture semantic similarity, in turn, affects model performance in several NLP applications where such knowledge is crucial. In particular, Natural Language Understanding tasks such as statistical dialog modeling, text simplification, or semantic text similarity Mrkšić et al. (2016); Kim et al. (2016); Ponti et al. (2019c), among others, suffer the most. As a consequence, resources providing clean information on semantic similarity are key in mitigating the side effects of the distributional signal. In particular, such databases can be employed for the intrinsic evaluations of specific WE models as a proxy of their reliability for downstream applications (Collobert and Weston, 2008; Baroni and Lenci, 2010; Hill, Reichart, and Korhonen, 2015); intuitively, the more WEs are misaligned with human judgments of similarity, the more their performance on actual tasks is expected to be degraded. Moreover, word representations can be specialized (a.k.a. retrofitted) by disentangling word relations of similarity and association. In particular, linguistic constraints sourced from external databases (such as synonyms from WordNet) can be injected into WEs (Faruqui et al., 2015; Wieting et al., 2015; Mrkšić et al., 2017; Lauscher et al., 2019; Kamath et al., 2019, inter alia) in order to enforce a particular relation in a distributional semantic space while preserving the original adjacency properties. ### 2.3 Similarity and Language Variation: Semantic Typology In this work, we tackle the concept of (true) semantic similarity from a multilingual perspective. While the same meaning representations may be shared by all human speakers at a deep cognitive level, there is no one-to-one mapping between the words in the lexicons of different languages. This makes the comparison of similarity judgments across languages difficult, since the meaning overlap of translationally equivalent words is sometimes far less than exact. This results from the fact that the way languages ‘partition’ semantic fields is partially arbitrary (Trier, 1931), although constrained cross- lingually by common cognitive biases Majid et al. (2007). For instance, consider the field of colors: English distinguishes between green and blue, whereas Murle (South Sudan) has a single word for both (Kay and Maffi, 2013). In general, semantic typology studies the variation in lexical semantics across the world’s languages. According to (Evans, 2011), the ways languages categorize concepts into the lexicon follow three main axes: 1) granularity: what is the number of categories in a specific domain?; 2) boundary location: where do the lines marking different categories lie?; 3) grouping and dissection: what are the membership criteria of a category; which instances are considered to be more prototypical? Different choices with respect to these axes lead to different lexicalization patterns.111More formally, colexification is a phenomenon when different meanings can be expressed by the same word in a language François (2008). For instance, the two senses which are distinguished in English as time and weather are co-lexified in Croatian: the word vrijeme is used in both cases. For instance, distinct senses in a polysemous word in English, such as skin (referring to both the body and fruit), may be assigned separate words in other languages such as Italian pelle and buccia, respectively (Rzymski et al., 2020). We later analyze whether similarity scores obtained from native speakers also loosely follow the patterns described by semantic typology. ## 3 Previous Work and Evaluation Data Word Pair Datasets. Rich expert-created resources such as WordNet Miller (1995); Fellbaum (1998), VerbNet Kipper Schuler (2005); Kipper et al. (2008), or FrameNet Baker, Fillmore, and Lowe (1998) encode a wealth of semantic and syntactic information, but are expensive and time-consuming to create. The scale of this problem gets multiplied by the number of languages in consideration. Therefore, crowd-sourcing with non-expert annotators has been adopted as a quicker alternative to produce smaller and more focused semantic resources and evaluation benchmarks. This alternative practice has had a profound impact on distributional semantics and representation learning Hill, Reichart, and Korhonen (2015). While some prominent English word pair datasets such as WordSim-353 Finkelstein et al. (2002), MEN Bruni, Tran, and Baroni (2014), or Stanford Rare Words Luong, Socher, and Manning (2013) did not discriminate between similarity and relatedness, the importance of this distinction was established by Hill, Reichart, and Korhonen (2015, see again the discussion in §2.1) through the creation of SimLex-999. This inspired other similar datasets which focused on different lexical properties. For instance, SimVerb-3500 Gerz et al. (2016) provided similarity ratings for 3,500 English verbs, whereas CARD-660 Pilehvar et al. (2018) aimed at measuring the semantic similarity of infrequent concepts. Semantic Similarity Datasets in Other Languages. Motivated by the impact of datasets such as SimLex-999 and SimVerb-3500 on representation learning in English, a line of related work focused on creating similar resources in other languages. The dominant approach is translating and reannotating the entire original English SimLex-999 dataset, as done previously for German, Italian, and Russian Leviant and Reichart (2015), Hebrew and Croatian Mrkšić et al. (2017), and Polish Mykowiecka, Marciniak, and Rychlik (2018). Venekoski:2017nodalida apply this process only to a subset of 300 concept pairs from the English SimLex-999. On the other hand, Camacho-Collados et al. (2017) sampled a new set of 500 English concept pairs to ensure wider topical coverage and balance across similarity spectra, and then translated those pairs to German, Italian, Spanish, and Farsi (SEMEVAL-500). A similar approach was followed by Ercan and Yıldız (2018) for Turkish, by Huang et al. (2019) for Mandarin Chinese, and by Sakaizawa and Komachi (2018) for Japanese. Netisopakul, Wohlgenannt, and Pulich (2019) translated the concatenation of SimLex-999, WordSim-353, and the English SEMEVAL-500 into Thai and then reannotated it. Finally, Barzegar et al. (2018) translated English SimLex-999 and WordSim-353 to 11 resource-rich target languages (German, French, Russian, Italian, Dutch, Chinese, Portuguese, Swedish, Spanish, Arabic, Farsi), but they did not provide details concerning the translation process and the resolution of translation disagreements. More importantly, they also did not reannotate the translated pairs in the target languages. As we discussed in § 2.3 and reiterate later in §5, semantic differences among languages can have a profound impact on the annotation scores; particulary, we show in §5.4 that these differences even roughly define language clusters based on language affinity. A core issue with the current datasets concerns a lack of one unified procedure that ensures the comparability of resources in different languages. Further, concept pairs for different languages are sourced from different corpora (e.g., direct translation of the English data versus sampling from scratch in the target language). Moreover, the previous SimLex-based multilingual datasets inherit the main deficiencies of the English original version, such as the focus on nouns and highly frequent concepts. Finally, prior work mostly focused on languages that are widely spoken and do not account for the variety of the world’s languages. Our long-term goal is devising a standardized methodology to extend the coverage also to languages that are resource-lean and/or typologically diverse (e.g., Welsh, Kiswahili as in this work). Multilingual Datasets for Natural Language Understanding. The Multi-SimLex initiative and corresponding datasets are also aligned with the recent efforts on procuring multilingual benchmarks that can help advance computational modeling of natural language understanding across different languages. For instance, pretrained multilingual language models such as multilingual bert Devlin et al. (2019) or xlm Conneau and Lample (2019) are typically probed on XNLI test data Conneau et al. (2018b) for cross-lingual natural language inference. XNLI was created by translating examples from the English MultiNLI dataset, and projecting its sentence labels Williams, Nangia, and Bowman (2018). Other recent multilingual datasets target the task of question answering based on reading comprehension: i) MLQA Lewis et al. (2019) includes 7 languages ii) XQuAD Artetxe, Ruder, and Yogatama (2019) 10 languages; iii) TyDiQA Clark et al. (2020) 9 widely spoken typologically diverse languages. While MLQA and XQuAD result from the translation from an English dataset, TyDiQA was built independently in each language. Another multilingual dataset, PAWS-X Yang et al. (2019), focused on the paraphrase identification task and was created translating the original English PAWS Zhang, Baldridge, and He (2019) into 6 languages. We believe that Multi-SimLex can substantially contribute to this endeavor by offering a comprehensive multilingual benchmark for the fundamental lexical level relation of semantic similarity. In future work, Multi-SimLex also offers an opportunity to investigate the correlations between word-level semantic similarity and performance in downstream tasks such as QA and NLI across different languages. ## 4 The Base for Multi-SimLex: Extending English SimLex-999 In this section, we discuss the design principles behind the English (eng) Multi-SimLex dataset, which is the basis for all the Multi-SimLex datasets in other languages, as detailed in §5. We first argue that a new, more balanced, and more comprehensive evaluation resource for lexical semantic similarity in English is necessary. We then describe how the 1,888 word pairs contained in the eng Multi-SimLex were selected in such a way as to represent various linguistic phenomena within a single integrated resource. Construction Criteria. The following criteria have to be satisfied by any high-quality semantic evaluation resource, as argued by previous studies focused on the creation of such resources (Hill, Reichart, and Korhonen, 2015; Gerz et al., 2016; Vulić et al., 2017a; Camacho-Collados et al., 2017, inter alia): (C1) Representative and diverse. The resource must cover the full range of diverse concepts occurring in natural language, including different word classes (e.g., nouns, verbs, adjectives, adverbs), concrete and abstract concepts, a variety of lexical fields, and different frequency ranges. (C2) Clearly defined. The resource must provide a clear understanding of which semantic relation exactly is annotated and measured, possibly contrasting it with other relations. For instance, the original SimLex-999 and SimVerb-3500 explicitly focus on true semantic similarity and distinguish it from broader relatedness captured by datasets such as MEN Bruni, Tran, and Baroni (2014) or WordSim-353 Finkelstein et al. (2002). (C3) Consistent and reliable. The resource must ensure consistent annotations obtained from non-expert native speakers following simple and precise annotation guidelines. In choosing the word pairs and constructing eng Multi-SimLex, we adhere to these requirements. Moreover, we follow good practices established by the research on related resources. In particular, since the introduction of the original SimLex-999 dataset Hill, Reichart, and Korhonen (2015), follow-up works have improved its construction protocol across several aspects, including: 1) coverage of more lexical fields, e.g., by relying on a diverse set of Wikipedia categories Camacho-Collados et al. (2017), 2) infrequent/rare words Pilehvar et al. (2018), 3) focus on particular word classes, e.g., verbs Gerz et al. (2016), 4) annotation quality control Pilehvar et al. (2018). Our goal is to make use of these improvements towards a larger, more representative, and more reliable lexical similarity dataset in English and, consequently, in all other languages. The Final Output: English Multi-SimLex. In order to ensure that the criterion C1 is satisfied, we consolidate and integrate the data already carefully sampled in prior work into a single, comprehensive, and representative dataset. This way, we can control for diversity, frequency, and other properties while avoiding to perform this time-consuming selection process from scratch. Note that, on the other hand, the word pairs chosen for English are scored from scratch as part of the entire Multi-SimLex annotation process, introduced later in §5. We now describe the external data sources for the final set of word pairs: 1) Source: SimLex-999. Hill, Reichart, and Korhonen (2015). The English Multi- SimLex has been initially conceived as an extension of the original SimLex-999 dataset. Therefore, we include all 999 word pairs from SimLex, which span 666 noun pairs, 222 verb pairs, and 111 adjective pairs. While SimLex-999 already provides examples representing different POS classes, it does not have a sufficient coverage of different linguistic phenomena: for instance, it contains only very frequent concepts, and it does not provide a representative set of verbs (Gerz et al., 2016). 2) Source: SemEval-17: Task 2 (henceforth SEMEVAL-500; Camacho-Collados et al., 2017). We start from the full dataset of 500 concept pairs to extract a total of 334 concept pairs for English Multi-SimLex a) which contain only single-word concepts, b) which are not named entities, c) where POS tags of the two concepts are the same, d) where both concepts occur in the top 250K most frequent word types in the English Wikipedia, and e) do not already occur in SimLex-999. The original concepts were sampled as to span all the 34 domains available as part of BabelDomains Camacho-Collados and Navigli (2017), which roughly correspond to the main high-level Wikipedia categories. This ensures topical diversity in our sub-sample. 3) Source: CARD-660 Pilehvar et al. (2018). 67 word pairs are taken from this dataset focused on rare word similarity, applying the same selection criteria a) to e) employed for SEMEVAL-500. Words are controlled for frequency based on their occurrence counts from the Google News data and the ukWaC corpus Baroni et al. (2009). CARD-660 contains some words that are very rare (logboat), domain-specific (erythroleukemia) and slang (2mrw), which might be difficult to translate and annotate across a wide array of languages. Hence, we opt for retaining only the concept pairs above the threshold of top 250K most frequent Wikipedia concepts, as above. 4) Source: SimVerb-3500 Gerz et al. (2016) Since both CARD-660 and SEMEVAL-500 are heavily skewed towards noun pairs, and nouns also dominate the original SimLex-999, we also extract additional verb pairs from the verb-specific similarity dataset SimVerb-3500. We randomly sample 244 verb pairs from SimVerb-3500 that represent all similarity spectra. In particular, we add 61 verb pairs for each of the similarity intervals: $[0,1.5),[1.5,3),[3,4.5),[4.5,6]$. Since verbs in SimVerb-3500 were originally chosen from VerbNet Kipper, Snyder, and Palmer (2004); Kipper et al. (2008), they cover a wide range of verb classes and their related linguistic phenomena. 5) Source: University of South Florida (USF; Nelson, McEvoy, and Schreiber, 2004) norms, the largest database of free association for English. In order to improve the representation of different POS classes, we sample additional adjectives and adverbs from the USF norms following the procedure established by Hill, Reichart, and Korhonen (2015); Gerz et al. (2016). This yields additional 122 adjective pairs, but only a limited number of adverb pairs (e.g., later – never, now – here, once – twice). Therefore, we also create a set of adverb pairs semi-automatically by sampling adjectives that can be derivationally transformed into adverbs (e.g. adding the suffix -ly) from the USF, and assessing the correctness of such derivation in WordNet. The resulting pairs include, for instance, primarily – mainly, softly – firmly, roughly – reliably, etc. We include a total of 123 adverb pairs into the final English Multi-SimLex. Note that this is the first time adverbs are included into any semantic similarity dataset. Fulfillment of Construction Criteria. The final eng Multi-SimLex dataset spans 1,051 noun pairs, 469 verb pairs, 245 adjective pairs, and 123 adverb pairs.222There is a very small number of adjective and verb pairs extracted from CARD-660 and SEMEVAL-500 as well. For instance, the total number of verbs is 469 since we augment the original 222 SimLex-999 verb pairs with 244 SimVerb-3500 pairs and 3 SEMEVAL-500 pairs; and similarly for adjectives. As mentioned above, the criterion C1 has been fulfilled by relying only on word pairs that already underwent meticulous sampling processes in prior work, integrating them into a single resource. As a consequence, Multi-SimLex allows for fine-grained analyses over different POS classes, concreteness levels, similarity spectra, frequency intervals, relation types, morphology, lexical fields, and it also includes some challenging orthographically similar examples (e.g., infection – inflection).333Unlike SEMEVAL-500 and CARD-660, we do not explicitly control for the equal representation of concept pairs across each similarity interval for several reasons: a) Multi-SimLex contains a substantially larger number of concept pairs, so it is possible to extract balanced samples from the full data; b) such balance, even if imposed on the English dataset, would be distorted in all other monolingual and cross-lingual datasets; c) balancing over similarity intervals arguably does not reflect a true distribution “in the wild” where most concepts are only loosely related or completely unrelated. We ensure that the criteria C2 and C3 are satisfied by using similar annotation guidelines as Simlex-999, SimVerb-3500, and SEMEVAL-500 that explicitly target semantic similarity. In what follows, we outline the carefully tailored process of translating and annotating Multi- SimLex datasets in all target languages. ## 5 Multi-SimLex: Translation and Annotation We now detail the development of the final Multi-SimLex resource, describing our language selection process, as well as translation and annotation of the resource, including the steps taken to ensure and measure the quality of this resource. We also provide key data statistics and preliminary cross-lingual comparative analyses. Language Selection. Multi-SimLex comprises eleven languages in addition to English. The main objective for our inclusion criteria has been to balance language prominence (by number of speakers of the language) for maximum impact of the resource, while simultaneously having a diverse suite of languages based on their typological features (such as morphological type and language family). Table 1 summarizes key information about the languages currently included in Multi-SimLex. We have included a mixture of fusional, agglutinative, isolating, and introflexive languages that come from eight different language families. This includes languages that are very widely used such as Chinese Mandarin and Spanish, and low-resource languages such as Welsh and Kiswahili. We hope to further include additional languages and inspire other researchers to contribute to the effort over the lifetime of this project. The work on data collection can be divided into two crucial phases: 1) a translation phase where the extended English language dataset with 1,888 pairs (described in §4) is translated into eleven target languages, and 2) an annotation phase where human raters scored each pair in the translated set as well as the English set. Detailed guidelines for both phases are available online at: https://multisimlex.com. Language | ISO 639-3 | Family | Type | # Speakers ---|---|---|---|--- Chinese Mandarin | cmn | Sino-Tibetan | Isolating | 1.116 B Welsh | cym | IE: Celtic | Fusional | 0.7 M English | eng | IE: Germanic | Fusional | 1.132 B Estonian | est | Uralic | Agglutinative | 1.1 M Finnish | fin | Uralic | Agglutinative | 5.4 M French | fra | IE: Romance | Fusional | 280 M Hebrew | heb | Afro-Asiatic | Introflexive | 9 M Polish | pol | IE: Slavic | Fusional | 50 M Russian | rus | IE: Slavic | Fusional | 260 M Spanish | spa | IE: Romance | Fusional | 534.3 M Kiswahili | swa | Niger-Congo | Agglutinative | 98 M Yue Chinese | yue | Sino-Tibetan | Isolating | 73.5 M Table 1: The list of 12 languages in the Multi-SimLex multilingual suite along with their corresponding language family (IE = Indo-European), broad morphological type, and their ISO 639-3 code. The number of speakers is based on the total count of L1 and L2 speakers, according to ethnologue.com. ### 5.1 Word Pair Translation Translators for each target language were instructed to find direct or approximate translations for the 1,888 word pairs that satisfy the following rules. (1) All pairs in the translated set must be unique (i.e., no duplicate pairs); (2) Translating two words from the same English pair into the same word in the target language is not allowed (e.g., it is not allowed to translate car and automobile to the same Spanish word coche). (3) The translated pairs must preserve the semantic relations between the two words when possible. This means that, when multiple translations are possible, the translation that best conveys the semantic relation between the two words found in the original English pair is selected. (4) If it is not possible to use a single-word translation in the target language, then a multi-word expression (MWE) can be used to convey the nearest possible semantics given the above points (e.g., the English word homework is translated into the Polish MWE praca domowa). Satisfying the above rules when finding appropriate translations for each pair–while keeping to the spirit of the intended semantic relation in the English version–is not always straightforward. For instance, kinship terminology in Sinitic languages (Mandarin and Yue) uses different terms depending on whether the family member is older or younger, and whether the family member comes from the mother’s side or the father’s side. In Mandarin, _brother_ has no direct translation and can be translated as either: 哥哥 (_older brother_) or 弟弟 (_younger brother_). Therefore, in such cases, the translators are asked to choose the best option given the semantic context (relation) expressed by the pair in English, otherwise select one of the translations arbitrarily. This is also used to remove duplicate pairs in the translated set, by differentiating the duplicates using a variant at each instance. Further, many translation instances were resolved using near- synonymous terms in the translation. For example, the words in the pair: _wood – timber_ can only be directly translated in Estonian to _puit_ , and are not distinguishable. Therefore, the translators approximated the translation for timber to the compound noun _puitmaterjal_ (literally: _wood material_) in order to produce a valid pair in the target language. In some cases, a direct transliteration from English is used. For example, the pair: _physician_ and _doctor_ both translate to the same word in Estonian (arst); the less formal word _doktor_ is used as a translation of _doctor_ to generate a valid pair. Languages: | cmn | cym | est | fin | fra | heb | pol | rus | spa | swa | yue | Avg ---|---|---|---|---|---|---|---|---|---|---|---|--- Nouns | 84.5 | 80.0 | 90.0 | 87.3 | 78.2 | 98.2 | 90.0 | 95.5 | 85.5 | 80.0 | 77.3 | 86.0 Adjectives | 88.5 | 88.5 | 61.5 | 73.1 | 69.2 | 100.0 | 84.6 | 100.0 | 69.2 | 88.5 | 84.6 | 82.5 Verbs | 88.0 | 74.0 | 82.0 | 76.0 | 78.0 | 100.0 | 74.0 | 100.0 | 74.0 | 76.0 | 86.0 | 82.5 Adverbs | 92.9 | 100.0 | 57.1 | 78.6 | 92.9 | 100.0 | 85.7 | 100.0 | 85.7 | 85.7 | 78.6 | 87.0 Overall | 86.5 | 81.0 | 82.0 | 82.0 | 78.0 | 99.0 | 85.0 | 97.5 | 80.5 | 81.0 | 80.5 | 84.8 Table 2: Inter-translator agreement (% of matched translated words) by independent translators using a randomly selected 100-pair English sample from the Multi-SimLex dataset, and the corresponding 100-pair samples from the other datasets. We measure the quality of the translated pairs by using a random sample set of 100 pairs (from the 1,888 pairs) to be translated by an independent translator for each target language. The sample is proportionally stratified according to the part-of-speech categories. The independent translator is given identical instructions to the main translator; we then measure the percentage of matched translated words between the two translations of the sample set. Table 2 summarizes the inter-translator agreement results for all languages and by part-of-speech subsets. Overall across all languages, the agreement is 84.8%, which is similar to prior work Camacho-Collados et al. (2017); Vulić, Ponzetto, and Glavaš (2019). ### 5.2 Guidelines and Word Pair Scoring Across all languages, 145 human annotators were asked to score all 1,888 pairs (in their given language). We finally collect at least ten valid annotations for each word pair in each language. All annotators were required to abide by the following instructions: 1\. Each annotator must assign an integer score between 0 and 6 (inclusive) indicating how semantically similar the two words in a given pair are. A score of 6 indicates very high similarity (i.e., perfect synonymy), while zero indicates no similarity. 2\. Each annotator must score the entire set of 1,888 pairs in the dataset. The pairs must not be shared between different annotators. 3\. Annotators are able to break the workload over a period of approximately 2-3 weeks, and are able to use external sources (e.g. dictionaries, thesauri, WordNet) if required. 4\. Annotators are kept anonymous, and are not able to communicate with each other during the annotation process. The selection criteria for the annotators required that all annotators must be native speakers of the target language. Preference to annotators with university education was given, but not required. Annotators were asked to complete a spreadsheet containing the translated pairs of words, as well as the part-of-speech, and a column to enter the score. The annotators did not have access to the original pairs in English. To ensure the quality of the collected ratings, we have employed an adjudication protocol similar to the one proposed and validated by Pilehvar:2018emnlp. It consists of the following three rounds: Round 1: All annotators are asked to follow the instructions outlined above, and to rate all 1,888 pairs with integer scores between 0 and 6. Round 2: We compare the scores of all annotators and identify the pairs for each annotator that have shown the most disagreement. We ask the annotators to reconsider the assigned scores for those pairs only. The annotators may chose to either change or keep the scores. As in the case with Round 1, the annotators have no access to the scores of the other annotators, and the process is anonymous. This process gives a chance for annotators to correct errors or reconsider their judgments, and has been shown to be very effective in reaching consensus, as reported by Pilehvar et al. (2018). We used a very similar procedure as Pilehvar et al. (2018) to identify the pairs with the most disagreement; for each annotator, we marked the $i$th pair if the rated score $s_{i}$ falls within: $s_{i}\geq\mu_{i}+1.5$ or $s_{i}\leq\mu_{i}-1.5$, where $\mu_{i}$ is the mean of the other annotators’ scores. Round 3: We compute the average agreement for each annotator (with the other annotators), by measuring the average Spearman’s correlation against all other annotators. We discard the scores of annotators that have shown the least average agreement with all other annotators, while we maintain at least ten annotators per language by the end of this round. The actual process is done in multiple iterations: (S1) we measure the average agreement for each annotator with every other annotator (this corresponds to the APIAA measure, see later); (S2) if we still have more than 10 valid annotators and the lowest average score is higher than in the previous iteration, we remove the lowest one, and rerun S1. Table 3 shows the number of annotators at both the start (Round 1) and end (Round 3) of our process for each language. We measure the agreement between annotators using two metrics, average pairwise inter-annotator agreement (APIAA), and average mean inter-annotator agreement (AMIAA). Both of these use Spearman’s correlation ($\rho$) between annotators scores, the only difference is how they are averaged. They are computed as follows: Languages: | cmn | cym | eng | est | fin | fra | heb | pol | rus | spa | swa | yue ---|---|---|---|---|---|---|---|---|---|---|---|--- R1: Start | 13 | 12 | 14 | 12 | 13 | 10 | 11 | 12 | 12 | 12 | 11 | 13 R3: End | 11 | 10 | 13 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 11 Table 3: Number of human annotators. R1 = Annotation Round 1, R3 = Round 3. $\displaystyle{1)\textsc{apiaa}}=\frac{2\sum_{i,j}\rho(s_{i},s_{j})}{N(N-1)}\hskip 17.07164pt{2)\textsc{amiaa}}=\frac{\sum_{i}\rho(s_{i},\mu_{i})}{N}\,\text{, where:}\;\mu_{i}=\frac{\sum_{j,j\neq i}s_{j}}{N-1}$ (1) where $\rho(s_{i},s_{j})$ is the Spearman’s correlation between annotators $i$ and $j$’s scores ($s_{i}$,$s_{j}$) for all pairs in the dataset, and $N$ is the number of annotators. APIAA has been used widely as the standard measure for inter-annotator agreement, including in the original SimLex paper Hill, Reichart, and Korhonen (2015). It simply averages the pairwise Spearman’s correlation between all annotators. On the other hand, AMIAA compares the average Spearman’s correlation of one held-out annotator with the average of all the other $N-1$ annotators, and then averages across all $N$ ‘held-out’ annotators. It smooths individual annotator effects and arguably serves as a better upper bound than APIAA (Gerz et al., 2016; Vulić et al., 2017a; Pilehvar et al., 2018, inter alia). Languages: | cmn | cym | eng | est | fin | fra | heb | pol | rus | spa | swa | yue ---|---|---|---|---|---|---|---|---|---|---|---|--- Nouns | 0.661 | 0.622 | 0.659 | 0.558 | 0.647 | 0.698 | 0.538 | 0.606 | 0.524 | 0.582 | 0.626 | 0.727 Adjectives | 0.757 | 0.698 | 0.823 | 0.695 | 0.721 | 0.741 | 0.683 | 0.699 | 0.625 | 0.64 | 0.658 | 0.785 Verbs | 0.694 | 0.604 | 0.707 | 0.58 | 0.644 | 0.691 | 0.615 | 0.593 | 0.555 | 0.588 | 0.631 | 0.76 Adverbs | 0.699 | 0.593 | 0.695 | 0.579 | 0.646 | 0.595 | 0.561 | 0.543 | 0.535 | 0.563 | 0.562 | 0.716 Overall | 0.68 | 0.619 | 0.698 | 0.583 | 0.646 | 0.697 | 0.572 | 0.609 | 0.53 | 0.576 | 0.623 | 0.733 Table 4: Average pairwise inter-annotator agreement (APIAA). A score of $0.6$ and above indicates strong agreement. Languages: | cmn | cym | eng | est | fin | fra | heb | pol | rus | spa | swa | yue ---|---|---|---|---|---|---|---|---|---|---|---|--- Nouns | 0.757 | 0.747 | 0.766 | 0.696 | 0.766 | 0.809 | 0.68 | 0.717 | 0.657 | 0.71 | 0.725 | 0.804 Adjectives | 0.800 | 0.789 | 0.865 | 0.79 | 0.792 | 0.831 | 0.754 | 0.792 | 0.737 | 0.743 | 0.686 | 0.811 Verbs | 0.774 | 0.733 | 0.811 | 0.715 | 0.757 | 0.808 | 0.72 | 0.722 | 0.69 | 0.71 | 0.702 | 0.784 Adverbs | 0.749 | 0.693 | 0.777 | 0.697 | 0.748 | 0.729 | 0.645 | 0.655 | 0.608 | 0.671 | 0.623 | 0.716 Overall | 0.764 | 0.742 | 0.794 | 0.715 | 0.76 | 0.812 | 0.699 | 0.723 | 0.667 | 0.703 | 0.71 | 0.792 Table 5: Average mean inter-annotator agreement (AMIAA). A score of $0.6$ and above indicates strong agreement. We present the respective APIAA and AMIAA scores in Table 4 and Table 5 for all part-of-speech subsets, as well as the agreement for the full datasets. As reported in prior work Gerz et al. (2016); Vulić et al. (2017a), AMIAA scores are typically higher than APIAA scores. Crucially, the results indicate ‘strong agreement’ (across all languages) using both measurements. The languages with the highest annotator agreement were French (fra) and Yue Chinese (yue), while Russian (rus) had the lowest overall IAA scores. These scores, however, are still considered to be ‘moderately strong agreement’. ### 5.3 Data Analysis Similarity Score Distributions. Across all languages, the average score (mean $=1.61$, median$=1.1$) is on the lower side of the similarity scale. However, looking closer at the scores of each language in Table 6, we indicate notable differences in both the averages and the spread of scores. Notably, French has the highest average of similarity scores (mean$=2.61$, median$=2.5$), while Kiswahili has the lowest average (mean$=1.28$, median$=0.5$). Russian has the lowest spread ($\sigma=1.37$), while Polish has the largest ($\sigma=1.62$). All of the languages are strongly correlated with each other, as shown in Figure 1, where all of the Spearman’s correlation coefficients are greater than 0.6 for all language pairs. Languages that share the same language family are highly correlated (e.g, cmn-yue, rus-pol, est-fin). In addition, we observe high correlations between English and most other languages, as expected. This is due to the effect of using English as the base/anchor language to create the dataset. In simple words, if one translates to two languages $L_{1}$ and $L_{2}$ starting from the same set of pairs in English, it is higly likely that $L_{1}$ and $L_{2}$ will diverge from English in different ways. Therefore, the similarity between $L_{1}$-eng and $L_{2}$-eng is expected to be higher than between $L_{1}$-$L_{2}$, especially if $L_{1}$ and $L_{2}$ are typologically dissimilar languages (e.g., heb-cmn, see Figure 1). This phenomenon is well documented in related prior work (Leviant and Reichart, 2015; Camacho-Collados et al., 2017; Mrkšić et al., 2017; Vulić, Ponzetto, and Glavaš, 2019). While we acknowledge this as a slight artifact of the dataset design, it would otherwise be impossible to construct a semantically aligned and comprehensive dataset across a large number of languages. Lang: cmn cym eng est fin fra heb pol rus spa swa yue Interval $[0,1)$ 56.99 52.01 50.95 35.01 47.83 17.69 28.07 49.36 50.21 43.96 61.39 57.89 $[1,2)$ 8.74 19.54 17.06 30.67 21.35 20.39 35.86 17.32 22.40 22.35 11.86 7.84 $[2,3)$ 13.72 11.97 12.66 16.21 12.02 22.03 16.74 11.86 11.81 14.83 9.11 11.76 $[3,4)$ 11.60 8.32 8.16 10.22 10.17 17.64 8.47 8.95 8.10 9.38 7.10 12.98 $[4,5)$ 6.41 5.83 6.89 6.25 5.61 12.55 6.62 7.57 5.88 6.78 6.30 6.89 $[5,6]$ 2.54 2.33 4.29 1.64 2.97 9.64 4.24 4.93 1.59 2.70 4.24 2.65 Table 6: Fine-grained distribution of concept pairs over different rating intervals in each Multi-SimLex language, reported as percentages. The total number of concept pairs in each dataset is 1,888. Figure 1: Spearman’s correlation coefficient ($\rho$) of the similarity scores for all languages in Multi-SimLex. We also report differences in the distribution of the frequency of words among the languages in Multi-SimLex. Figure 2 shows six example languages, where each bar segment shows the proportion of words in each language that occur in the given frequency range. For example, the 10K-20K segment of the bars represents the proportion of words in the dataset that occur in the list of most frequent words between the frequency rank of 10,000 and 20,000 in that language; likewise with other intervals. Frequency lists for the presented languages are derived from Wikipedia and Common Crawl corpora.444Frequency lists were obtained from fastText word vectors which are sorted by frequency: https://fasttext.cc/docs/en/crawl-vectors.html While many concept pairs are direct or approximate translations of English pairs, we can see that the frequency distribution does vary across different languages, and is also related to inherent language properties. For instance, in Finnish and Russian, while we use infinitive forms of all verbs, conjugated verb inflections are often more frequent in raw corpora than the corresponding infinitive forms. The variance can also be partially explained by the difference in monolingual corpora size used to derive the frequency rankings in the first place: absolute vocabulary sizes are expected to fluctuate across different languages. However, it is also important to note that the datasets also contain subsets of lower-frequency and rare words, which can be used for rare word evaluations in multiple languages, in the spirit of Pilehvar:2018emnlp’s English rare word dataset. Figure 2: A distribution over different frequency ranges for words from Multi- SimLex datasets for selected languages. Multi-word expressions are excluded from the analysis. Cross-Linguistic Differences. Table 7 shows some examples of average similarity scores of English, Spanish, Kiswahili and Welsh concept pairs. Remember that the scores range from 0 to 6: the higher the score, the more similar the participants found the concepts in the pair. The examples from Table 7 show evidence of both the stability of average similarity scores across languages (_unlikely – friendly_ , _book – literature_ , and _vanish – disappear_), as well as language-specific differences (_care – caution_). Some differences in similarity scores seem to group languages into clusters. For example, the word pair _regular – average_ has an average similarity score of 4.0 and 4.1 in English and Spanish, respectively, whereas in Kiswahili and Welsh the average similarity score of this pair is 0.5 and 0.8. We analyze this phenomenon in more detail in §5.4. Word Pair | POS | eng | spa | swa | cym ---|---|---|---|---|--- Similar average rating | | | | | unlikely – friendly | ADV | 0 | 0 | 0 | 0 book – literature | N | 2.5 | 2.3 | 2.1 | 2.3 vanish – disappear | V | 5.2 | 5.3 | 5.5 | 5.3 Different average rating | | | | | regular – average | ADJ | 4 | 4.1 | 0.5 | 0.8 care – caution | N | 4.1 | 5.7 | 0.2 | 3.1 One language higher | | | | | large – big | ADJ | 5.9 | 2.7 | 3.8 | 3.8 bank – seat | N | 0 | 5.1 | 0 | 0.1 sunset - evening | N | 1.6 | 1.5 | 5.5 | 2.8 purely – completely | ADV | 2.3 | 2.3 | 1.1 | 5.4 One language lower | | | | | woman – wife | N | 0.9 | 2.9 | 4.1 | 4.8 amazingly – fantastically | ADV | 5.1 | 0.4 | 4.1 | 4.1 wonderful – terrific | ADJ | 5.3 | 5.4 | 0.9 | 5.7 promise – swear | V | 4.8 | 5.3 | 4.3 | 0 Table 7: Examples of concept pairs with their similarity scores from four languages. For brevity, only the original English concept pair is included, but note that the pair is translated to all target languages, see §5.1. There are also examples for each of the four languages having a notably higher or lower similarity score for the same concept pair than the three other languages. For example, _large – big_ in English has an average similarity score of 5.9, whereas Spanish, Kiswahili and Welsh speakers rate the closest concept pair in their native language to have a similarity score between 2.7 and 3.8. What is more, _woman – wife_ receives an average similarity of 0.9 in English, 2.9 in Spanish, and greater than 4.0 in Kiswahili and Welsh. The examples from Spanish include _banco – asiento_ (_bank – seat_) which receives an average similarity score 5.1, while in the other three languages the similarity score for this word pair does not exceed 0.1. At the same time, the average similarity score of _espantosamente – fantásticamente_ (_amazingly – fantastically_) is much lower in Spanish (0.4) than in other languages (4.1 – 5.1). In Kiswahili, an example of a word pair with a higher similarity score than the rest would be _machweo – jioni_ (_sunset – evening_), having an average score of 5.5, while the other languages receive 2.8 or less, and a notably lower similarity score is given to _wa ajabu - mkubwa sana_ (_wonderful – terrific_), getting 0.9, while the other languages receive 5.3 or more. Welsh examples include _yn llwyr - yn gyfan gwbl_ (_purely – completely_), which scores 5.4 among Welsh speakers but 2.3 or less in other languages, while _addo – tyngu_ (_promise – swear_) is rated as 0 by all Welsh annotators, but in the other three languages 4.3 or more on average. There can be several explanations for the differences in similarity scores across languages, including but not limited to cultural context, polysemy, metonymy, translation, regional and generational differences, and most commonly, the fact that words and meanings do not exactly map onto each other across languages. For example, it is likely that the other three languages do not have two separate words for describing the concepts in the concept pair: _big – large_ , and the translators had to opt for similar lexical items that were more distant in meaning, explaining why in English the concept pair received a much higher average similarity score than in other languages. A similar issue related to the mapping problem across languages arose in the Welsh concept pair _yn llwye – yn gyfan gwbl_ , where Welsh speakers agreed that the two concepts are very similar. When asked, bilingual speakers considered the two Welsh concepts more similar than English equivalents _purely – completely_ , potentially explaining why a higher average similarity score was reached in Welsh. The example of _woman – wife_ can illustrate cultural differences or another translation-related issue where the word ‘wife’ did not exist in some languages (for example, Estonian), and therefore had to be described using other words, affecting the comparability of the similarity scores. This was also the case with the _football – soccer_ concept pair. The pair _bank – seat_ demonstrates the effect of the polysemy mismatch across languages: while ‘bank’ has two different meanings in English, neither of them is similar to the word ‘seat’, but in Spanish, ‘ _banco_ ’ can mean ‘bank’, but it can also mean ‘bench’. Quite naturally, Spanish speakers gave the pair _banco – asiento_ a higher similarity score than the speakers of languages where this polysemy did not occur. An example of metonymy affecting the average similarity score can be seen in the Kiswahili version of the word pair: _sunset – evening_ (_machweo – jioni_). The average similarity score for this pair is much higher in Kiswahili, likely because the word ‘sunset’ can act as a metonym of ‘evening’. The low similarity score of _wonderful – terrific_ in Kiswahili (_wa ajabu - mkubwa sana_) can be explained by the fact that while ‘ _mkubwa sana_ ’ can be used as ‘terrific’ in Kiswahili, it technically means ‘very big’, adding to the examples of translation- and mapping-related effects. The word pair _amazingly – fantastically_ (_espantosamente – fantásticamente_) brings out another translation-related problem: the accuracy of the translation. While ‘ _espantosamente_ ’ could arguably be translated to ‘amazingly’, more common meanings include: ‘frightfully’, ‘terrifyingly’, and ‘shockingly’, explaining why the average similarity score differs from the rest of the languages. Another problem was brought out by _addo – tyngu_ (_promise – swear_) in Welsh, where the ‘ _tyngu_ ’ may not have been a commonly used or even a known word choice for annotators, pointing out potential regional or generational differences in language use. Language | Word Pair | POS | Rating all participants agree with ---|---|---|--- eng | trial – test | N | 4-5 swa | archbishop – bishop | N | 4-5 spa, cym | start – begin | V | 5-6 eng | smart – intelligent | ADJ | 5-6 eng, spa | quick – rapid | ADJ | 5-6 spa | circumstance – situation | N | 5-6 cym | football – soccer | N | 5-6 swa | football – soccer | N | 6 swa | pause – wait | V | 6 swa | money – cash | N | 6 cym | friend – buddy | N | 6 Table 8: Examples of concept pairs with their similarity scores from four languages where all participants show strong agreement in their rating. Table 8 presents examples of concept pairs from English, Spanish, Kiswahili, and Welsh on which the participants agreed the most. For example, in English all participants rated the similarity of _trial – test_ to be 4 or 5. In Spanish and Welsh, all participants rated _start – begin_ to correspond to a score of 5 or 6. In Kiswahili, _money – cash_ received a similarity rating of 6 from every participant. While there are numerous examples of concept pairs in these languages where the participants agreed on a similarity score of 4 or higher, it is worth noting that none of these languages had a single pair where all participants agreed on either 1-2, 2-3, or 3-4 similarity rating. Interestingly, in English all pairs where all the participants agreed on a 5-6 similarity score were adjectives. ### 5.4 Effect of Language Affinity on Similarity Scores Based on the analysis in Figure 1 and inspecting the anecdotal examples in the previous section, it is evident that the correlation between similarity scores across languages is not random. To corroborate this intuition, we visualize the vectors of similarity scores for each single language by reducing their dimensionality to 2 via Principal Component Analysis (Pearson, 1901). The resulting scatter plot in Figure 3 reveals that languages from the same family or branch have similar patterns in the scores. In particular, Russian and Polish (both Slavic), Finnish and Estonian (both Uralic), Cantonese and Mandarin Chinese (both Sinitic), and Spanish and French (both Romance) are all neighbors. Figure 3: PCA of the language vectors resulting from the concatenation of similarity judgments for all pairs. In order to quantify exactly the effect of language affinity on the similarity scores, we run correlation analyses between these and language features. In particular, we extract feature vectors from URIEL (Littell et al., 2017), a massively multilingual typological database that collects and normalizes information compiled by grammarians and field linguists about the world’s languages. In particular, we focus on information about geography (the areas where the language speakers are concentrated), family (the phylogenetic tree each language belongs to), and typology (including syntax, phonological inventory, and phonology).555For the extraction of these features, we employed lang2vec: github.com/antonisa/lang2vec Moreover, we consider typological representations of languages that are not manually crafted by experts, but rather learned from texts. Malaviya, Neubig, and Littell (2017) proposed to construct such representations by training language-identifying vectors end- to-end as part of neural machine translation models. The vector for similarity judgments and the vector of linguistic features for a given language have different dimensionality. Hence, we first construct a distance matrix for each vector space, such that both columns and rows are language indices, and each cell value is the cosine distance between the vectors of the corresponding language pair. Given a set of L languages, each resulting matrix $S$ has dimensionality of $\mathbb{R}^{|L|\times|L|}$ and is symmetrical. To estimate the correlation between the matrix for similarity judgments and each of the matrices for linguistic features, we run a Mantel test (Mantel, 1967), a non-parametric statistical test based on matrix permutations that takes into account inter-dependencies among pairwise distances. The results of the Mantel test reported in Table 3 show that there exist statistically significant correlations between similarity judgments and geography, family, and syntax, given that $p<0.05$ and $z>1.96$. The correlation coefficient is particularly strong for geography ($r=0.647$) and syntax ($r=0.649$). The former result is intuitive, because languages in contact easily borrow and loan lexical units, and cultural interactions may result in similar cognitive categorizations. The result for syntax, instead, cannot be explained so easily, as formal properties of language do not affect lexical semantics. Instead, we conjecture that, while no causal relation is present, both syntactic features and similarity judgments might be linked to a common explanatory variable (such as geography). In fact, several syntactic properties are not uniformly spread across the globe. For instance, verbs with Verb–Object–Subject word order are mostly concentrated in Oceania (Dryer, 2013). In turn, geographical proximity leads to similar judgment patterns, as mentioned above. On the other hand, we find no correlation with phonology and inventory, as expected, nor with the bottom-up typological features from Malaviya, Neubig, and Littell (2017). Features | Dimension | Mantel r | Mantel p | Mantel z ---|---|---|---|--- geography | 299 | 0.647 | 0.007* | 3.443 family | 3718 | 0.329 | 0.023* | 2.711 syntax | 103 | 0.649 | 0.007* | 3.787 inventory | 158 | 0.155 | 0.459 | 0.782 phonology | 28 | 0.397 | 0.046 | 1.943 Malaviya, Neubig, and Littell (2017) | 512 | -0.431 | 0.264 | -1.235 Table 9: Mantel test on the correlation between similarity judgments from Multi-SimLex and linguistic features from typological databases. ## 6 Cross-Lingual Multi-SimLex Datasets A crucial advantage of having semantically aligned monolingual datasets across different languages is the potential to create cross-lingual semantic similarity datasets. Such datasets allow for probing the quality of cross- lingual representation learning algorithms Camacho-Collados et al. (2017); Conneau et al. (2018a); Chen and Cardie (2018); Doval et al. (2018); Ruder, Vulić, and Søgaard (2019); Conneau and Lample (2019); Ruder, Søgaard, and Vulić (2019) as an intrinsic evaluation task. However, the cross-lingual datasets previous work relied upon Camacho-Collados et al. (2017) were limited to a homogeneous set of high-resource languages (e.g., English, German, Italian, Spanish) and a small number of concept pairs (all less than 1K pairs). We address both problems by 1) using a typologically more diverse language sample, and 2) relying on a substantially larger English dataset as a source for the cross-lingual datasets: 1,888 pairs in this work versus 500 pairs in the work of Camacho:2017semeval. As a result, each of our cross- lingual datasets contains a substantially larger number of concept pairs, as shown in Table 11. The cross-lingual Multi-Simlex datasets are constructed automatically, leveraging word pair translations and annotations collected in all 12 languages. This yields a total of 66 cross-lingual datasets, one for each possible combination of languages. Table 11 provides the final number of concept pairs, which lie between 2,031 and 3,480 pairs for each cross-lingual dataset, whereas Table 10 shows some sample pairs with their corresponding similarity scores. The automatic creation and verification of cross-lingual datasets closely follows the procedure first outlined by Camacho:2015acl and later adopted by Camacho:2017semeval (for semantic similarity) and Vulic:2019acl (for graded lexical entailment). First, given two languages, we intersect their aligned concept pairs obtained through translation. For instance, starting from the aligned pairs attroupement – foule in French and rahvasumm – rahvahulk in Estonian, we construct two cross-lingual pairs attroupement – rahvaluk and rahvasumm – foule. The scores of cross-lingual pairs are then computed as averages of the two corresponding monolingual scores. Finally, in order to filter out concept pairs whose semantic meaning was not preserved during this operation, we retain only cross-lingual pairs for which the corresponding monolingual scores $(s_{s},s_{t})$ differ at most by one fifth of the full scale (i.e., $\mid s_{s}-s_{t}\mid\leq 1.2$). This heuristic mitigates the noise due to cross-lingual semantic shifts Camacho-Collados et al. (2017); Vulić, Ponzetto, and Glavaš (2019). We refer the reader to the work of Camacho:2015acl for a detailed technical description of the procedure. Pair | Concept-1 | Concept-2 | Score | Pair | Concept-1 | Concept-2 | Score ---|---|---|---|---|---|---|--- cym-eng | rhyddid | liberty | 5.37 | cmn-est | 可能 | optimistlikult | 0.83 cym-pol | plentynaidd | niemądry | 2.15 | fin-swa | psykologia | sayansi | 2.20 swa-eng | kutimiza | accomplish | 5.24 | eng-fra | normally | quotidiennement | 2.41 cmn-fra | 有弹性 | flexible | 4.08 | fin-spa | auto | bicicleta | 0.85 fin-spa | tietämättömyys | inteligencia | 0.55 | cmn-yue | 使灰心 | 使气馁 | 4.78 spa-fra | ganador | candidat | 2.15 | cym-swa | sefyllfa | mazingira | 1.90 est-yue | takso | 巴士 | 2.08 | est-spa | armee | legión | 3.25 eng-fin | orange | sitrushedelmä | 3.43 | fin-est | halveksuva | põlglik | 5.55 spa-pol | palabra | wskazówka | 0.55 | cmn-cym | 学生 | disgybl | 4.45 pol-swa | prawdopodobnie | uwezekano | 4.05 | pol-eng | grawitacja | meteor | 0.27 Table 10: Example concept pairs with their scores from a selection of cross- lingual Multi-SimLex datasets. To assess the quality of the resulting cross-lingual datasets, we have conducted a verification experiment similar to Vulic:2019acl. We randomly sampled 300 concept pairs in the English-Spanish, English-French, and English- Mandarin cross-lingual datasets. Subsequently, we asked bilingual native speakers to provide similarity judgments of each pair. The Spearman’s correlation score $\rho$ between automatically induced and manually collected ratings achieves $\rho\geq 0.90$ on all samples, which confirms the viability of the automatic construction procedure. | cmn | cym | eng | est | fin | fra | heb | pol | rus | spa | swa | yue ---|---|---|---|---|---|---|---|---|---|---|---|--- cmn | 1,888 | – | – | – | – | – | – | – | – | – | – | – cym | 3,085 | 1,888 | – | – | – | – | – | – | – | – | – | – eng | 3,151 | 3,380 | 1,888 | – | – | – | – | – | – | – | – | – est | 3,188 | 3,305 | 3,364 | 1,888 | – | – | – | – | – | – | – | – fin | 3,137 | 3,274 | 3,352 | 3,386 | 1,888 | – | – | – | – | – | – | – fra | 2,243 | 2,301 | 2,284 | 2,787 | 2,682 | 1,888 | – | – | – | – | – | – heb | 3,056 | 3,209 | 3,274 | 3,358 | 3,243 | 2,903 | 1,888 | – | – | – | – | – pol | 3,009 | 3,175 | 3,274 | 3,310 | 3,294 | 2,379 | 3,201 | 1,888 | – | – | – | – rus | 3,032 | 3,196 | 3,222 | 3,339 | 3,257 | 2,219 | 3,226 | 3,209 | 1,888 | – | – | – spa | 3,116 | 3,205 | 3,318 | 3,312 | 3,256 | 2,645 | 3,256 | 3,250 | 3,189 | 1,888 | – | – swa | 2,807 | 2,926 | 2,828 | 2,845 | 2,900 | 2,031 | 2,775 | 2,819 | 2,855 | 2,811 | 1,888 | – yue | 3,480 | 3,062 | 3,099 | 3,080 | 3,063 | 2,313 | 3,005 | 2,950 | 2,966 | 3,053 | 2,821 | 1,888 Table 11: The sizes of all monolingual (main diagonal) and cross-lingual datasets. (a) Rating distribution (b) Distribution over POS classes Figure 4: (a) Rating distribution and (b) distribution of pairs over the four POS classes in cross-lingual Multi-SimLex datasets averaged across each of the 66 language pairs ($y$ axes plot percentages as the total number of concept pairs varies across different cross-lingual datasets). Minimum and maximum percentages for each rating interval and POS class are also plotted. Score and Class Distributions. The summary of score and class distributions across all 66 cross-lingual datasets are provided in Figure 4(a) and Figure 4(b), respectively. First, it is obvious that the distribution over the four POS classes largely adheres to that of the original monolingual Multi-SimLex datasets, and that the variance is quite low: e.g., the eng-fra dataset contains the lowest proportion of nouns (49.21%) and the highest proportion of verbs (27.1%), adjectives (15.28%), and adverbs (8.41%). On the other hand, the distribution over similarity intervals in Figure 4(a) shows a much greater variance. This is again expected as this pattern resurfaces in monolingual datasets (see Table 6). It is also evident that the data are skewed towards lower-similarity concept pairs. However, due to the joint size of all cross- lingual datasets (see Table 11), even the least represented intervals contain a substantial number of concept pairs. For instance, the rus-yue dataset contains the least highly similar concept pairs (in the interval $[4,6]$) of all 66 cross-lingual datasets. Nonetheless, the absolute number of pairs (138) in that interval for rus-yue is still substantial. If needed, this makes it possible to create smaller datasets which are balanced across the similarity spectra through sub-sampling. ## 7 Monolingual Evaluation of Representation Learning Models After the numerical and qualitative analyses of the Multi-SimLex datasets provided in §§ 5.3–5.4, we now benchmark a series of representation learning models on the new evaluation data. We evaluate standard static word embedding algorithms such as fastText Bojanowski et al. (2017), as well as a range of more recent text encoders pretrained on language modeling such as multilingual BERT (Devlin et al., 2019). These experiments provide strong baseline scores on the new Multi-SimLex datasets and offer a first large-scale analysis of pretrained encoders on word-level semantic similarity across diverse languages. In addition, the experiments now enabled by Multi-SimLex aim to answer several important questions. (Q1) Is it viable to extract high-quality word-level representations from pretrained encoders receiving subword-level tokens as input? Are such representations competitive with standard static word-level embeddings? (Q2) What are the implications of monolingual pretraining versus (massively) multilingual pretraining for performance? (Q3) Do lightweight unsupervised post-processing techniques improve word representations consistently across different languages? (Q4) Can we effectively transfer available external lexical knowledge from resource-rich languages to resource-lean languages in order to learn word representations that distinguish between true similarity and conceptual relatedness (see the discussion in §2.3)? ### 7.1 Models in Comparison Static Word Embeddings in Different Languages. First, we evaluate a standard method for inducing non-contextualized (i.e., static) word embeddings across a plethora of different languages: fastText (ft) vectors Bojanowski et al. (2017) are currently the most popular and robust choice given 1) the availability of pretrained vectors in a large number of languages Grave et al. (2018) trained on large Common Crawl (CC) plus Wikipedia (Wiki) data, and 2) their superior performance across a range of NLP tasks Mikolov et al. (2018). In fact, fastText is an extension of the standard word-level CBOW and skip- gram word2vec models Mikolov et al. (2013) that takes into account subword- level information, i.e. the contituent character n-grams of each word Zhu, Vulić, and Korhonen (2019). For this reason, fastText is also more suited for modeling rare words and morphologically rich languages.666We have also trained standard word-level CBOW and skip-gram with negative sampling (SGNS) on full Wikipedia dumps for several languages, but our preliminary experiments have verified that they under-perform compared to fastText. This finding is consistent with other recent studies demonstrating the usefulness of subword- level information Vania and Lopez (2017); Mikolov et al. (2018); Zhu, Vulić, and Korhonen (2019); Zhu et al. (2019). Therefore, we do not report the results with CBOW and SGNS for brevity. We rely on $300$-dimensional ft word vectors trained on CC+Wiki and available online for 157 languages.777https://fasttext.cc/docs/en/crawl-vectors.html The word vectors for all languages are obtained by CBOW with position-weights, with character n-grams of length 5, a window of size 5, 10 negative examples, and 10 training epochs. We also probe another (older) collection of ft vectors, pretrained on full Wikipedia dumps of each language.888https://fasttext.cc/docs/en/pretrained-vectors.html. The vectors are 300-dimensional, trained with the skip-gram objective for 5 epochs, with 5 negative examples, a window size set to 5, and relying on all character n-grams from length 3 to 6. Following prior work, we trim the vocabularies for all languages to the 200K most frequent words and compute representations for multi-word expressions by averaging the vectors of their constituent words. Unsupervised Post-Processing. Further, we consider a variety of unsupervised post-processing steps that can be applied post-training on top of any pretrained input word embedding space without any external lexical semantic resource. So far, the usefulness of such methods has been verified only on the English language through benchmarks for lexical semantics and sentence-level tasks Mu, Bhat, and Viswanath (2018). In this paper, we assess if unsupervised post-processing is beneficial also in other languages. To this end, we apply the following post-hoc transformations on the initial word embeddings: 1) Mean centering (mc) is applied after unit length normalization to ensure that all vectors have a zero mean, and is commonly applied in data mining and analysis Bro and Smilde (2003); van den Berg et al. (2006). 2) All-but-the-top (abtt) Mu, Bhat, and Viswanath (2018); Tang, Mousavi, and de Sa (2019) eliminates the common mean vector and a few top dominating directions (according to PCA) from the input distributional word vectors, since they do not contribute towards distinguishing the actual semantic meaning of different words. The method contains a single (tunable) hyper- parameter $dd_{A}$ which denotes the number of the dominating directions to remove from the initial representations. Previous work has verified the usefulness of abtt in several English lexical semantic tasks such as semantic similarity, word analogies, and concept categorization, as well as in sentence-level text classification tasks Mu, Bhat, and Viswanath (2018). 3) uncovec Artetxe et al. (2018) adjusts the similarity order of an arbitrary input word embedding space, and can emphasize either syntactic or semantic information in the transformed vectors. In short, it transforms the input space $\bm{X}$ into an adjusted space $\bm{X}\bm{W}_{\alpha}$ through a linear map $\bm{W}_{\alpha}$ controlled by a single hyper-parameter $\alpha$. The $n^{\text{th}}$-order similarity transformation of the input word vector space $\bm{X}$ (for which $n=1$) can be obtained as $\bm{M}_{n}(\bm{X})=\bm{M}_{1}(\bm{X}\bm{W}_{(n-1)/2})$, with $\bm{W}_{\alpha}=\bm{Q}\bm{\Gamma}^{\alpha}$, where $\bm{Q}$ and $\bm{\Gamma}$ are the matrices obtained via eigendecomposition of $\bm{X}^{T}\bm{X}=\bm{Q}\bm{\Gamma}\bm{Q}^{T}$. $\bm{\Gamma}$ is a diagonal matrix containing eigenvalues of $\bm{X}^{T}\bm{X}$; $\bm{Q}$ is an orthogonal matrix with eigenvectors of $\bm{X}^{T}\bm{X}$ as columns. While the motivation for the uncovec methods does originate from adjusting discrete similarity orders, note that $\alpha$ is in fact a continuous real-valued hyper-parameter which can be carefully tuned. For more technical details we refer the reader to the original work of Artetxe et al. (2018). As mentioned, all post-processing methods can be seen as unsupervised retrofitting methods that, given an arbitrary input vector space $\bm{X}$, produce a perturbed/transformed output vector space $\bm{X}^{\prime}$, but unlike common retrofitting methods Faruqui et al. (2015); Mrkšić et al. (2017), the perturbation is completely unsupervised (i.e., self-contained) and does not inject any external (semantic similarity oriented) knowledge into the vector space. Note that different perturbations can also be stacked: e.g., we can apply uncovec and then use abtt on top the output uncovec vectors. When using uncovec and abtt we always length-normalize and mean-center the data first (i.e., we apply the simple mc normalization). Finally, we tune the two hyper-parameters $d_{A}$ (for abtt) and $\alpha$ (uncovec) on the English Multi-SimLex and use the same values on the datasets of all other languages; we report results with $dd_{A}=3$ or $dd_{A}=10$, and $\alpha=-0.3$. Contextualized Word Embeddings. We also evaluate the capacity of unsupervised pretraining architectures based on language modeling objectives to reason over lexical semantic similarity. To the best of our knowledge, our article is the first study performing such analyses. State-of-the-art models such as bert Devlin et al. (2019), xlm Conneau and Lample (2019), or roberta Liu et al. (2019b) are typically very deep neural networks based on the Transformer architecture Vaswani et al. (2017). They receive subword-level tokens as inputs (such as WordPieces Schuster and Nakajima (2012)) to tackle data sparsity. In output, they return contextualized embeddings, dynamic representations for words in context. To represent words or multi-word expressions through a pretrained model, we follow prior work Liu et al. (2019a) and compute an input item’s representation by 1) feeding it to a pretrained model in isolation; then 2) averaging the $H$ last hidden representations for each of the item’s constituent subwords; and then finally 3) averaging the resulting subword representations to produce the final $d$-dimensional representation, where $d$ is the embedding and hidden-layer dimensionality (e.g., $d=768$ with bert). We opt for this approach due to its proven viability and simplicity Liu et al. (2019a), as it does not require any additional corpora to condition the induction of contextualized embeddings.999We also tested another encoding method where we fed pairs instead of single words/concepts into the pretrained encoder. The rationale is that the other concept in the pair can be used as disambiguation signal. However, this method consistently led to sub-par performance across all experimental runs. Other ways to extract the representations from pretrained models Aldarmaki and Diab (2019); Wu et al. (2019); Cao, Kitaev, and Klein (2020) are beyond the scope of this work, and we will experiment with them in the future. In other words, we treat each pretrained encoder enc as a black-box function to encode a single word or a multi-word expression $x$ in each language into a $d$-dimensional contextualized representation $\mathbf{x}_{\textsc{enc}}\in\mathbb{R}^{d}=\textsc{enc}(x)$ (e.g., $d=768$ with bert). As multilingual pretrained encoders, we experiment with the multilingual bert model (m-bert) Devlin et al. (2019) and xlm (Conneau and Lample, 2019). m-bert is pretrained on monolingual Wikipedia corpora of 102 languages (comprising all Multi-SimLex languages) with a 12-layer Transformer network, and yields $768$-dimensional representations. Since the concept pairs in Multi-SimLex are lowercased, we use the uncased version of m-bert.101010https://github.com/google- research/bert/blob/master/multilingual.md m-bert comprises all Multi-SimLex languages, and its evident ability to perform cross-lingual transfer Pires, Schlinger, and Garrette (2019); Wu and Dredze (2019); Wang et al. (2020) also makes it a convenient baseline model for cross-lingual experiments later in §8. The second multilingual model we consider, xlm-100,111111https://github.com/facebookresearch/XLM is pretrained on Wikipedia dumps of 100 languages, and encodes each concept into a $1,280$-dimensional representation. In contrast to m-bert, xlm-100 drops the next-sentence prediction objective and adds a cross-lingual masked language modeling objective. For both encoders, the representations of each concept are computed as averages over the last $H=4$ hidden layers in all experiments, as suggested by Wu:2019arxiv.121212In our preliminary experiments on several language pairs, we have also verified that this choice is superior to: a) using the output of only the last hidden layer (i.e., $H=1$) and b) averaging over all hidden layers (i.e., $H=12$ for the bert-base architecture). Likewise, using the special prepended ‘[CLS]’ token rather than the constituent sub-words to encode a concept also led to much worse performance across the board. Besides m-bert and xlm, covering multiple languages, we also analyze the performance of “language-specific” bert and xlm models for the languages where they are available: Finnish, Spanish, English, Mandarin Chinese, and French. The main goal of this comparison is to study the differences in performance between multilingual “one-size-fits-all” encoders and language-specific encoders. For all experiments, we rely on the pretrained models released in the Transformers repository Wolf et al. (2019).131313github.com/huggingface/transformers. The full list of currently supported pretrained encoders is available here: huggingface.co/models. Unsupervised post-processing steps devised for static word embeddings (i.e., mean-centering, abtt, uncovec) can also be applied on top of contextualized embeddings if we predefine a vocabulary of word types $V$ that will be represented in a word vector space $\mathbf{X}$. We construct such $V$ for each language as the intersection of word types covered by the corresponding CC+Wiki fastText vectors and the (single-word or multi-word) expressions appearing in the corresponding Multi-SimLex dataset. Finally, note that it is not feasible to evaluate a full range of available pretrained encoders within the scope of this work. Our main intention is to provide the first set of baseline results on Multi-SimLex by benchmarking a sample of most popular encoders, at the same time also investigating other important questions such as performance of static versus contextualized word embeddings, or multilingual versus language-specific pretraining. Another purpose of the experiments is to outline the wide potential and applicability of the Multi-SimLex datasets for multilingual and cross-lingual representation learning evaluation. Languages: cmn cym eng est fin fra heb pol rus spa swa yue fastText (CC+Wiki) (272) (151) (12) (319) (347) (43) (66) (326) (291) (46) (222) (–) (1) ft:init .534 .363 .528 .469 .607 .578 .450 .405 .422 .511 .439 – (2) ft:+mc .539 .393 .535 .473 .621 .584 .480 .412 .424 .516 .469 – (3) ft:+abtt (-3) .557 .389 .536 .495 .642 .610 .501 .427 .459 .523 .473 – (4) ft:+abtt (-10) .583 .384 .551 .476 .651 .623 .503 .455 .500 .542 .462 – (5) ft:+uncovec .572 .387 .550 .465 .642 .595 .501 .435 .437 .525 .437 – (1)+(2)+(5)+(3) .574 .386 .549 .476 .655 .604 .503 .442 .452 .528 .432 – (1)+(2)+(5)+(4) .577 .376 .542 .455 .652 .613 .510 .466 .491 .540 .424 – fastText (Wiki) (429) (282) (6) (343) (345) (73) (62) (354) (343) (57) (379) (677) (1) ft:init .315 .318 .436 .400 .575 .444 .428 .370 .359 .432 .332 .376 (2) ft:+mc .373 .337 .445 .404 .583 .463 .447 .383 .378 .447 .373 .427 (3) ft:+abtt (-3) .459 .343 .453 .404 .584 .487 .447 .387 .394 .456 .423 .429 (4) ft:+abtt (-10) .496 .323 .460 .385 .581 .494 .460 .401 .400 .477 .406 .399 (5) ft:+uncovec .518 .328 .469 .375 .568 .483 .449 .389 .387 .469 .386 .394 (1)+(2)+(5)+(3) .526 .323 .470 .369 .564 .495 .448 .392 .392 .473 .388 .388 (1)+(2)+(5)+(4) .526 .307 .471 .355 .548 .495 .450 .394 .394 .476 .382 .396 m-bert (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) (1) m-bert:init .408 .033 .138 .085 .162 .115 .104 .069 .085 .145 .125 .404 (2) m-bert:+mc .458 .044 .256 .122 .173 .183 .128 .097 .123 .203 .128 .469 (3) m-bert:+abtt (-3) .487 .056 .321 .137 .200 .287 .144 .126 .197 .299 .135 .492 (4) m-bert:+abtt (-10) .456 .056 .329 .122 .164 .306 .121 .126 .183 .315 .136 .467 (5) m-bert:+uncovec .464 .063 .317 .144 .213 .288 .164 .144 .198 .287 .143 .464 (1)+(2)+(5)+(3) .464 .083 .326 .130 .201 .304 .149 .122 .199 .295 .148 .456 (1)+(2)+(5)+(4) .444 .086 .326 .112 .179 .305 .135 .127 .187 .285 .119 .447 Table 12: A summary of results (Spearman’s $\rho$ correlation scores) on the full monolingual Multi-SimLex datasets for 12 languages. We benchmark fastText word embeddings trained on two different corpora (CC+Wiki and only Wiki) as well the multilingual m-bert model (see §7.1). Results with the initial word vectors are reported (i.e., without any unsupervised post-processing), as well as with different unsupervised post-processing methods, described in §7.1. The language codes are provided in Table 1. The numbers in the parentheses (gray rows) refer to the number of OOV concepts excluded from the computation. The highest scores for each language and per model are in bold. ### 7.2 Results and Discussion The results we report are Spearman’s $\rho$ coefficients of the correlation between the ranks derived from the scores of the evaluated models and the human scores provided in each Multi-SimLex dataset. The main results with static and contextualized word vectors for all test languages are summarized in Table 12. The scores reveal several interesting patterns, and also pinpoint the main challenges for future work. State-of-the-Art Representation Models. The absolute scores of CC+Wiki ft, Wiki ft, and m-bert are not directly comparable, because these models have different coverage. In particular, Multi-SimLex contains some out-of- vocabulary (OOV) words whose static ft embeddings are not available.141414We acknowledge that it is possible to approximate word-level representations of OOVs with ft by summing the constituent n-gram embeddings as proposed by Bojanowski:2017tacl. However, we do not perform this step as the resulting embeddings are typically of much lower quality than non-OOV embeddings Zhu, Vulić, and Korhonen (2019). On the other hand, m-bert has perfect coverage. A general comparison between CC+Wiki and Wiki ft vectors, however, supports the intuition that larger corpora (such as CC+Wiki) yield higher correlations. Another finding is that a single massively multilingual model such as m-bert cannot produce semantically rich word-level representations. Whether this actually happens because the training objective is different—or because the need to represent 100+ languages reduces its language-specific capacity—is investigated further below. Languages: cmn cym eng est fin fra heb pol rus spa swa yue fastText (CC+Wiki) ft:init nouns (1,051) .561 .497 .592 .627 .709 .641 .560 .538 .526 .583 .544 .426 verbs (469) .511 .265 .408 .379 .527 .551 .458 .384 .464 .499 .391 .252 adj (245) .448 .338 .564 .401 .546 .616 .467 .284 .349 .401 .344 .288 adv (123) .622 .187 .482 .378 .547 .648 .491 .266 .514 .423 .172 .103 fastText (CC+Wiki) ft:+abtt (-3) nouns .601 .512 .599 .621 .730 .653 .592 .585 .578 .605 .553 .431 verbs .583 .305 .454 .379 .575 .602 .520 .390 .475 .526 .381 .314 adj .526 .372 .601 .427 .592 .646 .483 .316 .409 .411 .402 .312 adv .675 .150 .504 .397 .546 .695 .491 .230 .495 .416 .223 .081 m-bert m-bert:+abtt (-3) nouns .517 .091 .446 .191 .210 .364 .191 .188 .266 .418 .142 .539 verbs .511 .005 .200 .039 .077 .248 .038 .107 .181 .266 .091 .503 adj .227 .050 .226 .028 .128 .193 .044 .046 .002 .099 .192 .267 adv .282 .012 .343 .112 .173 .390 .326 .036 .046 .207 161 .049 xlm-100 xlm:+abtt (-3) all .498 .096 .270 .118 .203 .234 .195 .106 .170 .289 .130 .506 nouns .551 .132 .381 .193 .238 .234 .242 .184 .292 .378 .165 .559 verbs .544 .038 .169 .006 .190 .132 .136 .073 .095 .243 .047 .570 adj .356 .140 .256 .081 .179 .185 .150 .046 .022 .100 .220 .291 adv .284 .017 .040 .086 .043 .027 .221 .014 .022 .315 .095 .156 Table 13: Spearman’s $\rho$ correlation scores over the four POS classes represented in Multi-SimLex datasets. In addition to the word vectors considered earlier in Table 12, we also report scores for another contextualized model, xlm-100. The numbers in parentheses refer to the total number of POS-class pairs in the original eng dataset and, consequently, in all other monolingual datasets. The overall results also clearly indicate that (i) there are differences in performance across different monolingual Multi-SimLex datasets, and (ii) unsupervised post-processing is universally useful, and can lead to huge improvements in correlation scores for many languages. In what follows, we also delve deeper into these analyses. Impact of Unsupervised Post-Processing. First, the results in Table 12 suggest that applying dimension-wise mean centering to the initial vector spaces has positive impact on word similarity scores in all test languages and for all models, both static and contextualized (see the +mc rows in Table 12). Mimno and Thompson (2017) show that distributional word vectors have a tendency towards narrow clusters in the vector space (i.e., they occupy a narrow cone in the vector space and are therefore anisotropic Mu, Bhat, and Viswanath (2018); Ethayarajh (2019)), and are prone to the undesired effect of hubness Radovanović, Nanopoulos, and Ivanović (2010); Lazaridou, Dinu, and Baroni (2015).151515Hubness can be defined as the tendency of some points/vectors (i.e., “hubs”) to be nearest neighbors of many points in a high-dimensional (vector) space Radovanović, Nanopoulos, and Ivanović (2010); Lazaridou, Dinu, and Baroni (2015); Conneau et al. (2018a) Applying dimension-wise mean centering has the effect of spreading the vectors across the hyper-plane and mitigating the hubness issue, which consequently improves word-level similarity, as it emerges from the reported results. Previous work has already validated the importance of mean centering for clustering-based tasks Suzuki et al. (2013), bilingual lexicon induction with cross-lingual word embeddings Artetxe, Labaka, and Agirre (2018a); Zhang et al. (2019); Vulić et al. (2019), and for modeling lexical semantic change Schlechtweg et al. (2019). However, to the best of our knowledge, the results summarized in Table 12 are the first evidence that also confirms its importance for semantic similarity in a wide array of languages. In sum, as a general rule of thumb, we suggest to always mean-center representations for semantic tasks. The results further indicate that additional post-processing methods such as abtt and uncovec on top of mean-centered vector spaces can lead to further gains in most languages. The gains are even visible for languages which start from high correlation scores: for instance., cmn with CC+Wiki ft increases from 0.534 to 0.583, from 0.315 to 0.526 with Wiki ft, and from 0.408 to 0.487 with m-bert. Similarly, for rus with CC+Wiki ft we can improve from 0.422 to 0.500, and for fra the scores improve from 0.578 to 0.613. There are additional similar cases reported in Table 12. Overall, the unsupervised post-processing techniques seem universally useful across languages, but their efficacy and relative performance does vary across different languages. Note that we have not carefully fine-tuned the hyper- parameters of the evaluated post-processing methods, so additional small improvements can be expected for some languages. The main finding, however, is that these post-processing techniques are robust to semantic similarity computations beyond English, and are truly language independent. For instance, removing dominant latent (PCA-based) components from word vectors emphasizes semantic differences between different concepts, as only shared non- informative latent semantic knowledge is removed from the representations. In summary, pretrained word embeddings do contain more information pertaining to semantic similarity than revealed in the initial vectors. This way, we have corroborated the hypotheses from prior work Mu, Bhat, and Viswanath (2018); Artetxe et al. (2018) which were not previously empirically verified on other languages due to a shortage of evaluation data; this gap has now been filled with the introduction of the Multi-SimLex datasets. In all follow-up experiments, we always explicitly denote which post-processing configuration is used in evaluation. POS-Specific Subsets. We present the results for subsets of word pairs grouped by POS class in Table 13. Prior work based on English data showed that representations for nouns are typically of higher quality than those for the other POS classes Schwartz, Reichart, and Rappoport (2015, 2016); Vulić et al. (2017b). We observe a similar trend in other languages as well. This pattern is consistent across different representation models and can be attributed to several reasons. First, verb representations need to express a rich range of syntactic and semantic behaviors rather than purely referential features Gruber (1976); Levin (1993); Kipper et al. (2008). Second, low correlation scores on the adjective and adverb subsets in some languages (e.g., pol, cym, swa) might be due to their low frequency in monolingual texts, which yields unreliable representations. In general, the variance in performance across different word classes warrants further research in class-specific representation learning Baker, Reichart, and Korhonen (2014); Vulić et al. (2017b). The scores further attest the usefulness of unsupervised post- processing as almost all class-specific correlation scores are improved by applying mean-centering and abtt. Finally, the results for m-bert and xlm-100 in Table 13 further confirm that massively multilingual pretraining cannot yield reasonable semantic representations for many languages: in fact, for some classes they display no correlation with human ratings at all. Differences across Languages. Naturally, the results from Tables 12 and 13 also reveal that there is variation in performance of both static word embeddings and pretrained encoders across different languages. Among other causes, the lowest absolute scores with ft are reported for languages with least resources available to train monolingual word embeddings, such as Kiswahili, Welsh, and Estonian. The low performance on Welsh is especially indicative: Figure 1 shows that the ratings in the Welsh dataset match up very well with the English ratings, but we cannot achieve the same level of correlation in Welsh with Welsh ft word embeddings. Difference in performance between two closely related languages, est (low-resource) and fin (high- resource), provides additional evidence in this respect. The highest reported scores with m-bert and xlm-100 are obtained for Mandarin Chinese and Yue Chinese: this effectively points to the weaknesses of massively multilingual training with a joint subword vocabulary spanning 102 and 100 languages. Due to the difference in scripts, “language-specific” subwords for yue and cmn do not need to be shared across a vast amount of languages and the quality of their representation remains unscathed. This effectively means that m-bert’s subword vocabulary contains plenty of cmn- specific and yue-specific subwords which are exploited by the encoder when producing m-bert-based representations. Simultaneously, higher scores with m-bert (and xlm in Table 13) are reported for resource-rich languages such as French, Spanish, and English, which are better represented in m-bert’s training data. We also observe lower absolute scores (and a larger number of OOVs) for languages with very rich and productive morphological systems such as the two Slavic languages (Polish and Russian) and Finnish. Since Polish and Russian are known to have large Wikipedias and Common Crawl data Conneau et al. (2019) (e.g., their Wikipedias are in the top 10 largest Wikipedias worldwide), the problem with coverage can be attributed exactly to the proliferation of morphological forms in those languages. Finally, while Table 12 does reveal that unsupervised post-processing is useful for all languages, it also demonstrates that peak scores are achieved with different post-processing configurations. This finding suggests that a more careful language-specific fine-tuning is indeed needed to refine word embeddings towards semantic similarity. We plan to inspect the relationship between post-processing techniques and linguistic properties in more depth in future work. Multilingual vs. Language-Specific Contextualized Embeddings. Recent work has shown that—despite the usefulness of massively multilingual models such as m-bert and xlm-100 for zero-shot cross-lingual transfer Pires, Schlinger, and Garrette (2019); Wu and Dredze (2019)—stronger results in downstream tasks for a particular language can be achieved by pretraining language-specific models on language-specific data. In this experiment, motivated by the low results of m-bert and xlm-100 (see again Table 13), we assess if monolingual pretrained encoders can produce higher-quality word-level representations than multilingual models. Therefore, we evaluate language-specific bert and xlm models for a subset of the Multi- SimLex languages for which such models are currently available: Finnish Virtanen et al. (2019) (bert-base architecture, uncased), French Le et al. (2019) (the FlauBERT model based on xlm), English (bert-base, uncased), Mandarin Chinese (bert-base) Devlin et al. (2019) and Spanish (bert-base, uncased). In addition, we also evaluate a series of pretrained encoders available for English: (i) bert-base, bert-large, and bert-large with whole word masking (wwm) from the original work on BERT Devlin et al. (2019), (ii) monolingual “English-specific” xlm Conneau and Lample (2019), and (iii) two models which employ parameter reduction techniques to build more compact encoders: albert-b uses a configuration similar to bert-base, while albert-l is similar to bert-large, but with an $18\times$ reduction in the number of parameters Lan et al. (2020).161616All models and their further specifications are available at the following link: https://huggingface.co/models. From the results in Table 5, it is clear that monolingual pretrained encoders yield much more reliable word-level representations. The gains are visible even for languages such as cmn which showed reasonable performance with m-bert and are substantial on all test languages. This further confirms the validity of language-specific pretraining in lieu of multilingual training, if sufficient monolingual data are available. Moreover, a comparison of pretrained English encoders in Figure 5(b) largely follows the intuition: the larger bert-large model yields slight improvements over bert-base, and we can improve a bit more by relying on word-level (i.e., lexical-level) masking.Finally, light-weight albert model variants are quite competitive with the original bert models, with only modest drops reported, and albert-l again outperforms albert-b. Overall, it is interesting to note that the scores obtained with monolingual pretrained encoders are on a par with or even outperform static ft word embeddings: this is a very intriguing finding per se as it shows that such subword-level models trained on large corpora can implicitly capture rich lexical semantic knowledge. (a) Monolingual vs multilingual (b) Pretrained eng encoders Figure 5: (a) A performance comparison between monolingual pretrained language encoders and massively multilingual encoders. For four languages (cmn, eng, fin, spa), we report the scores with monolingual uncased bert-base architectures and multilingual uncased m-bert model, while for fra we report the results of the multilingual xlm-100 architecture and a monolingual French FlauBERT model Le et al. (2019), which is based on the same architecture as xlm-100. (b) A comparison of various pretrained encoders available for English. All these models are post-processed via abtt (-3). Similarity-Specialized Word Embeddings. Conflating distinct lexico-semantic relations is a well-known property of distributional representations Turney and Pantel (2010); Melamud et al. (2016). Semantic specialization fine-tunes distributional spaces to emphasize a particular lexico-semantic relation in the transformed space by injecting external lexical knowledge Glavaš, Ponti, and Vulić (2019). Explicitly discerning between true semantic similarity (as captured in Multi-SimLex) and broad conceptual relatedness benefits a number of tasks, as discussed in §2.1.171717For an overview of specialization methods for semantic similarity, we refer the interested reader to the recent tutorial Glavaš, Ponti, and Vulić (2019). Since most languages lack dedicated lexical resources, however, one viable strategy to steer monolingual word vector spaces to emphasize semantic similarity is through cross-lingual transfer of lexical knowledge, usually through a shared cross-lingual word vector space Ruder, Vulić, and Søgaard (2019). Therefore, we evaluate the effectiveness of specialization transfer methods using Multi-SimLex as our multilingual test bed. We evaluate a current state-of-the-art cross-lingual specialization transfer method with minimal requirements, put forth recently by Ponti:2019emnlp.181818We have also evaluated other specialization transfer methods, e.g., Glavaš and Vulić (2018); Ponti et al. (2018b), but they are consistently outperformed by the method of Ponti:2019emnlp. In a nutshell, their li-postspec method is a multi-step procedure that operates as follows. First, the knowledge about semantic similarity is extracted from WordNet in the form of triplets, that is, linguistic constraints $(w_{1},w_{2},r)$, where $w_{1}$ and $w_{2}$ are two concepts, and $r$ is a relation between them obtained from WordNet (e.g., synonymy or antonymy). The goal is to “attract” synonyms closer to each other in the transformed vector space as they reflect true semantic similarity, and “repel” antonyms further apart. In the second step, the linguistic constraints are translated from English to the target language via a shared cross-lingual word vector space. To this end, following Ponti:2019emnlp we rely on cross-lingual word embeddings (CLWEs) Joulin et al. (2018) available online, which are based on Wiki ft vectors.191919https://fasttext.cc/docs/en/aligned-vectors.html; for target languages for which there are no pretrained CLWEs, we induce them following the same procedure of Joulin:2018emnlp. Following that, a constraint refinement step is applied in the target language which aims to eliminate the noise inserted during the translation process. This is done by training a relation classification tool: it is trained again on the English linguistic constraints and then used on the translated target language constraints, where the transfer is again enabled via a shared cross-lingual word vector space.202020We again follow Ponti:2019emnlp and use a state-of-the-art relation classifier Glavaš and Vulić (2018). We refer the reader to the original work for additional technical details related to the classifier design. Finally, a state-of-the-art monolingual specialization procedure from Ponti:2018emnlp injects the (now target language) linguistic constraints into the target language distributional space. The scores are summarized in Table 14. Semantic specialization with li- postspec leads to substantial improvements in correlation scores for the majority of the target languages, demonstrating the importance of external semantic similarity knowledge for semantic similarity reasoning. However, we also observe deteriorated performance for the three target languages which can be considered the lowest-resource ones in our set: cym, swa, yue. We hypothesize that this occurs due to the inferior quality of the underlying monolingual Wikipedia word embeddings, which generates a chain of error accumulations. In particular, poor distributional word estimates compromise the alignment of the embedding spaces, which in turn results in increased translation noise, and reduced refinement ability of the relation classifier. On a high level, this “poor get poorer” observation again points to the fact that one of the primary causes of low performance of resource-low languages in semantic tasks is the sheer lack of even unlabeled data for distributional training. On the other hand, as we see from Table 13, typological dissimilarity between the source and the target does not deteriorate the effectiveness of semantic specialization. In fact, li-postspec does yield substantial gains also for the typologically distant targets such as heb, cmn, and est. The critical problem indeed seems to be insufficient raw data for monolingual distributional training. Languages: cmn cym eng est fin fra heb pol rus spa swa yue fastText (Wiki) (429) (282) (6) (343) (345) (73) (62) (354) (343) (57) (379) (677) ft:init .315 .318 – .400 .575 .444 .428 .370 .359 .432 .332 .376 li-postspec .584 .204 – .515 .619 .601 .510 .531 .547 .635 .238 .267 Table 14: The impact of vector space specialization for semantic similarity. The scores are reported using the current state-of-the-art specialization transfer li-postspec method of Ponti:2019emnlp, relying on English as a resource-rich source language and the external lexical semantic knowledge from the English WordNet. ## 8 Cross-Lingual Evaluation Similar to monolingual evaluation in §7, we now evaluate several state-of-the- art cross-lingual representation models on the suite of 66 automatically constructed cross-lingual Multi-SimLex datasets. Again, note that evaluating a full range of cross-lingual models available in the rich prior work on cross- lingual representation learning is well beyond the scope of this article. We therefore focus our cross-lingual analyses on several well-established and indicative state-of-the-art cross-lingual models, again spanning both static and contextualized cross-lingual word embeddings. ### 8.1 Models in Comparison Static Word Embeddings. We rely on a state-of-the-art mapping-based method for the induction of cross-lingual word embeddings (CLWEs): vecmap Artetxe, Labaka, and Agirre (2018b). The core idea behind such mapping-based or projection-based approaches is to learn a post-hoc alignment of independently trained monolingual word embeddings Ruder, Vulić, and Søgaard (2019). Such methods have gained popularity due to their conceptual simplicity and competitive performance coupled with reduced bilingual supervision requirements: they support CLWE induction with only as much as a few thousand word translation pairs as the bilingual supervision Mikolov, Le, and Sutskever (2013); Xing et al. (2015); Upadhyay et al. (2016); Ruder, Søgaard, and Vulić (2019). More recent work has shown that CLWEs can be induced with even weaker supervision from small dictionaries spanning several hundred pairs Vulić and Korhonen (2016); Vulić et al. (2019), identical strings Smith et al. (2017), or even only shared numerals Artetxe, Labaka, and Agirre (2017). In the extreme, fully unsupervised projection-based CLWEs extract such seed bilingual lexicons from scratch on the basis of monolingual data only (Conneau et al., 2018a; Artetxe, Labaka, and Agirre, 2018b; Hoshen and Wolf, 2018; Alvarez- Melis and Jaakkola, 2018; Chen and Cardie, 2018; Mohiuddin and Joty, 2019, inter alia). Recent empirical studies Glavaš et al. (2019); Vulić et al. (2019); Doval et al. (2019) have compared a variety of unsupervised and weakly supervised mapping-based CLWE methods, and vecmap emerged as the most robust and very competitive choice. Therefore, we focus on 1) its fully unsupervised variant (unsuper) in our comparisons. For several language pairs, we also report scores with two other vecmap model variants: 2) a supervised variant which learns a mapping based on an available seed lexicon (super), and 3) a supervised variant with self-learning (super+sl) which iteratively increases the seed lexicon and improves the mapping gradually. For a detailed description of these variants, we refer the reader to recent work Artetxe, Labaka, and Agirre (2018b); Vulić et al. (2019). We again use CC+Wiki ft vectors as initial monolingual word vectors, except for yue where Wiki ft is used. The seed dictionaries of two different sizes (1k and 5k translation pairs) are based on PanLex Kamholz, Pool, and Colowick (2014), and are taken directly from prior work Vulić et al. (2019),212121https://github.com/cambridgeltl/panlex-bli or extracted from PanLex following the same procedure as in the prior work. Contextualized Cross-Lingual Word Embeddings. We again evaluate the capacity of (massively) multilingual pretrained language models, m-bert and xlm-100, to reason over cross-lingual lexical similarity. Implicitly, such an evaluation also evaluates “the intrinsic quality” of shared cross-lingual word-level vector spaces induced by these methods, and their ability to boost cross- lingual transfer between different language pairs. We rely on the same procedure of aggregating the models’ subword-level parameters into word-level representations, already described in §7.1. As in monolingual settings, we can apply unsupervised post-processing steps such as abtt to both static and contextualized cross-lingual word embeddings. | cmn | cym | eng | est | fin | fra | heb | pol | rus | spa | swa | yue ---|---|---|---|---|---|---|---|---|---|---|---|--- cmn | | .076 | .348 | .139 | .154 | .392 | .190 | .207 | .227 | .300 | .049 | .484 cym | .041 | | .087 | .017 | .049 | .095 | .033 | .072 | .085 | .089 | .002 | .083 eng | .565 | .004 | | .168 | .159 | .401 | .171 | .182 | .236 | .309 | .014 | .357 est | .014 | .097 | .335 | | .143 | .161 | .100 | .113 | .083 | .134 | .025 | .124 fin | .049 | .020 | .542 | .530 | | .195 | .077 | .110 | .111 | .157 | .029 | .167 fra | .224 | .015 | .662 | .559 | .533 | | .191 | .229 | .297 | .382 | .038 | .382 heb | .202 | .110 | .516 | .465 | .445 | .469 | | .095 | .154 | .181 | .038 | .185 pol | .121 | .028 | .464 | .415 | .465 | .534 | .412 | | .139 | .183 | .013 | .205 rus | .032 | .037 | .511 | .408 | .476 | .529 | .430 | .390 | | .248 | .037 | .226 spa | .546 | .048 | .498 | .450 | .490 | .600 | .462 | .398 | .419 | | .055 | .313 swa | -.01 | .116 | .029 | .006 | .013 | -.05 | .033 | .052 | .035 | .045 | | .043 yue | .004 | .047 | .059 | .004 | .002 | .059 | .001 | .074 | .032 | .089 | -.02 | Table 15: Spearman’s $\rho$ correlation scores on all 66 cross-lingual datasets. 1) The scores below the main diagonal are computed based on cross- lingual word embeddings (CLWEs) induced by aligning CC+Wiki ft in all languages (except for yue where we use Wiki ft) in a fully unsupervised way (i.e., without any bilingual supervision). We rely on a standard CLWE mapping- based (i.e., alignment) approach: vecmap Artetxe, Labaka, and Agirre (2018b). 2) The scores above the main diagonal are computed by obtaining 768-dimensional word-level vectors from pretrained multilingual BERT (m-bert) following the procedure described in §7.1. For both fully unsupervised vecmap and m-bert, we report the results with unsupervised postprocessing enabled: all $2\times 66$ reported scores are obtained using the +abbt (-10) variant. (a) Average scores (b) Scores on eng-fra Figure 6: Further performance analyses of cross-lingual Multi-SimLex datasets. (a) Spearman’s $\rho$ correlation scores averaged over all 66 cross-lingual Multi-SimLex datasets for two pretrained multilingual encoders (m-bert and xlm). The scores are obtained with different configurations that exclude (init) or enable unsupervised post-processing. (b) A comparison of various pretrained encoders available for the English-French language pair, see the main text for a short description of each benchmarked pretrained encoder. ### 8.2 Results and Discussion Main Results and Differences across Language Pairs. A summary of the results on the 66 cross-lingual Multi-SimLex datasets are provided in Table 15 and Figure 6(a). The findings confirm several interesting findings from our previous monolingual experiments (§7.2), and also corroborate several hypotheses and findings from prior work, now on a large sample of language pairs and for the task of cross-lingual semantic similarity. First, we observe that the fully unsupervised vecmap model, despite being the most robust fully unsupervised method at present, fails to produce a meaningful cross-lingual word vector space for a large number of language pairs (see the bottom triangle of Table 15): many correlation scores are in fact no-correlation results, accentuating the problem of fully unsupervised cross-lingual learning for typologically diverse languages and with fewer amounts of monolingual data Vulić et al. (2019). The scores are particularly low across the board for lower-resource languages such as Welsh and Kiswahili. It also seems that the lack of monolingual data is a larger problem than typological dissimilarity between language pairs, as we do observe reasonably high correlation scores with vecmap for language pairs such as cmn-spa, heb- est, and rus-fin. However, typological differences (e.g., morphological richness) still play an important role as we observe very low scores when pairing cmn with morphologically rich languages such fin, est, pol, and rus. Similar to prior work of Vulic:2019we and doval2019onthe, given the fact that unsupervised vecmap is the most robust unsupervised CLWE method at present Glavaš et al. (2019), our results again question the usefulness of fully unsupervised approaches for a large number of languages, and call for further developments in the area of unsupervised and weakly supervised cross-lingual representation learning. The scores of m-bert and xlm-100222222The xlm-100 scores are not reported for brevity; they largely follow the patterns observed with m-bert. The aggregated scores between the two encoders are also very similar as indicated by Figure 6(a). lead to similar conclusions as in the monolingual settings. Reasonable correlation scores are achieved only for a small subset of resource-rich language pairs (e.g., eng, fra, spa, cmn) which dominate the multilingual m-bert training. Interestingly, the scores indicate a much higher performance of language pairs where yue is one of the languages when we use m-bert instead of vecmap. This boils down again to the fact that yue, due to its specific language script, has a good representation of its words and subwords in the shared m-bert vocabulary. At the same time, a reliable vecmap mapping between yue and other languages cannot be found due to a small monolingual yue corpus. In cases when vecmap does not yield a degenerate cross-lingual vector space starting from two monolingual ones, the final correlation scores seem substantially higher than the ones obtained by the single massively multilingual m-bert model. Finally, the results in Figure 6(a) again verify the usefulness of unsupervised post-processing also in cross-lingual settings. We observe improved performance with both m-bert and xlm-100 when mean centering (+mc) is applied, and further gains can be achieved by using abtt on the mean-centered vector spaces. A similar finding also holds for static cross-lingual word embeddings232323Note that vecmap does mean centering by default as one of its preprocessing steps prior to learning the mapping function Artetxe, Labaka, and Agirre (2018b); Vulić et al. (2019)., where applying abbt (-10) yields higher scores on 61/66 language pairs. Fully Unsupervised vs. Weakly Supervised Cross-Lingual Embeddings. The results in Table 15 indicate that fully unsupervised cross-lingual learning fails for a large number of language pairs. However, recent work Vulić et al. (2019) has noted that these sub-optimal non-alignment solutions with the unsuper model can be avoided by relying on (weak) cross-lingual supervision spanning only several thousands or even hundreds of word translation pairs. Therefore, we examine 1) if we can further improve the results on cross-lingual Multi-SimLex resorting to (at least some) cross-lingual supervision for resource-rich language pairs; and 2) if such available word-level supervision can also be useful for a range of languages which displayed near-zero performance in Table 15. In other words, we test if recent “tricks of the trade” used in the rich literature on CLWE learning reflect in gains on cross-lingual Multi-SimLex datasets. First, we reassess the findings established on the bilingual lexicon induction task Søgaard, Ruder, and Vulić (2018); Vulić et al. (2019): using at least some cross-lingual supervision is always beneficial compared to using no supervision at all. We report improvements over the unsuper model for all 10 language pairs in Table 16, even though the unsuper method initially produced strong correlation scores. The importance of self-learning increases with decreasing available seed dictionary size, and the +sl model always outperforms unsuper with 1k seed pairs; we observe the same patterns also with even smaller dictionary sizes than reported in Table 16 (250 and 500 seed pairs). Along the same line, the results in Table 17 indicate that at least some supervision is crucial for the success of static CLWEs on resource-leaner language pairs. We note substantial improvements on all language pairs; in fact, the vecmap model is able to learn a more reliable mapping starting from clean supervision. We again note large gains with self-learning. | cmn-eng | eng-fra | eng-spa | eng-rus | est-fin | est-heb | fin-heb | fra-spa | pol-rus | pol-spa ---|---|---|---|---|---|---|---|---|---|--- unsuper | .565 | .662 | .498 | .511 | .510 | .465 | .445 | .600 | .390 | .398 super (1k) | .575 | .602 | .453 | .376 | .378 | .363 | .442 | .588 | .399 | .406 +sl (1k) | .577 | .703 | .547 | .548 | .591 | .513 | .488 | .639 | .439 | .456 super (5k) | .587 | .704 | .542 | .535 | .518 | .473 | .585 | .631 | .455 | .463 +sl (5k) | .581 | .707 | .548 | .551 | .556 | .525 | .589 | .645 | .432 | .476 Table 16: Results on a selection of cross-lingual Multi-SimLex datasets where the fully unsupervised (unsuper) CLWE variant yields reasonable performance. We also show the results with supervised vecmap without self-learning (super) and with self-learning (+sl), with two seed dictionary sizes: 1k and 5k pairs; see §8.1 for more detail. Highest scores for each language pair are in bold. | cmn-fin | cmn-rus | cmn-yue | cym-fin | cym-fra | cym-pol | fin-swa ---|---|---|---|---|---|---|--- unsuper | .049 | .032 | .004 | .020 | .015 | .028 | .013 super (1k) | .410 | .388 | .372 | .384 | .475 | .326 | .206 +sl (1k) | .590 | .537 | .458 | .471 | .578 | .380 | .264 Table 17: Results on a selection of cross-lingual Multi-SimLex datasets where the fully unsupervised (unsuper) CLWE variant fails to learn a coherent shared cross-lingual space. See also the caption of Table 16. Multilingual vs. Bilingual Contextualized Embeddings. Similar to the monolingual settings, we also inspect if massively multilingual training in fact dilutes the knowledge necessary for cross-lingual reasoning on a particular language pair. Therefore, we compare the 100-language xlm-100 model with i) a variant of the same model trained on a smaller set of 17 languages (xlm-17); ii) a variant of the same model trained specifically for the particular language pair (xlm-2); and iii) a variant of the bilingual xlm-2 model that also leverages bilingual knowledge from parallel data during joint training (xlm-2++). We again use the pretrained models made available by Conneau:2019nips, and we refer to the original work for further technical details. The results are summarized in Figure 6(b), and they confirm the intuition that massively multilingual pretraining can damage performance even on resource- rich languages and language pairs. We observe a steep rise in performance when the multilingual model is trained on a much smaller set of languages (17 versus 100), and further improvements can be achieved by training a dedicated bilingual model. Finally, leveraging bilingual parallel data seems to offer additional slight gains, but a tiny difference between xlm-2 and xlm-2++ also suggests that this rich bilingual information is not used in the optimal way within the xlm architecture for semantic similarity. In summary, these results indicate that, in order to improve performance in cross-lingual transfer tasks, more work should be invested into 1) pretraining dedicated language pair-specific models, and 2) creative ways of leveraging available cross-lingual supervision (e.g., word translation pairs, parallel or comparable corpora) Liu et al. (2019a); Wu et al. (2019); Cao, Kitaev, and Klein (2020) with pretraining paradigms such as bert and xlm. Using such cross-lingual supervision could lead to similar benefits as indicated by the results obtained with static cross-lingual word embeddings (see Table 16 and Table 17). We believe that Multi-SimLex can serve as a valuable means to track and guide future progress in this research area. ## 9 Conclusion and Future Work We have presented Multi-SimLex, a resource containing human judgments on the semantic similarity of word pairs for 12 monolingual and 66 cross-lingual datasets. The languages covered are typologically diverse and include also under-resourced ones, such as Welsh and Kiswahili. The resource covers an unprecedented amount of 1,888 word pairs, carefully balanced according to their similarity score, frequency, concreteness, part-of-speech class, and lexical field. In addition to Multi-Simlex, we release the detailed protocol we followed to create this resource. We hope that our consistent guidelines will encourage researchers to translate and annotate Multi-Simlex -style datasets for additional languages. This can help and create a hugely valuable, large-scale semantic resource for multilingual NLP research. The core Multi-SimLex we release with this paper already enables researchers to carry out novel linguistic analysis as well as establishes a benchmark for evaluating representation learning models. Based on our preliminary analyses, we found that speakers of closely related languages tend to express equivalent similarity judgments. In particular, geographical proximity seems to play a greater role than family membership in determining the similarity of judgments across languages. Moreover, we tested several state-of-the-art word embedding models, both static and contextualized representations, as well as several (supervised and unsupervised) post-processing techniques, on the newly released Multi-SimLex. This enables future endeavors to improve multilingual representation learning with challenging baselines. In addition, our results provide several important insights for research on both monolingual and cross- lingual word representations: 1) Unsupervised post-processing techniques (mean centering, elimination of top principal components, adjusting similarity orders) are always beneficial independently of the language, although the combination leading to the best scores is language-specific and hence needs to be tuned. 2) Similarity rankings obtained from word embeddings for nouns are better aligned with human judgments than all the other part-of-speech classes considered here (verbs, adjectives, and, for the first time, adverbs). This confirms previous generalizations based on experiments on English. 3) The factor having the greatest impact on the quality of word representations is the availability of raw texts to train them in the first place, rather than language properties (such as family, geographical area, typological features). 4) Massively multilingual pretrained encoders such as m-bert (Devlin et al., 2019) and xlm-100 (Conneau and Lample, 2019) fare quite poorly on our benchmark, whereas pretrained encoders dedicated to a single language are more competitive with static word embeddings such as fastText (Bojanowski et al., 2017). Moreover, for language-specific encoders, parameter reduction techniques reduce performance only marginally. 5) Techniques to inject clean lexical semantic knowledge from external resources into distributional word representations were proven to be effective in emphasizing the relation of semantic similarity. In particular, methods capable of transferring such knowledge from resource-rich to resource-lean languages (Ponti et al., 2019c) increased the correlation with human judgments for most languages, except for those with limited unlabelled data. Future work can expand our preliminary, yet large-scale study on the ability of pretrained encoders to reason over word-level semantic similarity in different languages. For instance, we have highlighted how sharing the same encoder parameters across multiple languages may harm performance. However, it remains unclear if, and to what extent, the input language embeddings present in xlm-100 but absent in m-bert help mitigate this issue. In addition, pretrained language embeddings can be obtained both from typological databases (Littell et al., 2017) and from neural architectures (Malaviya, Neubig, and Littell, 2017). Plugging these embeddings into the encoders in lieu of embeddings trained end-to-end as suggested by prior work (Tsvetkov et al., 2016; Ammar et al., 2016; Ponti et al., 2019b) might extend the coverage to more resource-lean languages. Another important follow-up analysis might involve the comparison of the performance of representation learning models on multilingual datasets for both word-level semantic similarity and sentence-level Natural Language Understanding. In particular, Multi-SimLex fills a gap in available resources for multilingual NLP and might help understand how lexical and compositional semantics interact if put alongside existing resources such as XNLI Conneau et al. (2018b) for natural language inference or PAWS-X Yang et al. (2019) for cross-lingual paraphrase identification. Finally, the Multi-SimLex annotation could turn out to be a unique source of evidence to study the effects of polysemy in human judgments on semantic similarity: for equivalent word pairs in multiple languages, are the similarity scores affected by how many senses the two words (or multi-word expressions) incorporate? In light of the success of initiatives like Universal Dependencies for multilingual treebanks, we hope that making Multi-SimLex and its guidelines available will encourage other researchers to expand our current sample of languages. We particularly encourage creation and submission of comparable Multi-SimLex datasets for under-resourced and typologically diverse languages in future work. In particular, we have made a Multi-Simlex community website available to facilitate easy creation, gathering, dissemination, and use of annotated datasets: https://multisimlex.com/. ###### Acknowledgements. 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2024-09-04T02:54:58.838230

2003.04875

arxiv-papers

# Gravity field modelling for the Hannover $10\text{\,}\mathrm{m}$ atom interferometer Manuel Schilling German Aerospace Center (DLR), Institute for Satellite Geodesy and Inertial Sensing, c/o Leibniz Universität Hannover, DLR-Institut, Welfengarten 1, 30167 Hannover, Germany Leibniz Universität Hannover, Institut für Erdmessung, Schneiderberg 50, 30167 Hannover, Germany Étienne Wodey Leibniz Universität Hannover, Institut für Quantenoptik, Welfengarten 1, 30167 Hannover, Germany Ludger Timmen Leibniz Universität Hannover, Institut für Erdmessung, Schneiderberg 50, 30167 Hannover, Germany Dorothee Tell Leibniz Universität Hannover, Institut für Quantenoptik, Welfengarten 1, 30167 Hannover, Germany Klaus H. Zipfel Leibniz Universität Hannover, Institut für Quantenoptik, Welfengarten 1, 30167 Hannover, Germany Dennis Schlippert Leibniz Universität Hannover, Institut für Quantenoptik, Welfengarten 1, 30167 Hannover, Germany Christian Schubert German Aerospace Center (DLR), Institute for Satellite Geodesy and Inertial Sensing, c/o Leibniz Universität Hannover, DLR-Institut, Welfengarten 1, 30167 Hannover, Germany Leibniz Universität Hannover, Institut für Quantenoptik, Welfengarten 1, 30167 Hannover, Germany Ernst M. Rasel Leibniz Universität Hannover, Institut für Quantenoptik, Welfengarten 1, 30167 Hannover, Germany Jürgen Müller Leibniz Universität Hannover, Institut für Erdmessung, Schneiderberg 50, 30167 Hannover, Germany (This is a post-peer-review, pre-copyedit version of an article published in Journal of Geodesy 94:122. The final authenticated version is available online at: https://dx.doi.org/10.1007/s00190-020-01451-y) Absolute gravimeters are used in geodesy, geophysics, and physics for a wide spectrum of applications. Stable gravimetric measurements over timescales from several days to decades are required to provide relevant insight into geophysical processes. Users of absolute gravimeters participate in comparisons with a metrological reference in order to monitor the temporal stability of the instruments and determine the bias to that reference. However, since no measurement standard of higher-order accuracy currently exists, users of absolute gravimeters participate in key comparisons led by the International Committee for Weights and Measures. These comparisons provide the reference values of highest accuracy compared to the calibration against a single gravimeter operated at a metrological institute. The construction of stationary, large scale atom interferometers paves the way towards a new measurement standard in absolute gravimetry used as a reference with a potential stability up to $1\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ at $1\text{\,}\mathrm{s}$ integration time. At the Leibniz University Hannover, we are currently building such a very long baseline atom interferometer with a $10\text{\,}\mathrm{m}$ long interaction zone. The knowledge of local gravity and its gradient along and around the baseline is required to establish the instrument’s uncertainty budget and enable transfers of gravimetric measurements to nearby devices for comparison and calibration purposes. We therefore established a control network for relative gravimeters and repeatedly measured its connections during the construction of the atom interferometer. We additionally developed a 3D model of the host building to investigate the self-attraction effect and studied the impact of mass changes due to groundwater hydrology on the gravity field around the reference instrument. The gravitational effect from the building 3D model is in excellent agreement with the latest gravimetric measurement campaign which opens the possibility to transfer gravity values with an uncertainty below the $10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ level. Keywords: atom interferometry, gravity acceleration, absolute gravimetry, gravimeter reference ## Introduction A variety of applications in geodesy, geophysics and physics require the knowledge of local gravity g [67]. These applications include observing temporal variations of the mass distribution in the hydrosphere, atmosphere and cryosphere and furthermore the establishment and monitoring of height and gravity reference frames, the determination of glacial isostatic adjustment, and the realisation of SI111Système International d’unités units, e. g., of force and mass [32, 28, 54]. The absolute value of gravity g is usually measured by tracking the free-fall of a test mass using a laser interferometer [33]. The operation of an absolute gravimeter (AG), especially the combination of several instruments in a project, requires special consideration of the offset to _true g_ and the change thereof. In addition, the long-term stability of absolute gravimeters is of particular relevance when measuring small gravity trends. For example, the determination of the glacial isostatic adjustment (GIA) on regional scales of around $1000\text{\,}\mathrm{km}$ [63] requires an instrument stable to the $20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ level over several years. Extending this effort by deploying several AGs also requires the knowledge of the biases of all the instruments involved [36]. The lack of a calibration service with a $10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ uncertainty requires the participation in key comparisons [9, KC, e. g. ] where the reference values are determined with an uncertainty of approximately $10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. This uncertainty level requires the participation of multiple gravimeters and cannot be achieved by comparison against a single gravimeter operated at a metrological institute. However, the development of stationary atom interferometers, which can be operated as gravimeters, so-called quantum gravimeters (QG), may result in such a superior reference in the future available for regular comparisons or on demand by the user. A major requirement in this respect is the control of systematic effects like wavefront aberration or the Coriolis effect. In this paper, we focus on the modelling and measurement of the local gravity field. We start by discussing the typical approaches for monitoring the long-term stability of an AG and tracing the measurements back to the SI (section 2). Then, after briefly describing the working principle of atomic gravimeters and the case for very long baseline atom interferometry (section 3), we present a gravity model for the Hannover Very Long Baseline Atom Interferometry (Hannover-VLBAI) facility, a new $10\text{\,}\mathrm{m}$-scale baseline atom interferometer in commissioning at the Leibniz University Hannover (section 4). Finally, we present the micro-gravimetric surveys performed at the instrument’s site (section 5) to assess the accuracy of the gravity model (section 6). This paves the way towards control of the systematics in the atom interferometer and accurate transfers of measured g values between the VLBAI operating as a gravimeter and transportable AGs in a nearby laboratory. ## Gravimeter bias and SI traceability Micro-g LaCoste FG5(X) [34] instruments represent the current state of the art in absolute gravimetry. They track the trajectories of a free-falling test mass with corner cubes by means of laser interferometry to determine the local acceleration of gravity g. These types of absolute gravimeters are referred to as _classical absolute gravimeters_ in the following text. As described by the 2015 CCM-IAG222Consultative Committee for Mass and related quantities – International Association of Geodesy Strategy for Metrology in Absolute Gravimetry [5], there are two complementary paths for the traceability of absolute gravity measurements: a) calibration of incorporated frequency generators and b) additional gravimeter comparisons against a reference. The direct way of tracing absolute gravity measurements back to the SI goes through the calibration of their incorporated laser and oscillator to standards of length and time [68]. In high-accuracy instruments, the laser frequency is typically locked to a standard transition of molecular iodine [6, 44]. The time reference is usually given by a rubidium oscillator which needs to be regularly compared with a reference oscillator to ensure its accuracy as external higher-accuracy time sources are typically not available at measurement sites. In most cases, the oscillator’s frequency drift is linear $($<0.5\text{\,}\mathrm{mHz}\text{/}\mathrm{month}$\text{ or }$<1\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$/$\mathrm{month}$)$ and a few calibrations per year are sufficient. However, [30] and [53] report on sudden jumps in frequency333Current publications refer to the Microsemi (formerly Symmetricon) SA.22c rubidium oscillator equivalent to several tens of $\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ due to increased concentrations of gaseous helium [43] when measuring near superconducting gravimeters. Such higher concentrations might occur after installation, maintenance, or repair of a superconducting gravimeter and are unlikely during normal operation. The frequency drift changes to an exponential decrease after the helium event and may remain this way for years [51]. Figure 1: Degree of Equivalence (DoE) of joint participants of EURAMET.M.G-K1 [10, ], CCM.G-K2 [11, ], EURAMET.M.G-K2 [37, ] and EURAMET.M.G-K3 [9, ]. The participants are sorted by DoE of the first KC. The expanded uncertainty is given only for the last KC. Pilot Study (PS) indicates instruments of non NMI/DI institutions. All AGs shown are laser interferometers of which eight are FG5(X) type instruments. The equivalence of gravity measurement standards and the definition of the gravity reference are established by international comparisons in the framework of the CIPM MRA444Mutual Recognition Agreement of the Comité International des Poids et Mesures. Since no higher-order reference instrument is available, key comparisons are held in an approximately two-year interval, alternating between CIPM key comparisons and regional comparisons. There, the instruments operated by National Metrology Institutes (NMI) and Designated Institutes (DI) are used to determine the Key Comparison Reference Value (KCRV). The bias to the KCRV, or Degree of Equivalence (DoE) is then calculated for all individual instruments, including those without NMI/DI status participating in the so-called pilot study (PS), and serves as validation for their uncertainty. Figure 1 shows the common participants, out of a total number of 35 gravimeters participating in the comparisons, to the last four KC held in Europe [10, 11, 37, 9]. One observes that the spread of DoE over all instruments is around $\pm 75\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$, and at a similar level for the most extreme cases of individual instruments. Even though the DoEs of the instruments in these comparisons are typically within the uncertainties declared by the participants, figure 1 also shows the necessity of determining these biases of gravimeters, classical and quantum alike, to monitor an instrument’s stability in time. Biases can then be taken into account in gravimetric projects. The variation of the bias of an instrument can be explained by a variety of factors. For example, [35] show that a permanent change in the bias of a classical AG can occur during manufacturer service or unusual transport conditions (e. g. aviation transport). Also, [25, 26] identified, characterised and partially removed biases originating in the signal processing chain of FG5 gravimeters, e. g. due to cable length and fringe signal amplitude. Regional KCs are linked to a CIPM KC by a small number of common NMI/DI participants applying the so-called linking converter [22, typically around $\pm 10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$,]. The underlying assumption is that instrumental biases of the NMI/DI instruments remain stable [8]. Otherwise, this would introduce an additional shift in the bias of all participating instruments of the regional KC and PS. Quantum gravimeters, based on matter wave interferometry with cold atoms, offer a fully independent design. They have demonstrated stabilities and accuracies at levels comparable to those from state of the art classical AGs by participating in KCs [16, 23] or common surveys with other instruments at various locations [13, 51]. The availability of improved QGs as gravity references provides an opportunity to enhance the stability of reference values obtained during key comparisons and therefore lead to an international gravity datum of better stability in time. Just by that alone QGs could become a serious alternative to classical absolute gravimeters. ## Very long baseline atomic gravimetry ### Atominterferometric gravimetry Most atomic gravimeters use cold matter waves as free-falling test masses to measure absolute gravity. They exploit the coherent manipulation of the external degrees of freedom of these atomic test masses with light pulses to realise interferometers sensitive to inertial quantities and other forces. These techniques are for example used to perform precision measurements of fundamental constants [49, 3, 39], test fundamental physics [55, 48, 21], sense small forces [2] and perform gravimetry, gravity-gradiometry, and measure rotations with record instabilities and inaccuracies [31, 12, 15, 72, 50, 57]. Figure 2: Mach–Zehnder light-pulse atom interferometer geometry in a uniform acceleration field $\mathbf{a}$. At time $t_{0}$, the atomic matterwave is put in a superposition of momenta $p$ ( ) and $p+\hbar k_{\mathrm{eff}}$ ( ). The momenta are reversed at time $t_{0}+T$ to recombine the wave packets with a last light pulse at time $t_{0}+2T$. The populations in the two momentum classes after the last light pulse allow extracting the interferometric phase $\Delta\phi$. Atomic gravimeters typically realise the Mach–Zehnder light-pulse atom interferometer geometry [24] depicted in figure 2. In this analogon to the eponymous configuration for optical interferometers, the leading-order interferometric phase $\Delta\phi$ scales with the space-time area enclosed by the interferometer: $\Delta\phi=\mathbf{k}_{\mathrm{eff}}\cdot\mathbf{a}T^{2}$ (1) where $\hbar\mathbf{k}_{\mathrm{eff}}$ is the recoil transfered to the atomic wave packets by the atom-light interaction processes (cf. figure 2, $\hbar$ is the reduced Planck constant and $\mathbf{k}_{\mathrm{eff}}$ the effective optical wave vector), $\mathbf{a}$ the uniform acceleration experienced by the atoms during the interferometric sequence, and $T$ the pulse separation time. The full interferometer has a duration of $2T$. The knowledge of the instrument’s scale factor $k_{\mathrm{eff}}T^{2}$ and the measurement of the phase $\Delta\phi$ allow determining the projection of the acceleration $\mathbf{a}$ along $\mathbf{k}_{\mathrm{eff}}$. When $\mathbf{k}_{\mathrm{eff}}$ is parallel to $\mathbf{g}$, such an instrument can therefore be used as a gravimeter, measuring the total vertical acceleration of the matter waves used as test masses. The Mach–Zehnder light-pulse atom interferometer works as follows. For each interferometric sequence, a sample of cold atoms is prepared in a time $T_{p}$. Then, at time $t=t_{0}$, the first atom-light interaction pulse puts the matter wave in a superposition of quantum states with different momenta $\mathbf{p}$ and $\mathbf{p}+\hbar\mathbf{k}_{\mathrm{eff}}$, thus effectively creating two distinct semi-classical trajectories. At time $t=t_{0}+T$, a second atom-light interaction process redirects the two atomic trajectories to allow closing the interferometer at time $t=t_{0}+2T$ with a third light pulse. Counting the population of atoms in the two momentum states provides an estimation of the interferometric phase $\Delta\phi$. Finally, the cycle of preparation of the cold atoms, coherent manipulation of the matter waves, and detection is repeated. Since the atom-light interaction imprints the local phase of the light on the matter waves, the above measurement principle can be interpreted as measuring the successive positions of a free-falling matter wave at known times $t_{0}$, $t_{0}+T$, and $t_{0}+2T$ with respect to the light field. The inertial reference frame for the measurement system, similar to the superspring in FG5(X) gravimeters, is usually realised by a mirror retro-reflecting the light pulses, creating well-defined equiphase fronts. Practically, the interferometric phase $\Delta\phi$ is scanned by accelerating the optical wave fronts at a constant rate $\alpha$, effectively continuously tuning the differential velocity between the matter waves and the optical equiphase fronts. Assuming that $\mathbf{k}_{\mathrm{eff}}$ and $\mathbf{a}$ are parallel, the interferometric phase reads: $\Delta\phi=k_{\mathrm{eff}}\left(a-\frac{\alpha}{k_{\mathrm{eff}}}\right)T^{2}\ .$ (2) When $\alpha=k_{\mathrm{eff}}a$, the interferometric phase vanishes independently of the interferometer’s duration $2T$, allowing to unambiguously identify this operation point. Physically, $\alpha=k_{\mathrm{eff}}a$ exactly compensates the Doppler effect experienced by the atomic matter waves due to the acceleration $a$. Therefore, the measurement of the acceleration $a$ amounts to a measurement of the acceleration rate $\alpha$ which can be traced back to the SI since it corresponds to frequency generation in the radio- frequency domain. Assuming white noise at a level $\delta\phi$ for the detection of the interferometric phase, the instrument’s instability is given by: $\delta a(\tau)=\sqrt{2T+T_{p}}\cdot\frac{\delta\phi}{k_{\mathrm{eff}}T^{2}}\cdot\frac{1}{\sqrt{\tau}}\ .$ (3) where $\tau$ is the measurement’s integration time. This expression reveals the three levers for reducing the measurement instability: decreasing the single shot noise level $\delta\phi$, increasing the scale factor $k_{\mathrm{eff}}T^{2}$, and minimising the sample preparation time $T_{p}$, as it contributes to the total cycle time without providing phase information. In transportable devices, record instabilities have been achieved by [12] with $\delta a=$96\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$$ at $\tau=$1\text{\,}\mathrm{s}$$. Commercial instruments like the Muquans AQG [31] reached instabilities of $500\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ at $\tau=$1\text{\,}\mathrm{s}$$ with sample rates up to $2\text{\,}\mathrm{Hz}$. The dominant noise source is vibrations of the mirror realising the reference frame for the measurements. The accuracy of such quantum gravimeters stems from the well-controlled interaction between the test masses and their environment during the measurement sequence. The main sources of inaccuracy in such instruments originate from uncertainties in the atom-light interaction parameters (e. g. imperfections of the equiphase fronts of the light wave), stray electromagnetic field gradients creating spurious forces, thus breaking the free-fall assumption, and knowledge of the inhomogeneous gravity field along the trajectories. Extensive characterisation of these effects led to uncertainties in QGs below $40\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$, consistent with the results from CIPM key comparisons [15] or common surveys with classical AGs [12]. ### Very Long Baseline Atom Interferometry Very Long Baseline Atom Interferometry (VLBAI) represents a new class of ground-based atom interferometric platforms which extends the length of the interferometer's baseline from tens of centimetres like in typical transportable instruments [12, 15] to multiple meters. According to equation (1), the vertical acceleration sensitivity of a Mach–Zehnder type atom interferometer scales linearly with the length of the baseline ($\sim aT^{2}$). Therefore, an increase in the length of the baseline potentially enables a finer sensitivity for the atomic gravimeter through an increased scale factor $k_{\mathrm{eff}}T^{2}$. A $10\text{\,}\mathrm{m}$-long baseline instrument can for example extend the interferometric time $2T$ to around $1\text{\,}\mathrm{s}$ if the atoms are simply dropped along the baseline or up to $2.4\text{\,}\mathrm{s}$ if they are launched upwards in a fountain-like fashion. In the simple drop case, the velocity acquired by the atoms between their release from the source and the start of the interferometer leads to an interferometer duration shorter than half of the one for the launch case. For our apparatus, the distance between the top source chamber and the region of interest is around $2\text{\,}\mathrm{m}$ (see figure 3), constraining $T<$400\text{\,}\mathrm{ms}$$ for simple drops. Using realistic parameters ($T_{p}=$3\text{\,}\mathrm{s}$$, $\delta\phi=$10\text{\,}\mathrm{mrad}$$), equation (3) yields potential short- term instabilities for VLBAIs ($\tau=$1\text{\,}\mathrm{s}$$ integration time): $\begin{array}[]{ll}T=$400\text{\,}\mathrm{ms}$\text{:\leavevmode\nobreak\ }&\delta a=$8\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$\\\ T=$1.2\text{\,}\mathrm{s}$\text{:\leavevmode\nobreak\ }&\delta a=$1\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$\end{array}$ (4) competing with the noise level of superconducting gravimeters [45, 46] while providing absolute values of the gravity acceleration g. Nevertheless, the increased scale factor $k_{\mathrm{eff}}T^{2}$ gained by the expanded baseline comes at the price of a stationary device with added complexity due to its size, and a vibration noise sensitivity magnified by the same scale factor as the gravitational acceleration for frequencies below $\nicefrac{{1}}{{(2T)}}$. Hence, the use of VLBAIs as ultra stable gravimeters requires new developments in the control of environmental vibrations [19]. Also, time- and space-varying electromagnetic and gravity fields along the free-fall trajectories of the matter waves have a direct impact on the accuracy and stability of the instrument, as the corresponding spurious forces depart from the assumptions of equation (1), therefore leading to biases [7] and impacting the instrument’s effective height [60]. ### Effective height In order to compare measurements of a VLBAI gravimeter with other instruments, it is crucial to determine the effective height $z_{\mathrm{eff}}$ defined by: $g_{0}-\gamma z_{\mathrm{eff}}=\frac{\Delta\phi_{\mathrm{tot}}}{k_{\mathrm{eff}}T^{2}}$ (5) where $g_{0}\approx$9.81\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ is the value of gravity at $z=0$, $\gamma\approx$3\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-2}\text{\,}{\mathrm{m}}^{-1}$$ the magnitude of the linear gravity gradient, and $\Delta\phi_{\mathrm{tot}}$ the phase shift measured by the interferometer. The right-hand side is the value of gravity measured by the atom interferometer, including all bias sources. Restricting to first order in the gravity-gradient $\gamma$, and applying a path-integral formalism, one gets [40]: $z_{\mathrm{eff}}=z_{0}-\dfrac{\Delta g}{\gamma}\quad\text{with}\quad\Delta g=\frac{7}{12}\gamma g_{0}T^{2}-\gamma\bar{v}_{0}T$ (6) where $z_{0}$ is the height of the start of the interferometer and $\bar{v}_{0}=v_{0}+\nicefrac{{\hbar k_{\mathrm{eff}}}}{{(2m)}}$ the mean atomic velocity just after the interferometer opens ($v_{0}$ is the atomic velocity before the first beamsplitter, and $m$ is the atomic mass). This expression for $z_{\mathrm{eff}}$ is compatible with the one given for FG5 gravimeters by [38]. In particular, it only depends on the value of the gradient $\gamma$ through $v_{0}$ and $z_{0}$. Indeed, the interferometer is controlled in time and the initial position and velocity $z_{0}$ and $v_{0}$ are therefore given by the free-fall motion of the atoms between the source chamber and the region of interest. In general, $z_{\mathrm{eff}}$ depends on the geometry of the atom interferometer. For the simple drop case in the Hannover VLBAI facility (see section 3.4), $z_{\mathrm{eff}}\approx$9.2\text{\,}\mathrm{m}$$. Corrections to equation (6) must be taken into account to constrain the uncertainty on gravity at $z_{\mathrm{eff}}$ below $10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. On the one hand, terms of order $\gamma^{2}$ and higher in $\Delta\phi_{\mathrm{tot}}$ contribute at the sub-$\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ level. On the other hand, one can use perturbation theory [66] to estimate the effect of the non-homogeneous gravity gradient along the interferometer’s baseline. Using the data discussed here, we evaluate this effect below $5\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$, therefore lying within the model’s uncertainty (see section 6) and similar to the known contribution for FG5(X) gravimeters [60]. Finally, when using multiple concurrent interferometers at different heights, the effect of a homogeneous gravity gradient can be mitigated by measuring it simultaneously with the acceleration value [4]. In this case, the effective height corresponds to the position of the mirror giving the inertial reference. Detailed modelling is however still necessary to push the uncertainty budget in the sub-$10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ and calibrate the instrument to the level of its instability. ### The Hannover VLBAI facility We introduce the Hannover Very Long Baseline Atom Interferometry facility, an instrument developed at the newly founded Hannover Institute of Technology (HITec) of the Leibniz University Hannover, Germany. It builds on the concepts outlined in section 3.2 to provide a platform to tackle challenges in extended baseline atom interferometry. In the long term, it aims at tests of our physical laws and postulates like for example the universality of free fall [20], searches for new forces or phenomena, and the development of new methods for absolute gravimetry and gravity gradiometry [56]. Upper atomic sourceLower atomic sourceBaselineultra-high vacuum chamberand magnetic shieldInertial referencevibration isolated mirrorRegion of interestfor precisionatom interferometry$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$Height / $\mathrm{m}$ Figure 3: The Hannover Very Long Baseline Atom Interferometry (VLBAI) facility and its three main elements: source chambers , baseline , and inertial reference system and vacuum vessel (VTS) . The baseline and upper source chambers are supported by an aluminium structure (VSS, dark blue). The region of interest for atom interferometry is shaded in light blue. The Hannover VLBAI facility is built around three main elements shown in figure 3: 1. 1. Ultra cold samples of rubidium and ytterbium atoms are prepared in the two _source chambers_ , allowing for both drop (max $T=$400\text{\,}\mathrm{ms}$$) and launch (max $T=$1.2\text{\,}\mathrm{s}$$) modes of operation. Advanced atom-optics promise enhanced free-fall times by relaunching the wave packets during the interferometric sequence [1]; 2. 2. The reference frame for the inertial measurements is realised by a _seismically isolated mirror_ at the bottom of the apparatus. The seismic attenuation system (SAS) uses geometric anti-spring filters [69] to achieve vibration isolation above its natural resonance frequency of $320\text{\,}\mathrm{mHz}$. The isolation platform is operated under high vacuum conditions to reduce acoustic and thermal coupling. The vacuum vessel containing the SAS is denoted VTS in sections 4–6; 3. 3. The $10.5\text{\,}\mathrm{m}$-long _baseline_ consists of a $20\text{\,}\mathrm{cm}$ diameter cylindrical aluminium vacuum chamber and a high-performance magnetic shield [71]. The interferometric sequences take place along this baseline, in the $8\text{\,}\mathrm{m}$-long central _region of interest_ where the longitudinal magnetic field gradients fall below $2.5\text{\,}\mathrm{nT}\text{/}\mathrm{m}$. In order to decouple the instrument from oscillations of the walls of the building, the apparatus is only rigidly connected to the foundations of the building. The VTS (and SAS) and lower source chamber are mounted on a baseplate directly connected to the foundation. The baseline and upper source chamber are supported by a $10\text{\,}\mathrm{m}$ high aluminium tower, denoted as VLBAI support structure (VSS) in the following sections. The footprint of the device on the floor is $$2.5\text{\,}\mathrm{m}$\times$2.5\text{\,}\mathrm{m}$$. Traceability to the SI is ensured by locking the instrument’s frequency references to standards at the German NMI (PTB Braunschweig) via an optical link [42]. All heights are measured from the instrument’s baseplate. The altitude of this reference point in the German height datum is $50.545\text{\,}\mathrm{m}$. ## Environmental model (a) HITec cross-section (not to scale) (b) HITec top view Figure 4: Views of HITec: cross-section (4(a)) of the VLBAI laboratories with the gravimetric network of 2019 along two vertical profiles and region of interest (blue). The indicated groundwater variation (thick bar) refers to an average annual amplitude of $0.3\text{\,}\mathrm{m}$. The thin bar indicates extreme low and high levels. The height $z=$0\text{\,}\mathrm{m}$$ refers to the top of the baseplate. The top view of HITec (4(b)) shows the orientation of our coordinate system, the location of the VLBAI facility (blue) and the gravimetry lab including piers for gravimeters (light grey). The VLBAI facility is implemented in the laboratory building of the Hannover Institute of Technology. The building consists of three floors (one basement level, two above street level) and is divided into a technical part mainly containing the climate control systems, and a section with the laboratories (see figure 4). In the laboratory part, a so-called backbone gives laboratories access to the technical infrastructure and divides the building in two parts along its long axis. The backbone and southern row of laboratories have a footprint of $$13.4\text{\,}\mathrm{m}$\times$55.4\text{\,}\mathrm{m}$$ and extend approximately $5\text{\,}\mathrm{m}$ below surface level. The northern row of laboratories is fully above ground except for the gravimetry laboratory which is on an intermediate level, around $1.5\text{\,}\mathrm{m}$ below street level and $3.4\text{\,}\mathrm{m}$ above basement level (see figure 4(a)). The foundation of the building is $0.5\text{\,}\mathrm{m}$ thick except beneath the gravimetry laboratory, which has a separate and $0.8\text{\,}\mathrm{m}$ thick one. Figure 4(a) also shows the measurement points for the relative gravimeters along the VLBAI main axis and a second validation profile, occupied using tripods, next to the VLBAI which were used for the measurements presented in section 5. ### Physical model Following the methods described by [27], we discretise the HITec building into a model of rectangular prisms that accounts for more than $500$ elements. The geometry is extracted from the construction plans, and we verified all the heights by levelling, also including a benchmark with a known elevation in the German height datum. The building is embedded in a sedimentary ground of sand, clay, and marl ($2050\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$). For the edifice itself, we include all walls and floors made of reinforced concrete ($2500\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$), the $7\text{\,}\mathrm{cm}13\text{\,}\mathrm{cm}$ thick liquid flow screed covering the concrete floors in the labs ($2100\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$), and the gypsum drywalls ($800\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$). We also incorporate the insulation material ($150\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$) and gravel on the roof ($1350\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$). We use a simplified geometry to model the large research facilities in the surroundings. This is for example the case for the Einstein-Elevator [29], a free-fall simulator with a weight of $165\text{\,}\mathrm{t}$ and horizontal distances of $32\text{\,}\mathrm{m}$ and $16\text{\,}\mathrm{m}$ to the VLBAI facility and gravimetry laboratory, respectively. Finally, we account for laboratory equipment, e. g. optical tables ($550\text{\,}\mathrm{kg}$ each) according to the configuration at the time of the gravimetric measurement campaigns. During the first measurements (2017), the interior construction was still in progress, and the laboratories were empty. By the time of the second campaign (2019), the building was fully equipped. The VLBAI support structure (VSS) and the vacuum tank (VTS) for the seismic attenuation system were in place. The VLBAI instrument (atomic sources, magnetic shield, $10\text{\,}\mathrm{m}$ vacuum tube) and seismic attenuation system were completed after the second campaign. Due to their inclined or rounded surfaces, the VLBAI experimental apparatus and its support structure require a more flexible method than rectangular prisms to model their geometry. We apply the method described by [41] and divide the surface of the bodies to be modelled into polygonal faces to calculate the gravitational attraction from surface integrals. Contrary to the rectangular prisms method, there are only few restrictions on the underlying geometry. Most notably, all vertices of a face must lie in one plane and the normal vectors of all surfaces must point outward of the mass. For example, normal vectors of faces describing the outside surface of a hollow sphere must point away from the sphere and normal vectors on the inside surface must point towards the centre, away from the mass of the wall of the sphere. We extract the geometry of the VLBAI facility components from their tridimensional CAD model through an export in STL555Stereolithography or standard triangulation language format [47]. This divides the surface of the bodies into triangular faces, therefore ensuring planar faces by default. Moreover, the STL format encodes normal vectors pointing away from the object. Both prerequisites for the polygonal method by [41] are thus met. Using this method, the VSS (aluminium, $2650\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$, total weight $5825\text{\,}\mathrm{kg}$) consists of roughly $86000$ faces and the VTS and corresponding baseplates (stainless steel, $8000\text{\,}\mathrm{kg}\text{/}{\mathrm{m}}^{3}$, total weight $2810\text{\,}\mathrm{kg}$) contain $187000$ faces, mostly due to the round shape and fixtures of the VTS. As the overall computation time to extract the attraction of these components with a $\mathrm{c}\mathrm{m}$-resolution on both vertical profiles remains in the range of minutes on a desktop PC, we do not need to simplify the models. The Monte Carlo simulations described in section 6 nevertheless require the computing cluster of the Leibniz University Hannover (LUH). We use MATLAB666MATLAB Version 9.4.0.813654 (R2018a) to perform the numerical calculations. As a cross-check, we implemented both the rectangular prisms and polyhedral bodies methods for the calculation of the attraction effect of the main frame of the HITec building. Both approaches agree within floating point numerical accuracy. ### Time variable gravity changes Mostly for the benefit of the future operations of the VLBAI, we include the effects of groundwater level changes, atmospheric mass change, and Earth’s body and ocean tides in our modelling. This is necessary for the individual gravimetry experiment (and other physics experiments as well) in the VLBAI on one hand, and for comparing measurements from different epochs, e. g. with different groundwater levels, on the other hand. Previous investigations in the gravimetry lab of a neighbouring building showed a linear coefficient of $170\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ per meter change in the local groundwater table [64]. This corresponds to a porosity of $>30\text{\,}\mathrm{\char 37\relax}$ of the soil [17]. For our model, we adapt a porevolume of $30\text{\,}\mathrm{\char 37\relax}$, which has to be verified by gravimetric measurements and correlation with local groundwater measurements. Two automatic groundwater gauges are available around the building: one installed during the construction work and a second with records dating back several decades also used by [64]. The effect of atmospheric mass changes is calculated using the ERA5 atmospheric model provided by the European Centre for Medium-Range Weather Forecasts777https://www.ecmwf.int and the methods described by [51]. Tidal parameters are extracted from observational time series [58, 52]. Other temporal gravity changes are not in the scope of this work. Currently, time variable gravity is also monitored with the gPhone-98 gravimeter of the Institute of Geodesy (IfE) at the LUH. In the long term, we consider the addition of a superconducting gravimeter for this purpose when the VLBAI facility is fully implemented and the experimental work is beginning. The support of a superconducting gravimeter is also vital in the characterisation of new gravimeters [12]. ### Self-attraction results Figure 5 shows the vertical component of the gravitational acceleration generated by the building, equipment, VSS and VTS. The VLBAI main axis is in the centre of the left plot ($x=$0\text{\,}\mathrm{m}$$). The large structures around $5\text{\,}\mathrm{m}$ and $10\text{\,}\mathrm{m}$ correspond to the floor levels. Smaller structures are associated to, for example, optical tables or the VSS. The right panel of figure 5 highlights the attraction calculated for the main axis ($x=$0\text{\,}\mathrm{m}$$) and for a second profile along $x=$-1.8\text{\,}\mathrm{m}$$ and $y=$0\text{\,}\mathrm{m}$$. The first profile shows a smooth curve except for the bottom $2\text{\,}\mathrm{m}$, which are affected by the VTS. In this model, the part above $2\text{\,}\mathrm{m}$ on the main axis is empty space. The second profile, chosen as a sample from the xz-plane, passes through the floors, hence the zig-zag features around $5\text{\,}\mathrm{m}$ and $10\text{\,}\mathrm{m}$. While the main axis will later be occupied by the instrument’s baseline, this second profile, similar to the validation profile, represents a location that will always remain accessible to gravimeters. Figure 5: Calculated gravitational attraction from the building, large laboratory equipment, VSS and VTS in the xz-plane (left) and exemplarily on two profiles (right). ### Effect of groundwater level changes Based on the extensive groundwater level recordings from the gauge nearby the HITec building, we study the impact of groundwater level changes [67, see also] on gravitational attraction inside the building, specifically along the VLBAI main and validation profiles, as well as in the gravimetry laboratory. Due to the layout of the different basement levels in the building (see figure 4(a)), a change of the groundwater table affects gravity in the VLBAI laboratories differently than in the gravimetry lab. Depending on the groundwater level, the foundation beneath the VLBAI laboratories can be partially within the groundwater table, whereas this is never the case for the gravimetry laboratory. As shown on figure 4(a), the mean groundwater table is nevertheless below the level of the foundation below the VLBAI laboratories. Therefore, at certain points of the average annual cycle of amplitude $0.3\text{\,}\mathrm{m}$, the groundwater table will rise only around the foundation of the VLBAI laboratories, whereas its level will still increase below the gravimetry laboratory. This effect is even more stringent for years where the average cycle amplitude is exceeded (around one in four years). Figure 6: Effect of groundwater variations (all heights in the height system of the model, cf. figure 4(a)) on gravity in the gravimetry lab (left) and along the VLBAI axis (right) with respect to the mean groundwater level (dotted line ). The dashed line ( ) indicates the bottom of the foundation below the VLBAI. The coloured lines indicate the change of gravity $\delta g_{\mathrm{gw}}$ in various heights in the gravimetry and VLBAI laboratories. The height of the gravimetry piers in the height system of the model is $3.35\text{\,}\mathrm{m}$. Figure 6 illustrates the different influence of the groundwater table level on gravity in the VLBAI and gravimetry laboratories. The estimated change of gravity $\delta g_{\mathrm{gw}}$ due to the attraction corresponding to groundwater level variations is presented for different heights above the gravimetry pier and along the VLBAI main axis. As the groundwater level is always changing directly beneath the instrument piers in the gravimetry laboratory, we expect an almost linear change of gravity with changing groundwater level. The change of gravity is also almost independent of the height above the pier, as shown by the almost identical lines for $z=$3.35\text{\,}\mathrm{m}$$ directly on the pier and $1.4\text{\,}\mathrm{m}$ above the pier, covering the instrumental heights of transportable gravimeters. Therefore, AGs with various sensor heights, e. g., A-10 and FG5X, are affected in the same manner. The increase of $\delta g_{\mathrm{gw}}$ is $32\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ in an average year. This behaviour is different in the VLBAI laboratories. In current records, the groundwater level never fell below the foundation of the backbone (cf. figure 4(a)). This effect is seen in the small divergence (up to $3\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$) for groundwater levels below the foundation of the VLBAI (dashed line). Once the groundwater level reaches the lower edge of the VLBAI foundation, gravity will not increase linearly along the VLBAI main axis as the groundwater rises further. Moreover, in this situation, the effect has a different magnitude depending on the height in the room. In a year with the average amplitude of groundwater level variation, ca. $\pm 0.15\text{\,}\mathrm{m}$ around the line indicating the mean groundwater level, $\delta g_{\mathrm{gw}}$ will differ by $5\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ between basement and the top floor. In years exceeding the average groundwater variation, the difference between the basement and upper levels increases further. This effect is within $\pm 2\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ on the validation profile in the average groundwater cycle. These observations will be crucial when comparing AGs in the gravimetry laboratory to the VLBAI facility operated as a quantum gravimeter. Depending on the geometry of a specific atom interferometer realisation, the instrumental height of the VLBAI gravimeter changes and can introduce changes in the measured value of g of more than $10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ as a result of the groundwater effect in years with a higher than usual groundwater level. The magnitude of $10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ is larger than the targeted accuracy of the VLBAI and also a relevant size for classical AGs in comparisons. It should also be noted, that the model only calculates the gravitational attraction of the groundwater variation. A potential vertical displacement of the ground itself is currently not taken into account, leading to a possible underestimation of the effect. In order to track the effect of groundwater level changes more accurately, we plan to extend the findings of [64] by correlating periodic gravimetric measurements on the validation profile in the VLBAI laboratories with the recordings of the two groundwater level gauges around the building. This should in particular allow us to take into account that, due to capillarity effects, the groundwater level will probably not sink uniformly below the foundation beneath the VLBAI laboratories once it reaches that level. ## Gravimetric measurements In June 2017 and August 2019, we performed surveys using relative gravimeters to verify our model from section 4 along the VLBAI main and validation profiles. This approach was already demonstrated in [54], in which the gravity field impact of a $200\text{\,}\mathrm{kN}$ force standard machine at the Physikalisch-Technische Bundesanstalt in Braunschweig was modelled. That model was verified with gravimetric measurements prior and after the installation of the force machine. The difference between the modelled impact and the measurement was within the uncertainty of the gravimeters used. For each measurement point, we measured its connection to at least another point and applied the step method with ten connections [65]. A connection corresponds to one gravity difference observation between two points. Ten connections require five occupations of a measurement point with a gravimeter. We measured most connections with at least two different instruments, reducing the outcomes to a mean instrumental height of $0.22\text{\,}\mathrm{m}$ above ground or platform. We then performed a global least-squares adjustment using the Gravimetry Net Least Squares Adjustment software from IfE [70, GNLSA,]. The measurements are also calibrated in this process. We determined the individual calibration factors of the gravimeters on the Vertical Gravimeter Calibration Line in Hannover [61, 59] at least once in the week prior to the measurement campaigns. The software also corrects Earth tides, applying our observed parameters, and atmospheric mass changes by means of the linear factor of $3\text{\,}\mathrm{nm}\text{/}{\mathrm{s}}^{2}\text{/}\mathrm{hPa}$ with respect to normal air pressure at station elevation. In order to account for instrumental drift in the global adjustment, we treat each day and each instrument independently and use a variance component estimation to weight the measurements in the global network adjustment. The specific groundwater effect discussed in section 4.4, considering different magnitudes depending on height, does not apply for either 2017 or 2019 because the groundwater levels were below the foundation of the VLBAI in both years. ### 2017 Gravimetry campaign We first mapped the gravity profile along the VLBAI profiles in June 2017, when the HITec building was still under construction and the VLBAI experimental apparatus not yet installed. Using the Scintrex CG3M-4492 (short CG3M) and ZLS Burris B-144 (B-114) spring gravimeters [62, 52], we measured a total of $147$ connections between seven positions spaced by ca. $2\text{\,}\mathrm{m}$ along the VLBAI main axis, nine positions on the validation profile, and two points outside of the building. We used a scaffolding to access the measurement points on the main axis. However, although the scaffold was anchored against the walls, the uppermost platforms were too unstable to ensure reliable measurements. The B-114 was only able to measure on the bottom three positions, because the feedback system was not powerful enough to null the oscillating beam on the upper levels. The four upper levels were only occupied by the CG3M. We connected each point on the scaffold to another one on the same structure and to the closest fixed floor level, at a point part of the validation profile. As shown in figure 4(a), the validation profile included measurements on the floor and on different sized tripods to determine the gradients. The variance component estimation gives a posteriori standard deviations for a single gravity tie observation of $50\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ for the B-114 and $100\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ for the CG3M. The standard deviations for the adjusted gravity values range from $15\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}42\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ with a mean value of $28\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. The standard deviations of the adjusted gravity differences vary from $21\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ between fixed floor levels to $59\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ between consecutive levels on the scaffold. The transfer of height from the upper floor to the basement through the intermediate levels on the scaffold showed a $2\text{\,}\mathrm{mm}$ discrepancy compared to the heights from levelling. We included the corresponding $$2\text{\,}\mathrm{mm}$\cdot$3\text{\,}\mathrm{nm}\text{/}{\mathrm{s}}^{2}\text{/}\mathrm{mm}$=$6\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$$ as a systematic uncertainty for the adjusted gravity values for the values measured on the scaffold. We also account for a $1\text{\,}\mathrm{mm}$ uncertainty on the determination of the relative gravimeter sensor height. ### 2019 Gravimetry campaign Figure 7: Measurement at the VSS in 2019 with the B-64 (foreground) on the validation profile and the CG6 (background) inside the VSS on a platform with an operator wearing a security harness. The B-64 is operated on a small tripod to raise the sensor height closer to the CG6 sensor height. We mapped the gravity profile along the VLBAI axes in a more extensive manner in summer and fall 2019. Most measurements were performed in one week of August 2019, adding two days in October and November 2019. We used moveable platforms inside the VSS, installed in June 2019, and could measure on $16$ levels on the main axis, spaced by $0.45\text{\,}\mathrm{m}0.95\text{\,}\mathrm{m}$. The scheme for the validation profile did not change. The layout of the network is depicted in figure 4(a). For this campaign, we used the CG3M, the Scintrex CG6-0171 (CG6), and ZLS Burris B-64 (B-64) spring gravimeters [62, 59, 52]. Owing to the high mechanical stability of the VSS, measurements along the main axis were unproblematic for all instruments and the measurement noise was at a similar level on the moveable platforms and on the fixed floors (see figure 7). All but one position were occupied with at least two gravimeters, amounting to $439$ connections in the network adjustment. The a posteriori standard deviations (single gravity tie measurement) of the observations range from $15\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}60\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ with more than $50\text{\,}\mathrm{\char 37\relax}$ below $30\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. The higher standard deviations are a result of two days of measurements with the CG3M and connections to two particular positions outside of the region of interest of the VLBAI. The standard deviations of adjusted gravity values in the network range from $7\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}19\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ with a mean of $9\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. This improvement, compared to the previous campaign, can be attributed to the stability of the VSS, the addition of the CG6 and the total number of measurements performed. The height of the moveable platforms inside the VSS was determined by a combination of levelling and laser distance measurements888Leica Disto D210 to two fixed platforms and the ceiling. For the height determination of the platforms, the uncertainty is $1\text{\,}\mathrm{mm}$ due to the laser distance measurement. We also account for an $1\text{\,}\mathrm{mm}$ uncertainty in the determination of the instrumental height above the platforms. ## Combination of model and measurement The measurement and model results along the VLBAI main and validation profiles are presented in figure 8. Figure 8(a) shows the total variation of gravity along the main axis. The plot is dominated by the normal decrease of gravity with height. The effect of the building can be better seen when removing the change of gravity with height and visualising only the attraction effect of the building and laboratory equipment, as on figure 8(b). There, the model corresponds to the configuration for the 2019 campaign and is identical to the $x=$0\text{\,}\mathrm{m}$$, $y=$0\text{\,}\mathrm{m}$$ line in figure 5. Figure 8(d) shows the model and measurements along the validation profile. The models presented in figure 8 use the nominal values for the densities of building elements (concrete floors and walls, drywalls, etc.). Since these can have variations over the building, we performed a Monte Carlo simulation ($50000$ runs) varying the densities of the corresponding model elements by $\pm 5\text{\,}\mathrm{\char 37\relax}$ according to a normal distribution. This leads to a variation of attraction of $\pm 27\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}\pm 37\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ for heights between $4\text{\,}\mathrm{m}$ and $13\text{\,}\mathrm{m}$, as shown by the thin blue lines on figures 88(b)–8(d). Using a uniform distribution of the density parameters increases the variability by around $20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. The VSS and VTS are not part of the Monte Carlo simulation since their geometry and materials are well known. (a) central axis: gravity variation (b) central axis: model (c) central axis: residuals (d) validation profile: model Figure 8: Measurement and model results on the VLBAI central axis (8(a)–8(c)) and the validation profile (8(d)). The shaded area in (8(a)–8(c)) indicates the region of interest. The total variation of gravity along the central axis is shown in (8(a)). The modelled and measured attraction by the environment (with the change of gravity with height removed) on the central and validation profile is shown in (8(b)) and (8(d)). The errorbars indicate the standard deviations from the network adjustment and the model simulations according to equation (7). The maximum and minimum results of the $\pm 5\text{\,}\mathrm{\char 37\relax}$ density variations from Monte Carlo (MC) simulation of model parameters are indicated by the thin blue lines. The residuals of observations minus model $\delta g_{\mathrm{omc}}$ are given in (8(c)) along with the standard deviation of the model $\sigma_{\text{mod}}$ according to equation (8). The final location of the VLBAI facility and its main axis could only be approximated to the $\mathrm{cm}$-level during the measurement campaigns because of necessary installation tolerances. We estimated the effect of a horizontal variation of $\pm 3\text{\,}\mathrm{cm}$ and a vertical variation of $\pm 2\text{\,}\mathrm{mm}$ in a Monte Carlo simulation. The total amplitude of the variations at the locations of the gravimetric measurements is within $\pm 2\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ with a mean standard deviation of $0.3\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ for the horizontal and $0.4\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ for the vertical component along the main axis. The measurements, i. e. the markers in figure 8, are the result of the gravity network adjustment. Additionally, we removed the effect of the change of gravity with height for figures 88(b)–8(d). For this, the free air gradient is modified with a model of the soil surrounding HITec. As the density is only known to a certain degree, the Monte Carlo simulation also included the ground around HITec. The standard deviation of the simulation results for each gravimeter position is added to the measurements standard deviation by error propagation. The simulations’ standard deviations range from $10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ at the height of $4\text{\,}\mathrm{m}$ to $35\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ at the topmost position. This is also reflected in the increase in the standard deviations indicated by the errorbars in figure 8(c). The uncertainty of the measurements now consists of the following components: $\sigma_{\rm obs}=\sqrt{\sigma_{g}^{2}+\sigma_{h,\mathrm{geo}}^{2}+\sigma_{z,\mathrm{mod}}^{2}+\sigma_{\mathrm{grad}}^{2}}\ .$ (7) Here, the standard deviation of the network adjustment is $\sigma_{g}$. The contribution of the determination of the height of the gravimeter is $\sigma_{h,\mathrm{geo}}$. The result of the Monte Carlo simulations of the vertical component of geometric position of the central axis $\sigma_{z,\mathrm{mod}}$, and the modelling of the gravity gradient $\sigma_{\mathrm{grad}}$ are also attributed to the measurements. The standard deviation of the model consists of the following components: $\sigma_{\mathrm{mod}}=\sqrt{\sigma_{\mathrm{MC}}^{2}+\sigma_{hz,\mathrm{mod}}^{2}}\ ,$ (8) where $\sigma_{\mathrm{MC}}$ is the standard deviation of the Monte Carlo simulations of the model density, calculated in the heights of the gravimetric measurements, and $\sigma_{hz,\mathrm{mod}}$ is the standard deviation of the Monte Carlo simulations for the horizontal component of the geometric positions along the VLBAI main axis. $\sigma_{\mathrm{mod}}$ is shown in figure 8(c) with a range of $6\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}11\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ in the region of interest and about $8\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ at $z_{\mathrm{eff}}=$9.2\text{\,}\mathrm{m}$$ (see section 3.3). Furthermore, a single parameter is estimated to reduce the gravity values from the magnitude of $9.81\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-2}$ to the order of magnitude of the model values for the attraction. This parameter is the mean difference of observed minus computed results at the location of the observation in the region of interest. The measurements of 2017 are also corrected for the changes within the building with respect to 2019. No additional parameters were estimated to fit the measurements to the model or vice versa. The remaining signal should now contain the effect of the HITec building on gravity. In general, the 2017 measurements and the main axis model do not show a good agreement [51, see also] due to the instability of the scaffolding used as a platform [18, see also]. The agreement on the validation profile is better, and only the two topmost points do not agree with the model and simulation. These earlier measurements serve as a proof of concept and are given for the sake of completeness. The following discussion concerns only the 2019 measurements. The 2019 campaign provides a clear improvement considering the number of stations along the VLBAI main axis, the stability of the platforms in the VSS and therefore data quality. Consequently, the agreement between measurement and model is significantly improved. The measurement scheme on the validation profile remained unchanged compared to the 2017 campaign. Figure 8(c) shows the difference between the measurements and the model on the central axis. The region of interest for experiments in the VLBAI is approximately between $4\text{\,}\mathrm{m}$ and $13\text{\,}\mathrm{m}$ (see figure 3). Within this region, only the second-highest point is not within the simulation’s $\pm 5\text{\,}\mathrm{\char 37\relax}$ density variations. The two tailed statistical test ($\alpha=0.05$) on the equality of model $\delta g_{\mathrm{mod},i}$ and measurement $\delta g_{\mathrm{obs},i}$ at point $i$ according to Null hypothesis: $\displaystyle\delta g_{\mathrm{omc},i}=$ $\displaystyle\delta g_{\mathrm{obs},i}-\delta g_{\mathrm{mod},i}=0$ Alternative hypothesis: $\displaystyle\delta g_{\mathrm{omc},i}\neq$ $\displaystyle 0$ Test statistics: $\displaystyle t_{i}=$ $\displaystyle\frac{\left|\delta g_{\mathrm{omc},i}\right|}{\sqrt{\sigma_{\mathrm{obs},i}^{2}+\sigma_{\mathrm{mod},i}^{2}}}$ passes for all but three points. The null hypothesis, considering the symmetry of the normal distribution, is rejected if $t_{i}>N_{(0,1,1-\nicefrac{{\alpha}}{{2}})}$. The test fails for the points at $z=$1.72\text{\,}\mathrm{m}5.55\text{\,}\mathrm{m}12.99\text{\,}\mathrm{m}$$. The lowest point at $z=$1.72\text{\,}\mathrm{m}$$, directly on the VTS, was challenging to measure, as the pump of the vacuum tank was active during the measurements causing high-frequency vibrations. As this position is outside of the experimental region of interest, no additional measurements were taken. The cause for the significant deviation from the model at $z=$12.99\text{\,}\mathrm{m}$$, which was measured with only one gravimeter, is unknown. The height difference to the point above is only $0.16\text{\,}\mathrm{m}$ of free space, so a real gravity variation appears unlikely. Treating this point as an outlier, and repeating the test after calculating the offset between adjusted gravity values and model without this measurement, the test also passes for the point at $z=$5.55\text{\,}\mathrm{m}$$. All points on the validation profile pass the statistical test. The standard deviation of observations minus model is $20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ ($31\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ if the second-highest point is included) for the central axis in the region of interest and $34\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ on the validation profile. The density of the different model components, chosen initially from technical documentation, are sufficient to generate a model which is identical to in situ measurements at a $95\text{\,}\mathrm{\char 37\relax}$ confidence level. Modelling a $5\text{\,}\mathrm{\char 37\relax}$ normally distributed variation of these densities results in a narrow range of possible model variations, which covers almost all measurements used to verify the model. We expect that using individual densities for each floor instead of one common density value for all concrete components in the building would improve the agreement between model and observations on the validation profile. Such extra modelling step should however be constrained not to deteriorate the model accuracy in the experimental region of interest. As a final step, the VLBAI magnetic shield and vacuum system [71] will be added to the model. Similarly to the VSS and VTS, this component was designed using CAD, built with known materials, and can be exported into the required format for our model. While the assembly is significantly more complex, we expect the octagonal symmetry of the magnetic shield to simplify the numerical calculations and allow us to reach the same level of accuracy in the gravity model as for the VSS and VTS. It will however only be possible to check the quality of the extended model with measurements on the validation profile, as the main axis is obstructed by the instrument’s vacuum chamber. Nevertheless, the understanding of environmental variations (mostly hydrology) outlined in section 4.4 will render this possible with good accuracy. Due to the work associated with the installation of the VLBAI baseline components, this last model extension and its corresponding validation have not been done yet. Extending our model with the VLBAI baseline components will allow us to connect gravimetric measurements along the validation profile and future data acquired by a VLBAI quantum gravimeter along its main axis in our adjusted gravimetric network. Since the measurement positions along the validation profile will remain free during operation of the VLBAI facility, this will for example enable comparisons of the VLBAI QG with FG5(X)-type classical AGs positioned in the VLBAI laboratories. In this specific setup, contributions of time variable gravity to the measurements are minimal for the VLBAI and instrument under test. To further minimize the height dependency due to the groundwater effect, the atom interferometer could be realised with an effective height close to the instrumental height of the classical AG, e. g. with the AG on the groundfloor. Taking into consideration the mean standard deviation of the relative gravimeter network of $9\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$, we expect to be able to transfer g with an uncertainty of $10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ and possibly below from the VLBAI baseline. Furthermore, creating a similar network including stations along the validation profile and in the HITec gravimetry laboratory would permit gravimetric comparisons between the VLBAI QG and instruments operated on the gravimetric piers. The estimates so far exclude the inevitable contribution of the VLBAI gravity measurement. The determination and validation of the VLBAI uncertainty budget will be published in a separate study. ## Conclusions We established a gravimetric control network for the Hannover VLBAI facility, a novel $10\text{\,}\mathrm{m}$-scale atom interferometer. The network consists of $439$ connections measured by relative gravimeters. A least squares adjustment of the network results in a mean standard deviation of the adjusted gravity values of $9\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$. In addition, we developed a structural model of the building hosting the VLBAI facility and its surroundings. When compared, the model and the measurements agree with $95\text{\,}\mathrm{\char 37\relax}$ confidence, with standard deviations of the residuals of $20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ along the atom interferometer’s baseline, and $34\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ on a second, parallel profile. Moreover, we gained insight on some dynamical aspects of the gravity field around the instrument, namely the effect of groundwater level variations. We anticipate this gravimetric network to contribute to the assessment of the quantum gravimeter’s uncertainty budget, which is currently not included in our study. The current work is also essential to help determining the effective instrumental height (g-value reference position) and enable transfers of g values from the atom interferometer’s baseline to the validation profile, accessible to mobile gravimeters for comparison and possibly calibration purposes, at the $10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ repeatability level (relative to the VLBAI deduced g-values). Completing the model by including the VLBAI baseline, refining the description of the soil surrounding the host building, and including better estimates for the building material densities, we expect to shift the possibility for gravity field measurement transfers and mobile instrument calibration towards the $5\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-2}$ level, improving the temporal stability of the current state of the art, which is still largely based on gravimeter comparisons. This paves the way for the realisation of a new gravity standard based on atom interferometry. Finally, the knowledge of the dynamical gravity field and its gradients is key to reaching new frontiers in fundamental physics tests with very long baseline atom interferometry. ###### Acknowledgements. The Hannover Very Long Baseline Atom Interferometry facility is a major research equipment funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation). This work was supported by the DFG Collaborative Research Center 1128 “geo-Q” (project A02, Contract Number 239994235) and is supported by the CRC 1227 “DQ-mat” (project B07, Contract Number 274200144), Germany's Excellence Strategy – EXC-2123 “QuantumFrontiers” – 390837967, and the computing cluster of the Leibniz University Hannover under patronage of the Lower Saxony Ministry of Science and Culture (MWK) and the DFG. M. S., É. W., and C. S. acknowledge support from “Niedersächsisches Vorab” through the “Quantum- and Nano-Metrology (QUANOMET)” initiative (project QT3), and for initial funding of research in the DLR-SI institute. D. S. acknowledges funding from the German Federal Ministry of Education and Research (BMBF) through the funding program Photonics Research Germany (contract number 13N14875). The VLBAI support structure was conceived by the engineering office Heinz Berlin (Wennigsen, Germany) in collaboration with the VLBAI science team, and produced by Aljo Aluminium-Bau Jonuscheit GmbH (Berne, Germany). We thank W. Ertmer for his vision and long lasting support on very long baseline atom interferometry and the acquisition of funding for the Hannover Institute of Technology. We are grateful to T. Froböse and A. Wanner for their assistance during the installation of the vacuum tank and support structure. We thank the three reviewers for their valuable input to improve this article. ###### Author contributions. M.S., É.W., L.T. planned geometric and gravimetric measurements, evaluated the data and prepared the initial draft. É.W., D.T., D.S., C.S., E.M.R. conceptualised VSS, VTS. É.W., D.T., K.H.Z. designed and built measurement platforms for VSS. M.S., É.W., L.T., D.T., K.H.Z. carried out the measurements. M.S. developed and implemented the gravity model. D.T., K.H.Z., D.S., C.S., E.M.R., J.M. provided critical input to the manuscript and approved the final version. ###### Data availability statement. Data of absolute gravimeter key comparisons is available in the Key Comparison Database (https://www.bipm.org/kcdb) and cited literature. Gravimetric measurements in instrument specific ascii data formats and datasets generated in this study are available from the corresponding author on reasonable request. ## References * [1] S. Abend, M. Gebbe, M. Gersemann, H. Ahlers, H. Müntinga, E. Giese, N. Gaaloul, C. Schubert, C. Lämmerzahl, W. Ertmer, W.. Schleich and E.. Rasel “Atom-chip fountain gravimeter” In _Phys. Rev. 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2024-09-04T02:54:58.853231

2003.04878

arxiv-papers

# Calculating Fluctuations and Self-Correlations Numerically for Causal Charge Diffusion in Relativistic Heavy-Ion Collisions Aritra De School of Physics & Astronomy, University of Minnesota, Minneapolis, MN 55455, USA Christopher Plumberg Department of Astronomy and Theoretical Physics, Lund University, Sölvegatan 14A, SE-223 62 Lund, Sweden Joseph I. Kapusta School of Physics & Astronomy, University of Minnesota, Minneapolis, MN 55455, USA ###### Abstract We study the propagation and diffusion of electric charge fluctuations in the Bjorken hydrodynamic model with both white and Catteneo noise using purely numerical methods. We show that a global lattice of noise fluctuations is required to fully calculate the two-point correlators of charge. We solve the stochastic differential equations that arise from the charge conservation equation on the lattice. We explicitly identify the self-correlation term in the case of Catteneo noise and provide a physical interpretation. We provide a numerical recipe to remove this contribution from the full two-point correlators. Finally, we calculate the balance functions for charged hadrons. By limiting the speed of signal propagation, we observe the expected narrowing of the balance functions after removing the self-correlations. ## I Introduction Relativistic hydrodynamics is used to study not only the equation of state but also dynamical quantities, such as the transport coefficients, of the quark- gluon plasma. The applicability of hydrodynamics is justified if the mean free paths of the particles are small compared to the distances over which thermodynamic quantities vary. It turns out that hydrodynamics is very successful in modeling high energy nuclear collisions. There are experimental facilities which produce and study quark-gluon plasma: the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory and the Large Hadron Collider (LHC) at CERN. Fluid equations describe conservation of energy, momentum, baryon number, electric charge, and strangeness. Anisotropic particle production, such as elliptic flow, in heavy ion collisions gives credence to the use of hydrodynamics in simulating these collisions. It has been successful in describing various properties like particle spectra, particle correlations, and in obtaining values of transport quantities like the ratio of shear viscosity to entropy density $\eta/s$. By comparing particle spectra with experimental data, hydrodynamical simulations also help in understanding the initial state, its fluctuations, and hence properties of strongly interacting matter in general. Initially, the assumption of ideal hydrodynamics worked very well in describing the data which indicated that the system was strongly interacting. Later, important aspects like the lattice- computed QCD equation of state and viscous properties were taken into account to study transport properties of quark-gluon plasma with more precision. The fluctuation-dissipation theorem relates the dissipative properties of a system to its hydrodynamical fluctuations. In particular, it allows us to infer quantities like shear and bulk viscosity, and electrical conductivity, from the magnitude of fluctuations. Hydrodynamical fluctuations have also been used to study static critical phenomena torres near a possible critical point. More recently, there have been studies of dynamic critical phenomena near a QCD critical point teaney ; yiyin ; nahrgang ; Bluhm . Critical points are characterized by large fluctuations. This led to the suggestion to study fluctuations in conserved quantities, such as electric charge, baryon number, and strangeness on an event-by-event basis stephanov_shuryak . It has also been suggested to study non-gaussianities (higher order cumulants) of the fluctuations near critical points as they are more sensitive to large correlation lengths stephanovprl . Correlation lengths theoretically diverge near the critical points, but in the scenario of the heavy ion collisions are limited by the finite system size stephanov_shuryak as well as by the finite lifetime of the system berdnikov . Thus it becomes imperative to study the hydrodynamics of fluctuations in the context of heavy ion collisions. The relativistic theory of hydrodynamic fluctuations in the context of heavy-ion collisions was introduced in Ref. kapusta2012 . In the current work, we focus on calculating the two-point correlations of charge fluctuations and the resulting balance functions of pions. The balance function measures the difference in probability of finding a particle of opposite charge in another fluid cell versus a particle of same charge given a charged particle in a given fluid cell spratt . This problem was studied analytically in the 1+1 dimensional Bjorken model in Ref. plumbergkapusta . Analytic calculations are not possible for state of the art 3+1 dimensional, non-boost invariant, hydrodynamics. In preparation for extensions to modern hydrodynamic models, we develop numerical methods to solve the relevant stochastic differential equations numerically. Particular attention is paid to the physical interpretation of self-correlations and how they can be subtracted to make comparison to experimental data. The outline of the paper is as follows. In Sec. II we review the normal diffusion, the Cattaneo, and the Gurtin-Pipkin equations, and discuss how self-correlations arise. In Sec. III we outline the application of the relevant stochastic differential equations in the context of the Bjorken hydrodynamic model for heavy-ion collisions. In Sec. IV we present solutions to those equations. In Sec. V we show how self-correlations can be clearly identified. In Sec. VI we calculate the balance functions that relate theory to experiment. Our conclusions are presented in Sec. VII. Details of how the stochastic differential equations are solved are presented in the Appendices. The numerical method is readily transferrable to heavy-ion collisions which have no spatial symmetries and thus useful for future calculations. ## II Noise, Fluctuations and Self-Correlations Since the usual diffusion equation leads to instantaneous signal propagation, which is inconsistent with special relativity, one needs a diffusion equation which is the same order in spatial and temporal derivatives with characteristic relaxation times and lengths. In this paper we solve the simplest diffusion equation satisfying this condition, called the Catteneo equation catteneo , numerically. The resulting differential equation is a stochastic differential equation (SDE) because it contains random noise terms. The way to solve SDEs is to solve the differential equation for a large number of events (here on the order of 1 million or more) and study the correlation functions. A finite difference method is used to solve the SDE. Consider the ordinary diffusion equation with white noise. In the context of the Bjorken model, which has boost invariance and no dependence on transverse coordinates, the two variables are proper time $\tau=\sqrt{t^{2}-z^{2}}$ and spatial rapidity $\xi={\textstyle{\frac{1}{2}}}\ln[(t+z)/(t-z)]$, where the beam axis is along the $z$ direction. The noise $f$, appropriately defined (see below), is a dimensionless random variable with correlator $\langle f(\tau_{1},\xi_{1})f(\tau_{2},\xi_{2})\rangle=\frac{N(\tau_{2})}{2\pi}\delta(\tau_{1}-\tau_{2})\delta(\xi_{1}-\xi_{2})\,,$ (1) which is a product of Dirac $\delta$-functions in time and space with normalization determined by the fluctuation-dissipation theorem $N(\tau)=\frac{4\pi\sigma_{Q}(\tau)T(\tau)}{A\tau s^{2}(\tau)}\,.$ (2) Here $\sigma_{Q}$ is the charge conductivity, $T$ is the temperature, $s$ is the entropy density, and $A$ is the transverse area. To generate this numerically on a discrete lattice with spacings $\Delta\xi$ and $\Delta\tau$, we sample $f$ from a normal distribution with zero mean and variance $N(\tau)/(2\pi\Delta\xi\Delta\tau)$. The analysis of how finite difference methods work computationally for solving SDEs is discussed in Appendix A. Consider the difference between white and colored noise. The standard two- point function for white noise in frequency and momentum space is $\displaystyle\langle\tilde{f}(\omega_{1},k_{1})\tilde{f}(\omega_{2},k_{2})\rangle$ $\displaystyle=$ $\displaystyle\int d\tau_{1}\,d\tau_{2}\,d\xi_{1}\,d\xi_{2}\,e^{-i(k_{1}\xi_{1}+k_{2}\xi_{2})}\,e^{-i(\omega_{1}\tau_{1}+\omega_{2}\tau_{2})}\langle f(\tau_{1},\xi_{1})f(\tau_{2},\xi_{2})\rangle$ (3) $\displaystyle=$ $\displaystyle\delta(k_{1}+k_{2})\tilde{N}(\omega_{1}+\omega_{2})$ where $\tilde{N}$ is the Fourier transform of $N$. Generalizing this to Catteneo noise (which is an example of colored noise), we recall that the two- point function for the noise obeys kapustayoung . $\langle(\tau_{Q}\,\partial\tau_{1}+1)\tilde{f}(k_{1},\tau_{1})(\tau_{Q}\,\partial\tau_{2}+1)\tilde{f}(k_{2},\tau_{2})\rangle=N(\tau_{1})\delta(\tau_{1}-\tau_{2})\delta(k_{1}+k_{2})\,$ (4) where $\tau_{Q}$ is a relaxation time. In frequency and momentum space this becomes $\langle\tilde{f}(\omega_{1},k_{1})\tilde{f}(\omega_{2},k_{2})\rangle=\frac{\delta(k_{1}+k_{2})\tilde{N}(\omega_{1}+\omega_{2})}{(i\tau_{Q}\omega_{1}+1)(i\tau_{Q}\omega_{2}+1)}\,.$ (5) The noise correlator is no longer a Dirac $\delta$-function in time anymore; instead, it is smeared out, hence the name colored noise. The following three figures will help illustrate some of the physics to come. Figure 1 shows a fluctuation, represented by a star, in a particular spacetime cell. The signal, represented by bursts, is transmitted to the two adjacent spatial cells in the next time step. Hence those two cells have correlated fluctuations. This type of correlation can arise from either white or colored noise. Figure 2 shows a fluctuation in one spacetime cell with its signal transmitted to two spacetime cells two time steps later. This type of correlation can also happen with either white or colored noise. Figure 3 shows a situation that only happens with colored noise. The two stars are correlated, and their signals lead to correlations between the same two cells as shown in the previous figures. Self-correlations arise from correlations in the same spatial cell. For white noise this means the star and the burst are in the same spacetime cell. In discretized spacetime this leads to a Kronecker $\delta$-function in $\xi$, while in the continuum limit this leads to a Dirac $\delta$-function. The latter is somewhat unphysical, since all correlations have some finite extent. For colored noise, the self-correlation begins in the cell hosting the original fluctuation, and then continues in subsequent time steps but always in the same spatial cell due to the time-correlated nature of colored noise. Noise generated at a previous time in the same spatial cell will hydrodynamically evolve to a correlated charge fluctuation in a different spatial fluid cell. Hence the self-correlation will be non-trivial for colored noise. Figure 1: An example of either white or colored noise. A fluctuation in one cell, represented by a star, causes a correlation between two cells in the next time step, represented by bursts, separated in space from each other and from the original fluctuation. (color online) Figure 2: An example of either white or colored noise. A fluctuation in one cell, represented by a star, causes a correlation between two cells two time steps later, represented by bursts, but only one is separated in space from the original fluctuation. (color online) Figure 3: An example of colored noise. Fluctuations at the same point in space but at different times are correlated, as represented by the stars. This results in a correlation between the two cells, represented by bursts. (color online) Figure 4 shows another way to visualize the colored Cattaneo noise. At a fixed spatial cell, correlations arise at different times due to $\tau_{Q}>0$. Correlations also propagate to other spatial cells with increasing time via a Green’s function. The mathematical formalism and details of how it is implemented numerically with be presented in the following sections. Figure 4: Schematic of the lattice setup for Catteneo noise. The final charge correlations are determined at some $\tau_{f}$. One must integrate over all prior times $\tau_{i}\leq\tau\leq\tau_{f}$ to obtain the final time charge correlators. Thus one can define self-correlations as the correlation of a charge fluctuation generated in $\xi_{1}$ at final time $\tau_{f}$ with another charge fluctuation generated at the same $\xi_{1}$ but at a previous time and hence had time to travel to a different $\xi_{2}$ at $\tau_{f}$. It is non-trivial for colored noise because colored noise generated in same $\xi$ are correlated in time. One can go further and consider the Gurtin-Pipkin noise gurtin which introduces a noise correlation in spatial rapidity in addition to the correlation in proper time. Gurtin-Pipkin noise has been dealt with analytically in Ref. kapustayoung . In Cartesian coordinates Gurtin-Pipkin noise results in the following diffusion equation $\left[\frac{\partial}{\partial t}-D_{Q}\nabla^{2}+\tau_{Q}\frac{\partial^{2}}{\partial t^{2}}+\tau_{2}^{2}\frac{\partial^{3}}{\partial t^{3}}-\tau_{3}D_{Q}\frac{\partial}{\partial t}\nabla^{2}\right]n_{Q}=0\,.$ (6) Numerical simulation of Gurtin-Pipkin noise is deferred to a future work. ## III Diffusion in Boost Invariant 1+1 Hydrodynamics This section is a mini-review of the problem addressed previously in Ref. plumbergkapusta to help setup the use of numerical methods for solving the resulting SDE. We will work in 1+1 dimensional boost-invariant Bjorken hydrodynamics. The longitudinal boost-invariance implies that the initial conditions for local variables are only functions of the proper time $\tau$. We neglect the bulk and shear viscosities in order to focus on charge transport. The energy-momentum tensor for an ideal fluid is $T^{\mu\nu}=wu^{\mu}u^{\nu}-pg^{\mu\nu}\,.$ (7) We take the Landau-Lifshitz approach where $u^{\mu}$ is the velocity of energy transport. The electric current takes the form $J_{Q}^{\mu}=n_{Q}u^{\mu}+\Delta J^{\mu}$ (8) where $n_{Q}$ is the proper charge density and $\Delta J^{\mu}$ is the dissipative part. In first-order viscous fluid dynamics $\Delta J^{\mu}$ takes the form $\Delta J^{\mu}=D_{Q}\Delta^{\mu}n_{Q}=\sigma_{Q}\Delta^{\mu}\mu_{Q}$ (9) where $\mu_{Q}$ is the charge chemical potential, $\sigma_{Q}$ is the charge conductivity and $\Delta^{\mu}$ is the transverse derivative $\Delta^{\mu}=\partial^{\mu}-u^{\mu}(u\cdot\partial)\,.$ (10) Conventional charge diffusion follows the usual diffusion equation $\left(\frac{\partial}{\partial t}-D_{Q}\nabla^{2}\right)n_{Q}=0\,.$ (11) The diffusion constant $D_{Q}$ and charge conductivity are related by the Einstein relation $D_{Q}=\sigma_{Q}/\chi_{Q}$, where $\chi_{Q}$ is the electric charge susceptibility defined by $\chi_{Q}=\frac{\partial n_{Q}(T,\mu_{Q})}{\partial\mu_{Q}}\,.$ (12) The diffusion equation leads to an infinite speed of propagation which is unphysical and not suitable for hydrodynamic simulations of heavy-ion collision. Therefore the usual diffusion equation is replaced by one with a double derivative in time with a relaxation time factor $\tau_{Q}$. $\left(\frac{\partial}{\partial t}-D_{Q}\nabla^{2}+\tau_{Q}\frac{\partial^{2}}{\partial t^{2}}\right)n_{Q}=0$ (13) This equation is called the Cattaneo equation catteneo . It is a combination of the diffusion equation with the wave equation. The dissipative current gets modified to $\Delta J^{\mu}=D_{Q}\Delta^{\mu}\left[\frac{1}{1+\tau_{Q}(u\cdot\partial)}\right]n_{Q}$ (14) One can show that high frequency waves travel at a speed of $v_{Q}=\sqrt{D_{Q}/\tau_{Q}}$ kapustayoung . The fluctuation-dissipation theorem relates the two-point function, which provides a measure of the variance of fluctuations, to the dissipation from diffusion. A stochastic noise term $I^{\mu}$ is therefore added to the charge current. $J^{\mu}=n_{Q}u^{\mu}+\Delta J^{\mu}+I^{\mu}$ (15) One-point functions vanish and the two-point functions are determined by the fluctuation-dissipation theorem. For the usual diffusion equation $\langle I^{\mu}(x)\rangle=0\qquad\langle I^{\mu}(x_{1})I^{\nu}(x_{2})\rangle=2\sigma_{Q}T\,h^{\mu\nu}\delta(x_{1}-x_{2})$ (16) where $h^{\mu\nu}=u^{\mu}u^{\nu}-g^{\mu\nu}$ is the transverse projector. This is white noise. In the Catteneo equation the fluctuations are $\langle I^{i}(x_{1})I^{j}(x_{2})\rangle=\frac{\sigma_{Q}T}{\tau_{Q}}\delta(\mbox{\boldmath$x$}_{1}-\mbox{\boldmath$x$}_{2})\,e^{-|t_{1}-t_{2}|/\tau_{Q}}\,\delta_{ij}$ (17) The delta function in time is replaced by an exponential decay function. In the limit $\tau_{Q}\rightarrow 0$ this two-point function becomes the Dirac $\delta$\- function for white noise. The following are the relations between the Cartesian coordinates and the proper time and spatial rapidity appropriate for the Bjorken model. $\displaystyle\begin{aligned} t&=&\tau\cosh\xi\qquad z&=\tau\sinh\xi\\\ \tau&=&\sqrt{t^{2}-z^{2}}\qquad\xi&=\tanh^{-1}\left(\frac{z}{t}\right)\\\ \end{aligned}$ (18) The flow velocity is $u^{0}=\cosh\xi\quad u^{z}=\sinh\xi\,.$ (19) The transverse derivatives are $\displaystyle\Delta^{0}=-\frac{\sinh\xi}{\tau}\frac{\partial}{\partial\xi}\qquad\Delta^{3}=-\frac{\cosh\xi}{\tau}\frac{\partial}{\partial\xi}\quad\text{with}\quad u\cdot\partial=\frac{\partial}{\partial\tau}\,.$ (20) The fluctuating contribution to the current is written as $\displaystyle I^{0}$ $\displaystyle=$ $\displaystyle s(\tau)f(\xi,\tau)\sinh\xi$ (21) $\displaystyle I^{3}$ $\displaystyle=$ $\displaystyle s(\tau)f(\xi,\tau)\cosh\xi\,.$ (22) The entropy density $s$ is factored out to make $f$ dimensionless. The background fluid equations for the proper charge density and entropy density are $\displaystyle\frac{ds}{d\tau}+\frac{s}{\tau}=0\;\;$ $\displaystyle\Rightarrow$ $\displaystyle\;\;s(\tau)=\frac{s_{i}\tau_{i}}{\tau}$ (23) $\displaystyle\frac{dn_{Q}}{d\tau}+\frac{n_{Q}}{\tau}=0\;\;$ $\displaystyle\Rightarrow$ $\displaystyle\;\;n_{Q}(\tau)=\frac{n_{i}\tau_{i}}{\tau}\,.$ (24) These are a manifestation of the conservation of entropy and charge, respectively. The $s_{i}$ and $n_{i}$ are the densities at some initial time $\tau_{i}$. We take the initial proper charge density $n_{i}$ to be zero, hence the average charge density for subsequent times is zero as well. Now let us look at the charge current conservation equation $\partial_{\mu}J^{\mu}=0$. It is convenient to define the variable $X=\tau\delta n$ because, in the absence of fluctuations, this quantity is conserved during the hydrodynamic evolution. After a few steps of algebra the full charge conservation equation becomes $\left[\frac{\tau}{D_{Q}\chi_{Q}T}+\tau_{Q}\frac{\partial}{\partial\tau}\left(\frac{\tau}{D_{Q}\chi_{Q}T}\right)\right]\frac{\partial X}{\partial\tau}+\frac{\tau_{Q}\tau}{D_{Q}\chi_{Q}T}\frac{\partial^{2}X}{\partial\tau^{2}}-\frac{1}{\tau\chi_{Q}T}\frac{\partial^{2}X}{\partial\xi^{2}}$ $+\left[\frac{\tau s}{D_{Q}\chi_{Q}T}+\tau_{Q}\frac{\partial}{\partial\tau}\left(\frac{\tau s}{D_{Q}\chi_{Q}T}\right)\right]\frac{\partial f}{\partial\xi}+\frac{\tau_{Q}\tau s}{D_{Q}\chi_{Q}T}\frac{\partial^{2}f}{\partial\xi\partial\tau}=0\,.$ (25) For the case $\tau_{Q}=0$ (usual diffusion equation) this simplifies to $\frac{\partial X}{\partial\tau}-\frac{D_{Q}}{\tau^{2}}\frac{\partial^{2}X}{\partial\xi^{2}}+s\frac{\partial f}{\partial\xi}=0\,.$ (26) Due to boost invariance it is useful to use the Fourier transform $X(\xi,\tau)=\int_{-\infty}^{\infty}\frac{dk}{2\pi}e^{ik\xi}\tilde{X}(k,\tau)\,,$ (27) and similarly for $f$. Then the SDE for white noise is $\frac{\partial}{\partial\tau}\tilde{X}+\frac{D_{Q}k^{2}}{\tau^{2}}\tilde{X}=-iks\tilde{f}$ (28) and for colored Cattaneo noise $\left[\frac{\tau}{D_{Q}\chi_{Q}T}+\tau_{Q}\frac{\partial}{\partial\tau}\left(\frac{\tau}{D_{Q}\chi_{Q}T}\right)\right]\frac{\partial\tilde{X}}{\partial\tau}+\frac{\tau_{Q}\tau}{D_{Q}\chi_{Q}T}\frac{\partial^{2}\tilde{X}}{\partial\tau^{2}}+\frac{k^{2}}{\tau\chi_{Q}T}\tilde{X}$ $=-ik\left[\frac{\tau s}{D_{Q}\chi_{Q}T}+\tau_{Q}\frac{\partial}{\partial\tau}\left(\frac{\tau s}{D_{Q}\chi_{Q}T}\right)\right]\tilde{f}-i\frac{k\tau_{Q}\tau s}{D_{Q}\chi_{Q}T}\frac{\partial\tilde{f}}{\partial\tau}\,.$ (29) For the sake of comparison and for definiteness, we follow Ref. plumbergkapusta and assume both $D_{Q}$ and $\tau_{Q}$ are constant within the range of temperature to be considered. This means that high frequency waves propagate with a constant value of $v_{Q}$. For the same reasons we assume that $s\sim T^{3}$ and $\chi\sim T^{2}$. Hence $T\sim\tau^{-1/3}$. ## IV Solving the Stochastic Differential Equations We start by solving the stochastic differential equation for white noise. As explained earlier, we will solve it on a spacetime lattice and choose spacings $\Delta\xi=0.09$ and $\Delta\tau=10^{-4}$ fm/c. We set the parameters such that $(\chi(\tau_{f})T_{f})/(\tau_{f}\Delta\xi)=0.5122$ MeV3 fm-3. We source the noise function $f$ from a normal distribution with mean $0$ and variance $1/\sqrt{\Delta t\Delta\xi}$. The density-density correlator arising from the noise fluctuation which is a solution to the SDE in our discretized system, evaluted at the final time $\tau_{f}$, has the analytical form $\langle\delta n(\xi_{1},\tau_{f})\delta n(\xi_{2},\tau_{f})\rangle=\frac{\chi_{Q}(\tau_{f})T_{f}}{A\tau_{f}}\left[\frac{\delta_{\xi_{1},\xi_{2}}}{\Delta\xi}-\frac{1}{\sqrt{\pi w^{2}}}e^{-(\xi_{1}-\xi_{2})^{2}/w^{2}}\right]$ (30) where $w^{2}=8D_{Q}\left(\frac{1}{\tau_{i}}-\frac{1}{\tau_{f}}\right)\,.$ (31) In the continuum limit $\delta_{\xi_{1},\xi_{2}}/\Delta\xi\rightarrow\delta(\xi_{1}-\xi_{2})$. The parameters chosen for this work are the same as in Ref. plumbergkapusta , namely $\tau_{i}=0.5$ fm/c, $\tau_{f}=6.352$ fm/c, $T_{i}=350$ MeV, and $T_{f}=150$ MeV. We use diffusion constant $D_{Q}=0.162\>\text{fm}$ which is an average over the temperature interval from 150 to 350 MeV taken from Ref. Aarts . The equation of state used is the same as in Ref. torres , which is $\chi_{Q}=\frac{2}{3}T^{2}$ (including up, down and strange quarks). The details of how we solve an SDE are discussed in Appendix A. The solution is presented in Fig. 5. The dots represent the result of the SDE simulation for ten million random events. The solid curve is the Gaussian from Eq. (30); it overlays the dots within the width of the line. The Kronecker $\delta$-function at $\xi=0$ is clearly evident. Figure 5: White noise density-density correlation function for 10 million events. The solid curve is the Gaussian from Eq. 30. (color online) Next we turn to colored noise. We have to generate a noise that has the desired correlation in proper time but is uncorrelated in rapidity. The way we do that is by solving another SDE which is called the Langevin equation. $f+\tau_{Q}\frac{\partial f}{\partial\tau}=\zeta$ (32) Here $\zeta$ is the regular white noise. The relaxation time $\tau_{Q}$ smoothens the Dirac $\delta$-correlation in proper time. The $\tau_{Q}$ also introduces the maximum mode velocity to be $v_{Q}^{2}=D_{Q}/\tau_{Q}$, thereby removing instantaneous signal propagation. The analytical solution to the Langevin equation (with rapidity dependences suppressed) is $\langle f(\tau_{1})f(\tau_{2})\rangle=\frac{N(\tau_{2})}{4\pi\tau_{Q}}\left[e^{-|\tau_{1}-\tau_{2}|/\tau_{Q}}-e^{(2\tau_{i}-\tau_{1}-\tau_{2})/\tau_{Q}}\right]\equiv{\cal N}(\tau_{1},\tau_{2})\,.$ (33) The derivation is given in Appendix B. The numerically computed two-point function is plotted in Fig. 6. The expected result (33) and the numerical result are consistent for ten million simulated events. Figure 6: Comparison of numerical and analytical results for $v_{Q}^{2}=0.16$ with rapidity dependences suppressed. (color online) The grid sizes chosen ensures that they obey the Courant Friedrichs Lewy (CFL) condition CFL . This condition states that the numerical domain of dependence of any point in space and time must include the analytical domain of dependence. Physically, this condition amounts to a signal propagating no more than one spatial cell away during one time step. For speed $v_{Q}$ being a constant, this amounts to the condition $\Delta\tau/\tau<\Delta\xi/v_{Q}$. Figure 7 shows the dependence of the two-point correlator for two very different values of the propogation speed, or equivalently the relaxation time $\tau_{Q}$. Figure 7: Variation of the density-density correlator with the propogation speed $v^{2}_{Q}$. (color online) ## V Characterizing Self-Correlations The self-correlations are trivial for the case of white noise; it’s a Dirac $\delta$-function. Even the two-point correlation function of a free Boltzmann gas has a $\delta$\- function term landau1 ; landau2 . $\langle\delta n(\textbf{x}_{1})\delta n(\textbf{x}_{2})\rangle=\chi T\delta^{3}(\textbf{x}_{1}-\textbf{x}_{2})+\cdots$ (34) This is explained in the Ref. landau2 where $\langle(\Delta N)^{2}\rangle=\chi TV$. One can see this in Eq. (30), where the denomintor $\tau A$ is the Jacobian factor from the Bjorken expansion instead of stationary Cartesian coordinates. Experiments measure just the two-particle correlation and hence we have to subtract the self-correlation spratt . The challenge is to characterize the self-correlations in the presence of colored noise, since it is no longer a Dirac $\delta$-function. Figure 8 shows the numerically computed self-correlation. If we subtract the two-point correlation in Fig. 5 from that presented in this figure, we get the expected Gaussian. This is shown in Fig. 9 where it is compared with the analytical Gaussian function in Eq. (30) for 1 million events. Figure 8: The self-correlation. Figure 9: The solid curve is the expected Gaussian while the dots represent the result of the SDE simulation for 1 million random events. (color online) Now we move on to the meaning of self-correlation for colored noise. Based on the prescription of self-correlation that we discussed in the introduction, we consider the schematic diagram in Fig. 10. We are interested in noise sources at one particular $\xi$ because noise generated at any other $\xi$ would be uncorrelated. Figure 10: Schematic of the self-correlation. The star denotes a noise source and the bursts are the charge fluctuations resulting from noise. (color online) Let us try to understand what the analytical formula for this would look like. We start with the following expression for the charge fluctuation in $k$-space. $\delta\tilde{n}(k,\tau)=-\frac{1}{\tau}\int_{\tau_{i}}^{\tau}d\tau^{\prime}s(\tau^{\prime})\tilde{G}(k,\tau,\tau^{\prime})\tilde{f}(k,\tau^{\prime})\,.$ (35) Here $\tilde{G}$ is the Green’s function for the homogeneous part of the SDE (29), which can be written down in terms of Kummer’s function for the temperature dependences listed after that equation plumbergkapusta . This gives the full form of the two-point correlation function as in Eq. (49) of Ref. plumbergkapusta . $\displaystyle\langle\delta n(\xi_{1},\tau_{f})\delta n(\xi_{2},\tau_{f})\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\tau_{f}^{2}}\int\frac{dk}{2\pi}e^{ik(\xi_{1}-\xi_{2})}\int d\tau^{\prime}s(\tau^{\prime})\int d\tau^{\prime\prime}s(\tau^{\prime\prime})$ (36) $\displaystyle\times$ $\displaystyle\tilde{G}(k,\tau_{f},\tau^{\prime})\;\tilde{G}(-k,\tau_{f},\tau^{\prime\prime})\,{\cal N}(\tau^{\prime},\tau^{\prime\prime})\,.$ Following Eqs. (54) and (55) of Ref. plumbergkapusta , we can write the self- correlation term as $\displaystyle\langle\delta n(\xi_{1},\tau_{f})$ $\displaystyle\delta n$ $\displaystyle(\xi_{2},\tau_{f})\rangle_{\text{self}}$ (37) $\displaystyle=$ $\displaystyle\frac{\chi_{Q}(\tau_{f})T_{f}}{A\tau_{Q}}\int\frac{d\tau^{\prime\prime}}{\tau^{\prime\prime}}\left[e^{-(\tau_{f}-\tau_{2})/\tau_{Q}}-e^{-(\tau_{f}+\tau_{2}-2\tau_{i})/\tau_{Q}}\right]\int\frac{dk}{2\pi}e^{ik(\xi_{1}-\xi_{2})}\frac{\tilde{G}(-k,\tau_{f},\tau^{\prime\prime})}{ik}$ $\displaystyle=$ $\displaystyle\frac{s(\tau_{f})}{D_{Q}}\int\frac{dk}{2\pi}e^{ik(\xi_{1}-\xi_{2})}\int d\tau^{\prime\prime}s(\tau^{\prime\prime})\;\frac{\tilde{G}(k,\tau_{f},\tau_{f})}{ik}\;\frac{\tilde{G}(-k,\tau_{f},\tau^{\prime\prime})}{ik}\,{\cal N}(\tau_{f},\tau^{\prime\prime})\,.$ Recall from Ref. plumbergkapusta that $\tilde{G}(k,\tau_{f},\tau_{f})=ik$. It denotes a noise fluctuation that was generated at the final time and didn’t have to move anywhere. The $\tilde{G}(-k,\tau_{f},\tau^{\prime\prime})$ is a noise fluctuation generated at a time $\tau^{\prime\prime}<\tau_{f}$ and then moved to $\xi_{2}$ at $\tau_{f}$. For white noise, fluctuations generated at two separate spacetime points can’t be correlated, so the fluctuation generated at $\tau_{f}$ is only correlated to itself. Hence for white noise, $\tau_{Q}\to 0$, and the self-correlation is just a Dirac $\delta$-function. As $\tau_{Q}$ increases, the more backward in time the noise sources would be correlated. Once generated the noise will travel and give rise to a correlated electric charge fluctuation further away in spacetime rapidity. Hence we expect the self-correlation term to be more spread out in spacetime rapidity. One can use the same SDE solver for generating the self-correlation. The only change is that the Green’s function $\tilde{G}$ is replaced by $\tilde{G}/(ik)$ when solving for the charge density fluctuation. $\displaystyle\langle$ $\displaystyle\delta n$ $\displaystyle(\xi_{2},\tau_{f})f(\xi_{1},\tau_{f})\rangle$ (38) $\displaystyle=$ $\displaystyle\left\langle\int\frac{dk}{2\pi}e^{ik\xi_{2}}\delta\tilde{n}(k,\tau_{f})\int\frac{dk_{1}}{2\pi}e^{ik_{1}\xi_{1}}\tilde{f}(k,\tau_{f})\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\tau_{f}}\int^{\tau_{f}}_{\tau_{i}}d\tau^{\prime}s(\tau^{\prime})\int\frac{dk}{2\pi}e^{ik\xi_{2}}\frac{\tilde{G}(k,\tau^{\prime},\tau_{f})}{ik}\int\frac{dk_{1}}{2\pi}e^{ik_{1}\xi_{1}}\langle\tilde{f}(k,\tau^{\prime})\tilde{f}(k_{1},\tau_{f})\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\tau_{f}}\int^{\tau_{f}}_{\tau_{i}}d\tau^{\prime}s(\tau^{\prime})\int\frac{dk}{2\pi}e^{ik(\xi_{2}-\xi_{1})}\frac{\tilde{G}(k,\tau^{\prime},\tau_{f})}{ik}\mathcal{N}(\tau^{\prime},\tau_{f})$ $\displaystyle=$ $\displaystyle\frac{D_{Q}}{s_{f}\tau_{f}}\,\frac{\chi_{f}T_{f}}{A\tau_{Q}}\int^{\tau_{f}}_{\tau_{i}}\frac{d\tau^{\prime}}{\tau^{\prime}}\left[e^{-|\tau_{f}-\tau^{\prime}|/\tau_{Q}}-e^{(2\tau_{i}-\tau_{f}-\tau^{\prime})/\tau_{Q}}\right]\int\frac{dk}{2\pi}e^{ik(\xi_{2}-\xi_{1})}\frac{\tilde{G}(k,\tau^{\prime},\tau_{f})}{ik}$ Note that in going to the second step from the first step, we have used $\tilde{G}/(ik)$ and not $\tilde{G}$. The implication is that for generating the self-correlations, we will be using the differential equation whose Green’s function is going to be $\tilde{G}/(ik)$, instead of $\tilde{G}$. Thus, we arrive at the following relation for self-correlations. $\langle\delta n(\xi_{1},\tau_{f})\delta n(\xi_{2},\tau_{f})\rangle_{\text{self}}=\frac{s(\tau_{f})\tau_{f}}{D_{Q}}\langle\delta n(\xi_{1},\tau_{f})f(\xi_{2},\tau_{f})\rangle\,.$ (39) If $\tilde{G}/(ik)$ is our desired Green’s function, then $Z\equiv(\tau\delta n(\xi,\tau))/(\tau_{f}s(\tau_{f}))=\delta n(\xi,\tau)/s(\tau)$ satisfies the following equation $\left(z^{2}+2z\frac{\tau_{Q}}{\tau_{i}}\right)\frac{\partial Z}{\partial z}+z^{2}\frac{\tau_{Q}}{\tau_{i}}\frac{\partial^{2}Z}{\partial z^{2}}-v_{Q}^{2}\frac{\tau_{Q}}{\tau_{i}}\frac{\partial^{2}Z}{\partial\xi^{2}}+\left(z+\frac{\tau_{Q}}{\tau_{i}}\right)f+z\frac{\tau_{Q}}{\tau_{i}}\frac{\partial f}{\partial z}=0\,,$ (40) where $z=\tau/\tau_{i}$. This is the same as Eq. (25) except that $\partial f/\partial\xi$ is replaced by $f$ and $\partial^{2}f/\partial\xi\partial\tau$ is replaced by $\partial f/\partial\tau$. The justification is discussed in the Appendix C. In Fig. 11 we show the self-correlation at the final time $\tau_{f}$ for various values of $\tau_{Q}$. As the speed of propagation decreases, the height of the self-correlation decreases and widens. As a check, the limit $\tau_{Q}\to 0$ is shown in Fig. 12. Figure 11: Numerical results for self-correlations for colored noise. (color online) Figure 12: Numerical results for self-correlation for white noise. ## VI Balance Functions Balance functions are described in Ref. spratt . The width of a balance function plotted against particle rapidity is a measure of the diffusion. Balance functions have been experimentally studied by the ALICE and STAR collaborations balance1 ; balance2 ; balance3 ; balance4 . Reference ling_springer_stephanov studied the effect of white noise in balance functions and compared their analytical results with experimental data. Reference plumbergkapusta calculated balance functions for colored noise. We will see how the widths of balance functions change if we vary the speed of propagation of signals in case of Catteneo noise. To see the effect of charge fluctuations in particle spectra we have to calculate how the fluctuations freeze-out. The freeze-out happens when the system has expanded and cooled to the extent that thermal equilibrium can no longer be maintained. Then hadrons freeze-out and free-stream to the detectors. The standard procedure to calculate freeze-out abundances of particles is the Cooper-Frye prescription cooper_frye . This formula gives us the distribution of emitted particles on a freeze-out hypersurface $\Sigma_{f}$. This procedure has already been performed for this hydrodynamical model in Refs. kapusta2012 ; plumbergkapusta ; ling_springer_stephanov ; torres We will just give the salient features of that calculation here. $E\frac{dN}{d^{3}p}=d\int_{\Sigma_{f}}\frac{d^{3}\sigma_{\mu}}{(2\pi)^{3}}p^{\mu}f({\mbox{\boldmath$x$},\mbox{\boldmath$p$}})$ (41) Here $d$ is the degeneracy of the particle species under consideration. We take the distribution function to be the relativistic Boltzmann $f({\mbox{\boldmath$x$},\mbox{\boldmath$p$}})=e^{-(u\cdot p-\mu)/T}\,,$ (42) where $\mu$ is the chemical potential for that particle. The energy flux through an infinitesimal freeze-out fluid cell is given by $d^{3}\sigma_{\mu}p^{\mu}=\tau_{f}\,d\xi\,d^{2}x_{\perp}m_{\perp}\cosh(y-\xi)\,.$ (43) The variable $y$ represents the particle rapidity $p^{\mu}=(m_{\perp}\cosh y,p_{x},p_{y},m_{\perp}\sinh y)$ (44) with $m_{\perp}=\sqrt{m^{2}+p_{\perp}^{2}}$ the transverse mass. The average number of particles per unit rapidity at the final freeze-out time is $\left\langle\frac{dN}{dy}\right\rangle=\frac{dA\tau_{f}T_{f}^{3}}{4\pi^{2}}\int^{\infty}_{-\infty}\frac{dx}{\cosh^{2}x}\Gamma\left(3,\frac{m}{T_{f}}\cosh x\right)\,.$ (45) Reference plumbergkapusta calculates the fluctuation in this quantity due to a $\mu$ around the freeze-out $\mu_{f}=0$. After a few more steps of algebra, the fluctuation in $\frac{dN}{dy}$ reads $\delta\left(\frac{dN}{dy}\right)=\frac{dA\tau_{f}T_{f}^{2}}{4\pi^{2}}\int d\xi\,\delta n\,F_{n}(y-\xi)$ (46) where $F_{n}$ is the smearing function $F_{n}(x)=\frac{1}{\chi_{Q}\cosh^{2}x}\Gamma\left(3,\frac{m}{T_{f}}\cosh x\right)\,.$ (47) Using this in the definition of the charge balance function we arrive at the Eq. (74) in Ref. plumbergkapusta . $B(\Delta y)=\frac{\langle\delta\left(dN/dy_{1}\right)\delta\left(dN/dy_{2}\right)\rangle}{\langle dN/dy\rangle}=\frac{dA\tau_{f}T_{f}}{4\pi^{2}}\frac{C(\Delta y)}{Q(m/T_{f})}\,.$ (48) Here $C(\Delta y)=2\pi\int d\xi_{1}d\xi_{2}\,F_{n}(y_{1}-\xi_{1})\,F_{n}(y_{2}-\xi_{2})\,C_{nn}(\xi_{1}-\xi_{2},\tau_{f})\,.$ (49) The two-point correlator for the charge fluctuation is $C_{nn}(\xi_{1}-\xi_{2},\tau_{f})$ which is obtained from the solution of the SDE. The function $Q$ is given by $Q\left(\frac{m}{T_{f}}\right)=\int^{\infty}_{-\infty}\frac{dx}{\cosh^{2}x}\Gamma\left(3,\frac{m}{T_{f}}\cosh x\right)\,.$ (50) Let us first demonstrate the trivial self-correlation for white noise in terms of the balance function for pions. Figure 13 shows the balance function for the full unsubtracted correlation function for white noise. Notice the positive and negative part of the curve; this is because the full two-point correlation for white noise is composed of a positive self-correlation and a negative piece which does not include any self-correlation. The balance function for the self-correlation part only is shown in Fig. 14. When this is subtracted from Fig. 13 one obtains the so-called subtracted balance function shown in Fig. 15. Figure 13: Balance function for the full white noise two-point function. Figure 14: Balance function for the self-correlation of white noise. Figure 15: Balance function for the pure two-point function of white noise. We follow the same procedure for carrying out cancellations of the contributions arising from the self-correlations for colored noise to the balance function. Figure 16 shows the full unsubtracted balance functions for various values of $v_{Q}$ for colored noise. Figure 17 shows the self- correlation part only, and Fig. 18 shows the subtracted balance functions. The width of the subtracted balance function denotes the diffusion distance. That width increases with increasing $v_{Q}$, as expected, since it represents the rapidity interval over which the average charge pair has diffused to by freeze out. Figure 16: Balance function for the full unsubtracted two-point function. (color online) Figure 17: Balance function for the self-correlation part of two-point function. (color online) Figure 18: Balance function for the subtracted two-point correlation function. (color online) We estimated the error in our numerical simulations using the jackknife method. This method estimates the error of statistics without making any assumptions about the distribution that generated the data. It only uses the sample provided. We create jackknife samples over the whole data set which are “leave-one-out” data sets. In our case, we consider the two-point correlation statistic $S$ on the original sample size of $10^{7}$ events. We leave out the $i_{th}$ event to create the $i_{th}$ jackknife statistic $S_{i}$. The average of the jackknife sample is $S_{\rm avg}=\sum_{i}S_{i}/n$. The jackknife error is then estimated as $\sigma_{\rm jack}=\sqrt{\frac{n-1}{n}\sum_{i}(S_{i}-S_{\rm avg})^{2}}$ (51) The error we observe on $\langle\delta n\delta n\rangle$ is of the order of $10^{-2}\;\text{MeV}^{3}\>\text{fm}^{-3}$. This amounts to $\sigma_{\langle\delta n\delta n\rangle}/\langle\delta n\delta n\rangle\approx 10^{-3}$. We give a representative plot of the error bounds for $v_{Q}^{2}=1/10$ in Fig. 19. The bounds are visible only when zoomed in. This shows that for $10^{7}$ events, the statistical error in our simulations turn out to be negligible. Figure 19: Jackknife error bounds for $v_{Q}^{2}=1/10$. (color online) ## VII Conclusions State-of-the-art modeling of high energy nuclear collisions uses relativistic 2nd order viscous hydrodynamics. The fluctuation-dissipation theorem says that viscosity and thermal fluctuations are intricately connected. Although thousands of particles are produced in these collisions, that is still immensely smaller than Avogadro’s number. Therefore it has become apparent that thermal fluctuations really ought to be part of the standard model for heavy ion collisions kapusta2012 . Fully 3+1 dimensional hydrodynamic simulations are required which presents a major challenge for implementation of thermal noise. The goal of this paper is to understand the numerical methods necessary to do this and, furthermore, how to subract self- correlations from the numerically computed two-point correlators in order to compare with experiment. We chose to study causal electric charge diffusion in the boost-invariant 1+1 dimensional Bjorken model for two reasons. First, a Cattaneo description of diffusion propagates signal at a finite speed which is a necessity in heavy ion collisions. Second, this simple model was studied and solved with essentially analytic methods plumbergkapusta against which we can compare to verify the validity of the purely numerical approach. We introduced the noise term in the dissipative charge conservation equation, which in our case is the Catteneo noise. We simulated the stochastic differential equations that arise from the electric charge conservation equations. The way we solve the stochastic differential equations is by using normal random number generators with a specific, well-defined variance and then interpreting the derivatives of the noise in terms of what they mean when integrating by parts. The whole machinery on how to handle the noise is discussed in Appendix A. We used this methodology in simulating the white noise charge conservation equation and obtained the expect result. Then we generated colored Catteneo noise using a Langevin equation. We solved the full colored noise charge conservation equation and again obtained the expected result. The two-point charge correlator consists of two pieces. The first is the self-correlation, which is a manifestation of the stochastic nature of the dynamics. Once this piece is subtracted off, we are left with the physically relevant two-point correlation function. The self-correlation is a trivial Dirac $\delta$-function in the case of white noise, but is more complicated for colored noise. In this work, we gave a physically insightful interpretation of the meaning of self-correlation in the case of colored noise. This interpretation allows us to use the stochastic differential equation solver we developed to generate the self-correlations. In the case of the white noise, we populated the whole spacetime lattice with noise source terms uncorrelated to each other. It is obvious that all the individual noise terms are not required to calculate the final two-point correlation function, but more than a single noise term is necessary. Hence Monte-Carlo simulations will be insufficient to reproduce the results for colored noise. One can, however, speed up the stochastic differential equation solving procedure by removing noise terms that are outside the causal past of the spacetime points for which we want to calculate the twopoint correlations. We used the results obtained to compute the balance functions for pions within the context of this model. As one would expect, reducing the speed of propagation of signal leads to narrowing of the balance functions and to a corresponding increase in their height at small rapidities. As done previously in Ref. plumbergkapusta we neglect the contributions from resonance decays to the measured particle spectra used in the balance functions. Our results are in good quantitative agreement with that previous study. The numerical method used in this paper is verified. Future work entails furthering the current methodology to a full 3+1 dimensional fluid dynamical models of heavy ion collisions such as MUSIC music . Further, the prescription for self-correlations given in this paper for Catteneo noise can be straightforwardly extended to the case of shear and bulk viscosity and thermal conductivity, the details of which are deferred to a future work. Since baryon charge conductivity diverges near a critical point, this study can be extended to study charge fluctuations near the purported QCD critical point, which is also deferred to future work. Another possible direction of future work would be to study the higher order cumulants in the presence of colored noise. Since two-point correlations and higher order cumulants are expected to diverge near a QCD critical point, the ultimate culmination of the present work would be to characterize the noisy hydrodynamics of near-critical point behavior of heavy ion collisions. ## VIII Acknowledgements A. D. thanks Gaurav Nirala for enlightening discussions. We thank Chun Shen for suggesting the jackknife method. This work was supported by the U.S. DOE Grant No. DE-FG02-87ER40328. C. P. acknowledges support from the CLASH project (KAW 2017-0036). The authors acknowledge the Minnesota Supercomputing Institute (MSI) at the University of Minnesota for providing resources that contributed to the research results reported within this paper. ## Appendix A Numerical Simulation of SDEs In this appendix, we show our procedure for representing the Dirac delta function and its derivatives on a discrete lattice. White noise is defined as $\langle f(x)f(x^{\prime})\rangle=\delta(x-x^{\prime})$ and $\langle f(x)\rangle=0$. This implies that $\int dx\>g(x)\langle f(x)f(x^{\prime})\rangle=g(x^{\prime})$ (52) On a discrete lattice, the integral becomes a sum over lattice points and $dx$ becomes the lattice spacing $\Delta x$. Hence $g(x_{i})=\sum_{i^{\prime}}g(x_{i^{\prime}})\langle f(x_{i})f(x_{i^{\prime}})\rangle\Delta x=\sum_{i^{\prime}}g(x_{i^{\prime}})\left(\frac{\delta_{ii^{\prime}}}{\Delta x}\right)\Delta x$ (53) From the above, we can conclude $\langle f(x_{i})f(x_{i^{\prime}})\rangle=\frac{\delta_{ii^{\prime}}}{\Delta x}$ (54) The $\delta_{ii^{\prime}}/\Delta x$ becomes a Dirac-delta function in the limit $\Delta x\rightarrow 0$ which is the continuous case. Therefore we sample the white noise function $f$ from a Normal distribution of mean $0$ and standard deviation $1/\sqrt{\Delta x}$. We use a random number generator for a large number of instances ($10^{6}$) and compute the two-point function. It gives us the variance as the peak of a Kronecker delta at $x=0$. This is illustrated in the following figure. We used $\Delta x=0.09$. Figure 20: Two-point function of $f$ for 1 million events. Next, we investigate the correlation between white noise $f$ and its derivative $df/dx$. The two-point function $\langle f(x)f^{\prime}(x^{\prime})\rangle$ must then satisfy $\int dx\,g(x)\langle f(x)f^{\prime}(x^{\prime})\rangle=\int dx\,g(x)\frac{\partial}{\partial x^{\prime}}\delta(x-x^{\prime})=\frac{\partial}{\partial x^{\prime}}\int dx\,g(x)\delta(x-x^{\prime})=g^{\prime}(x^{\prime})$ (55) The derivative is $g^{\prime}(x)=(g_{i+1}-g_{i})/\Delta x$ in a discrete lattice. Replacing the integral by the sum, we get $g^{\prime}(x_{i})=\sum_{i^{\prime}}g(x_{i^{\prime}})\langle f(x_{i})f^{\prime}(x_{i^{\prime}})\rangle\Delta x=\frac{g_{i+1}-g_{i}}{\Delta x}=\sum_{i^{\prime}}g(x_{i^{\prime}})\left(\frac{\delta_{i+1,i^{\prime}}-\delta_{i,i^{\prime}}}{\Delta x^{2}}\right)\Delta x$ (56) Hence $\langle f(x_{i})f^{\prime}(x_{i^{\prime}})\rangle=\frac{\delta_{i+1,i^{\prime}}-\delta_{i,i^{\prime}}}{\Delta x^{2}}$ (57) If we again use the previous random number generator and calculate the two point function we get the results shown in Fig. 21. Figure 21: The correlation between $f$ and its derivative for 1 million events. Similarly, we can look into the correlation of the derivative of white noise with itself. $\langle f^{\prime}(x)f^{\prime}(x^{\prime})\rangle=\frac{\partial^{2}}{\partial x\partial x^{\prime}}\delta(x-x^{\prime})$ (58) $\int dx\,g(x)\langle f^{\prime}(x)f^{\prime}(x^{\prime})\rangle=\int dx\,g(x)\frac{\partial^{2}}{\partial x\partial x^{\prime}}\delta(x-x^{\prime})=-\int dx\,g(x)\frac{\partial^{2}}{\partial x^{2}}\delta(x-x^{\prime})=-g^{\prime\prime}(x^{\prime})$ (59) In the second step we performed an integration by parts. The second derivative is defined in the discrete case as $g^{\prime\prime}(x)=(g_{i+1}+g_{i-1}-2g_{i})/\Delta x^{2}$. Substituting the discrete sum in place of the integral, we get $-g^{\prime\prime}(x^{\prime})=-\sum_{i}g_{i}\frac{\delta_{i,i^{\prime}+1}+\delta_{i,i^{\prime}-1}-2\delta_{i,i^{\prime}}}{\Delta x^{3}}\Delta x=\sum_{i^{\prime}}g(x_{i^{\prime}})\langle f^{\prime}(x_{i})f^{\prime}(x_{i^{\prime}})\rangle\Delta x$ (60) $\langle f^{\prime}(x_{i})f^{\prime}(x_{i^{\prime}})\rangle=-\frac{\delta_{i,i^{\prime}+1}+\delta_{i,i^{\prime}-1}-2\delta_{i,i^{\prime}}}{\Delta x^{3}}$ (61) Figure 22 shows what we get numerically. Figure 22: The two-point correlation of the derivative of $f$ for 1 million events. Integration of white noise is called a random walk $W(z)$ which is a succession of random steps as a function of $z$. It is defined by $W=\int_{z_{i}}^{z}f(z)dz$ (62) We can easily calculate the variance of $W$: $\langle W^{2}(z)\rangle=\int_{z_{0}}^{z}dz^{\prime}\int_{z_{0}}^{z}dz^{\prime\prime}\langle f(z^{\prime})f(z^{\prime\prime})\rangle=\int_{z_{0}}^{z}dz^{\prime}\int_{z_{0}}^{z}dz^{\prime\prime}\delta(z^{\prime}-z^{\prime\prime})=\int_{z_{0}}^{z}dz^{\prime}=z-z_{0}$ (63) On a discrete lattice, $W_{i+1}=W_{i}+f\Delta z$ where we source $f$ from a normal distribution of mean $0$ and standard deviation $1/\sqrt{\Delta z}$. $W_{n}=\sum_{i=1}^{n}f\Delta z$ (64) Hence the variance is $\displaystyle\langle W^{2}\rangle=\langle(\sum_{i=1}^{n}\Delta zf)^{2}\rangle=\langle\sum_{i=1}^{n}(\Delta zf)^{2}\rangle=\sum_{i=1}^{n}(\Delta z)^{2}\langle f^{2}\rangle=\sum_{i=1}^{n}(\Delta z)^{2}\frac{1}{\Delta z}=\sum_{i=1}^{n}\Delta z=z-z_{0}$ Figure 23 shows the numerical results. Figure 23: Two point correlation of $W$ for 1 million events with $z-z_{i}=5$. We are ready to take up a simple stochastic differential equation to solve. Consider $\frac{dX}{dz}=-\frac{\partial f}{\partial\xi}$ (66) where $z$ has dimensions of time and $\xi$ is dimensionless. Let us define the following two-point function $\langle f(z_{1},\xi_{1})f(z_{2},\xi_{2})\rangle=M\delta(z_{1}-z_{2})\delta(\xi_{1}-\xi_{2})$ Here $M$ has dimensions of time to make $f$ dimensionless. We calculate the two-point function in $\xi$. $\displaystyle\langle X(z_{f},\xi_{1})X(z_{f},\xi_{2})\rangle$ $\displaystyle=$ $\displaystyle\left\langle\int^{z_{f}}_{z_{i}}\frac{\partial f}{\partial\xi}(\xi_{1})dz\int^{z_{f}}_{z_{i}}\frac{\partial f}{\partial\xi}(\xi_{2})dz^{\prime}\right\rangle$ (67) $\displaystyle=$ $\displaystyle\int^{z_{f}}_{z_{i}}\int^{z_{f}}_{z_{i}}dzdz^{\prime}\left\langle\frac{\partial f}{\partial\xi}(z,\xi_{1})\frac{\partial f}{\partial\xi}(z^{\prime},\xi_{2})\right\rangle$ $\displaystyle=$ $\displaystyle-\int^{z_{f}}_{z_{i}}\int^{z_{f}}_{z_{i}}dzdz^{\prime}M\left(\frac{\delta_{i+1}+\delta_{i-1}-2\delta_{i}}{\Delta\xi^{3}}\right)\delta(z-z^{\prime})$ $\displaystyle=$ $\displaystyle-M(z_{f}-z_{i})\left(\frac{\delta_{i+1}+\delta_{i-1}-2\delta_{i}}{\Delta\xi^{3}}\right)$ We used Eq. (61) in the above calculation. The two-point function has dimensions of time-squared and so is the expression on the right. On a discrete lattice, $X(z+\Delta z)=X(z)-\Delta z\times\frac{\Delta f}{\Delta\xi}$ (68) Figure 24 shows the numerical results using $z_{f}-z_{i}=10$. Figure 24: Two point correlation of $X$ for a million events. ## Appendix B Analytical Solution of Langevin Equation The Langevin equation can be written as $\frac{df(\tau)}{d\tau}=-\frac{1}{\tau_{Q}}f(\tau)+\frac{1}{\tau_{Q}}\zeta(\tau)$ (69) Here $\zeta$ is white noise and $f$ is the Catteneo noise. $\langle\zeta(\tau)\rangle=0\qquad\langle\zeta(\tau_{1})\zeta(\tau_{2})\rangle=N(\tau_{1})\delta(\tau_{1}-\tau_{2})$ (70) It does not matter whether we use $N(\tau_{1})$ or $N(\tau_{2})$ because of the Dirac-delta function. Let us multiply both sides by the factor $e^{\tau/\tau_{Q}}$. $\int_{\tau_{i}}^{\tau}\frac{d}{d\tau}(e^{\tau/\tau_{Q}}f)d\tau=\int_{\tau_{i}}^{\tau}\frac{e^{\tau/\tau_{Q}}}{\tau_{Q}}\zeta d\tau$ (71) $e^{\tau/\tau_{Q}}f(\tau)-e^{\tau_{i}/\tau_{Q}}f(\tau_{i})=\frac{1}{\tau_{Q}}\int^{\tau}_{\tau_{i}}\zeta(\tau)e^{(\tau^{\prime}-\tau)/\tau_{Q}}d\tau^{\prime}$ (72) Let us set $f(\tau_{i})=0$. Another way to see this is in an equilibrium system, the system does not have any initial conditions to be sensitive to. Any fluctuations in $f(\tau)$ will then be solely due to the action of $\zeta(\tau)$. Now we consider two separate times $\tau_{1}$, $\tau_{2}$. $\displaystyle\begin{aligned} \langle f(\tau_{1})f(\tau_{2})\rangle&=\frac{N}{\tau_{Q}^{2}}\int_{\tau_{i}}^{\tau_{1}}e^{(\tau^{\prime\prime}-\tau_{1})/\tau_{Q}}d\tau^{\prime\prime}\int_{\tau_{i}}^{\tau_{2}}e^{(\tau^{\prime}-\tau_{2})/\tau_{Q}}d\tau^{\prime}\delta(\tau_{1}-\tau_{2})\\\ &=\frac{N}{\tau_{Q}^{2}}\int_{\tau_{i}}^{\text{min}(\tau_{1},\tau_{2})}e^{(2\tau^{\prime\prime}-\tau_{1}-\tau_{2})/\tau_{Q}}d\tau^{\prime\prime}=\frac{N}{2\tau_{Q}}\left[e^{|\tau_{1}-\tau_{2}|/\tau_{Q}}-e^{(2\tau_{i}-\tau_{1}-\tau_{2})/\tau_{Q}}\right]\end{aligned}$ (73) ## Appendix C Constructing the self-correlations Self-correlations are defined by Eq. (37). As discussed above, their dynamics can be modeled by an equation whose (Fourier transformed) Green’s function is related to the original Green’s function by $\tilde{G}_{\mathrm{self}}(k,\tau,\tau^{\prime})=\frac{\tilde{G}(k,\tau,\tau^{\prime})}{ik}$ (74) The original Green’s function is defined schematically by the stochastic differential equation $D_{1}X(\tau,\xi)=D_{2}\frac{\partial f}{\partial\xi}(\tau,\xi),$ (75) where $D_{1}$ and $D_{2}$ are differential operators which contain no explicit $\xi$-dependence (other than $\xi$-derivatives) and $f$ is the noisy source. Fourier transforming the $\xi$-dependence to $k$ as before, this equation becomes $\tilde{D}_{1}\tilde{X}(\tau,k)=ik\tilde{D}_{2}\tilde{f}(\tau,k)$ (76) and its solution is written in terms of the original Green’s function as $\tilde{X}(k,\tau)=-\int_{\tau_{0}}^{\tau}d\tau^{\prime}\tilde{G}(k;\tau,\tau^{\prime})\tilde{f}(k,\tau^{\prime})$ (77) We therefore seek an ‘unphysical’ field $X_{\mathrm{self}}$ whose two-point function corresponds to the self-correlations which need to be subtracted out. This field solution will be generated by the expression $\displaystyle\tilde{X}_{\mathrm{self}}(k,\tau)$ $\displaystyle=$ $\displaystyle-\int_{\tau_{0}}^{\tau}d\tau^{\prime}\tilde{G}_{\mathrm{self}}(k;\tau,\tau^{\prime})\tilde{f}(k,\tau^{\prime})$ (78) $\displaystyle=$ $\displaystyle-\int_{\tau_{0}}^{\tau}d\tau^{\prime}\frac{\tilde{G}(k;\tau,\tau^{\prime})}{ik}\tilde{f}(k,\tau^{\prime})$ $\displaystyle=$ $\displaystyle-\int_{\tau_{0}}^{\tau}d\tau^{\prime}\tilde{G}(k;\tau,\tau^{\prime})\left(\frac{\tilde{f}(k,\tau^{\prime})}{ik}\right)$ The self-correlations can be therefore straightforwardly generated by replacing $\tilde{f}\to\tilde{f}/(ik)$ in $k$-space, which amounts to discarding the $\xi$-derivative in Eq. (75). Thus, the unphysical self- correlation field is generated by solving the modified equation $D_{1}X(\tau,\xi)=D_{2}f(\tau,\xi)$ (79) ## References * (1) J. I. Kapusta and J. M. Torres-Rincon, Phys. Rev. C. 86, 054911 (2012). * (2) Y. Akamatsu, D. Teaney, F. Yan, and Y. Yin, Phys. Rev. C 100, 044901 (2019). * (3) M. Stephanov and Y. Yin, Phys. Rev. D 98, 036006 (2018). * (4) M. Nahrgang, M. Bluhm, T. Schäfer, and S. A. Bass, Phys. Rev. D 99, 116015 (2019). * (5) M. Bluhm et al., arXiv:2001.08831 [nucl-th]. * (6) M. Stephanov, K. Rajagopal, and E. Shuryak, Phys. Rev. D 60, 114028 (1999). * (7) M. A. Stephanov, Phys. Rev. Lett. 102, 032301 (2009) * (8) B. Berdnikov and K. Rajagopal, Phys. Rev. D 61, 105017 (2000). * (9) J. I. Kapusta, B. Müller, and M. Stephanov, Phys. Rev. C. 85, 054906 (2012). * (10) S. Pratt, Phys. Rev. Lett. 108, 212301 (2012). * (11) J. I. Kapusta and C. Plumberg, Phys. Rev. C 97, 014906 (2018). * (12) C. Cattaneo, Atti Del Semin. Mat. e Fis. Univ. Modena 3, 3 (1948); C. R. Acad. Sci. 247, 431 (1958). * (13) J. I. Kapusta and C. Young, Phys. Rev. C 90, 044902 (2014). * (14) M. E. Gurtin and A. C. Pipkin, Arch. Ration. Mech. Anal. 31, 113 (1968). * (15) G. Aarts, C. Allton, A. Amato, P. Giudice, S. Hands, and J.-I. Skullerud, JHEP 02, 186 (2015). * (16) R.Courant, K. Friedrichs, and H. Lewy, Mathematische Annalen, Vol. 100, pp. 32-74 (1928). * (17) L. D. Landau and E. M. Lifshitz, Statistical Physics: Part 1, Vol. 5, Course of Theoretical Physics (Pergamon, 1980). * (18) E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics: Part 2, Vol. 9, Course of Theoretical Physics (Pergamon, 1980). * (19) B. Abelev, et al. (STAR Collaboration), Phys. Rev. Lett. 105, 022301 (2010). * (20) A. R. Timmins (for the ALICE Collaboration), J. Phys. G: Nucl. Part. Phys. 38, 124093 (2011). * (21) M. Aggarwal, et al. (STAR Collaboration), Phys. Rev. C 82, 024905 (2010). * (22) B. Abelev, et al. (ALICE Collaboration), Phys. Lett. B 723, 267 (2013). * (23) B. Ling, T. Springer, and M. Stephanov, Phys. Rev. C 89, 064901 (2014). * (24) F. Cooper and G. Frye, Phys. Rev. D 10, 186 (1974). * (25) B. Schenke, S. Jeon, and C. Gale, Phys. Rev. C 82, 014903 (2010). * (26) https://github.com/audide12/HeavyIon

2024-09-04T02:54:58.871503

2003.04956

arxiv-papers

# SQUIRL: Robust and Efficient Learning from Video Demonstration of Long- Horizon Robotic Manipulation Tasks Bohan Wu, Feng Xu, Zhanpeng He, Abhi Gupta, and Peter K. Allen This work is supported by NSF Grant CMMI-1734557. Authors are with Columbia University Robotics Group, New York, NY, 10027, USA ###### Abstract Recent advances in deep reinforcement learning (RL) have demonstrated its potential to learn complex robotic manipulation tasks. However, RL still requires the robot to collect a large amount of real-world experience. To address this problem, recent works have proposed learning from expert demonstrations (LfD), particularly via inverse reinforcement learning (IRL), given its ability to achieve robust performance with only a small number of expert demonstrations. Nevertheless, deploying IRL on real robots is still challenging due to the large number of robot experiences it requires. This paper aims to address this scalability challenge with a robust, sample- efficient, and general meta-IRL algorithm, SQUIRL, that performs a new but related long-horizon task robustly given only a single video demonstration. First, this algorithm bootstraps the learning of a task encoder and a task- conditioned policy using behavioral cloning (BC). It then collects real-robot experiences and bypasses reward learning by directly recovering a Q-function from the combined robot and expert trajectories. Next, this algorithm uses the Q-function to re-evaluate all cumulative experiences collected by the robot to improve the policy quickly. In the end, the policy performs more robustly (90%+ success) than BC on new tasks while requiring no trial-and-errors at test time. Finally, our real-robot and simulated experiments demonstrate our algorithm’s generality across different state spaces, action spaces, and vision-based manipulation tasks, e.g., pick-pour-place and pick-carry-drop. ## I Introduction We aspire robots to become generalists who acquire new complex skills robustly and quickly. The robotic system, whether planned or learned, needs to leverage its existing knowledge to solve a new but related task in an efficient yet high-performance manner. Thanks to recent advances in machine learning and sim-to-real transfer mechanisms, short-horizon robotic manipulation such as grasping has improved in performance. However, many real-world robotic manipulation tasks are long-horizon, diverse, and abundant in volume. In the absence of a scalable and systematic way to construct simulation environments for a large number of tasks, the robot needs to learn a new task directly in the physical world from only a handful of trials, due to the high cost of collecting real-robot trial-and-errors and experiences. Figure 1: Learning from a single video demonstration of a long-horizon manipulation task via Soft Q-functioned Meta-IRL (SQUIRL). In the pick-pour- place example above, the robot needs to approach, pick-up and carry the grey bottle, pour the iron pebble inside the bottle into a specific container, and finally place the bottle back on the table. During training, the robot is given a single video demonstration for each of the 117 training tasks. After learning from these 117 videos, the robot also practices 90 trial-and-errors in total on these tasks. From such combined expert and robot trajectories, the robot learns the general skills of pouring robustly. At test time, given a single video demonstration of pouring into a new, unseen red container at a new position, the robot successfully replicates this new task without the need for any trial-and-errors. We observe that real-world robotic skill acquisition can become more sample- efficient in several important ways. First, we notice that humans learn tasks quickly by watching others perform similar tasks. Among many forms of task representations such as rewards, goal images, and language instructions, human demonstrations guide exploration effectively and can lead to significant sample efficiency gains. Furthermore, learning from video demonstrations sidesteps hand-designing a proper reward function for every new task. In the case of a vision-based task, video demonstrations also conveniently provide the same pixel state space for the robot. In learning from demonstrations (LfD), the robot should be sample-efficient in two dimensions – it should use as few expert demonstrations (“demonstrations” hereafter) as possible and take as few trial-and-errors (practices) as possible on its own to learn a robust policy. Among LfD methods, behavioral cloning (“BC” hereafter) is sample-efficient but susceptible to compounding errors. Here, compounding errors refer to the problem in which every time a behavioral-cloned robot makes a small error, it makes a larger error down the road as it drifts away from the expert state distribution. In contrast, IRL alleviates compounding errors by allowing the robot to try the tasks out in the real world and measure its behavior against the expert. However, due to the need to learn a reward function, IRL can require many trial-and-errors in the real world, while BC does not require such robot experiences. We posit that leveraging off-policy experiences of trial-and-errors is essential to making IRL sample-efficient enough for real robots. Here, “off-policy experiences” refer to the cumulative experiences that the robot has collected thus far during training. In contrast, “on-policy experiences” are the most recent experiences that the robot has collected using its current policy. Humans leverage lifelong, cumulative experiences to learn quickly at present. We envision robots to acquire new skills more quickly by learning from off- policy (i.e., cumulative) experiences. Finally, many real-world tasks are related and share structures and knowledge that can be exploited to solve a new but similar task later. For example, humans can quickly learn to pick and place a new object after learning to pick and place many known objects. Meta-learning, explicitly utilizing this property, aims to learn a new but related task quickly if it has already learned several similar tasks in the past. Figure 2: Fig.2: Pick-Pour-Place Robot Setup at Test Time. Given an RGB image from the top-down (black) or 45°camera (also black), the UR5-Seed robot is tasked to approach and pick-up the grey cylindrical bottle, pour the iron pebble already inside the bottle into a specific container on the table and finally place the bottle back on the table. With these motivations, we introduce SQUIRL, a meta-IRL algorithm that learns long-horizon tasks quickly and robustly by learning from 1) video demonstrations, 2) off-policy robot experiences, and 3) a set of related tasks. Fig.1 explains this algorithm using the example of a set of long- horizon pick-pour-place tasks, using the UR5-Seed111Site: www.seedrobotics.com/rh8d-dexterous-hand.html robot setup shown in Fig.2. In this task, we have the containers (green, yellow, orange, and red), a cylindrical bottle (grey), and an iron pebble inside the bottle. The robot needs to first approach and pick-up the grey bottle, pour the iron pebble inside the bottle into a specific container on the table, and then finally place the bottle back on the table, as shown in each row of images in Fig.1. At the beginning of each task, the bottle is not yet in hand, but the iron pebble is already in the bottle. At training time, the robot is given a single video demonstration for each of the 117 pick-pour-place tasks, as shown in the first two rows of images in Fig.1. Every new combination of container positions represents a different pick-pour-place task. Furthermore, the robot only needs to pour into one of the containers in a single task. Therefore, pouring into different containers also represents different tasks. After learning from these 117 demonstrations, the robot also practices 90 trial-and- errors on these tasks in total. From such a combination of expert and robot trajectories, the robot learns the general skills of pick-pour-place robustly. In all 117 training tasks, only two of the four containers appear on the table: the green and yellow containers, as shown in the first two rows of images in Fig.1. The orange and red containers are excluded during training and only appear at test time, as shown in the last row of images in Fig.1. We do so to evaluate our algorithm’s generalizability to unseen containers at test time. As shown in the last row of images in Fig.1, the robot successfully pours into a new container (red) at test time, at a new position never seen before during training, without the need for any trials or practices. To achieve such fast generalization to new tasks, our algorithm learns a task encoder network and a task-conditioned policy. The task encoder generates a 32-dimensional task embedding vector that encodes task-specific information. The policy network then learns to generalize to new tasks by accepting this task embedding vector as input, thus becoming “task-conditioned”. During training, our algorithm first bootstraps learning by training both the task encoder and the policy jointly via the BC loss. The robot then collects 10 trials across 10 tasks using the warmed-up policy and the task encoder. Next, using the combined experiences of the expert and the robot, our algorithm bypasses reward learning by directly learning a task-conditioned Q-function. Using this Q-function, our algorithm then reuses and re-evaluates all cumulative experiences of the robot to improve the policy quickly. This cycle repeats until the $90^{th}$ trial. Finally, at test time, the task encoder generates a new task embedding from a single video demonstration of a new task. This embedding is then inputted into the task-conditioned policy to solve the new task without any trial-and-errors and yet in a high-performance manner. In summary, our contributions are: 1. 1. A robust meta-IRL algorithm that outperforms ($90\%$\+ success) its behavioral cloning counterpart in real-robot and simulated vision-based long-horizon manipulation; 2. 2. A novel Q-functioned IRL formulation that circumvents reward learning and improves IRL sample efficiency; 3. 3. An efficient method that leverages off-policy robot experiences for training and requires no trials at test time; 4. 4. A general approach that tackles various long-horizon robotic manipulation tasks and works with both vision and non-vision observations and different action spaces. ## II Related Work ### II-A Inverse Reinforcement Learning (IRL) and Meta-IRL Inverse reinforcement learning (IRL) models another agent’s (typically the expert’s) reward function, given its policy or observed behavior. Previous works have approached the IRL problem with maximum margin methods [1][2] and maximum entropy methods [3][4][5]. In particular, maximum entropy methods recover a distribution of trajectories that have maximum entropy among all distributions and match the demonstrated policy’s behaviors. While these methods have shown promising results in continuous control problems, they suffer from low sample efficiency due to the need for evaluating the robot’s policy, which can be alleviated by meta-learning (i.e., meta-IRL). SMILe [6] and PEMIRL [7] are two meta-IRL algorithms based on AIRL [8] that leverage a distribution of tasks to learn a continuous task-embedding space to encode task information and achieve fast adaptation to a new but similar task. Our work differs from [6][7] in four crucial ways. First, our meta-IRL algorithm works with real robots and image observations. Second, instead of a reward function, we directly model a Q-function that the policy can optimize, in order to increase IRL sample efficiency. Third, we train the task encoder with the behavioral cloning (BC) gradient as opposed to the IRL gradient for stabler and more efficient learning. Lastly, we bootstrap policy and task encoder learning using BC before training via meta-IRL. ### II-B Real-robot Learning from Demonstrations (LfD) Our work is related to real-robot LfD [9], such as [10][11][12]. In particular, [13] developed IRL on real robots without learning from raw pixels. Other works (e.g., [14][15][16][17]) used BC for real-robot LfD. Another work [18] developed goal-conditioned BC on a simulated robot to learn long-horizon tasks by playing with objects in the scene. While enjoying efficient learning by casting imitation learning into a supervised learning problem, BC suffers from the covariate shift between the train and test data. In comparison, IRL achieves robust performance by modeling the state-action joint distribution instead of the conditional action distribution in BC [19]. Different from previous works, our meta-IRL algorithm works on real-robot vision-based tasks, and its Q-functioned IRL policy gradient can be directly combined with the BC gradient signal to approach both the sample efficiency of BC and the robustness of IRL. ### II-C One-shot Meta-imitation Learning on Real Robots Our algorithm attempts to enable robots to quickly and robustly imitate a single unseen video demonstration by learning from a distribution of tasks with shared structure, i.e., one-shot robot meta-imitation learning. For example, [20] combines gradient-based meta-learning and BC on a real robot to learn quickly from video demonstrations. [21] then extends [20] to enable robots to learn from human-arm demonstrations directly. [22] then improves [21] to meta-imitation-learn multi-stage real-robot visuomotor tasks in a hierarchical manner. However, constrained by the covariate shift problem of BC, these works show limited task performance (e.g., around $50\%$ success rate for the training tasks). In contrast, our algorithm learns a vision-based manipulation task robustly ($90\%+$ success rates) and efficiently (117 videos, 90 trials) by utilizing the generalization ability of task embeddings [23] and a novel Q-functioned IRL formulation. ## III Preliminaries ### III-A Off-policy Reinforcement Learning via Soft Actor-Critic Standard RL models a task $\mathcal{M}$ as an MDP defined by a state space $\mathcal{S}$, an initial state distribution $\rho_{0}\in\Pi(\mathcal{S})$, an action space $\mathcal{A}$, a reward function $\mathcal{R}:\mathcal{S}\times\mathcal{A}\to\mathbb{R}$, a dynamics model $\mathcal{T}:\mathcal{S}\times\mathcal{A}\to\Pi(\mathcal{S})$, a discount factor $\gamma\in[0,1)$, and a horizon $H$. Here, $\Pi(\cdot)$ defines a probability distribution over a set. The robot acts according to stochastic policy $\pi:\mathcal{S}\to\Pi(\mathcal{A})$, which specifies action probabilities for each $s$. Each policy $\pi$ has a corresponding $Q^{\pi}:\mathcal{S}\times\mathcal{A}\to\mathbb{R}$ function that defines the expected discounted cumulative reward for taking an action $a$ from $s$ and following $\pi$ onward. Off-policy RL, particularly Soft Actor-Critic (SAC) [24], reuses historical experiences to improve learning sample efficiency by recovering a “soft” Q-function estimator $Q_{\theta}$. A policy can then be learned by minimizing the KL divergence between the policy distribution and the exponential-Q distribution: $\pi^{*}=\operatorname*{arg\,min}_{\pi\in\Pi}D_{KL}(\pi(a\mid s)\;\|\;\frac{\exp(Q^{\pi_{old}}_{\theta}(s,a))}{Z(s)})$ ### III-B Timestep-centric IRL as Adversarial Imitation Learning The purpose of IRL is to learn the energy function $f_{\theta}$ implicit in the provided expert demonstrations and use $f_{\theta}$ to learn a policy that robustly matches the expert performance. In particular, timestep-centric IRL aims to recover an energy function $f_{\theta}(s,a)$ to rationalize and match the demonstrated expert’s action conditional distribution: $p_{\theta}(a\mid s)=\frac{\exp(f_{\theta}(s,a))}{Z_{\theta}}\propto\exp(f_{\theta}(s,a))=\overline{p_{\theta}}(a\mid s)$, where $Z_{\theta}$ is the partition function, an integral over all possible actions given state $s$. In other words, IRL minimizes the KL divergence between the actual and predicted expert conditional action distributions: $\pi_{E}(a\mid s)$ and $p_{\theta}(a\mid s)$. Adversarial IRL [8][25] provides a sampling-based approximation to MatEntIRL [4] in an adversarial manner. Specifically, AIRL [8] learns a generative policy $\pi_{\psi}$ and a binary discriminator $D_{\theta}$ derived from energy function $f_{\theta}$: $\displaystyle D_{\theta}(s,a)=P((s,a)\text{ is generated by expert})$ $\displaystyle=\frac{\overline{p_{\theta}}(a\mid s)}{\overline{p_{\theta}}(a\mid s)+\pi_{\psi}(a\mid s)}=\frac{\exp(f_{\theta}(s,a))}{\exp(f_{\theta}(s,a))+\pi_{\psi}(a\mid s)}$ (1) and $\theta$ is trained to distinguish state-action pairs sampled from the expert vs. the policy, using binary cross entropy loss: $\displaystyle\mathcal{L}^{IRL}=-\mathbb{E}$ ${}_{(s,a)\sim\pi_{\psi},\pi_{E}}[y(s,a)\log(D_{\theta}(s,a))$ $\displaystyle+(1-y(s,a))\log(1-D_{\theta}(s,a))]$ (2) where $y(s,a)=\mathds{1}\\{(s,a)\text{ is generated by expert }\pi_{E}\\}$. Meanwhile, the policy is trained to maximize the MaxEntIRL Objective [4], or equivalently, to match the expert’s state-action joint distribution via reverse KL divergence [19]. ### III-C One-shot Meta-imitation Learning from A Single Video In one-shot meta-imitation learning, the robot is trained to solve a large number of tasks drawn from a task distribution $p(\mathcal{M})$. The total number of tasks in this task distribution can be finite or infinite. Each imitation task $\mathcal{M}_{train}^{i}$ consists of a single video demonstration $\mathcal{D}^{i}_{\pi_{E}}$. During training, the robot can also generate limited practice trajectories (e.g., 90). For example, in the Pick- Pour-Place experiment in Fig.1, the robot receives a single video demonstration for each of the 117 tasks. Each task is characterized by a different combination of container positions, or pouring into the green vs. the yellow container. At test time, the robot receives a single video of a new task $\mathcal{M}_{test}^{i}$ drawn from $p(\mathcal{M})$. For example, a new Pick-Pour-Place test task can be a new combination of container positions or pouring into a new container (e.g., the red or orange container). The robot then needs to solve this task the first time without trial-and-error. ### III-D Embedding-based Meta-learning Embedding-based meta-learning [7][23] learns a task-specific embedding vector $z$ that contains task-level abstraction to adapt to a new but related task quickly. This method aims to learn a task-conditioned policy $\pi(a|s,z)$ that maximizes task-conditioned expected returns: $\max_{\pi}\mathbb{E}_{(s_{t},a_{t})\sim\pi,\rho_{0}}[\sum_{t=1}^{T}r(s_{t},a_{t}|c)+\alpha\mathcal{H}(\pi(a_{t}|s_{t},c))]$, by learning an embedding space $Z$ that maximizes the mutual information between $z$ and task context $c$. The goal is to make this learned embedding space generalizable to new tasks so that at test time, the policy can quickly adapt to unseen tasks with no or few practices. A key advantage of embedding- based meta-learning is the ability to learn from off-policy experiences. However, current methods are mostly if not only demonstrated in non-vision tasks in simulation. ## IV Mathematical Formulation for SQUIRL ### IV-A SQUIRL: Timestep-centric IRL as Soft Q-Learning Previous works in timestep-centric IRL such as [6][7][8] have interpreted the energy function $f_{\theta}$ in Eq.III-B as a reward function $r_{\theta}$ and later recover a Q or advantage function based on reward $r_{\theta}$ for policy improvement. To improve IRL sample efficiency, we propose to bypass this reward learning and directly interpret $f_{\theta}(s,a)$ as the soft Q-function [24] $Q^{\pi_{mix}}_{\theta}(s,a)$. This soft Q-function models the expert’s behavior as maximizing both the Q-value and its entropy (i.e., randomness) simultaneously. It also encourages the robot to explore the real world to imitate the expert more robustly. Under this formulation, approximating the expert’s conditional action distribution is equivalent to recovering a soft Q-function under which the expert is soft Q-optimal: $\displaystyle\operatorname*{arg\,min}_{\theta}D_{KL}(\pi_{E}(a\mid s)\;\|\;p_{\theta}(a\mid s))$ $\displaystyle=$ $\displaystyle\operatorname*{arg\,max}_{\theta}\mathbb{E}_{a\sim\pi_{E}(a\mid s)}[Q^{\pi_{mix}}_{\theta}(s,a)]-\log Z_{\theta}$ (3) Eq.3 rationalizes the expert behavior intuitively because the expert should be optimal with respect to the cumulative reward [3], not the immediate reward. Here, $Q^{\pi_{mix}}_{\theta}$ is under a mixture policy $\pi_{mix}$ between the robot and expert’s policies. ### IV-B SQUIRL as Expert Imitation and Adversarial Learning Under SQUIRL, the policy learning objective (Eq.4) is also equivalent (derivations on website) to matching: 1) the exponential-Q distribution of the discriminator $\theta$ (Eq.5), 2) the generator’s objective in Generative Adversarial Networks (GANs) [26] (Eq.6), and 3) the joint state-action distribution of expert [19] (Eq.7): $\pi^{*}=\operatorname*{arg\,min}_{\pi\in\Pi}\mathcal{L}^{RL}(\pi)$, where $\displaystyle\mathcal{L}^{RL}(\pi)=D_{KL}(\pi_{\psi}(a\mid s)\;\|\;\frac{\exp{Q^{\pi_{mix}}_{\theta}(s,a)}}{Z(s)})$ (4) $\displaystyle=D_{KL}(\pi_{\psi}(a\mid s)\;\|\;p_{\theta}(a\mid s))$ (5) $\displaystyle=\mathbb{E}_{(s,a)\sim\pi_{mix}}[\log(1-D_{\theta}(s,a))-\log(D_{\theta}(s,a))]$ (6) $\displaystyle=D_{KL}(\rho_{\pi_{\psi}}(s,a)\;\|\;\rho_{\pi_{E}}(s,a))$ (7) Meanwhile, the discriminator $\theta$ is matching its Q-function to the log- distribution of the expert’s conditional action distribution (Section III-B). Therefore, when this Q-function is optimal: $Q^{\pi_{mix}}_{\theta}=Q^{\pi_{mix}}_{\theta^{*}}$, the robot’s policy objective (Eq.4) is also matching the expert’s conditional action distribution: $\psi^{*}=\operatorname*{arg\,min}_{\psi}E_{\rho_{\pi_{mix}}(s)}[D_{KL}(\pi_{\psi}(a\mid s)\;\|\;\pi_{E}(a\mid s))]$ (8) ### IV-C Comparison to the Behavioral Cloning (BC) Objective While BC attempts to learn a policy that also matches the expert’s conditional action distribution [19], the fundamental difference is that the KL-divergence in BC’s case is computed under the expert’s narrow state distribution $\rho_{\pi_{E}}(s)$: $\psi_{BC}^{*}=\operatorname*{arg\,min}_{\psi}E_{\rho_{\pi_{E}}(s)}[D_{KL}(\pi_{E}(a\mid s)\;\|\;\pi_{\psi}(a\mid s))]$. In contrast, ours (Eq.8) is computed under $\rho_{\pi_{mix}}(s)$: the state distribution of the combined cumulative experience of the robot and the expert, which is a much wider distribution than the expert distribution. We hypothesize that this, along with matching the joint state-action distribution of the expert (Eq.7), makes our algorithm less susceptible to compounding errors than BC, as experimentally tested in Section VI. Figure 3: SQUIRL: Soft Q-functioned Meta-IRL. To begin, our algorithm bootstraps learning for the policy (orange) and the task encoder (yellow) via behavioral cloning (the left third of Fig.3). Next, our algorithm uses the warmed-up policy and task encoder to generate 10 trials in the physical world (not in simulation). Using the combined expert and robot trajectories, our algorithm learns a task-conditioned soft Q-function (green) that rationalizes the expert’s behaviors as maximizing both cumulative reward and entropy (i.e., randomness). Using this Q-function, our algorithm then quickly improves the policy using all cumulative robot and expert timesteps. This cycle repeats until convergence, totaling 90 trials (the middle third of Fig.3). Finally, at test time (the right third Fig.3), our algorithm generates a new embedding $z$ for the new task, and inputs this embedding into the task-conditioned policy to solve the new task without any practices. ## V SQUIRL: Soft Q-functioned Meta-IRL Shown in Fig.3, our algorithm learns three neural networks jointly – a task encoder (yellow), a task-conditioned policy (orange), and a task-conditioned soft Q-function (green): 1. 1. $\Psi_{\phi}(c)$: a task encoder that encodes a sampled batch of $C=64$ expert state-action pairs $c=\\{s^{i}_{1:C},a^{i}_{1:C}\\}$ from a task $i$ into a single 32-dim embedding vector $z^{i}\in\mathbb{R}^{32}$ (by computing the mean vector across 64 embeddings) that enables generalization to new tasks. This batch of expert state-action pairs is randomly sampled and thus does not encode time information. Both the policy and the Q-function accept this embedding vector as input. 2. 2. $\pi_{\psi}(s,z^{i})$: a task-conditioned policy the robot uses to perform a task $i$ given state $s$ and the task embedding vector $z^{i}\in\mathbb{R}^{32}$ outputted by the task encoder $\Psi_{\phi}(c)$. 3. 3. $Q_{\theta}(s,a,z^{i})$: a task-conditioned soft Q-function used to train the policy $\pi_{\psi}(s,z^{i})$ to more robustly mimic the expert’s behavior for the robotic manipulation task $i$. To begin, the robot is given an expert trajectory of state-action pairs $\mathcal{D}_{\pi_{E}}$ for each of the 117 training tasks. The robot first uses these expert trajectories to bootstrap training for both its policy $\pi_{\psi}$, and the task encoder $\Psi_{\phi}$ via behavioral cloning (Eq.9). This way, the robot can distinguish the train tasks better and learn more quickly in the real world. Next, the robot generates 10 trials (state- action pairs) $\overline{\mathcal{D}}_{\pi_{\psi}}$ in the physical world (not simulation) using its warmed-up policy and task encoder. Then, the robot uses both the expert’s and its state-action pairs to train a discriminator $\theta$. This discriminator classifies which state-action pairs come from the expert $\pi_{E}$ vs. the robot $\pi_{\psi}$. At first, the robot is distinctively worse than the expert at performing the tasks. This makes it easy for the discriminator to classify. By doing so, the discriminator learns a Q-function $Q^{\pi_{mix}}_{\theta}$ using Eq.3. Using the learned Q-function $Q^{\pi_{mix}}_{\theta}$, the robot trains its policy $\pi_{\psi}$ via Eq.4. Meanwhile, the robot also has the option to continue updating its task-conditioned policy and task encoder via behavioral cloning (Eq.9). Since training the policy via Eq.4 is equivalent to indirectly imitating the expert (Eq.7 and 8), as derived in Section IV-B, the trajectories generated by the policy gradually become more similar to the expert. This makes the state-action pairs more difficult for the discriminator to classify. This difficulty, in turn, forces the discriminator to learn a more precise Q-function, which then encourages the policy to mimic the expert even more closely. This cycle repeats until convergence (90 trials in total), at which point: 1) the policy matches the expert performance, 2) the task encoder learns to generalize to new tasks, and 3) the discriminator continues to struggle to distinguish state-action pairs correctly despite having learned an accurate Q-function. ### V-A Rationale for Bypassing Reward Learning via SQUIRL SQUIRL learns a Q-function without rewards because 1) the policy is ultimately trained by the Q-function, not rewards, thus bypassing reward learning improves IRL sample efficiency, and 2) circumventing reward learning avoids off-policy Q-learning from a constantly changing reward function and makes training easier and more stable empirically. ### V-B Architectures for Policy, Task Encoder, and Q-function For all non-vision tasks, we parameterize $\pi_{\psi},\Psi_{\phi},Q_{\theta}$ with five fully-connected (FC) layers. For vision tasks, we use a 5-layer CNN followed by a spatial-softmax activation layer for the RGB image. This activation vector is then concatenated with the non-vision input vector and together passed through five FC layers. Our algorithm is general and works with many other network architectures, state, and action spaces. ### V-C Incorporating BC to Bootstrap and Accelerate Learning Since our algorithm’s IRL objective (Eq.8) is compatible with BC, as explained in Section IV-C, our algorithm can jointly be trained with BC to stabilize and accelerate learning without conflicting gradient issues (line 16 in Algorithm 1): $\mathcal{L}^{BC}=\mathbb{E}_{(s,a)\sim\pi_{E}}[\left\lVert\pi_{\psi}(s,\Psi_{\phi}(c))-a\right\rVert^{2}]$ (9) This, combined with the off-policy nature of our algorithm, also allows the robot to bootstrap learning by first “pre-training” via BC (Eq.9) using the expert demonstrations, before improving performance further via meta-IRL training. Algorithm 1 SQUIRL: Soft Q-functioned Meta-IRL (Train) Input: One expert video demonstration trajectory of state-action pairs $\mathcal{D}^{i}_{\pi_{E}}=\\{s^{i}_{1:H},a^{i}_{1:H}\\}$ for each of the $n$ training tasks $i=1:n$, where $H$ is the horizon of the task (e.g., $n=117,H=100$) 1: Initialize soft Q-function $Q_{\theta}$, policy $\pi_{\psi}$, task encoder $\Psi_{\phi}$, and an empty buffer of off-policy robot trajectories $\mathcal{D}^{i}_{\pi_{\psi}}\leftarrow\\{\\}$ for each training task $i=1:n$ 2: Warm-up policy and task encoder via $\mathcal{L}^{BC}$ (Eq.9) 3: while not converged do 4: Sample a batch of $m$ task indices $\\{i^{1:m}\\}$ from all training tasks $i=1:n$, (e.g., $m=10$) 5: for $i=i^{1:m}$ do 6: Infer task embedding $z^{i}=\mathbb{R}^{\mathcal{Z}}\leftarrow\Psi_{\phi}(c)$, where $c=\\{s^{i}_{1:C},a^{i}_{1:C}\\}\sim\mathcal{D}^{i}_{\pi_{E}}$ (e.g., $\mathcal{Z}=32,C=64$) 7: Generate a robot trajectory of state-action pairs $\overline{\mathcal{D}}^{i}_{\pi_{\psi}}=\\{s^{i}_{1:H},a^{i}_{1:H}\\}$ from task $i$ using $\pi_{\psi},z^{i}$ 8: $\mathcal{D}^{i}_{\pi_{\psi}}\leftarrow\mathcal{D}^{i}_{\pi_{\psi}}\cup\overline{\mathcal{D}}^{i}_{\pi_{\psi}}$ 9: end for 10: for $j=1:J$ (e.g., $J=400$) do 11: Sample another batch of $m$ task indices $\\{i^{1:m}\\}$ 12: $\theta\leftarrow\theta-\nabla_{\theta}\mathcal{L}^{IRL}$ (Eq.III-B) using a combined batch of $\mathcal{B}=128$ robot and expert timesteps: $\overline{\mathcal{D}}^{i}_{\pi_{\psi}}\cup\overline{\mathcal{D}}^{i}_{\pi_{E}}$ and $z^{i}$, where $\overline{\mathcal{D}}^{i}_{\pi_{\psi}}\sim\mathcal{D}^{i}_{\pi_{\psi}}$, $\overline{\mathcal{D}}^{i}_{\pi_{E}}\sim\mathcal{D}^{i}_{\pi_{E}}$, $i=\\{i^{1:m}\\}$ 13: end for 14: for $k=1:K$ (e.g., $K=2000$) do 15: Sample another batch of $m$ task indices $\\{i^{1:m}\\}$ 16: if necessary then $\\{\psi,\phi\\}\leftarrow\\{\psi,\phi\\}-\nabla_{\psi,\phi}\mathcal{L}^{BC}$ (Eq.9) using a batch of $\mathcal{B}$ expert timesteps $\overline{\mathcal{D}}^{i}_{\pi_{E}}\sim\mathcal{D}^{i}_{\pi_{E}},z^{i}$, $i=\\{i^{1:m}\\}$ end if 17: $\psi\leftarrow\psi-\nabla_{\psi}\mathcal{L}^{RL}$ (Eq.4) using a combined batch of $\mathcal{B}$ robot and expert timesteps: $\overline{\mathcal{D}}^{i}_{\pi_{\psi}}\cup\overline{\mathcal{D}}^{i}_{\pi_{E}}$ and $z^{i}$, where $\overline{\mathcal{D}}^{i}_{\pi_{\psi}}\sim\mathcal{D}^{i}_{\pi_{\psi}}$, $\overline{\mathcal{D}}^{i}_{\pi_{E}}\sim\mathcal{D}^{i}_{\pi_{E}}$, $i=\\{i^{1:m}\\}$ 18: end for 19: end while 20: return soft Q-function $Q_{\theta}$, policy $\pi_{\psi}$, task encoder $\Psi_{\phi}$ Algorithm 2 SQUIRL: Soft Q-functioned Meta-IRL (Test) Input: $\pi_{\psi}$, $\Psi_{\phi}$, $Q_{\theta}$, and a single expert video demonstration of state-action pairs $\mathcal{D}^{i}_{\pi_{E}}=\\{s^{i}_{1:H}$, $a^{i}_{1:H}\\}$ from a new task $i$ unseen during training 1: Infer task embedding vector $z^{i}=\mathbb{R}^{\mathcal{Z}}\leftarrow\Psi_{\phi}(c)$, where $c=\\{s^{i}_{1:C},a^{i}_{1:C}\\}\sim\mathcal{D}^{i}_{\pi_{E}}$ (e.g., $\mathcal{Z}=32,C=64$) 2: Rollout robot trajectory in the real world using $\pi_{\psi}$, $z^{i}$ Approach Box Lower to Box Grasp Box Pick up Box Carry Box Drop Box Figure 4: Pick-Carry-Drop Experiment. The robot needs to approach, lower to, grasp, pick-up, carry, and drop the box to solve the task. ### V-D Using Expert Demonstration as Both the Input Task Context Variables and Training Signal for the Task Encoder Learning robust task embeddings enables robots to generalize to new tasks quickly [23]. To this end, our algorithm proposes to use 64 expert timesteps as the input task context variable $c$ into the task encoder, as opposed to 64 robot timesteps. This is because context variables should explore the task and environment sufficiently well to expose the key information of the task, and expert demonstration timesteps are an ideal candidate compared to the timesteps from the robot’s suboptimal policy. As a result, the context variable $c$ input into the task encoder only includes the states and actions of the expert, but not the rewards or the next states. In addition, we choose the BC loss $\mathcal{L}^{BC}$ in Eq.9 as the training loss for learning the task encoder $\Psi_{\phi}$. This BC loss is stable since the expert timesteps are fixed. In contrast, the IRL loss $\mathcal{L}^{IRL}$ (Eq.III-B) and the policy loss $\mathcal{L}^{RL}$ (Eq.4) are less stable because the training data distribution for both losses are non-stationary. This design choice also allows us to learn a robust task embeddings first via BC pre-training before performing meta-IRL training via SQUIRL. We empirically observe that such pre-training can improve the training stability and the sample efficiency of SQUIRL, but the final policy performance is similar with or without BC pre-training. In summary, our algorithm is detailed in Algorithm 1 (train) and Algorithm 2 (test), with hyperparameters detailed here222Hyperparameters in Algorithm 1 and 2. Policy gradient batch size $\mathcal{B}$: 1024 (non-vision), 128 (vision); task embedding batch size $C$: 64; all learning rates: $3e^{-4}$; starting SAC alpha: $1e^{-5}$; SAC target entropy: $-300$; IRL updates per epoch $J$: $400$; policy updates per epoch $K$: $2000$; task embedding size $\mathcal{Z}$: 32; meta-batch size $m$: 10; discount rate $\gamma$: 0.99. ## VI Experiments and Results Analysis We evaluate the generality and robustness of our algorithm across long-horizon vision and non-vision tasks with continuous state and action spaces in both simulation (Pick-Carry-Drop, a horizon of 1024 timesteps, 30 train tasks) and real-world (Pick-Pour-Place, a horizon of 100 timesteps, 117 train tasks). There is only a single expert demonstration for each of the train or test tasks. We compare with the PEARL-BC baseline, which is the behavioral cloning version of PEARL [23]. Evaluation: We evaluate real-robot and simulation experiments on 50 and 500 trials respectively across 50 seen and unseen tasks. We report mean and standard deviation (“stdev” hereafter). The performance difference between different experiments is statistically significant if the difference in mean is at least either standard deviation away. Experimental video is at http://crlab.cs.columbia.edu/squirl. Approach Bottle Grasp Bottle Carry Bottle Pour Orange Cup Carry Bottle Place Bottle Figure 5: Pick-Pour-Place at Test Time. To solve this task, the robot needs to first approach, grasp and carry the grey bottle, pour the iron pebble inside the bottle into a specific container, and carry and place the bottle back on the table. At the beginning of each task, the bottle is not in hand, and but the iron pebble is already in the bottle. Top row: top-down camera images. Bottom row: 45°camera images. ### VI-A Simulation Experiment: Pick-Carry-Drop Description. We modify the planar Stacker task [27] to create “Pick-Carry- Drop”. Shown in Fig.4, a robot is tasked to approach, pick, carry, and drop the black box into the stack marked in green. The task is successful if the box is dropped into the stack within 1024 timesteps, and failed otherwise. State Space. We evaluate our algorithm on both the vision and the non-vision version of the task, to demonstrate that SQUIRL is general across different state space modalities. The state space for the vision version includes 1) the joint angles and velocities for its 5-DOFs, 2) a one-hot vector indicating the current stage of the task, and 3) an RGB image shown in Fig.4. The non-vision version’s state space replaces the RGB image with the position of the black box. Action Space. The robot controls its 5-DOF joint torques. Task Definition. There are a total of 30 training tasks in this experiment, each corresponding to a different drop location: $x\in\\{-0.15,-0.14,\ldots,0.14\\}$. During test time, we randomly sample a new, real-valued drop location from the maximum valid range: $x\in[-0.25,0.25]$. The green drop location is invisible in both the vision and the non-vision version of the task. Therefore, the robot needs to infer the green drop location (i.e., task information) solely from the provided expert video demonstration. On the other hand, the starting pose of the robot and the location of the black box are all initialized randomly at the beginning of each task. Robot Trials. The robot uses 150 training trials in total. Expert Demonstration. We trained an expert policy from scratch via RL to provide expert demonstrations. The reward function used to train the expert policy comprises of six stages, each stage with a reward of 10. Designing this reward function has taken significant human effort, which exhibits the value of directly learning from video demonstrations. TABLE I: Pick-Carry-Drop Results (% Drop Success$\pm$Stdev) Tasks | Seen | Unseen | Seen | Unseen ---|---|---|---|--- | Vision | Non-Vision SQUIRL (BC + IRL) | 95.8$\pm$1.7 | 95.0$\pm$1.5 | 97.3$\pm$3.0 | 96.9$\pm$2.0 Baseline (PEARL-BC) | 77.8$\pm$1.6 | 76.5$\pm$0.7 | 90.8$\pm$2.5 | 89.5$\pm$1.6 Ablation: No BC Joint Training or BC Pre-training SQUIRL (IRL Only) | 93.8$\pm$1.8 | 93.2$\pm$1.6 | 94.7$\pm$1.7 | 93.9$\pm$1.4 Simulation Results and Analysis. As shown in Table I, our algorithm, “SQUIRL (BC + IRL)”, pre-trains via BC and then trains the policy using both the BC loss (Eq.9) and the IRL policy gradient loss (Eq.4). It statistically significantly outperforms the PEARL-BC baseline in both the vision (95.8%$\pm$1.7 vs. 77.8%$\pm$1.6) and non-vision (97.3%$\pm$3.0 vs. 90.8%$\pm$2.5) version of the task for seen tasks. For unseen tasks, we observed similar outperformance (95.0%$\pm$1.5 vs. 76.5%$\pm$0.7 in the vision case and 96.9%$\pm$2.0 vs. 89.5%$\pm$1.6 in the non-vision case). Qualitatively, in the PEARL-BC’s case, the robot sometimes misses the drop location as it attempts to drop the box or fails to pick up the box when the box gets stuck by the walls of the stack (kindly see website). The performance drop of the baseline from the non-vision version (90.8%$\pm$2.5 and 89.5%$\pm$1.6 for seen and unseen tasks) to the vision version (77.8%$\pm$1.6 and 76.5%$\pm$0.7 for seen and unseen tasks) is mainly because vision-based manipulation tends to suffer from larger compounding errors. Nevertheless, as evident in the statistical similarities between seen and unseen tasks for SQUIRL (95.8%$\pm$1.7 vs. 95.0%$\pm$1.5 for vision) and PEARL-BC (77.8%$\pm$1.6 vs. 76.5%$\pm$0.7 for vision), both algorithms can generalize to unseen tasks, due to the generalizability of task embeddings. Ablation: IRL Gradient Only. To compare the performance contribution of SQUIRL’s meta-IRL core training procedure directly against PEARL-BC, we created “SQUIRL (IRL only)”, which trains the policy using only the policy gradient loss in Eq.4 (no BC joint training or pre-training). This ablated version still outperforms the PEARL-BC baseline (93.8%$\pm$1.8 vs. 77.8%$\pm$1.6 for seen vision tasks, 93.2%$\pm$1.6 vs. 76.5%$\pm$0.7 for unseen vision tasks). Nevertheless, by combining BC and IRL gradients, “SQUIRL (BC + IRL)” improves performance slightly further (95.8%$\pm$1.7 and 95.0%$\pm$1.5). Intuitively, while BC only matches the expert’s conditional action distribution under the expert’s state distribution, BC’s supervised learning signal is stabler than IRL. Joint training with BC and IRL gradients can be interpreted as combining the stability of BC and the robustness of Q-functioned IRL, by matching the conditional action distribution of the expert under the broader state distribution of the expert-robot mixture experience (Eq.8), in addition to matching the expert’s joint state-action distribution (Eq.7). ### VI-B Real-Robot Experiment: Pick-Pour-Place Description. We evaluated our algorithm on the UR5-Seed robot (Fig.2) to perform a set of long-horizon pick-pour-place tasks. As shown in Fig.2, in each task, there is a grey cylindrical bottle, an iron pebble that is already in the bottle, and more than one container on the table. The robot is tasked to approach and pick-up the grey bottle, pour the iron pebble into a specific container, and place the bottle back on the table. The task is a success only if the pebble is poured into the correct container and the bottle is placed upright on the table within $H=100$ timesteps, and a failure otherwise. State Space. The state space contains a top-down or 45°camera’s RGB image (Fig.5), and 2 binary indicators for whether the robot has poured or closed the hand, respectively. Action Space. The action space includes the Cartesian unit directional vector for the end-effector movement. During each timestep, the robot can adjust the end-effector by 2cm along any 3D direction. The action space also includes a binary indicator to control the arm vs. the hand and a trinary indicator to close, open, or rotate the hand for pouring. Orthogonality to State and Action Representions. While Pick-Pour-Place can be tackled by first localizing the correct container via object detection (alternative state space) and then executing motion-planning trajectories to pour (alternative action space), our algorithm is general across and orthogonal to alternative state and action spaces. Task Definition. As shown in each row of images in Fig.1, each task is defined by the positions and colors of the containers, and by the correct container to pour into. There are always only the green and yellow containers in the 117 train tasks. 25 of the 50 test tasks have the green and yellow containers at new positions. The remaining 25 test tasks add the red and the orange unseen containers, or either. Since there is always more than one container in the RGB image, the robot will not know which container to pour into without the expert demonstration. Therefore, the robot needs to depend solely on the task encoder’s ability to extract the correct task information from the expert demonstration. Robot Trials. The robot collects 90 training trials in total. Expert Demonstration. We collect demonstrations via teleoperation using a Flock of Birds sensor333Flock of Birds is a 6D pose tracker from Ascension Technologies Corp.. Using the human wrist pose detected by the sensor in real- time, we move, open, close, or rotate the robot hand for pouring. We collected $117$ video demonstrations across 117 tasks for training. It takes 1-2 minutes to collect one demonstration. TABLE II: Pick-Pour-Place Results (% Pour Success$\pm$Stdev) Tasks | RGB Image | Seen | Unseen ---|---|---|--- SQUIRL (BC + IRL) | Top-Down (90°) | 92.0$\pm$4.5 | 90.0$\pm$7.1 Baseline (PEARL-BC) | 70.0$\pm$7.1 | 68.0$\pm$11.0 Baseline (Standard-BC) | 60.0$\pm$10.0 | 56.0$\pm$11.4 SQUIRL (BC + IRL) | $45\degree$ (Ablation) | 90.0$\pm$7.1 | 88.0$\pm$8.4 Real-robot Results and Analysis. As shown in Table II, our algorithm outperforms the PEARL-BC baseline statistically significantly in both seen tasks (92.0%$\pm$4.5 vs. 70.0%$\pm$7.1) and unseen tasks (90.0%$\pm$7.1 vs. 68.0%$\pm$11.0). This observed outperformance mainly originates from our soft Q-functioned IRL formulation, which forces the robot to imitate the expert under a much wider state distribution provided by the expert-robot mixture trajectories, instead of the narrow state distribution of the expert demonstrations. This helps reduce compounding errors during task execution. The low performance of the PEARL-BC baseline is mainly due to additional compounding errors induced by real-world sensory noises such as unstable lighting conditions and small perturbation to camera positions. Qualitatively, the PEARL-BC baseline sometimes pours into the wrong container, misses the target container by a few centimeters, or moves past the target container while failing to pour in time (kindly see website for examples). Nevertheless, from the statistical similarity between seen and unseen tasks for both our algorithm (92.0%$\pm$4.5 vs. 90.0%$\pm$7.1) and PEARL-BC (70.0%$\pm$7.1 vs. 68.0%$\pm$11.0), we see that the learned task encoder is still effectively generalizing to a new, related task. Comparison to the “Standard-BC” Baseline. We also compared to “Standard-BC” (60.0%$\pm$10.0 and 56.0%$\pm$11.4 for seen and unseen tasks), which performs no meta-learning and learns every train or test task independently from scratch via BC. As a result, the neural network overfits to the single demonstration and fails to generalize to real-world sensory (camera) noises at test time. Note that Standard-BC’s unseen-task performance is slightly lower than seen tasks since the unseen tasks are more challenging with at most 4 containers on the table, compared to only 2 containers in seen tasks. Ablation: Non-top-down Camera. We also tested our algorithm with a $45\degree$ RGB image (90.0%$\pm$7.1 and 88.0%$\pm$8.4 for seen and unseen tasks) against a top-down RGB image (92.0%$\pm$4.5 and 90.0%$\pm$7.1 for seen and unseen tasks). The statistical similarity between the two shows that SQUIRL is general and can accept a non-top-down RGB input image. ## VII Conclusion We introduced SQUIRL, a robust, efficient, and general Soft Q-functioned meta- IRL algorithm, towards enabling robots to learn from limited expert (one per task) and robot (90 in total) trajectories. 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2024-09-04T02:54:58.885703

2003.04960

arxiv-papers

# Curriculum Learning for Reinforcement Learning Domains: A Framework and Survey Sanmit Narvekar<EMAIL_ADDRESS> Department of Computer Science University of Texas at Austin Bei Peng<EMAIL_ADDRESS> Department of Computer Science University of Oxford Matteo Leonetti<EMAIL_ADDRESS> School of Computing University of Leeds Jivko Sinapov<EMAIL_ADDRESS> Department of Computer Science Tufts University Matthew E. Taylor<EMAIL_ADDRESS> Alberta Machine Intelligence Institute Department of Computing Science University of Alberta Peter Stone<EMAIL_ADDRESS> Department of Computer Science University of Texas at Austin and Sony AI ###### Abstract Reinforcement learning (RL) is a popular paradigm for addressing sequential decision tasks in which the agent has only limited environmental feedback. Despite many advances over the past three decades, learning in many domains still requires a large amount of interaction with the environment, which can be prohibitively expensive in realistic scenarios. To address this problem, transfer learning has been applied to reinforcement learning such that experience gained in one task can be leveraged when starting to learn the next, harder task. More recently, several lines of research have explored how tasks, or data samples themselves, can be sequenced into a _curriculum_ for the purpose of learning a problem that may otherwise be too difficult to learn from scratch. In this article, we present a framework for curriculum learning (CL) in reinforcement learning, and use it to survey and classify existing CL methods in terms of their assumptions, capabilities, and goals. Finally, we use our framework to find open problems and suggest directions for future RL curriculum learning research. Keywords: curriculum learning, reinforcement learning, transfer learning ## 1 Introduction Curricula are ubiquitous throughout early human development, formal education, and life-long learning all the way to adulthood. Whether learning to play a sport, or learning to become an expert in mathematics, the training process is organized and structured so as to present new concepts and tasks in a sequence that leverages what has previously been learned. In a variety of human learning domains, the quality of the curricula has been shown to be crucial in achieving success. Curricula are also present in animal training, where it is commonly referred to as shaping (Skinner, 1958; Peterson, 2004). As a motivating example, consider the game of Quick Chess (shown in Figure 1), a game designed to introduce children to the full game of chess, by using a sequence of progressively more difficult “subgames.” For example, the first subgame is played on a 5x5 board with only pawns, where the player learns how pawns move, get promoted, and take other pieces. Next, in the second subgame, the king piece is added, which introduces a new objective: keeping the king alive. In each successive subgame, new elements are introduced (such as new pieces, a larger board, or different configurations) that require learning new skills and building upon knowledge learned in previous games. The final game is the full game of chess. The idea of using such curricula to train artificial agents dates back to the early 1990s, where the first known applications were to grammar learning (Elman, 1993; Rohde and Plaut, 1999), robotics control problems (Sanger, 1994), and classification problems (Bengio et al., 2009). Results showed that the order of training examples matters and that generally, incremental learning algorithms can benefit when training examples are ordered in increasing difficulty. The main conclusion from these and subsequent works in curriculum learning is that starting small and simple and gradually increasing the difficulty of the task can lead to faster convergence as well as increased performance on a task. Recently, research in reinforcement learning (RL) (Sutton and Barto, 1998) has been exploring how agents can leverage transfer learning (Lazaric et al., 2008; Taylor and Stone, 2009) to re-use knowledge learned from a source task when attempting to learn a subsequent target task. As knowledge is transferred from one task to the next, the sequence of tasks induces a curriculum, which has been shown to improve performance on a difficult problem and/or reduce the time it takes to converge to an optimal policy. Figure 1: Different subgames in the game of Quick Chess, which are used to form a curriculum for learning the full game of Chess. Many groups have been studying how such a curriculum can be generated automatically to train reinforcement learning agents, and many approaches to do so now exist. However, what exactly constitutes a curriculum and what precisely qualifies an approach as being an example of curriculum learning is not clearly and consistently defined in the literature. There are many ways of defining a curriculum: for example, the most common way is as an ordering of tasks. At a more fundamental level, a curriculum can also be defined as an ordering of individual experience samples. In addition, a curriculum does not necessarily have to be a simple linear sequence. One task can build upon knowledge gained from multiple source tasks, just as courses in human education can build off of multiple prerequisites. Methods for curriculum generation have separately been introduced for areas such as robotics, multi-agent systems, human-computer and human-robot interaction, and intrinsically motivated learning. This body of work, however, is largely disconnected. In addition, many landmark results in reinforcement learning, from TD-Gammon (Tesauro, 1995) to AlphaGo (Silver et al., 2016) have implicitly used curricula to guide training. In some domains, researchers have successfully used methodologies that align with our definition of curriculum learning without explicitly describing it that way (e.g., self-play). Given the many landmark results that have utilized ideas from curriculum learning, we think it is very likely that future landmark results will also rely on curricula, perhaps more so than researchers currently expect. Thus, having a common basis for discussion of ideas in this area is likely to be useful for future AI challenges. ### Overview The goal of this article is to provide a systematic overview of curriculum learning (CL) in RL settings and to provide an over-arching framework to formalize this class of methods. We aim to define classification criteria for computational models of curriculum learning for RL agents, that describe the curriculum learning research landscape over a broad range of frameworks and settings. The questions we address in this survey include: * • What is a _curriculum_ , and how can it be represented for reinforcement learning tasks? At the most basic level, a curriculum can be thought of as an ordering over experience samples. However, it can also be represented at the task level, where a set of tasks can be organized into a sequence or a directed acyclic graph that specifies the order in which they should be learned. We address this question in detail in Section 3.1. * • What is the _curriculum learning_ method, and how can such methods be evaluated? We formalize this class of methods in Section 3.2 as consisting of three parts, and extend metrics commonly used in transfer learning (introduced in Section 2) to the curriculum setting to facilitate evaluation in Section 3.3. * • How can tasks be constructed for use in a curriculum? The quality of a curriculum is dependent on the quality of tasks available to select from. Tasks can either be generated in advance, or dynamically and on-the-fly with the curriculum. Section 4.1 surveys works that examine how to automatically generate good intermediate tasks. * • How can tasks or experience samples be sequenced into a curriculum? In practice, most curricula for RL agents have been manually generated for each problem. However, in recent years, automated methods for generating curricula have been proposed. Each makes different assumptions about the tasks and transfer methodology used. In Section 4.2, we survey these different automated approaches, as well as describe how humans have approached curriculum generation for RL agents. * • How can an agent transfer knowledge between tasks as it learns through a curriculum? Curriculum learning approaches make use of transfer learning methods when moving from one task to another. Since the tasks in the curriculum can vary in state/action space, transition function, or reward function, it’s important to transfer relevant and reusable information from each task, and effectively combine information from multiple tasks. Methods to do this are enumerated and discussed in Section 4.3. The next section provides background in reinforcement learning and transfer learning. In Section 3, we define the curriculum learning method, evaluation metrics, and the dimensions along which we will classify curriculum learning approaches. Section 4, which comprises the core of the survey, provides a detailed overview of the existing state of the art in curriculum learning in RL, with each subsection considering a different component of the overall curriculum learning approach. Section 5 discusses paradigms related to curriculum learning for RL, such as curriculum learning for supervised learning and for human education. Finally, in Section 6, we identify gaps in the existing literature, outline the limitations of existing CL methods and frameworks, and provide a list of open problems. ## 2 Background In this section, we provide background on Reinforcement Learning (RL) and Transfer Learning (TL). ### 2.1 Reinforcement Learning Reinforcement learning considers the problem of how an agent should act in its environment over time, so as to maximize some scalar reward signal. We can formalize the interaction of an agent with its environment (also called a _task_) as a Markov Decision Process (MDP). In this article, we restrict our attention to _episodic_ MDPs:111In continuing tasks, a discount factor $\gamma$ is often included. For simplicity, and due to the fact that tasks typically terminate in curriculum learning settings, we present the undiscounted case. But unless otherwise noted, our definitions and discussions can easily apply to the discounted case as well. ###### Definition 1 An episodic MDP $M$ is a 6-tuple $(\mathcal{S},\mathcal{A},p,r,\Delta s_{0},\mathcal{S}_{f})$, where $\mathcal{S}$ is the set of states, $\mathcal{A}$ is the set of actions, $p(s^{\prime}|s,a)$ is a transition function that gives the probability of transitioning to state $s^{\prime}$ after taking action $a$ in state $s$, and $r(s,a,s^{\prime})$ is a reward function that gives the immediate reward for taking action $a$ in state $s$ and transitioning to state $s^{\prime}$. In addition, we shall use $\Delta s_{0}$ to denote the initial state distribution, and $\mathcal{S}_{f}$ to denote the set of terminal states. We consider time in discrete time steps. At each time step $t$, the agent observes its state and chooses an action according to its _policy_ $\pi(a|s)$. The goal of the agent is to learn an _optimal policy_ $\pi^{*}$, which maximizes the expected _return_ $G_{t}$ (the cumulative sum of rewards $R$) until the episode ends at timestep $T$: $G_{t}=\sum_{i=1}^{T-t}R_{t+i}$ There are three main classes of methods to learn $\pi^{*}$: value function approaches, policy search approaches, and actor-critic methods. In _value function approaches_ , a value $v_{\pi}(s)$ is first learned for each state $s$, representing the expected return achievable from $s$ by following policy $\pi$. Through policy evaluation and policy improvement, this value function is used to derive a policy better than $\pi$, until convergence towards an optimal policy. Using a value function in this process requires a model of the reward and transition functions of the environment. If the model is not known, one option is to learn an action-value function instead, $q_{\pi}(s,a)$, which gives the expected return for taking action $a$ in state $s$ and following $\pi$ after: $q_{\pi}(s,a)=\sum_{s^{\prime}}p(s^{\prime}|s,a)[r(s,a,s^{\prime})+q_{\pi}(s^{\prime},a^{\prime})]\textrm{ , where }a^{\prime}\sim\pi(\cdot|s^{\prime})$ The action-value function can be iteratively improved towards the optimal action-value function $q_{*}$ with on-policy methods such as SARSA (Sutton and Barto, 1998). The optimal action-value function can also be learned directly with off-policy methods such as $Q$-learning (Watkins and Dayan, 1992). An optimal policy can then be obtained by choosing action $\text{argmax}_{a}q_{*}(s,a)$ in each state. If the state space is large or continuous, the action-value function can instead be estimated using a function approximator (such as a neural network), $q(s,a;\bm{w})\approx q_{*}(s,a)$, where $\bm{w}$ are the weights of the network. In contrast, _policy search methods_ directly search for or learn a parameterized policy $\pi_{\bm{\theta}}(a|s)$, without using an intermediary value function. Typically, the parameter $\bm{\theta}$ is modified using search or optimization techniques to maximize some performance measure $J(\bm{\theta})$. For example, in the episodic case, $J(\bm{\theta})$ could correspond to the expected value of the policy parameterized by $\bm{\theta}$ from the starting state $s_{0}\sim\Delta s_{0}$: $v_{\pi_{\theta}}(s_{0})$. A third class of methods, _actor-critic methods_ , maintain a parameterized representation of both the current policy and value function. The actor is a parameterized policy that dictates how the agent selects actions. The critic estimates the (action-)value function for the actor using a policy evaluation method such as temporal-difference learning. The actor then updates the policy parameter in the direction suggested by the critic. An example of actor-critic methods is Deterministic Policy Gradient (Silver et al., 2014). ### 2.2 Transfer Learning In the standard reinforcement learning setting, an agent usually starts with a random policy, and directly attempts to learn an optimal policy for the target task. When the target task is difficult, for example due to adversarial agents, poor state representation, or sparse reward signals, learning can be very slow. Transfer learning is one class of methods and area of research that seeks to speed up training of RL agents. The idea behind transfer learning is that instead of learning on the _target task_ tabula rasa, the agent can first train on one or more _source task_ MDPs, and _transfer_ the knowledge acquired to aid in solving the target. This knowledge can take the form of samples (Lazaric et al., 2008; Lazaric and Restelli, 2011), options (Soni and Singh, 2006), policies (Fernández et al., 2010), models (Fachantidis et al., 2013), or value functions (Taylor and Stone, 2005). As an example, in value function transfer (Taylor et al., 2007), the parameters of an action-value function $q_{source}(s,a)$ learned in a source task are used to initialize the action- value function in the target task $q_{target}(s,a)$. This biases exploration and action selection in the target task based on experience acquired in the source task. Some of these methods assume that the source and target MDPs either share state and action spaces, or that a _task mapping_ (Taylor et al., 2007) is available to map states and actions in the target task to known states and actions in the source. Such mappings can be specified by hand, or learned automatically (Taylor et al., 2008; Ammar et al., 2015). Other methods assume the transition or reward functions do not change between tasks. The best method to use varies by domain, and depends on the relationship between source and target tasks. Finally, while most methods assume that knowledge is transferred from one source task to one target task, some methods have been proposed to transfer knowledge from several source tasks directly to a single target (Svetlik et al., 2017). See Taylor and Stone (2009) or Lazaric (2012) for a survey of transfer learning techniques. ### 2.3 Evaluation Metrics for Transfer Learning There are several metrics to quantify the benefit of transferring from a source task to a target task (Taylor and Stone, 2009). Typically, they compare the learning trajectory on the target task for an agent after transfer, with an agent that learns directly on the target task from scratch (see Figure 2a). One metric is _time to threshold_ , which computes how much faster an agent can learn a policy that achieves expected return $G_{0}\geq\delta$ on the target task if it transfers knowledge, as opposed to learning the target from scratch, where $\delta$ is some desired performance threshold. Time can be measured in terms of CPU time, wall clock time, episodes, or number of actions taken. Another metric is _asymptotic performance_ , which compares the final performance after convergence in the target task of learners when using transfer versus no transfer. The _jumpstart_ metric instead measures the initial performance increase on the target task as a result of transfer. Finally, the _total reward_ ratio compares the total reward accumulated by the agent during training up to a fixed stopping point, using transfer versus not using transfer. $\begin{array}[]{cc}\includegraphics[height=113.81102pt]{img/weakTL.png}&\includegraphics[height=113.81102pt]{img/strongTL.png}\\\ (a)&(b)\end{array}$ Figure 2: Performance metrics for transfer learning using (a) weak transfer and (b) strong transfer with offset curves. An important evaluation question is whether to include time spent _learning in source tasks_ into the cost of using transfer. The transfer curve in Figure 2a shows performance on the target task, and starts at time 0, even though time has already been spent learning one or more source tasks. Thus, it does not reflect time spent training in source tasks before transferring to the target task. This is known in transfer learning as the _weak transfer_ setting, where time spent training in source tasks is treated as a sunk cost. On the other hand, in the _strong transfer_ setting, the learning curves must account for time spent in all source tasks. One way to do this is to offset the curves to reflect time spent in source tasks, as shown in Figure 2b. Another option is to freeze the policy while learning on source tasks, and plot that policy’s performance on the target task. ## 3 The Curriculum Learning Method A _curriculum_ serves to sort the experience an agent acquires over time, in order to accelerate or improve learning. In the rest of this section we formalize this concept and the methodology of _curriculum learning_ , and describe how to evaluate the benefits and costs of using a curriculum. Finally, we provide a list of attributes which we will use to categorize curriculum learning approaches in the rest of this survey. ### 3.1 Curricula A curriculum is a general concept that encompasses both schedules for organizing past experiences, and schedules for acquiring experience by training on tasks. As such, we first propose a fully general definition of curriculum, and then follow it with refinements that apply to special cases common in the literature. We assume a _task_ is modeled as a Markov Decision Process, and define a curriculum as follows: ###### Definition 2 (Curriculum) Let $\mathcal{T}$ be a set of tasks, where $m_{i}=(\mathcal{S}_{i},\mathcal{A}_{i},p_{i},r_{i})$ is a task in $\mathcal{T}$. Let $\mathcal{D}^{\mathcal{T}}$ be the set of all possible transition samples from tasks in $\mathcal{T}$: $\mathcal{D}^{\mathcal{T}}=\\{(s,a,r,s^{\prime})\>|\>\exists\,m_{i}\in\mathcal{T}\;\mathrm{s.t.}\;s\in\mathcal{S}_{i},a\in\mathcal{A}_{i},s^{\prime}\sim p_{i}(\cdot|s,a),r\leftarrow r_{i}(s,a,s^{\prime})\\}$. A _curriculum_ $C=(\mathcal{V},\mathcal{E},g,\mathcal{T})$ is a directed acyclic graph, where $\mathcal{V}$ is the set of vertices, $\mathcal{E}\subseteq\\{(x,y)\;|\;(x,y)\in\mathcal{V}\times\mathcal{V}\>\land x\neq y\\}$ is the set of directed edges, and $g:\mathcal{V}\to\mathcal{P}(\mathcal{D}^{\mathcal{T}})$ is a function that associates vertices to subsets of samples in $\mathcal{D}^{\mathcal{T}}$, where $\mathcal{P}(\mathcal{D}^{\mathcal{T}})$ is the power set of $\mathcal{D}^{\mathcal{T}}$. A directed edge $\langle v_{j},v_{k}\rangle$ in $C$ indicates that samples associated with $v_{j}\in\mathcal{V}$ should be trained on before samples associated with $v_{k}\in\mathcal{V}$. All paths terminate on a single sink node $v_{t}\in\mathcal{V}$.222In theory, a curriculum could have multiple sink nodes corresponding to different target tasks. For the purpose of exposition, we assume a separate curriculum is created and used for each task. A curriculum can be created online, where edges are added dynamically based on the learning progress of the agent on the samples at a given vertex. It can also be designed completely offline, where the graph is generated before training, and edges are selected based on properties of the samples associated with different vertices. Creating a curriculum graph at the sample level can be computationally difficult for large tasks, or large sets of tasks. Therefore, in practice, a simplified representation for a curriculum is often used. There are 3 common dimensions along which this simplification can happen. The first is the single-task curriculum, where all samples used in the curriculum come from a single task: ###### Definition 3 (Single-task Curriculum) A _single-task curriculum_ is a curriculum $C$ where the cardinality of the set of tasks considered for extracting samples $|\mathcal{T}|=1$, and consists of only the target task $m_{t}$. A single-task curriculum essentially considers how best to organize and train on experience acquired from a single task. This type of curriculum is common in experience replay methods (Schaul et al., 2016). A second common simplification is to learn a curriculum at the task level, where each vertex in the graph is associated with samples from a single task. At the task level, a curriculum can be defined as a directed acyclic graph of _intermediate_ tasks: ###### Definition 4 (Task-level Curriculum) For each task $m_{i}\in\mathcal{T}$, let $\mathcal{D}^{\mathcal{T}}_{i}$ be the set of all samples associated with task $m_{i}$: $\mathcal{D}^{\mathcal{T}}_{i}=\\{(s,a,r,s^{\prime})\>|\>s\in\mathcal{S}_{i},a\in\mathcal{A}_{i},s^{\prime}\sim p_{i}(\cdot|s,a),r\leftarrow r_{i}(s,a,s^{\prime})\\}$. A _task-level curriculum_ is a curriculum $C=(\mathcal{V},\mathcal{E},g,\mathcal{T})$ where each vertex is associated with samples from a single task in $\mathcal{T}$. Thus, the mapping function $g$ is defined as $g:\mathcal{V}\to\\{\mathcal{D}^{\mathcal{T}}_{i}\;|\;m_{i}\in\mathcal{T}\\}$. In reinforcement learning, the entire set of samples from a task (or multiple tasks) is usually not available ahead of time. Instead, the samples experienced in a task depend on the agent’s behavior policy, which can be influenced by previous tasks learned. Therefore, while generating a task-level curriculum, the main challenge is how to order tasks such that the behavior policy learned is useful for acquiring good samples in future tasks. In other words, selecting and training on a task $m$ induces a mapping function $g$, and determines the set of samples $\mathcal{D}_{i}^{\mathcal{T}}$ that will be available at the next vertex based on the agent’s behavior policy as a result of learning $m$. The same task is allowed to appear at more than one vertex, similar to how in Definition 2 the same set of samples can be associated with more than one vertex. Therefore, tasks can be revisited when the agent’s behavior policy has changed. Several works have considered learning task-level curricula over a graph of tasks (Svetlik et al., 2017; MacAlpine and Stone, 2018). An example can be seen in Figure 3b. Finally, another simplification of the curriculum is the linear _sequence_. This is the simplest and most common structure for a curriculum in existing work: ###### Definition 5 (Sequence Curriculum) A _sequence curriculum_ is a curriculum $C$ where the indegree and outdegree of each vertex $v$ in the graph $C$ is at most 1, and there is exactly one source node and one sink node. These simplifications can be combined to simplify a curriculum along multiple dimensions. For example, the sequence simplification and task-level simplification can be combined to produce a task-level sequence curriculum. This type of curriculum can be represented as an ordered list of tasks $[m_{1},m_{2},...m_{n}]$. An example can be seen in Figure 3a (Narvekar et al., 2017). A final important question when designing curricula is determining the stopping criteria: that is, how to decide _when_ to stop training on samples or tasks associated with a vertex, and move on to the next vertex. In practice, typically training is stopped when performance on the task or set of samples has converged. Training to convergence is not always necessary, so another option is to train on each vertex for a fixed number of episodes or epochs. Since more than one vertex can be associated with the same samples/tasks, this experience can be revisited later on in the curriculum. ### 3.2 Curriculum Learning _Curriculum learning_ is a methodology to _optimize_ the order in which experience is accumulated by the agent, so as to increase performance or training speed on a set of final tasks. Through generalization, knowledge acquired quickly in simple tasks can be leveraged to reduce the exploration of more complex tasks. In the most general case, where the agent can acquire experience from multiple intermediate tasks that differ from the final MDP, there are 3 key elements to this method: * • Task Generation. The quality of a curriculum is dependent on the quality of tasks available to choose from. Task generation is the process of creating a good set of intermediate tasks from which to obtain experience samples. In a task-level curriculum, these tasks form the nodes of the curriculum graph. This set of intermediate tasks may either be pre-specified, or dynamically generated during the curriculum construction by observing the agent. * • Sequencing. Sequencing examines how to create a partial ordering over the set of experience samples $\mathcal{D}$: that is, how to generate the edges of the curriculum graph. Most existing work has used manually defined curricula, where a human selects the ordering of samples or tasks. However, recently automated methods for curriculum sequencing have begun to be explored. Each of these methods make different assumptions about the tasks and transfer methodology used. These methods will be the primary focus of this survey. * • Transfer Learning. When creating a curriculum using multiple tasks, the intermediate tasks may differ in state/action space, reward function, or transition function from the final task. Therefore, transfer learning is needed to extract and pass on reusable knowledge acquired in one task to the next. Typically, work in transfer learning has examined how to transfer knowledge from one or more source tasks directly to the target task. Curriculum learning extends the transfer learning scenario to consider training sessions in which the agent must repeatedly transfer knowledge from one task to another, up to a set of final tasks. $\begin{array}[]{cc}\includegraphics[height=156.49014pt]{img/sequences.png}&\includegraphics[height=156.49014pt]{img/dag.png}\\\ (a)&(b)\end{array}$ Figure 3: Examples of structures of curricula from previous work. (a) Linear sequences in a gridworld domain (Narvekar et al., 2017) (b) Directed acyclic graphs in block dude (Svetlik et al., 2017). ### 3.3 Evaluating Curricula Curricula can be evaluated using the same metrics as for transfer learning (cf. Section 2.3), by comparing performance on the target task after following the complete curriculum, versus performance following no curriculum (i.e., learning from scratch). If there are multiple final tasks, the metrics can easily be extended: for example, by comparing the average asymptotic performance over a set of tasks, or the average time to reach a threshold performance level over a set of tasks. Similarly, it is possible to distinguish between weak and strong transfer. However, in curriculum learning, there is the additional expense required to _build_ the curriculum itself, in addition to training on intermediate tasks in the curriculum, which can also be factored in when evaluating the cost of the curriculum. As in the transfer learning case, cost can be measured in terms of wall clock time, or data/sample complexity. Most existing applications of curricula in reinforcement learning have used curricula created by humans. In these cases, it can be difficult to assess how much time, effort, and prior knowledge was used to design the curriculum. Automated approaches to generate a curriculum also typically require some prior knowledge or experience in potential intermediate tasks, in order to guide the sequencing of tasks. Due to these difficulties, these approaches have usually treated curriculum generation as a sunk cost, focusing on evaluating the performance of the curriculum itself, and comparing it versus other curricula, including those designed by people. The best set of evaluation criteria to use ultimately depends on the specific problem and settings being considered. For example, how expensive is it to collect data on the final task compared to intermediate tasks? If intermediate tasks are relatively inexpensive, we can treat time spent in them as sunk costs. Is it more critical to improve initial performance, final performance, or reaching a desired performance threshold? If designing the curriculum will require human interaction, how will this time be factored into the cost of using a curriculum? Many of these questions depend on whether we wish to evaluate the utility of a specific curriculum (compared to another curriculum), or whether we wish to evaluate the utility of using a curriculum design approach versus training without one. ### 3.4 Dimensions of Categorization We categorize curriculum learning approaches along the following seven dimensions, organized by attributes (in bold) and the values (in italics) they can take. We use these dimensions to create a taxonomy of surveyed work in Section 4. 1. 1. Intermediate task generation: _target / automatic / domain experts / naive users_. In curriculum learning, the primary challenge is how to sequence a set of tasks to improve learning speed. However, finding a good curriculum depends on first having useful source tasks to select from. Most methods assume the set of possible source tasks is fixed and given ahead of time. In the simplest case, only samples from the _target_ task are used. When more than one intermediate task is used, typically they are manually designed by humans. We distinguish such tasks as designed by either _domain experts_ , who have knowledge of the agent and its learning algorithm, or _naive users_ , who do not have this information. On the other hand, some works consider _automatically_ creating tasks online using a set of rules or generative process. These approaches may still rely on some human input to control/tune hyper-parameters, such as the number of tasks generated, or to verify that generated tasks are actually solvable. 2. 2. Curriculum representation: _single / sequence / graph_. As we discussed previously, the most general form of a curriculum is a directed acyclic graph over subsets of samples. However, in practice, simplified versions of this representation are often used. In the simplest case, a curriculum is an ordering over samples from a _single_ task. When multiple tasks can be used in a curriculum, curricula are often created at the task-level. These curricula can be represented as a linear chain, or _sequence_. In this case, there is exactly one source for each intermediate task in the curriculum. It is up to the transfer learning algorithm to appropriately retain and combine information gathered from previous tasks in the chain. More generally, they can be represented as a full directed acyclic _graph_ of tasks. This form supports transfer learning methods that transfer from many-to-one, one-to- many, and many-to-many tasks. 3. 3. Transfer method: _policies / value function / task model / partial policies / shaping reward / other / no transfer_. Curriculum learning leverages ideas from transfer learning to transfer knowledge between tasks in the curriculum. As such, the transfer learning algorithm used affects how the curriculum will be produced. The type of knowledge transferred can be low-level knowledge, such as an entire _policy_ , an _(action-)value function_ , or a full _task model_ , which can be used to directly initialize the learner in the target task. It can also be high-level knowledge, such as _partial policies_ (e.g. options) or _shaping rewards_. This type of information may not fully initialize the learner in the target task, but it could be used to guide the agent’s learning process in the target task. We use partial policies as an umbrella term to represent closely related ideas such as options, skills, and macro-actions. When samples from a single task are sequenced, _no transfer_ learning algorithm is necessary. Finally, we use _other_ to refer to other types of transfer learning methods. We categorize papers along this dimension based on what is transferred between tasks in the curriculum in each paper’s experimental results. 4. 4. Curriculum sequencer: _automatic / domain experts / naive users_. Curriculum learning is a three-part method, consisting of task generation, sequencing, and transfer learning. While much of the attention of this survey is on automated sequencing approaches, many works consider the other parts of this method, and assume the sequencing is done by a human or oracle. Thus, we identify and categorize the type of sequencing approach used in each work similar to task generation: it can be done _automatically_ by a sequencing algorithm, or manually by humans that are either _domain experts_ or _naive users_. 5. 5. Curriculum adaptivity: _static / adaptive_. Another design question when creating a curriculum is whether it should be generated in its entirety before training, or dynamically adapted during training. We refer to the former type as _static_ and to the latter as _adaptive_. Static approaches use properties of the domain and possibly of the learning agent, to generate a curriculum before any task is learned. Adaptive methods, on the other hand, are influenced by properties that can only be measured during learning, such as the learning progress by the agent on the task it is currently facing. For example, learning progress can be used to guide whether subsequent tasks should be easier or harder, as well as how relevant a task is for the agent at a particular point in the curriculum. 6. 6. Evaluation metric: _time to threshold / asymptotic / jumpstart / total reward_. We discussed four metrics to quantify the effectiveness of learned curricula in Section 3.3. When calculating these metrics, one can choose whether to treat time spent generating the curriculum and training on the curriculum as a sunk cost, or whether to account for both of these for performance. Specifically, there are three ways to measure the cost of learning and training via a curriculum. 1) The cost of generating and using the curriculum is treated as a sunk cost, and the designer is only concerned with performance on the target task after learning. This case corresponds to the weak transfer setting. 2) The cost of training on intermediate tasks in the curriculum is accounted for, when comparing to training directly on the target task. This case is most common when it is hard to evaluate the cost of generating the curriculum itself, for example if it was hand-designed by a human. 3) Lastly, the most comprehensive case accounts for the cost of generating the curriculum as well as training via the curriculum. We will refer to the last two as strong transfer, and indicate it by bolding the corresponding metric. Note that achieving asymptotic performance improvements implies strong transfer. 7. 7. Application area: _toy / sim robotics / real robotics / video games / other_. Curriculum learning methods have been tested in a wide variety of domains. _Toy_ domains consist of environments such as grid worlds, cart-pole, and other low dimensional environments. _Sim robotics_ environments simulate robotic platforms, such as in MuJoCo. _Real robotics_ papers test their method on physical robotic platforms. _Video games_ consist of game environments such as Starcraft or the Arcade Learning Environment (Atari). Finally, _other_ is used for custom domains that do not fit in these categories. We list these so that readers can better understand the scalability and applicability of different approaches, and use these to inform what methods would be suitable for their own problems. ## 4 Curriculum Learning for Reinforcement Learning Agents In this section, we systematically survey work on each of the three central elements of curriculum learning: task generation (Section 4.1), sequencing (Section 4.2), and transfer learning (Section 4.3). For each of these subproblems, we provide a table that categorizes work surveyed according to the dimensions outlined in Section 3. The bulk of our attention will be devoted to the subproblem most commonly associated with curriculum learning: sequencing. ### 4.1 Task Generation Task generation is the problem of creating intermediate tasks specifically to be part of a curriculum. In contrast to the life-long learning scenario, where potentially unrelated tasks are constantly proposed to the agent (Thrun, 1998), the aim of task generation is to create a set of tasks such that knowledge transfer through them is beneficial. Therefore, all the generated tasks should be relevant to the final task(s) and avoid _negative transfer_ , where using a task for transfer hurts performance. The properties of the research surveyed in this section are reported in Table 1. Very limited work has been dedicated to formally studying this subproblem in the context of reinforcement learning. All known methods assume the domain can be parameterized using some kind of representation, where different instantiations of these parameters create different tasks. For instance, Narvekar et al. (2016) introduce a number of methods to create intermediate tasks for a specific final task. The methods hinge on a definition of a domain as a set of MDPs identified by a _task descriptor_ , which is a vector of parameters specifying the _degrees of freedom_ in the domain. For example, in the quick chess example (see Section 1), these parameters could be the size of the board, number of pawns, etc. By varying the degrees of freedom and applying task _restrictions_ , the methods define different types of tasks. Methods introduced include: _task simplification_ , which directly changes the degrees of freedom to reduce the task dimensions; _promising initialization_ , which modifies the set of initial states by adding states close to high rewards; _mistake learning_ , which rewinds the domain to a state a few steps before a mistake is detected and resumes learning from there; and several other methods. The set of methods determine different kinds of possible tasks, which form a space of tasks in which appropriate intermediate tasks can be chosen. Da Silva and Reali Costa (2018) propose a similar partially automated task generation procedure in their curriculum learning framework, based on Object- Oriented MDPs. Each task is assumed to have a class _environment_ parameterized by a number of attributes. A function, which must be provided by the designer, creates simpler versions of the final task by instantiating the attributes with values that make the tasks easier to solve. For example, continuing the quick chess example, the attributes could be the types of pieces, and the values are the number of each type of piece. The presence of different kinds and numbers of objects provide a range of tasks with different levels of difficulty. However, since the generation is mostly random, the designer has to make sure that the tasks are indeed solvable. Citation | Intermediate Task Generation | Curriculum Representation | Transfer Method | Curriculum Sequencer | Curriculum Adaptivity | Evaluation Metric | Application Area ---|---|---|---|---|---|---|--- Da Silva and Reali Costa (2018) | automatic | graph | value function | automatic | static | time to threshold, total reward | toy, video games Narvekar et al. (2016) | automatic | sequence | value function | domain experts | adaptive | asymptotic | video games Schmidhuber (2013) | automatic | sequence | partial policies | automatic | adaptive | asymptotic | other Stone and Veloso (1994) | automatic | sequence | other | domain experts | adaptive | time to threshold | other Table 1: The papers discussed in Section 4.1, categorized along the dimensions presented in Section 3.4. Bolded values under evaluation metric indicate strong transfer. Generating auxiliary intermediate tasks is a problem that has been studied in non-RL contexts as well. For instance, Stone and Veloso (1994) consider how to semiautomatically create subproblems to aid in learning to solve difficult _planning_ problems. Rather than using a static analysis of the domain’s properties, they propose to use a partially completed search trajectory of the target task to identify what makes a problem difficult, and suggest auxiliary tasks. For example, if the task took too long and there are multiple goals in the task, try changing the order of the goals. Other methods they propose include reducing the number of goals, creating tasks to solve difficult subgoals, and changing domain operators and objects available for binding. Lastly, Schmidhuber (2013) introduced Powerplay, a framework that focuses on inventing new problems to train a more and more general problem solver in an unsupervised fashion. The system searches for both a new task and a modification of the current problem solver, such that the modified solver can solve all previous tasks, plus the new one. The search acts on a domain- dependent encoding of the problem and the solver, and has been demonstrated on pattern recognition and control tasks (Srivastava et al., 2013). The generator of the task and new solver is given a limited computational budget, so that it favors the generation of the simplest tasks that could not be solved before. Furthermore, a possible task is to solve all previous tasks, but with a more compact representation of the solver. The resulting iterative process makes the system increasingly more competent at different tasks. The task generation process effectively creates a curriculum, although in this context there are no final tasks, and the system continues to generate pairs of problems and solvers indefinitely, without any specific goal. ### 4.2 Sequencing Given a set of tasks, or samples from them, the goal of sequencing is to order them in a way that facilitates learning. Many different sequencing methods exist, each with their own set of assumptions. One of the fundamental assumptions of curriculum learning is that we can configure the environment to create different tasks. For the practitioner attempting to use curriculum learning, the amount of control one has to shape the environment affects the type of sequencing methods that could be applicable. Therefore, we categorize sequencing methods by the degree to which intermediate tasks may differ. Specifically, they form a spectrum, ranging from methods that simply reorder experience in the final task without modifying any property of the corresponding MDP, to ones that define entirely new intermediate tasks, by progressively adjusting some or all of the properties of the final task. In this subsection, we discuss the different sequencing approaches. First, in Section 4.2.1, we consider methods that reorder samples in the target task to derive a curriculum. Experience replay methods are one such example. In Section 4.2.2, we examine multi-agent approaches to curriculum generation, where the cooperation or competition between two (typically evolving) agents induces a sequence of progressively challenging tasks, like a curriculum. Then, in Section 4.2.3, we begin describing methods that explicitly use intermediate tasks, starting with ones that vary in limited ways from the target task. In particular, these methods only change the reward function and/or the initial and terminal state distributions to create a curriculum. In Section 4.2.4, we discuss methods that relax this assumption, and allow intermediate tasks that can vary in any way from the target task MDP. Finally, in Section 4.2.5, we discuss work that explores how humans sequence tasks into a curriculum. #### 4.2.1 Sample Sequencing First we consider methods that reorder samples from the final task, but do not explicitly change the domain itself. These ideas are similar to curriculum learning for supervised learning (Bengio et al., 2009), where training examples are presented to a learner in a specific order, rather than completely randomly. Bengio et al. (2009) showed that ordering these examples from simple to complex can improve learning speed and generalization ability. An analogous process can be used for reinforcement learning. For example, many current reinforcement learning methods, such as Deep Q Networks (DQN) (Mnih et al., 2015) use a replay buffer to store past state-action-reward experience tuples. At each training step, experience tuples are sampled from the buffer and used to train DQN in minibatches. The original formulation of DQN performed this sampling uniformly randomly. However, as in the supervised setting, samples can be reordered or “prioritized,” according to some measure of usefulness or difficulty, to improve learning. Citation | Intermediate Task Generation | Curriculum Representation | Transfer Method | Curriculum Sequencer | Curriculum Adaptivity | Evaluation Metric | Application Area ---|---|---|---|---|---|---|--- Sample Sequencing (Section 4.2.1) Andrychowicz et al. (2017) | target | single | no transfer | automatic | adaptive | asymptotic | sim robotics Fang et al. (2019) | target | single | no transfer | automatic | adaptive | asymptotic | sim robotics Kim and Choi (2018) | target | single | no transfer | automatic | adaptive | asymptotic | toy, other Lee et al. (2019) | target | single | no transfer | automatic | adaptive | time to threshold | toy, video games Ren et al. (2018) | target | single | no transfer | automatic | adaptive | asymptotic | video games Schaul et al. (2016) | target | single | no transfer | automatic | adaptive | asymptotic | video games Co-learning (Section 4.2.2) Baker et al. (2020) | automatic | sequence | policies | automatic | adaptive | asymptotic, time to threshold | other Bansal et al. (2018) | automatic | sequence | policies | automatic | adaptive | asymptotic | sim robotics Pinto et al. (2017) | automatic | sequence | policies | automatic | adaptive | time to threshold | sim robotics Sukhbaatar et al. (2018) | automatic | sequence | policies | automatic | adaptive | time to threshold, asymptotic | toy, video games Vinyals et al. (2019) | automatic | sequence | policies | automatic | adaptive | asymptotic | video games Reward and Initial/Terminal State Distribution Changes (Section 4.2.3) Asada et al. (1996) | domain experts | sequence | value function | automatic | adaptive | asymptotic | sim/real robotics Baranes and Oudeyer (2013) | automatic | sequence | partial policies | automatic | adaptive | asymptotic | sim/real robotics Florensa et al. (2017) | automatic | sequence | policies | automatic | adaptive | asymptotic | sim robotics Florensa et al. (2018) | automatic | sequence | policies | automatic | adaptive | asymptotic | sim robotics Ivanovic et al. (2019) | automatic | sequence | policies | automatic | adaptive | asymptotic | sim robotics Racaniere et al. (2019) | automatic | sequence | policies | automatic | adaptive | asymptotic | toy, video games Riedmiller et al. (2018) | domain experts | sequence | policies | automatic | adaptive | time to threshold | sim/real robotics Wu and Tian (2017) | domain experts | sequence | task model | automatic | both | asymptotic | video games No Restrictions (Section 4.2.4) Bassich et al. (2020) | domain experts | sequence | policies | automatic | adaptive | asymptotic, time to threshold | toy Da Silva and Reali Costa (2018) | automatic | graph | value function | automatic | static | time to threshold, total reward | toy, video games Foglino et al. (2019a) | domain experts | sequence | value function | automatic | static | time to threshold, asymptotic, total reward | toy Foglino et al. (2019b) | domain experts | sequence | value function | automatic | static | total reward | toy Foglino et al. (2019c) | domain experts | sequence | value function | automatic | static | total reward | toy Jain and Tulabandhula (2017) | domain experts | sequence | value function | automatic | adaptive | time to threshold, total reward | toy Matiisen et al. (2017) | domain experts | sequence | policies | automatic | adaptive | asymptotic | toy, video games Narvekar et al. (2017) | automatic | sequence | value function | automatic | adaptive | time to threshold | toy Narvekar and Stone (2019) | domain experts | sequence | value function, shaping reward | automatic | adaptive | time to threshold | toy, video games Svetlik et al. (2017) | domain experts | graph | shaping reward | automatic | static | asymptotic, time to threshold | toy, video games Human-in-the-loop Curriculum Generation (Section 4.2.5) Hosu and Rebedea (2016) | target | single | no transfer | automatic | adaptive | asymptotic | video games Khan et al. (2011) | domain experts | sequence | no transfer | naive users | static | N/A | other MacAlpine and Stone (2018) | domain experts | graph | policies | domain experts | static | asymptotic | sim robotics Peng et al. (2018) | domain experts | sequence | task model | naive users | static | time to threshold | other Stanley et al. (2005) | domain experts | sequence | partial policies | domain experts | adaptive | asymptotic | video games Table 2: The papers discussed in Section 4.2, categorized along the dimensions presented in Section 3.4. Bolded values under evaluation metric indicate strong transfer. The first to do this type of sample sequencing in the context of deep learning were Schaul et al. (2016). They proposed Prioritized Experience Replay (PER), which prioritizes and replays _important_ transitions more. Important transitions are those with high expected learning progress, which is measured by their temporal difference (TD) error. Intuitively, replaying samples with larger TD errors allows the network to make stronger updates. As transitions are learned, the distribution of important transitions changes, leading to an implicit curriculum over the samples. Alternative metrics for priority/importance have been explored as well. Ren et al. (2018) propose to sort samples using a complexity index (CI) function, which is a combination of a self-paced prioritized function and a coverage penalty function. The self-paced prioritized function selects samples that would be of appropriate difficulty, while the coverage function penalizes transitions that are replayed frequently. They provide one specific instantiation of these functions, which are used in experiments on the Arcade Learning Environment (Bellemare et al., 2013), and show that it performs better than PER in many cases. However, these functions must be designed individually for each domain, and designing a broadly applicable domain- independent priority function remains an open problem. Kim and Choi (2018) consider another extension of prioritized experience replay, where the weight/priority of a sample is jointly learned with the main network via a secondary neural network. The secondary network, called ScreenerNet, learns to predict weights according to the error of the sample by the main network. Unlike PER, this approach is memoryless, which means it can directly predict the significance of a training sample even if that particular example was not seen. Thus, the approach could potentially be used to actively request experience tuples that would provide the most information or utility, creating an online curriculum. Instead of using sample importance as a metric for sequencing, an alternative idea is to restructure the training process based on trajectories of samples experienced. For example, when learning, typically easy to reach states are encountered first, whereas harder to reach states are encountered later on in the learning cycle. However, in practical settings with sparse rewards, these easy to reach states may not provide a reward signal. Hindsight Experience Replay (HER) (Andrychowicz et al., 2017) is one method to make the most of these early experiences. HER is a method that learns from “undesired outcomes,” in addition to the desired outcome, by replaying each episode with a goal that was actually achieved rather than the one the agent was trying to achieve. The problem is set up as learning a Universal Value Function Approximator (UVFA) (Schaul et al., 2015), which is a value function $v_{\pi}(s,g)$ defined over states $s$ and goals $g$ . The agent is given an initial state $s_{1}$ and a desired goal state $g$. Upon executing its policy, the agent may not reach the goal state $g$, and instead land on some other terminal state $s_{T}$. While this trajectory does not help to learn to achieve $g$, it does help to learn to achieve $s_{T}$. Thus, this trajectory is added to the replay buffer with the goal state substituted with $s_{T}$, and used with an off-policy RL algorithm. HER forms a curriculum by taking advantage of the implicit curriculum present in exploration, where early episodes are likely to terminate on easy to reach states, and more difficult to reach states are found later in the training process. One of the issues with vanilla HER is that all goals in seen trajectories are replayed evenly, but some goals may be more useful at different points of learning. Thus, Fang et al. (2019) later proposed Curriculum-guided HER (CHER) to adaptively select goals based on two criteria: curiosity, which leads to the selection of diverse goals, and proximity, which selects goals that are closer to the true goal. Both of these criteria rely on a measure of distance or similarity between goal states. At each minibatch optimization step, the objective selects a subset of goals that maximizes the weighted sum of a diversity and proximity score. They manually impose a curriculum that starts biased towards diverse goals and gradually shifts towards proximity based goals using a weighting factor that is exponentially scaled over time. Other than PER and HER, there are other works that reorder/resample experiences in a novel way to improve learning. One example is the episodic backward update (EBU) method developed by Lee et al. (2019). In order to speed up the propagation of delayed rewards (e.g., a reward might only be obtained at the end of an episode), Lee et al. (2019) proposed to sample a whole episode from the replay buffer and update the values of all transitions within the sampled episode in a backward fashion. Starting from the end of the sampled episode, the $\max$ Bellman operator is applied recursively to update the target $Q$-values until the start of the sampled episode. This process basically reorders all the transitions within each sampled episode from the last timestep of the episode to the first, leading to an implicit curriculum. Updating highly correlated states in a sequence while using function approximation is known to suffer from cumulative overestimation errors. To overcome this issue, a diffusion factor $\beta\in(0,1)$ was introduced to update the current $Q$-value using a weighted sum of the new bootstrapped target value and the pre-existing $Q$-value estimate. Their experimental results show that in 49 Atari games, EBU can achieve the same mean and median human normalized performance of DQN by using significantly fewer samples. Methods that sequence experience samples have wide applicability and found broad success in many applications, since they can be applied directly on the target task without needing to create intermediate tasks that alter the environment. In the following sections, we consider sequencing approaches that progressively alter how much intermediate tasks in the curriculum may differ. #### 4.2.2 Co-learning Co-learning is a multi-agent approach to curriculum learning, in which the curriculum emerges from the interaction of several agents (or multiple versions of the same agent) in the same environment. These agents may act either cooperatively or adversarially to drive the acquisition of new behaviors, leading to an implicit curriculum where both sets of agents improve over time. Self-play is one methodology that fits into this paradigm, and many landmark results such as TD-Gammon (Tesauro, 1995) and more recently AlphaGo (Silver et al., 2016) and AlphaStar (Vinyals et al., 2019) fall into this category. Rather than describing every work that uses self-play or co- learning, we describe a few papers that focus on how the objectives of the multiple agents can be set up to facilitate co-learning. Sukhbaatar et al. (2018) proposed a novel method called asymmetric self-play that allows an agent to learn about the environment without any external reward in an unsupervised manner. This method considers two agents, a teacher and a student, using the paradigm of “the teacher proposing a task, and the student doing it.” The two agents learn their own policies simultaneously by maximizing interdependent reward functions for goal-based tasks. The teacher’s task is to navigate to an environment state that the student will use either as 1) a goal, if the environment is resettable, or 2) as a starting state, if the environment is reversible. In the first case, the student’s task is to reach the teacher’s final state, while in the second case, the student starts from the teacher’s final state with the aim of reverting the environment to its original initial state. The student’s goal is to minimize the number of actions it needs to complete the task. The teacher, on the other hand, tries to maximize the difference between the actions taken by the student to execute the task, and the actions spent by the teacher to set up the task. The teacher, therefore, tries to identify a state that strikes a balance between being the simplest goal (in terms of number of teacher actions) for itself to find, and the most difficult goal for the student to achieve. This process is iterated to automatically generate a curriculum of intrinsic exploration. Another example of jointly training a pair of agents adversarially for policy learning in single-agent RL tasks is Robust Adversarial RL (RARL) by Pinto et al. (2017). Unlike asymmetric self-play (Sukhbaatar et al., 2018), in which the teacher defines the goal for the student, RARL trains a protagonist and an adversary, where the protagonist learns to complete the original RL task while being robust to the disturbance forces applied by the adversarial agent. RARL is targeted at robotic systems that are required to generalize effectively from simulation, and learn robust policies with respect to variations in physical parameters. Such variations are modeled as disturbances controlled by an adversarial agent, and the adversarial agent’s goal is to learn the optimal sequence of destabilizing actions via a zero-sum game training procedure. The adversarial agent tries to identify the hardest conditions under which the protagonist agent may be required to act, increasing the agent’s robustness. Learning takes place in turns, with the protagonist learning against a fixed antagonist’s policy, and then the antagonist learning against a fixed protagonist’s policy. Each agent tries to maximize its own return, and the returns are zero-sum. The set of “destabilizing actions” available to the antagonist is assumed to be domain knowledge, and given to the adversary ahead of time. For multi-agent RL tasks, several works have shown how simple interaction between multiple learning agents in an environment can result in emergent curricula. Such ideas were explored early on in the context of evolutionary algorithms by Rosin and Belew (1997). They showed that competition between 2 groups of agents, dubbed hosts and parasites, could lead to an “arms race,” where each group drives the other to acquire increasingly complex skills and abilities. Similar results have been shown in the context of RL agents by Baker et al. (2020). They demonstrated that increasingly complex behaviors can emerge in a physically grounded task. Specifically, they focus on a game of hide and seek, where there are two teams of agents. One team must hide with the help of obstacles and other items in the environment, while the other team needs to find the first team. They were able to show that as one team converged on a successful strategy, the other team was pressured to learn a counter-strategy. This process was repeated, inducing a curriculum of increasingly competitive agents. A similar idea was explored by Bansal et al. (2018). They proposed to use multi-agent curriculum learning as an alternative to engineering dense shaping rewards. Their method interpolates between dense “exploration” rewards, and sparse multi-agent competitive rewards, with the exploration reward gradually annealed over time. In order to prevent the adversarial agent from getting too far ahead of the learning agent and making the task impossible, the authors propose to additionally sample older versions of the opponent. Lastly, in order to increase robustness, the stochasticity of the tasks is increased over time. Curriculum learning approaches have also been proposed for cooperative multi- agent systems (Wang et al., 2020; Yang et al., 2020). In these settings, there is a natural curriculum created by starting with a small number of agents, and gradually increasing them in subsequent tasks. The schedule with which to increase the number of agents is usually manually defined, and the emphasis instead is on how to perform transfer when the number of agents change. Therefore, we discuss these approaches in more detail in Section 4.3. Finally, while self-play has been successful in a wide variety of domains, including solving games such as Backgammon (Tesauro, 1995) and Go (Silver et al., 2016), such an approach alone was not sufficient for producing strong agents in a complex, multi-agent, partially-observable game like Starcraft. One of the primary new elements of Vinyals et al. (2019) was the introduction of a Starcraft League, a group of agents that have differing strategies learned from a combination of imitation learning from human game data and reinforcement learning. Rather than have every agent in the league maximize their own probability of winning against all other agents like in standard self play, there were some agents that did this, and some whose goal was to optimize against the main agent being trained. In effect, these agents were trained to exploit weaknesses in the main agent and help it improve. Training against different sets of agents over time from the league induced a curriculum that allowed the main agents to achieve grandmaster status in the game. #### 4.2.3 Reward and Initial/Terminal State Distribution Changes Thus far, the curriculum consisted of ordering experience from the target task or modifying agents in the target environment. In the next two sections, we begin to examine approaches that explicitly create different MDPs for intermediate tasks, by changing some aspect of the MDP. First we consider approaches that keep the state and action spaces the same, as well as the environment dynamics, but allow the reward function and initial/terminal state distributions to vary. One of the earliest examples of this type of method was _learning from easy missions_. Asada et al. (1996) proposed this method to train a robot to shoot a ball into a goal based on vision inputs. The idea was to create a series of tasks, where the agent’s initial state distribution starts close to the goal state, and is progressively moved farther away in subsequent tasks, inducing a curriculum of tasks. In this work, each new task starts one “step” farther away from the goal, where steps from the goal is measured using a domain specific heuristic: a state is closer to the terminal state if the goal in the camera image gets larger. The heuristic implicitly requires that the state space can be categorized into “substates,” such as goal size or ball position, where the ordering of state transitions in a substate to a goal state is known. Thus, each substate has a dimension for making the task simpler or more complex. Source tasks are manually created to vary along these dimensions of difficulty. Recently, Florensa et al. (2017) proposed more general methods for performing this reverse expansion. They proposed reverse curriculum generation, an algorithm that generates a distribution of starting states that get increasingly farther away from the goal. The method assumes at least one goal state is known, which is used as a seed for expansion. Nearby starting states are generated by taking a random walk from existing starting states by selecting actions with some noise perturbation. In order to select the next round of starting states to expand from, they estimate the expected return for each of these states, and select those that produce a return between a manually set minimum and maximum interval. This interval is tuned to expand states where progress is possible, but not too easy. A similar approach by Ivanovic et al. (2019) considered combining the reverse expansion phase for curriculum generation with physics-based priors to accelerate learning by continuous control agents. An opposite “forward” expansion approach has also been considered by Florensa et al. (2018). This method allows an agent to automatically discover different goals in the state space, and thereby guide exploration of the space. They do this discovery with a Generative Adversarial Network (GAN) (Goodfellow et al., 2014), where the generator network proposes goal regions (parameterized subsets of the state space) and the discriminator evaluates whether the goal region is of appropriate difficulty for the current ability of the agent. Goal regions are specified using an indicator reward function, and policies are conditioned on the goal in addition to the state, like in a universal value function approximator (Schaul et al., 2015). The agent trains on tasks suggested by the generator. In detail, the approach consists of 3 parts: 1) First, goal regions are labelled according to whether they are of appropriate difficulty. Appropriate goals are those that give a return between hyperparameters $R_{min}$ and $R_{max}$. Requiring at least $R_{min}$ ensures there is a signal for learning progress. Requiring less than $R_{max}$ ensures that it is not too easy. 2) They use the labeled goals to train a Goal GAN. 3) Goals are sampled from the GAN as well as a replay buffer, and used for training to update the policy. The goals generated by the GAN shift over time to reflect the difficulty of the tasks, and gradually move from states close to the starting state to those farther away. Racaniere et al. (2019) also consider an approach to automatically generate a curriculum of goals for the agent, but for more complex goal-conditioned tasks in dynamic environments where the possible goals vary between episodes. The idea was to train a “setter” model to propose a curriculum of goals for a “solver” agent to attempt to achieve. In order to help the setter balance its goal predictions, they proposed three objectives which lead to a combination of three losses to train the setter model: goal validity (the goal should be valid or achievable by the current solver), goal feasibility (the goal should match the feasibility estimates for the solver with current skill), and goal coverage (encourage the setter to choose more diverse goals to encourage exploration in the space of goals). In addition, a “judge” model was trained to predict the reward the current solver agent would achieve on a goal (the feasibility of a goal) proposed by the setter. Their experimental results demonstrate the necessity of all three criteria for building useful curricula of goals. They also show that their approach is more stable and effective than the goal GAN method (Florensa et al., 2018) on complex tasks. An alternative to modifying the initial or terminal state distribution is to modify the reward function. Riedmiller et al. (2018) introduce SAC-X (Scheduled Auxiliary Control), an algorithm for scheduling and executing auxiliary tasks that allow the agent to efficiently explore its environment and also make progress towards solving the final task. Auxiliary tasks are defined to be tasks where the state, action, and transition function are the same as the original MDP, but where the reward function is different. The rewards they use in auxiliary tasks correspond to changes in raw or high level sensory input, similar to Jaderberg et al. (2017). However, while Jaderberg et al. (2017) only used auxiliary tasks for improving learning of the state representation, here they are used to guide exploration, and are sequenced. The approach is a hierarchical RL method: they need to 1) learn intentions, which are policies for the auxiliary tasks, and 2) learn the scheduler, which sequences intention policies and auxiliary tasks. To learn the intentions, they learn to maximize the action-value function of each intention from a starting state distribution that comes as a result of following each of the other intention policies. This process makes the policies compatible. The scheduler can be thought of as a meta-agent that performs sequencing, whose goal is to maximize the return on the target task MDP. The scheduler selects intentions, whose policy is executed on the extrinsic task, and is used to guide exploration. Heuristic-based methods have also been designed to sequence tasks that differ in their reward functions. One such approach is SAGG-RIAC (Self-Adaptive Goal Generation - Robust Intelligent Adaptive Curiosity) (Baranes and Oudeyer, 2013). They define _competence_ as the distance between the achieved final state and the goal state, and _interest_ as the change in competence over time for a set of goals. A region of the task space is deemed more _interesting_ than others, if the latest tasks in the region have achieved a high increase in competence. The approach repeatedly selects goals by first picking a region with a probability proportional to its interest, and then choosing a goal at random within that region. With a smaller probability the system also selects a goal at random over the whole task set or a goal close to a previously unsuccessful task. The bias towards interesting regions causes the goals to be more dense in regions where the competence increases the fastest, creating a curriculum. Because of the stochastic nature of the goal generating process, however, not every task is necessarily beneficial in directly increasing the agent’s ability on the target task, but contributes to updating the competence and interest measures. Since the intermediate tasks are generated online as the agent learns, in this approach both sequencing and generation result from the same sampling process. Finally, Wu and Tian (2017) also consider changing the transition dynamics and the reward functions of the intermediate tasks. They propose a novel framework for training an agent in a partially observable 3D Doom environment. Doom is a First-Person Shooter game, in which the player controls the agent to fight against enemies. In their experiment, they first train the agent on some simple maps with several curricula. Each curriculum consists of a sequence of progressively more complex environments with varying domain parameters (e.g., the movement speed or initial health of the agent). After learning a capable initial task model, the agent is then trained on more complicated maps and more difficult tasks with a different reward function. They also design an adaptive curriculum learning strategy in which a probability distribution over different levels of curriculum is maintained. When the agent performs well on the current distribution, the probability distribution is shifted towards more difficult tasks. #### 4.2.4 No restrictions Next, there is a class of methods that create a curriculum using intermediate tasks, but make no restrictions on the MDPs of these intermediate tasks. We categorize them in three ways by how they address the task sequencing problem: treating sequencing 1) as an MDP/POMDP, 2) as a combinatorial optimization over sequences, and 3) as learning the connections in a directed acyclic task graph. Because there are no limitations on the types of intermediate tasks allowed, some assumptions are usually made about the transfer learning algorithm, and additional information about the intermediate tasks (such as task descriptors) is typically assumed. Finally, we also discuss work on an auxiliary problem to sequencing: how long to spend on each task. #### MDP-based Sequencing The first formalization of the sequencing problem is as a Markov Decision Process. These methods formulate curriculum generation as an interaction between 2 types of MDPs. The first is the standard MDP, which models a _learning agent_ (i.e., the student) interacting with a task. The second is a higher level meta-MDP for the _curriculum agent_ (i.e., the teacher), whose goal is to select tasks for the learning agent. Narvekar et al. (2017) denote the meta-MDP as a curriculum MDP (CMDP), where the state space $\mathcal{S}$ is the set of policies the learning agent can represent. These can be represented parametrically using the weights of the learning agent. The action space $\mathcal{A}$ is the set of tasks the learning agent can train on next. Learning a task updates the learning agent’s policy, and therefore leads to a transition in the CMDP via a transition function $p$. Finally, the reward function $r$ is the time in steps or episodes that it took to learn the selected task. Under this model, a curriculum agent typically starts in an initial state corresponding to a random policy for the learning agent. The goal is to reach a terminal state, which is defined as a policy that can achieve some desired performance threshold on the target task, as fast as possible. Matiisen et al. (2017) consider a similar framework, where the interaction is defined as a POMDP. The state and action spaces of the meta-POMDP are the same as in Narvekar et al. (2017), but access to the internal parameters of the learning agent is not available. Instead, an observation of the current score of the agent on each intermediate task is given. The reward is the change in the score on the task from this timestep to the previous timestep when the same task was trained on. Thus, while Narvekar et al. (2017) focused on minimizing time to threshold performance on the target task, the design of Matiisen et al. (2017) aims to maximize the sum of performance in all tasks encountered. While both approaches are formalized as POMDPs, learning on these POMDPs is computationally expensive. Thus, both propose heuristics to guide the selection of tasks. Narvekar et al. (2017) take a sample-based approach, where a small amount of experience samples gathered on the target and intermediate tasks are compared to identify relevant intermediate tasks. The task that causes the greatest change in policy as evaluated on the target task samples is selected. In contrast, Matiisen et al. (2017) select tasks where the absolute value of the slope of the learning curve is highest. Thus it selects tasks where the agent is making the most progress or where the agent is forgetting the most about tasks it has already learned. Initially tasks are sampled randomly. As one task starts making progress, it will be sampled more, until the learning curve plateaus. Then another will be selected, and the cycle will repeat until all the tasks have been learned. Subsequently, Narvekar and Stone (2019) explored whether learning was possible in a curriculum MDP, thus avoiding the need for heuristics in task sequencing. They showed that you can represent a CMDP state using the weights of the knowledge transfer representation. For example, if the agent uses value function transfer, the CMDP state is represented using the weights of the value function. By utilizing function approximation over this state space, they showed it is possible to learn a policy over this MDP, termed a curriculum policy, which maps from the current status of learning progress of the agent, to the task it should learn next. In addition, the approach addresses the question of how long to train on each intermediate task. While most works have trained on intermediate tasks until learning plateaus, this is not always necessary. Narvekar and Stone (2019) showed that training on each intermediate task for a few episodes, and letting the curriculum policy reselect tasks that require additional time, results in faster learning. However, while learning a curriculum policy is possible, doing so independently for each agent and task is still very computationally expensive. #### Combinatorial Optimization and Search A second way of approaching sequencing is as a combinatorial optimization problem: given a fixed set of tasks, find the permutation that leads to the best curriculum, where best is determined by one of the CL metrics introduced in Section 3.3. Finding the optimal curriculum is a computationally difficult black-box optimization problem. Thus, typically fast approximate solutions are preferred. One such popular class of methods are metaheuristic algorithms, which are heuristic methods that are not tied to specific problem domains, and thus can be used as black boxes. Foglino et al. (2019a) adapt and evaluate four representative metaheuristic algorithms to the task sequencing problem: beam search (Ow and Morton, 1988), tabu search (Glover and Laguna, 1998), genetic algorithms (Goldberg, 1989), and ant colony optimization (Dorigo et al., 1991). The first two are trajectory-based, which start at a guess of the solution, and search the neighborhood of the current guess for a better solution. The last two are population-based, which start with a set of candidate solutions, and improve them as a group towards areas of increasing performance. They evaluate these methods for 3 different objectives: time to threshold, maximum return (asymptotic performance), and cumulative return. Results showed that the trajectory-based methods outperformed their population-based counterparts on the domains tested. While metaheuristic algorithms are broadly applicable, it is also possible to create specific heuristic search methods targeted at particular problems, such as task sequencing with a specific transfer metric objective. Foglino et al. (2019b) introduce one such heuristic search algorithm, designed to optimize for the cumulative return. Their approach begins by computing transferability between all pairs of tasks, using a simulator to estimate the cumulative return attained by using one task as a source for another. The tasks are then sorted according to their potential of being a good source or target, and iteratively chained in curricula of increasing length. The algorithm is anytime, and eventually exhaustively searches the space of all curricula with a predefined maximum length. Jain and Tulabandhula (2017) propose 4 different online search methods to sequence tasks into a curriculum. Their methods also assume a simulator is available to evaluate learning on different tasks, and use the learning trajectory of the agent on tasks seen so far to select new tasks. The 4 approaches are: 1) Learn each source task for a fixed number of steps, and add the one that gives the most reward. The intuition is that high reward tasks are the easiest to make progress on. 2) Calculate a transferability matrix for all pairs of tasks, and create a curriculum by chaining tasks backwards from the target tasks greedily with respect to it. 3) Extract a feature vector for each task (as in Narvekar et al., 2016), and learn a regression model to predict transferability using the feature vector. 4) Extract pair wise feature vectors between pairs of tasks, and learn a regression model to predict transferability. Finally, instead of treating the entire problem as a black box, it has also been treated as a gray box. Foglino et al. (2019c) propose such an approach, formulating the optimization problem as the composition of a white box scheduling problem and black box parameter optimization. The scheduling formulation partially models the effects of a given sequence, assigning a utility to each task, and a penalty to each pair of tasks, which captures the effect on the objective of learning two tasks one after the other. The white- box scheduling problem is an integer linear program, with a single optimal solution that can be computed efficiently. The quality of the solution, however, depends on the parameters of the model, which are optimized by a black-box optimization algorithm. This external optimization problem searches the optimal parameters of the internal scheduling problem, so that the output of the two chained optimizers is a curriculum that maximizes cumulative return. #### Graph-based Sequencing Another class of approaches explicitly treats the curriculum sequencing problem as connecting nodes with edges into a directed acyclic task graph. Typically, the task-level curriculum formulation is used, where nodes in the graph are associated with tasks. A directed edge from one node to another implies that one task is a source task for another. Existing work has relied on heuristics and additional domain information to determine how to connect different task nodes in the graph. For instance, Svetlik et al. (2017) assume the set of tasks is known in advance, and that each task is represented by a task feature descriptor. These features encode properties of the domain. For example, in a domain like Ms. Pac-Man, features could be the number of ghosts or the type of maze. The approach consists of three parts. First, a binary feature vector is extracted from the feature vector to represent non-zero elements. This binary vector is used to group subsets of tasks that share similar elements. Second, tasks within each group are connected into subgraphs using a novel heuristic called _transfer potential_. Transfer potential is defined for discrete state spaces, and trades off the applicability of a source task against the cost needed to learn it. Applicability is defined as the number of states that a value function learned in the source can be applied to a target task. The cost of a source task is approximated as the size of its state space. Finally, once subgraphs have been created, they are linked together using directed edges from subgraphs that have a set of binary features to subgraphs that have a superset of those features. Da Silva and Reali Costa (2018) follow a similar procedure, but formalize the idea of task feature descriptors using an object-oriented approach. The idea is based on representing the domain as an object-oriented MDP, where states consist of a set of objects. A task OO-MDP is specified by the set of specific objects in this task, and the state, action, transition, and reward functions of the task. With this formulation, source tasks can be generated by selecting a smaller set of objects from the target task to create a simpler task. To create the curriculum graph, they adapt the idea of transfer potential to the object-oriented setting: instead of counting the number of states that the source task value function is applicable in, they compare the sets of objects between the source and target tasks. While the sequencing is automated, human input is still required to make sure the tasks created are solvable. #### Auxiliary Problems Finally, we discuss an additional approach that tackles an auxiliary problem to sequencing: how long to spend on each intermediate task in the curriculum. Most existing work trains on intermediate tasks until performance plateaus. However, as we mentioned previously, Narvekar and Stone (2019) showed that this is unnecessary, and that better results can be obtained by training for a few episodes, and reselecting or changing tasks dynamically as needed. Bassich et al. (2020) consider an alternative method for this problem based on _progression_ functions. Progression functions specify the pace at which the difficulty of the task should change over time. The method relies on the existence of a task-generation function, which maps a desired complexity $c_{t}\in[0,1]$ to a task of that complexity. The most complex task, for which $c_{t}=1$, is the final task. After every episode, the progression function returns the difficulty of the task that the agent should face at that time. The authors define two types of progression functions: fixed progressions, for which the learning pace is predefined before learning takes place; and adaptive progressions, which adjust the learning pace online based on the performance of the agent. Linear and exponential progressions are two examples of fixed progression functions, and increase the difficulty of the task linearly and exponentially, respectively, over a prespecified number of time steps. The authors also introduce an adaptive progression based on a friction model from physics, which increases $c_{t}$ as the agent’s performance is increasing, and slows down the learning pace if performance decreases. Progression functions allow the method to change the task at every episode, solving the problem of deciding how long to spend in each task, while simultaneously creating a continually changing curriculum. #### 4.2.5 Human-in-the-Loop Curriculum Generation Thus far, all the methods discussed in Section 4.2 create a curriculum _automatically_ using a sequencing algorithm, which either reorders samples from the final task or progressively alters how much intermediate tasks in the curriculum may differ. Bengio et al. (2009) and Taylor (2009) both emphasize the importance of better understanding how _humans_ approach designing curricula. Humans may be able to design good curricula by considering which intermediate tasks are “too easy” or “too hard,” given the learner’s current ability to learn, similar to how humans are taught with the zone of proximal development (Vygotsky, 1978). These insights could then be leveraged when designing automated curriculum learning systems. Therefore, in this section, we consider curriculum sequencing approaches that are done _manually_ by humans who are either _domain experts_ , who have specialized knowledge of the problem domain, or _naive users_ , who do not necessarily know about the problem domain and/or machine learning. One example of having domain experts manually generate the curriculum is the work done by Stanley et al. (2005), in which they explore how to keep video games interesting by allowing agents to change and to improve through interaction with the player. They use the NeuroEvolving Robotic Operatives (NERO) game, in which simulated robots start the game with no skills and have to learn complicated behaviors in order to play the game. The human player takes the role of a trainer and designs a curriculum of training scenarios to train a team of simulated robots for military combat. The player has a natural interface for setting up training exercises and specifying desired goals. An ideal curriculum would consist of exercises with increasing difficulty so that the agent can start with learning basic skills and gradually building on them. In their experiments, the curriculum is designed by several NERO programmers who are familiar with the game domain. They show that the simulated robots could successfully be trained to learn different sophisticated battle tactics using the curriculum designed by these domain experts. It is unclear whether the human player who is not familiar with the game can design good curriculum. A more recent example is by MacAlpine and Stone (2018). They use a very extensive manually constructed curriculum to train agents to play simulated robot soccer. The curriculum consists of a training schedule over 19 different learned behaviors. It encompasses skills such as moving to different positions on the field with different speeds and rotation, variable distance kicking, and accessory skills such as getting up when fallen. Optimizing these skills independently can lead to problems at the intersection of these skills. For example, optimizing for speed in a straight walk can lead to instability if the robot needs to turn or kick due to changing environment conditions. Thus, the authors of this work hand-designed a curriculum to train related skills together using an idea called overlapping layered learning. This curriculum is designed using their domain knowledge of the task and agents. While domain experts usually generate good curricula to facilitate learning, most existing work does not explicitly explore their curriculum design process. It is unclear what kind of design strategies people follow when sequencing tasks into a curriculum. Published research on Interactive Reinforcement Learning (Thomaz and Breazeal, 2006; Knox and Stone, 2009; Suay and Chernova, 2011; Knox and Stone, 2012; Griffith et al., 2013; Subramanian et al., 2016; Loftin et al., 2016; MacGlashan et al., 2017) has shown that RL agents can successfully speed up learning using human feedback, demonstrating the significant role can humans play in teaching an agent to learn a (near-) optimal policy. This large body of work mainly focuses on understanding how human teachers want to teach the agent and how to incorporate these insights into the standard RL framework. Similarly, the way we define curriculum design strategies still leaves a lot to be defined by human teachers. As pointed out by Bengio et al. (2009), the notion of simple and complex tasks is often based on human intuition, and there is value in understanding how humans identify “simple” tasks. Along these lines, some work has been done to study whether curriculum design is a prominent teaching strategy that naive users choose to teach the agent and how they approach designing curricula. $\begin{array}[]{cc}\includegraphics[height=79.6678pt]{img/dog_training_given_final_task.png}&\includegraphics[width=303.53267pt]{img/dog_training_curriculum.png}\\\ (a)&(b)\end{array}$ Figure 4: One example of curricula designed by human users. (a) Given final task. (b) A curriculum designed by one human participant. To study the teaching strategies followed by naive users, Khan et al. (2011) conduct behavioral studies in which human participants need to teach a robot the concept of whether an object can be grasped with one hand. In their experiment, participants are provided with 31 cards with photos of common objects (e.g., food, furniture, and animals) for them to select. The experiment consists of two subtasks. In the first subtask, participants sort the objects on the table based on their subjective ratings of their graspability. In the second subtask, participants pick up the cards from the table and show them to the robot while teaching the robot the concept of graspability, using as few cards as possible. While teaching the robot the object’s graspability, participants can either use any natural language or say either “graspable” or “not graspable,” depending on one of the two conditions they are randomly assigned. They observe that participants follow three distinct teaching strategies, one of which is consistent with the curriculum learning principle, i.e., starting simple and gradually increasing the difficulty of the task. Furthermore, they propose a novel theoretical framework as a potential explanation for the teaching strategy that follows the curriculum learning principle, which shows that it is the result of minimizing per-iteration expected error of the learner. Peng et al. (2018) also explore how naive users design a curriculum of tasks for an agent, but in a more complex sequential decision-making task. Specifically, a simple simulated home environment is used, where the agent must learn to perform tasks in a variety of environments. The tasks are specified via text commands and the agent is trained to perform the task via reinforcement and punishment feedback from a human trainer. It uses the goal- directed Strategy-Aware Bayesian Learning (SABL) algorithm (Loftin et al., 2016) for learning from human feedback. In the user study, participants are asked to design a set of training assignments for the agent to help it quickly learn to complete the given final assignment (shown in Figure 4a). A set of source tasks are provided for human participants to select and sequence. One example of curricula designed by human participants is shown in Figure 4b. Their empirical results show that, compared to directly learning the pre- specified final task from scratch, non-expert humans can successfully design curricula that result in better overall agent performance on learning both the entire curriculum and the final task. They also discover that humans are more likely to select commands for intermediate tasks that include concepts that are important for the final task, and that doing so results in curricula that lead to better overall agent performance. Furthermore, they demonstrate that by taking advantage of this type of non-expert guidance, their curriculum- learning algorithm can be adapted to learn the human-generated curricula more efficiently. There is also some work that does not explicitly ask humans to design a curriculum, but uses human data to help generate the curriculum. One example is the work done by Hosu and Rebedea (2016), in which they propose a deep RL method that combines online agent experiences with offline human experiences to train the agent more efficiently. In some sparse-reward Atari games such as Montezuma’s Revenge and Private Eye, the agent needs to execute a long sequence of specific actions to receive the first positive reward from the environment, which makes the exploration problem much harder. Thus, the commonly used $\epsilon$-greedy strategy could not find any game paths to reach a first state with positive reward, preventing the neural network from learning relevant features to good states. Inspired by curriculum learning and the human starts evaluation metric used for testing Atari agents, they use checkpoints sampled from a human player’s game experience as starting points for the learning process. The main intuition behind this approach is that at least some of the checkpoints will be an “easier” starting point, which is closer to some states with positive reward that the agent can benefit from. While this method belongs to the class of sequencing approaches, as discussed in Section 4.2.1, that reorders samples in the final task to derive a curriculum, it additionally considers more informative sample data generated by naive human users in order to build a more efficient curriculum. We find that very limited work has been done on investigating how humans design curricula. While the work discussed in this section enriches our empirical understanding of human teaching and gives us some insights into the development of new machine-learning algorithms and interfaces that can better accommodate machine- or human-created curricula, we believe more work needs to be done along this line. ### 4.3 Knowledge Transfer While we view sequencing, as covered in Section 4.2, to be the core concept of curriculum learning, the whole premise of CL depends on an agent’s ability to transfer knowledge among tasks. While a full discussion of transfer learning for RL is beyond the scope of this survey, this subsection is designed to provide the reader a brief introduction to the area so that they can effectively leverage it as part of their own explorations in curriculum learning. In curriculum learning, transfer learning methods are used to allow the agent to reuse knowledge learned from one intermediate task to another within the curriculum. It is worth noting that when creating a curriculum using only samples from the target task (discussed in Section 4.2.1), there is no transfer as there is only a single task (the target task) and correspondingly no change in the environment. However, when creating a curriculum using multiple intermediate tasks, which may differ in state/action space, reward function, or transition function from the final task, transfer learning is needed to extract and pass on reusable knowledge acquired in one intermediate task to the next. The type of knowledge transferred also directly affects the type of learner that is applicable to the learning process. Transferred knowledge can be low-level, such as an entire policy, a value function, a full task model, or some training instances, which can be directly used to initialize the learner in the target task. The knowledge can also be high-level, such as partial policies or options, skills, shaping rewards, or subtask definitions. This type of information may not fully initialize the learner in the target task, but it could be used to guide the agent’s learning process in the target task. In this subsection, we discuss different transfer learning approaches used in curricula. In policy transfer, a policy learned in a source or intermediate task is used to initialize the policy in the target task. When transferring policies between different tasks, the tasks may differ in some aspect of the MDP, such as starting states (Florensa et al., 2017), reward functions (Florensa et al., 2018; Riedmiller et al., 2018), or transition functions (Clegg et al., 2017). For instance, Clegg et al. (2017) demonstrate that an arm-like manipulator can successfully learn the control policy for a simulated dressing task, by transferring policies between tasks with different transition functions. In a dressing task, the goal is to achieve a desired relative positioning of the garment and the limb. To do this, they first train a sphere to move through a funnel-like geometry to reach some target location. They then directly apply the learned policy to a different scenario in which a manipulator with arbitrary shape navigates through a simulated garment. The main trick is to train multiple spheres using a curriculum learning strategy and then aggregate them to control the manipulator in the dressing task. Citation | Intermediate Task Generation | Curriculum Representation | Transfer Method | Curriculum Sequencer | Curriculum Adaptivity | Evaluation Metric | Application Area ---|---|---|---|---|---|---|--- Clegg et al. (2017) | domain experts | sequence | policies | domain experts | static | asymptotic, time to threshold | sim robotics Fujii et al. (1998) | domain experts | sequence | partial policies | domain experts | static | asymptotic | real robotics Karpathy and Van De Panne (2012) | domain experts/target | sequence/single | partial policies /no transfer | domain experts/automatic | static/adaptive | time to threshold | sim robotics Rusu et al. (2016) | domain experts | sequence | policies | domain experts | static | asymptotic | video games Shao et al. (2018) | domain experts | sequence | task model | domain experts | static | asymptotic, total reward | video games Sinapov et al. (2015) | automatic | sequence | value function | automatic | static | jump start | video games Tessler et al. (2017) | domain experts | sequence | partial policies | domain experts | static | asymptotic | video games Vezhnevets et al. (2016) | automatic | sequence | partial policies | automatic | static | asymptotic, total reward | video games Wang et al. (2020) | domain experts | sequence | policies | domain experts | static | asymptotic | video games Yang and Asada (1996) | domain experts | sequence | partial policies | automatic | adaptive | asymptotic, time to threshold | real robotics Yang et al. (2020) | domain experts | sequence | policies | domain experts | static | asymptotic, time to threshold | toy, other Zimmer et al. (2018) | domain experts | sequence | partial policies | domain experts | static | asymptotic, total reward | sim robotics Table 3: The papers discussed in Section 4.3, categorized along the dimensions presented in Section 3.4. Bolded values under evaluation metric indicate strong transfer. In Shao et al. (2018), a learned task model is transferred between tasks, which is used to initialize the policy network. Thus, it is similar to transferring policies. Their work aims to solve the problem of multi-agent decision making in StarCraft micromanagement, where the goal is to control a group of units to destroy the enemy under certain terrain conditions. A parameter sharing multi-agent gradient-descent Sarsa($\lambda$) (PS-MAGDS) method is proposed to train the units to learn an optimal policy, which is parametrized by a feed-forward neural network. PS-MAGDS simply extends the traditional Sarsa($\lambda$) to multiple units by sharing parameters of the policy network among units to encourage cooperative behaviors. A reward function including small immediate rewards is also designed to accelerate the learning process. When using transfer learning in their experiments, the agents are first trained in some small scale source scenarios using PS-MAGDS. The well-trained model is then used to initialize the policy network to learn micromanagement in the target scenarios. To scale the combat to a large scale scenario, they combine curriculum learning and transfer learning where the agents are trained with a sequence of progressively more complex micromanagement tasks. The difficulty of the micromanagement task is controlled by changing the number and type of units. Value function transfer is another common method for transferring low-level knowledge between intermediate tasks within a curriculum. In most existing work (Sinapov et al., 2015; Narvekar et al., 2017; Da Silva and Reali Costa, 2018), value function transfer is achieved by using the parameters of a value function learned in one intermediate task to initialize the value function in the next intermediate task in the curriculum, such that the agent learns the final task with some initial policy that is better than random exploration. For example, Sinapov et al. (2015) focus on addressing the task selection problem in curriculum learning using value function transfer, under the assumption that no samples from the final tasks are available. They propose to use meta-data (i.e., a fixed-length feature vector that describes the task) associated with each task to identify suitable intermediate tasks. The main idea is to use such meta-data to learn the benefits of transfer between different ‘source-target’ task pairs, and have this generalize to new unseen task pairs to guide task selection. When transferring low-level policies or value functions across tasks, there are several challenges that arise, particularly in the modern context of deep reinforcement learning. First is the problem of catastrophic forgetting, where knowledge from previously learned tasks is lost as information on a new task is incorporated. This effect occurs because the weights of the neural network optimized for a first task must be changed to meet the objectives of a new task, often resulting in poorer performance on the original task. Typically, in the curriculum setting, we only care about performance in the final tasks. However, if information from two orthogonal tasks needs to be combined (such as two independent skills), this challenge needs to be addressed. One approach is progressive neural networks (Rusu et al., 2016), which trains a new network “column” for each new task, and leverages lateral connections to previously learned network columns to achieve transfer. When training subsequent columns, parameters from previous columns are frozen, which prevents catastrophic forgetting. The limitation is that the number of parameters grows with the number of tasks, and at inference time, the task label is needed to know which column to extract output from. A second problem is the case where the state and action spaces differ between tasks. One alternative is to transfer higher-level knowledge across tasks, such as partial policies or options. A partial policy is a policy that is not necessarily defined for all states in the state space of an MDP. We use partial policies as an umbrella term to represent closely related ideas such as options, skills, and macro-actions. Yang and Asada (1996) transfer learned control parameters between tasks, which are similar to partial policies. To solve the impedance learning problem for high-speed robotic assembly, they allow the system to learn impedance parameters associated with different dynamic motions separately, rather than to learn all the control parameters simultaneously. For instance, they first learn only the parameters associated with quasistatic motion by driving the system slowly, leaving other parameters unlearned. After the quasistatic parameters have been learned, they then slightly increase the motion speed, and use the learned values to initialize the quasistatic parameters when learning other parameters. Another example of transferring partial policies between tasks is the work done by Zimmer et al. (2018). Their main idea is to progressively increase the dimensionality of the tackled problem by increasing the (continuous) state and action spaces of the MDP, while an agent is learning a policy. The agent first learns to solve the source task with reduced state and action spaces until the increase in performance stagnates. Then, the partial policy learned by the agent is used as an initialization to learn the full policy in the target task with full state and action spaces. A developmental layer (like a dropout layer) is added to the network to filter dimensions of the states and actions. Similarly, Fujii et al. (1998) transfer options between tasks. To train mobile robots to learn collision avoidance behaviors in multi-robot systems more efficiently, they develop a multi-layered RL mechanism. Rather than gradually increasing the level of task complexity based on the learner’s performance as in Yang and Asada (1996), their learning process consists of four stages like a curriculum in which each stage learns a pre-defined controller. Each controller learns an option to solve a pre-defined sub-task. For instance, the first controller learns to move toward a specific goal. Then the output (goal- directed behavior) of the first controller is used as input for the second controller, which aims to learn to avoid the collision to a single robot, and so on. Vezhnevets et al. (2016) also transfer high-level macro-actions between tasks, which are simpler instances of options. In their experiment, the agent is trained with a curriculum where the goal state is first set to be very close to the start state and is then moved further away during learning process. Although the task gets progressively harder, the temporally abstracted macro- actions remain the same. The macro-actions learned early on can also be easily adapted using their proposed architecture. Specifically, a deep recurrent neural network architecture is used to maintain a multi-step action plan. The network learns when to commit to the action plan to generate macro-actions and when to update the plan based on observations. Another mechanism for transfer are skills. Tessler et al. (2017) propose a deep RL method that effectively retains and transfers learned skills to solve lifelong learning in MineCraft. In their work, a set of $N$ skills are trained a priori on various sub-tasks, which are then reused to solve the harder composite task. In their MineCraft experiment, the agent’s action space includes the original primitive actions as well as the set of pre-learned skills (e.g., navigate and pickup). A hierarchical architecture is developed to learn a policy that determines when to execute primitive actions and when to reuse pre-learned skills, by extending the vanilla DQN architecture (Mnih et al., 2015). The skills could be sub-optimal when they are directly reused for more complex tasks, and this hierarchical architecture allows the agent to learn to refine the policy by using primitive actions. They also show the potential for reusing the pre-learned skill to solve related tasks without performing any additional learning. Rather than selectively reusing pre-learned skills, Karpathy and Van De Panne (2012) focus on learning motor skills in an order of increasing difficulty. They decompose the acquisition of skills into a two-level curriculum: a _high-level_ curriculum specifies the order in which different motor skills should be learned, while the _low-level_ curriculum defines the learning process for a specific skill. The high-level curriculum orders the skills in a way such that each skill is relatively easy to learn, using the knowledge of the previously learned skills. For instance, the Acrobot first learns the Hop (easy to learn from scratch) and Flip (similar to hopping very slowly) skills, and then learns the more complex Hop-Flip skill. The learned skill-specific task parameters for easier skills will highly constrain the states that the Acrobat could be in, making it easier to learn more complex skills. For example, the Hop-Flip skills begin from a hopping gait of some speed, which can be reached by repeatedly executing the previously learned Hop skill. In multi-agent settings, several specific methods have been designed for curricula that progressively scale the number of agents between tasks. In these settings, the state and action spaces often scale based on the number of agents present. One common assumption in many of these methods is that the state space can be factored into elements for the environment $s^{env}$, the agent $s^{n}$, and all other agents $s^{-n}$. For example, Yang et al. (2020) propose CM3, which takes a two-stage approach. In the first stage, a single agent is trained without the presence of other agents. This is done by inducing a new MDP that removes all dependencies on agent interactions (i.e., removing $s^{-n}$) and training a network on this subspace. Then in the second stage, cooperation is learned by adding the parameters for the other agents into the network. Wang et al. (2020) propose 3 different approaches for multi-agent settings. The first is buffer reuse, which saves the replay buffers from all previous tasks, and samples experience from all of them to train in the current task. Samples from lower dimensional tasks are padded with zeros. The second is curriculum distillation, which adds a distillation loss based on KL divergence between policies/q-values between tasks. The third is transferring the model using a new network architecture called Dynamic Agent-number Network (DyAN). In this architecture, the state space elements related to the agent and environment go through a fully connected network, while the observations for each teammate agent are passed through a graph neural network (GNN) and then aggregated. These networks are subsequently combined to produce q-values or policies. ## 5 Related Areas and Paradigms Curriculum learning is an idea that has been studied in other areas of machine learning and human education, and is similar to several existing paradigms in reinforcement learning. In this section, we first relate curriculum learning to approaches in reinforcement learning that aim to improve sample complexity, and that consider learning multiple sets of tasks (Section 5.1). Then we describe approaches to learn curricula in supervised learning (Section 5.2) and for teaching and human education (Section 5.3). We include these approaches with the idea that the insights discovered in these areas could be adapted to apply to the reinforcement learning setting with autonomous agents. ### 5.1 Related Paradigms in Reinforcement Learning One of the central challenges in applying reinforcement learning to real world problems is sample complexity. Due to issues such as a sparse reward signal or complex dynamics, difficult problems can take an RL agent millions of episodes to learn a good policy, with many suboptimal actions taken during the course of learning. Many different approaches have been proposed to deal with this issue. To name a few, one method is imitation learning (Schaal, 1997), which uses demonstrations from a human as labels for supervised learning to bootstrap the learning process. Another example is off-policy learning (Hanna et al., 2017), which uses existing data from an observed behavior policy, to estimate the value of a desired target policy. Model-based approaches (Sutton and Barto, 1998) first learn a model of the environment, which can then be used for planning the optimal policy. Each of these methods come with their advantages and disadvantages. For imitation learning, the assumption is that human demonstrations are available. However, these are not always easy to obtain, especially when a good policy for the task is not known. In off-policy learning, in order to make full use of existing data, it is assumed that the behavior policy has a nonzero probability of selecting each action, and typically that every action to be evaluated or the target policy has been seen at least once. Finally, model- based approaches typically first learn a model of the environment, and then use it for planning. However, any inaccuracies in the learned model can compound as the planning horizon increases. Curriculum learning takes a different approach, and makes a different set of assumptions. The primary assumption is that the environment can be configured to create different subtasks, and that it is easier for the agent to discover _on its own_ reusable pieces of knowledge in these subtasks that can be used for solving a more challenging task. Within reinforcement learning, there are also several paradigms that consider learning on a set of tasks so as to make learning more efficient. Multitask learning, lifelong/continuous learning, active learning, and meta-learning are four such examples. In _multitask learning_ , the goal is to learn how to solve _sets_ of prediction or decision making tasks. Formally, given a set of tasks $m_{1},m_{2},\ldots m_{n}$, the goal is to _co-learn_ all of these tasks, by optimizing the performance over all $n$ tasks simultaneously. Typically, this optimization is facilitated by learning over some shared basis space. For example, Caruana (1997) considers multitask learning for supervised learning problems, and shares layers of a neural network between tasks. In supervised learning, these tasks are different classification or regression problems. Similar ideas have been applied in a reinforcement learning context by Wilson et al. (2007). In reinforcement learning, different tasks correspond to different MDPs. _Lifelong learning_ and _continual learning_ can be viewed as an online version of multitask learning. Tasks are presented one at a time to the learner, and the learner must use shared knowledge learned from previous tasks to more efficiently learn the presented task. As in multitask learning, typically the goal is to optimize performance over all tasks given to the learner. Lifelong and continual learning have been examined in both the supervised setting (Ruvolo and Eaton, 2013a) and the reinforcement learning setting (Ring, 1997; Ammar et al., 2014). The distinguishing feature of curriculum learning compared to these works is that in curriculum learning, we have full control over the _order_ in which tasks are selected. Indeed, we may have control over the _creation_ of tasks as well. In addition, the goal is to optimize performance for a specific target task, rather than all tasks. Thus, source tasks in curriculum learning are designed solely to improve performance on the target task—we are not concerned with optimizing performance in a source. In _active learning_ , the learner chooses which task or example to learn or ask about next, from a given set of tasks. Typically, active learning has been examined in a semi-supervised learning setting: a small amount of labeled data exists whereas a larger amount of unlabeled data is present. The labeled data is used to learn a classifier to infer labels for unlabeled data. Unlabeled data that the classifier is not confident about is requested for a label from a human user. For example, Ruvolo and Eaton (2013b) consider active learning in a lifelong learning setting, and show how a learner can actively select tasks to improve learning speed for all tasks in a set, or for a specific target task. The selection of which task to be learned next is similar to the _sequencing_ aspect of curriculum learning. However, the full method of curriculum learning is much broader, as it also encompasses creating the space of tasks to consider. Ruvolo and Eaton (2013b) and similar active learning work typically assume the set of tasks to learn and select from are already given. In addition, typically active learning has been examined for supervised prediction tasks, whereas we are concerned with reinforcement learning tasks. Finally, in _meta-learning_ (Finn et al., 2017), the goal is to train an agent on a variety of tasks such that it can quickly adapt to a new task within a small number of gradient descent steps. Typically, the agent is not given information identifying the task it is training on. In contrast, in curriculum learning, the learning agent may or may not have information identifying the task. However, the process that designs the curriculum by sequencing tasks usually does have this information. Like in the lifelong setting, there is no significance attached to the order in which tasks are presented to the learner. In addition, the objective in meta-learning is to train for fast adaptability, rather than for a specific final task as is the case in curriculum learning. ### 5.2 Curricula in Supervised Machine Learning In addition to reinforcement learning, curriculum learning has been examined for supervised learning. While it is beyond the scope of this article to extensively survey supervised CL methods, we would like to highlight a few that could inspire ideas and draw parallels to the RL setting. Bengio et al. (2009) first formalized the idea of curriculum learning in the context of supervised machine learning. They conducted case studies examining when and why training with a curriculum can be beneficial for machine learning algorithms, and hypothesized that a curriculum serves as both a continuation method and a regularizer. A continuation method is an optimization method for non-convex criteria, where a smoothed version of the objective is optimized first, with the smoothing gradually reduced over training iterations. Typically, “easy” examples in a curriculum correspond to a smoother objective. Using a simple shape recognition and language domain, they showed that training with a curriculum can improve both learning speed and performance. While many papers before Bengio et al. (2009) _used_ the idea of a curriculum to improve training of machine learning algorithms, most work considering how to systematically _learn_ a curriculum came after. One recent example is work by Graves et al. (2017). They introduced measures of _learning progress_ , which indicate how well the learner is currently improving from the training examples it is being given. They introduce 2 main measures based on 1) rate of increase in prediction accuracy and 2) rate of increase of network complexity. These serve as the reward to a non-stationary multi-armed bandit algorithm, which learns a stochastic policy for selecting tasks. These signals of learning progress could in theory be applied or adapted to the reinforcement learning setting as well. Graves et al. (2017) also make an interesting observation, which is that using a curriculum is similar to changing the step size of the learning algorithm. Specifically, in their experiments, they found that a random curriculum still serves as a strong baseline, because all tasks in the set provide a gradient333Note however that in the reinforcement learning setting, because the policy affects the distribution of states an agent encounters, random training can be significantly worse.. Easier tasks provide a stronger gradient while harder tasks provide a gradient closer to 0. Thus, choosing easy, useful tasks allows the algorithm to take larger steps and converge faster. More recently, Fan et al. (2018) frame curriculum learning as “Learning to Teach,” where a teacher agent learned to train a learning agent using a curriculum. The process is formulated as an MDP between these two interacting agents, similar to the MDP approaches discussed in Section 4.2.4: the teacher agent selects the training data, loss function, and hypothesis space, while the learning agent trains given the parameters specified by the teacher. The state space of the MDP is represented as a combination of features of the data, features of the student model, and features that represent the combination of both data and learner models. The reward signal is the accuracy on a held-out development set. Training a teacher agent can be computationally expensive. They amortize this cost by using a learned teacher agent to teach a new student with the same architecture. For example, they train the teacher using the first half of MNIST, and use the learned teacher to train a new student from the second half of MNIST. Another way they amortize the cost is to train a new student with a different architecture (e.g., changing from ResNet32 to ResNet110). Similar ideas have been explored in the reinforcement learning setting. However, the test set distribution is different from the training set distribution, which makes performing these kind of evaluations more challenging. However, showing that the cost for training a teacher can be amortized is an important direction for future work. Finally, Jiang et al. (2015) explore the idea of self-paced curriculum learning for supervised learning, which unifies and takes advantage of the benefits of self-paced learning and curriculum learning. In their terminology, curriculum learning uses prior knowledge, but does not adapt to the learner. Specifically, a curriculum is characterized by a ranking function, which orders a dataset of samples by priority. This function is usually derived by predetermined heuristics, and cannot be adjusted by feedback from the learner. In contrast, self-paced learning (SPL) adjusts to the learner, but does not incorporate prior knowledge and leads to overfitting. In SPL, the curriculum design is implicitly embedded as a regularization term into the learning objective. However, during learning, the training loss usually dominates over the regularization, leading to overfitting. This paper proposes a framework that unifies these two ideas into a concise optimization problem, and discusses several concrete implementations. The idea is to replace the regularization term in SPL with a self-paced function, such that the weights lie within a predetermined curriculum region. In short, the curriculum region induces _a weak ordering_ over the samples, and the self-paced function determines the actual learning scheme within that ordering. The idea has parallels to a task-level curriculum for RL, where the curriculum induces a weak ordering over samples from all tasks, and with the learning algorithm determining the actual scheme within that ordering. ### 5.3 Algorithmically Designed Curricula in Education Curriculum learning has also been widely used for building effective Intelligent Tutoring Systems (ITS) for human education (Iglesias et al., 2003, 2009; Green et al., 2011; Brunskill and Russell, 2011; Doroudi et al., 2016). An ITS system involves a student interacting with an intelligent tutor (a computer-based system), with the goal of helping the student to master all skills quickly, using as little learning content as possible. Given that students have different learning needs, styles, and capabilities, the intelligent tutor should be able to provide customized instructions to them. To achieve this goal, one common strategy is called _curriculum sequencing_ , which aims to provide the learning materials in a meaningful order that maximizes learning of the students with different knowledge levels. The main problem this strategy must solve is to find the most effective lesson to propose next, given the student’s current learning needs and capabilities. Reinforcement learning is one of the machine learning techniques that has been used with intelligent tutors to partially automate construction of the student model and to automatically compute an optimal teaching policy (Woolf, 2007). One advantage of using RL methods in tutoring is that the model can learn adaptive teaching actions based on each individual student’s performance in real time, without needing to encode complex pedagogical rules that the system requires to teach effectively (e.g., how to sequence the learning content, when and how to provide an exercise). Another advantage is that it is a general domain-independent technique that can be applied in any ITS. As a concrete example, Iglesias et al. (2003, 2009) adapt $Q$-learning (Watkins, 1989) to an adaptive and intelligent educational system to allow it to automatically learn how to teach each student. They formulate the learning problem as an RL problem, where the state is defined as the description of the student’s knowledge, indicating whether the student has learned each knowledge item. The set of actions the intelligent tutor can execute includes selecting and showing a knowledge item to the student. A positive reward is given when all required content has been learned, otherwise no reward is given. The system evaluates the student’s knowledge state through tests, which shows how much the student knows about each knowledge item. The $Q$-value estimates the usefulness of executing an action when the student is in a particular knowledge state. Then, the tutoring problem can be solved using the traditional $Q$-learning algorithm. Green et al. (2011) propose using a multi-layered Dynamic Bayes Net (DBN) to model the teaching problem in an ITS system. The main idea is to model the dynamics of a student’s skill acquisition using a DBN, which is normally used in RL to represent transition functions for state spaces. More specifically, they formulate the problem as a factored MDP, where the state consists of one factor for each skill, corresponding to the student’s proficiency on that particular skill. The actions are to either provide a hint or to pose a problem about a particular skill to the student. From a history of teacher- student interaction, the teacher can model the student’s proficiency state, with the goal of teaching the student to achieve the highest possible proficiency value on each skill, using as few problems and hints as possible. Subsequently, the learned DBN model is used by a planning algorithm to search for the optimal teaching policy, mapping proficiency states of student knowledge to the most effective problem or hint to pose next. To allow the automated teacher to select a sequence of pedagogical actions in cases where learner’s knowledge may be unobserved, a different problem formulation is posed by Rafferty et al. (2016). They formulate teaching as a partially observable Markov decision process (POMDP), where the learner’s knowledge state is considered as a hidden state, corresponding to the learner’s current understanding of the concept being taught. The actions the automated teacher can select is a sequence of pedagogical choices, such as examples or short quizzes. The learner’s next knowledge state is dependent on her current knowledge state and the pedagogical action the teacher chooses. Changes in the learner’s knowledge state reflect learning. In this framework, the automated teacher makes some assumptions about student learning, which is referred to as the learner model: it specifies the space of possible knowledge states and how the knowledge state changes. Then the teacher can update its beliefs about the learner’s current knowledge state based on new observations, given this learner model. Using this POMDP framework, they explore how different learner models affect the teacher’s selection of pedagogical actions. While most approaches seek to solely maximize overall learning gains, Ramachandran and Scassellati (2014) propose an RL-based approach that uses a personalized social robot to tutor children, that maximizes learning gains and sustained engagement over the student-robot interaction. The main goal of the social robot is to learn the ordering of questions presented to a child, based on difficulty level and the child’s engagement level in real time. To represent the idea that children with different knowledge levels need a different curriculum, each child is categorized into a given group based on knowledge level at the start of the one-on-one tutoring interaction. An optimal teaching policy is then learned specific to each group. In particular, their approach consists of a training phase and an interaction phase. In the training phase, participants are asked to complete a tutoring exercise. A pretest and post-test will be used to evaluate the participant’s relative learning gains, which will also be used as the reward function to learn an optimal policy during the training phase. Subsequently, in the interaction phase, the child’s real-time engagement will be detected, serving as another reward signal for the RL algorithm to further optimize the teaching policy. Non-RL-based algorithms have been considered as well. Ballera et al. (2014) leverage the roulette wheel selection algorithm (RWSA) to perform personalized topic sequencing in e-learning systems. RWSA is typically used in genetic algorithms to arrange the chromosomes based on their fitness function, such that individuals with higher fitness value will have higher probability of being selected (Goldberg, 1989). Similarly, in an e-learning system, a chromosome is denoted by a lesson. Each lesson has a fitness value that dynamically changes based on the student’s learning performance. This fitness value indicates how well the topic was learned by the student, depending on three performance parameters: exam performance, study performance, and review performance of the learner. A lower fitness value means that the student has a poorer understanding of the topic. Thus, a reversed mechanism of RWSA is implemented, so as to select the lessons with lower fitness values more often for reinforcement. Then, this reversed RWSA algorithm is combined with linear ranking algorithm to sort the lessons. ## 6 Open Questions Through our survey of the literature, we have identified several open problems that have not been sufficiently studied in past work, and could be useful avenues for future research. ### 6.1 Fully Automated Task Creation Task creation is an important piece of the method of curriculum learning. Whether tasks are created “on-demand” or all in advance, the quality of the pool of tasks generated directly affects the quality of curricula that can be produced. In addition, the _quantity_ of tasks produced affect the search space and efficiency of curriculum sequencing algorithms. Despite this, very limited work (see Section 4.1) has been done on the problem of automatically generating tasks. Existing work either assumes the pool of tasks are manually crafted and specified beforehand, or defines a set of rules for semi- automatically creating tasks. However, these rules often have hyper-parameters that control how many tasks are created, and are also usually manually tuned. Reducing the amount of manual input required by these methods remains an important area for future work. ### 6.2 Transferring Different Types of Knowledge Between each pair of tasks in a curriculum, knowledge must be transferred from one task to the subsequent task. In virtually all of the works surveyed, the type of knowledge transferred has been fixed. For example, a value function was always transferred between tasks by Narvekar et al. (2017) while a shaping reward was always transferred by Svetlik et al. (2017). However, this limitation opens the question of whether different tasks could benefit from extracting different types of knowledge. For instance, it may be useful to extract an option from one task, and a model from another. Thus, in addition to deciding _which_ task to transfer from, we could also ask _what_ to extract and transfer from that task. Past transfer learning literature has shown that many forms of transfer are possible. The best type of knowledge to extract may differ based on task, and techniques will need to be developed to effectively combine these different types of knowledge. ### 6.3 Reusing Curricula and Sim-to-Real Curriculum Learning Another limitation of many curriculum learning approaches is that the time to generate a curriculum can be greater than the time to learn the target task outright. This shortcoming stems from the fact that curricula are typically learned independently for each agent and target task. However, in areas such as human education, curricula are used to train multiple students in multiple subjects. Thus, one way to amortize the cost would be to learn a curriculum to train multiple different agents, or to solve multiple different target tasks (Narvekar and Stone, 2020). Another option for amortizing the cost is to learn curricula for a sim-to-real setting on physical robots, where a curriculum is learned in simulation and then used to train a physical robot. While the exact weights of the policy learned in simulation would not apply in the real world, the semantics of the curriculum tasks might. Therefore, the physical robot could go through the same training regimen, but learn using the physics and dynamics of the real world. ### 6.4 Combining Task Generation and Sequencing The curriculum learning method can be thought of as consisting of 3 parts: task generation, sequencing, and transfer learning. For the most part, previous work has tackled each of these pieces independently. For example, sequencing methods typically assume the tasks are prespecified, or a task generation method exists. However, an interesting question is whether the task generation and task sequencing phases can be done simultaneously, by directly generating the next task in the curriculum. Some very preliminary work has been done in this direction in the context of video game level generation. For example, Green et al. (2019) used an evolutionary algorithm to generate maps for a gridworld, where each tile had a different element. The generator was optimized to maximize the loss of deep RL agent’s network, inducing a training curriculum. Combining task generation and sequencing has additional challenges, such as specifying the space of possible maps, ensuring those maps are valid/solvable, and creating maps that are challenging, but not too difficult to solve. In addition, training the generator can be very expensive. However, it promises an end-to-end solution that could reduce the amount of human intervention needed to design curricula. ### 6.5 Theoretical Results There have been many practical applications of curricula to speed up learning in both supervised and reinforcement learning. However, despite empirical evidence that curricula are beneficial, there is a lack of theoretical results analyzing when and why they are useful, and how they should be created. An initial analysis in the context of supervised learning was done by Weinshall et al. (2018) and Weinshall and Amir (2018). They analyzed whether reordering samples in linear regression and binary classification problems could improve the ability to learn new concepts. They did this analysis by formalizing the idea of an Ideal Difficulty Score (IDS), which is the loss of the example with respect to the optimal hypothesis, and the Local Difficulty Score (LDS), which is the loss of the example with respect to the current hypothesis. These are 2 ways to classify the difficulty of a sample, which can be used as a means to sequence samples. They showed that the convergence of an algorithm like stochastic gradient descent monotonically decreases with the IDS, and monotonically increases with the LDS. An open question is whether similar grounded metrics for difficulty of tasks can be identified in reinforcement learning, and what kind of convergence guarantees we can draw from them. ### 6.6 Understanding General Principles for Curriculum Design Determining the difficulty of a training example for an agent, and ensuring that each example presented to the agent is suitable given its current ability, is a major challenge in curriculum learning. In most existing work, the curriculum is generated either automatically (see Section 4.2), by ordering samples from the target tasks or iteratively selecting intermediate tasks with increasing difficulty tailored to the current ability of the learner; or manually by domain experts, who will typically have specialized knowledge of the problem domain. Very limited work (see Section 4.2.5) has been done to better understand how non-expert humans design curricula. The way we define curriculum design strategies still leaves a lot to be defined by human teachers. Can non-expert humans design effective curricula for a given final task? What kind of curriculum design strategies do they tend to follow when building curricula? If we could find some general principles non-expert humans follow for designing and/or sequencing more “interesting” intermediate tasks into a curriculum, we could incorporate these insights into the automatic process of generating useful source tasks for any task domain. Furthermore, can we adapt curriculum learning algorithms to better take advantage of this type of non- expert guidance to learn more efficiently? We believe a better understanding of the curriculum-design strategies used by non-expert humans may help us to 1) understand the general principles that make some curriculum strategies work better than others, and 2) inspire the design of new machine-learning algorithms and interfaces that better accommodate the natural tendencies of human trainers. ## 7 Conclusion This survey formalized the concept of a curriculum, and the method of curriculum learning in the context of reinforcement learning. Curriculum learning is a 3-part approach consisting of 1) task generation, 2) sequencing, and 3) transfer learning. We systematically surveyed existing work addressing each of these parts, with a particular focus on sequencing methods. We broke down sequencing methods into five categories, based on the assumptions they make about intermediate tasks in the curriculum. The simplest of these are sample sequencing methods, which reorder samples from the final task itself, but do not explicitly change the domain. These were followed by co-learning methods, where a curriculum emerges from the interaction of several agents in the same environment. Next we considered methods that explicitly changed the MDP to produce intermediate tasks. Some of these assumed that the environment dynamics stay the same, but that the initial/terminal state distribution and reward function can change. Others made no restrictions on the differences allowed from the target task MDP. Finally, we also discussed how humans approach sequencing, to shed light on manually designed curricula in existing work. Our survey of the literature concluded with a list of open problems, which we think will serve as worthwhile directions for future work. As a budding area in reinforcement learning, we hope that this survey will provide a common foundation and terminology to promote discussion and advancement in this field. Acknowledgments We would like to sincerely thank Brad Knox, Garrett Warnell, and the anonymous reviewers for helpful comments and suggestions that improved the presentation of many ideas in this article. Part of this work has taken place in the Learning Agents Research Group (LARG) at the Artificial Intelligence Laboratory, The University of Texas at Austin. LARG research is supported in part by grants from the National Science Foundation (CPS-1739964, IIS-1724157, NRI-1925082), the Office of Naval Research (N00014-18-2243), Future of Life Institute (RFP2-000), Army Research Office (W911NF-19-2-0333), DARPA, Lockheed Martin, General Motors, and Bosch. The views and conclusions contained in this document are those of the authors alone. Peter Stone serves as the Executive Director of Sony AI America and receives financial compensation for this work. The terms of this arrangement have been reviewed and approved by the University of Texas at Austin in accordance with its policy on objectivity in research. Part of this work has taken place in the Sensible Robots Research Group at the University of Leeds, which is partially supported by the Engineering and Physical Sciences Research Council of the UK (EP/R031193/1, EP/S005056/1), and the British Council. Part of this work has taken place in the Control, Robotics, Identification and Signal Processing (CRISP) Laboratory at Tufts University which is partially supported by DARPA (W911NF-19-2-0006), the Verizon Foundation, PTC Inc., and the Center for Applied Brain and Cognitive Sciences (CABCS). 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2024-09-04T02:54:58.913772

2003.04978

arxiv-papers

# Fake News Detection with Different Models Sairamvinay Vijayaraghavan <EMAIL_ADDRESS> &Zhiyuan Guo <EMAIL_ADDRESS> &Ye Wang <EMAIL_ADDRESS> John Voong <EMAIL_ADDRESS> &Wenda Xu <EMAIL_ADDRESS> Armand Nasseri <EMAIL_ADDRESS> &Jiaru Cai <EMAIL_ADDRESS> Linda Li <EMAIL_ADDRESS> &Kevin Vuong <EMAIL_ADDRESS> &Eshan Wadhwa <EMAIL_ADDRESS> ###### Abstract Problem: The problem we intend to solve is modelled as a binary classification problem. We intend to find the relation in the words and the context in which the words appear within the text and how it could be used to classify texts as real (negative cases) or fake (positive). High-level description: Many news sources contain false information and are therefore “fake news.” Because there is a lot of “fake news” articles and fabricated, misleading information on the web, we would like to determine which texts are legitimate (real) and which are illegitimate (fake). To solve this as a binary classification problem, we investigate the effectiveness of different Natural Language Processing models which are used to convert character based texts into numeric representations such as TFIDF, CountVectorizer and Word2Vec models and find out which model is able to preserve most of the contextual information about the text used in a fake news data set and how helpful and effective it is in detecting whether the text is a fake news or not. Results:We find that out of the three pre-training vectorizing algorithms, Word2Vec performs comparatively the worst in general and the CountVectorizer performs slightly better than the TF-IDF models in most of the cases. Out of the five fine-tuning algorithms, neural networks (ANNs and LSTMs) perform better. A combination of cv with LSTM achieves the best performance. Contribution to the machine learning field: We presented a simple model which can be used to classify a given text as “real” or “fake” mostly accurately. This form of pre-training embedding algorithms and then fine-tuning on the downstream supervised task (of binary classification) proves to be efficient and effective in classifying susceptible news text. ## 1 Introduction For this report, we are exploring the field of natural language processing, which is the broad study of how computers and machines can understand human to human communication and how texts are analyzed based on contextual information by machines. In particular, we are using natural language processing to classify news articles as real news or “fake news”. Fake news is misinformation masked under the guise of a real news article, and is used to deceptively influence people’s beliefs. For this report, we are classifying news articles as “real” or “fake”, which will be a binary classification problem - classifying the samples as a positive (with fake news) or negative (not fake news) sample. Many studies have used machine learning algorithms and build classifiers based on features like content, the author’s name and job-title, using lots of models like the convolutional neural network (CNN), recurrent neural network (RNN), feed- forward neural network (FFNN), long-short term memory (LSTM) and logistic regression to find the most optimal model and return its results. In [1], the author built a classifier using natural language processing and used models like CNN, RNN, FFNN, and Logistic Regression and concluded that the CNN classifiers could not be as competitive as the RNN classifiers. The authors in [2] think that their study can be improved by having more features like knowing the history of lies spoken by the news reporter or the speaker. Moreover, apart from the traditional machine learning methods, new models have also been developed. One of the newer models, TraceMiner, creates an LSTM-RNN model inferring from the embedding of social media users in the social network structure to propagate through the path of messages and has provided high classification accuracy5. FAKEDETECTOR is another inference model developed to detect the credibility of the fake news which is considered to be quite reliable and accurate7. There also have been studies that have a different approach. A paper surveys the current state-of-the-art technologies that are imperative when adopting and developing fake news detection and provides a classification of several accurate assessment methods that analyze the text and detect anomalies3. These previous approaches lack a clear contextual analysis used in NLP. We considered the semantic meaning of each word and we feel that the presence of particular words influence the meaning. We reckoned this important since we felt the contextual meaning of the text needs to be preserved and analyzed for better classification. Other studies emphasize the user and features related to them. In [4], “45 features…[were used] for predicting accuracy…across four types: structural, user, content, and temporal,” so features included characteristics beyond the text. Article [6] "learn[s] the representations of news articles, creators and subjects simultaneously." In our project, we emphasize the content by working with articles whose labels only relate to the text, nothing outside that scope, and have used SVM, Logistic Regression, ANN, LSTM, and Random Forest. We had devised this problem into 3 different phases: pre-processing, text-to- numeric representation conversion using pre-trained algorithms, and then evaluate the models using state-of-the-art machine learning algorithms. We had analysed the data set and in particular the text part of the data explaining how it is distributed and then we converted each text into numeric representation using pre-training models such as TFIDF, CV and W2V for vector representation. Finally, we evaluated our numeric conversion data using significant machine learning algorithms such as neural networks, classification algorithms etc to perform the classification. ## 2 Methods ### 2.1 The Dataset The training data set has five features: ID, title, author, text, and label. The ID uniquely identifies the news article. The title and author are the title and author of the news article respectively. The text is the content of the article, and may be incomplete. The label indicates whether the article is reliable (real) or not (fake): label = $\begin{cases}0&\textrm{if reliable news}\\\ 1&\textrm{if fake news}\end{cases}$ The training data set contains 20800 odd number of samples. The test data set does not have labels, so we do not use it. The test data set will be selected from the training data set randomly when we are evaluating our models. In our project, since we hypothesized that the text and the words used within the text are key to distinguish between real and fake news samples, we decided to investigate only the text column. ### 2.2 Data Pre-processing #### 2.2.1 Removed numbers Within the context of a news article title or text, numbers simply quantify claims and do not change the meaning of the text. Therefore it is best to remove all numbers to minimize noise in our data. We use the string.digits string constant in Python as well as the translate and maketrans methods from Python’s string module to convert all numerical digits to an empty string, effectively removing all digits. #### 2.2.2 Removed punctuation and special characters In addition of pre-processing the textual data, we removed all characters that are not textual (not alphabets such as punctuation, extra delimiters etc.). We used the string.punctuation module in Python to find all punctuation characters. We remove all those punctuation characters from every word in the texts, with the exception of the symbols ‘#’ and ‘@’. Because these are characters used for Twitter hashtags and mentions, we handle these later. Next, we removed an assortment of special characters that don’t appear on traditional American keyboards and don’t contribute to the meaning of the tweets. The long dash (“–”), single and double Asian quotations, ellipse characters (…), and bullet points (•) all were removed for this reason. After removing all special characters, there are still a couple of pre- processing cases we account for. For these cases, we used regular expressions to detect certain patterns we wish to remove. One of the patterns is Twitter hashtags and mentions. In a news setting, Twitter hashtags and mentions are often added to try to obtain more search results and relevance, but often distract from the overall meaning of the news content itself. In our problem, we are primarily concerned with words and mostly their contextual meanings used in the text and we assumed that these unnecessary characters. To detect the hashtags and mentions, we simply use regular expressions to remove all text after a hashtag (#) or @ symbol, and stop removing text when we reach the next space. We also use regular expressions to handle em dashes (—) and more than two consecutive spaces. Em dashes are used in various linguistic contexts like joining independent clauses. They do not add to the meaning of the text, however they are surrounded by two words of different clauses, so we replaced all em dashes with a single space to maintain the integrity of each phrase. Lastly, we replace any set of two or more consecutive spaces with just one space. Proceeding further, we make all of our texts lowercase and then remove all rows that have foreign language characters in their text, since we are only interested in identifying fake news in English. To do this we used the package langid in Python to identify the language of all texts, and removed all rows with foreign characters. This finally ensures the text we preserve is only with English words with no non-alpha character. #### 2.2.3 Removed stop words Stop words are a list of the most common words in a language, such as “a”, “be”, “quite”, “should”…etc. They are often void of meaning, and does not add anything to the content. They are also most frequently present in every text. Hence, we presumed removal of stop words can have multiple advantages. For once, it decreases memory overhead, since we cut down a huge amount of text (and hence narrows down the number of features to train our models on). Second, it reduces noise, since by eliminating stop words, we are able to focus on more meaningful contents (the more distinct features between these two classes). Although it is not often the case that removing stop words are the most optimal, sometimes the information that we are looking for may be included in the stop words that we removed. For example, in most cases of language modeling, or translation, where it is important that we keep all the stop words. However, in our circumstances, we are using the semantics of the text to make a decision. In this case, we can safely remove stop words to observe the more meaningful context words. ### 2.3 Data Distribution We performed some data analysis on the text and wanted to understand how the text is distributed. We had analyzed and represented our data (text) distribution in a few different perspectives. We first analyzed the data through graphing its sentiment polarity, most popular unigram and bigram, as well as looking at the distribution of the word types. We will be comparing the graphs before and after preprocessing, which includes, stop word removal, removing punctuation and special characters, and numbers. #### 2.3.1 Sentiment Polarity Polarity Graphs before pre-processing Polarity Graphs after pre-processing For both before and after pre-processing, the distribution of the polarity of fake news sentiment and real news sentiment are mostly the same. For both fake news and real news, there are slightly more positive news than the negatives. However, there is a noticeable difference between the polarity. We can see that although not by much, fake news are a little bit more polar than real news. There are more outliers, and the data are a little bit more spread out. #### 2.3.2 Part of Speech Distribution Part of Speech Graphs before pre-processing Part of Speech Graphs after pre-processing Although the differences are slight, there is a difference in part of speech distribution between real and fake news. In fake news, there are a higher percentage of adverbs and adjectives compared to all the other parts of speech, while there is a lower percentage of proper pronoun; however, in real news, there are a higher percentage of pronoun. We can interpret this as there are more adverbs and adjectives in fakes new, and there are more pronoun in real news. Perhaps, this is indicating that fake news are more likely to use adverbs and adjectives to embellish their sentences, while real news use more pronouns to establish as reference to their legitimacy. #### 2.3.3 Unigram and Bigram Unigrams Real News Fake News Before After Before After the nt the nt to trump to Trump of people of people and clinton and clinton in hillary in hillary that said that said for like is like on new for new he time it time is World on world it state as state was election with election said government are government mr president this preseident with war by war as years before years his states was states at american you american by obama have obama from media they media Bigrams Real News Fake News Before After Before After of the mr trump of the hillary clinton in the united states in the donald trump to the new york to the united states on the mr trumps on the white house mr trump white house and the new york at the donald trump that the hillary clintons and the mrs clinton to be clinton campaign that the said mr for the clinton foundation to be york times it is secretary state he said islamic state with the nt know with the mr obama from the american people from the breitbart news by the mainstream media by the president trump at the foreign policy it was years ago hillary clinton bill clinton The comparison between the result of the top unigram and bigram before and after preprocessing demonstrates that our decision to remove stop words is the correct choice. The top unigram and bigram are all consisted of words, in other words, filler words that does supply us with any explanation. After removing the stop words, we can see that the top unigrams and bigrams become much more specific. ### 2.4 Unsupervised Pre-training to encode our texts into numeric representations #### 2.4.1 Natural Language Processing Models After text have been cleaned, they are mapped into numeric representations in form of vectors of the textual data using three pre-training algorithms (i.e. CountVectorizer, TF-IDFVectorizer, and Word2Vec). Each sample, originally consisting of all text, is converted into a vector of features. Since only the text is passed into these pre-training algorithm, this stage is unsupervised. In the cases of CountVectorizer and TfidfVectorizer, the number of features is clipped at 10000 to avoid memory overrun and overfitting (because of the large number of features (the vocabulary)). #### 2.4.2 CountVectorizer The CountVectorizer provides a simple way to both tokenize a collection of text documents and build a vocabulary of known distinct words, but also to encode new documents using that vocabulary13. Given a collection of text documents, $S$ , CountVectorizer will generate a sparse matrix $A$ of size $m$ by $n$, where $m=$ total number of documents, $n=$ total number of distinct words used in $S$. $A=\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\\ \vdots&\vdots&\vdots&\vdots\\\ a_{m1}&a_{m2}&\cdots&a_{mn}\end{pmatrix}$ This matrix is the one hot encoded representation of the different words present in the corpus. Entry $a_{ij}=$ total number of times $j$th word appears in the $i$th document. We had converted the sparse matrix into a dense one since we found that there are plenty of distinct words in the corpus which may not even be present in some of the samples and hence they may be populated with zeros. Hence, we felt that since zeros may be entirely populated, we decided to convert it to a dense matrix using the todense() method call which a dense representation of the sparse matrix. #### 2.4.3 TF-IDFVectorizer Although TF-IDF is an old algorithm, it is simple and effective to be used in the phase of pre-training11. The computation of TfidfVectorizer involves computing the product of term frequency and inverse document frequency. As the term implies, TF-IDF calculates values for each word in a document through an inverse proportion of the frequency of the word in a particular document to the percentage of documents the word appears in12. The term frequency $tf(t,d)$ calculates the proportion of times that the term $t\in V(d)$ appears in the document $d$. The vocabulary $V(d)=\sum_{t}n(t,d)$ is constructed by the document $d$. Thus, if a word $w^{\prime}$ does not appear in a document $d^{\prime}$, the term frequency $tf(t^{\prime},d^{\prime})$ in this case would be zero. The idea of the term frequency is essentially the same as CountVectorizer. $tf(t,d)=\frac{n(t,d)}{V(d)}$ $n(t,d)=\textrm{ occurrence of the word }t\textrm{ in the document }d$ Given a document collection $D$, the inverse document frequency $idf(t,D)$ is the log of the number of documents $N$ divided by $df(t,D)$, the number of documents $d\in D$ containing the term $t$. As a result, common words in $D$ will have a low term frequency score, while infrequent words will have a high term frequency. Thus, the term frequency will be very likely to separate fake news that often have less common words (even ungrammatical) from real news that usually consist of common words. $idf(t,D)=\log\Big{(}\frac{N}{df(t,D)}\Big{)}$ As a summary, TF-IDF score $w(t,d)$ for a word increases with its count, but will be counteracted if the word appears in too many documents. $w(t,d)=tf(t,d)\times idf(t,D)$ Similar to CountVectorizer, we found that most of the entries within the matrix were 0. Hence, we used the dense (todense() call) to return the dense representation of the sparse TFIDF matrix representation. #### 2.4.4 Word2Vec Word2Vec is another state of the art model used to represent words into vectors. Word2Vec is a simple neural network which basically tries to predict the next word within a context given a set of words provided. Word2Vec basically represents a vector for each word within the context and the vector representation is the weights of the particular connection from the input layer node into one of the hidden layer neurons. This information is mainly encoding the contextual information of the particular word within the corpus (collection of texts) on which we train our word2vec model. In this project, all we did was we trained the word2vec model on our current corpus. We did this because we felt that the corpus contained very specific words which had a contextual meaning completely different from what is used in general. Hence, we chose to train the corpus on the existing texts in our corpus texts over the pre-trained word2vec models such as google models. For training our word2vec models, we chose the minimum count as the average number of words in each of the texts in general, since we believed that texts which are shorter than the mean length have less context and hence we rejected those sentences to train on. We then used the number of features as the default number of features as 100 since we wanted to analyze on a short number of features. For this project, we decided on a very simple and plain approach. We obtained the vector for each sentence by summing all the vector representations for each word in the sentence only if the word belongs to the word2vec model. The summed up vector is finally divided with the number of words in the sentence since we wanted to make sure that the size of the text doesn’t affect the vector embeddings and hence we normalized our word2vec embedding. ### 2.5 Outlier Removal During outlier removal, the Isolation Forest algorithm isolates observations by randomly selecting a feature and then randomly selecting a split value between the maximum and minimum values of selected features. In Isolation Forest, an anomaly score can be calculated as the number of conditions required to separate given observation. In our outlier detections and removals, Isolation Forest has been applied to three different features. Generated from TFIDF, CV, WV. Percentages of outlier in each feature set is calculated, bar graph of percentage of training outliers are included. ### 2.6 Fine-tuning Once the representations of text are pre-trained from previous unsupervised learning, the representations are then fed into 5 different models to perform supervised learning on the downstream task. In this case, the downstream task is a binary classification of the fake news as either real or fake. A k-fold prediction error is obtained from each of the 5 models, and since we have 3 different pre-training models, we have a total of 15 models to compare. #### 2.6.1 Artificial Neural Network (ANN) We trained simple Artificial Neural Networks which contains an input layer, particular number of output layers (specified by a hyperparameter) in which each hidden layer contains the same number of neurons and the same activation function, and an output layer with just one node for the classification (real or fake) which uses sigmoid as an activation function. We chose sigmoid as the output layer activation and the binary_crossentropy as the loss since it is a binary classification problem and the use of softmax normalizes the results which is not needed for this problem and since we use only one output node to return the activation, we applied sigmoid for the output layer activation. We performed Grid Search strategy to find the best hyper-parameters such as activations, optimizers, number of hidden layers and number of hidden neurons. We had used Keras Sequential model and we used Dense Layers which contains connections to every hidden node in the next layer. Due to the limitation of computing resource, the grid search for Neural Networks is divided into three sequential steps. Instead of performing grid search on all the hyperparameters all at once, we chose to do grid search for the activations for the hidden layers, optimizers and the number of hidden layers and hidden neurons (done together). We coupled the number of hidden layers and the number of neurons since we believed that each of these hyperparameters interact with each other in improving the model training. We also did a K-fold Split for 3 splits at each step and picked the best hyperparameters which renders the highest accuracy. #### 2.6.2 Long Short Term Memory networks (LSTMs) Long Short Term Memory networks (LSTMs) is a special recurrent neural network (RNN) introduced by Hochreiter & Schmidhuber (1997)8. (Christopher Olah. “Understanding LSTM Networks.”) The chain-like nature of an RNN allows information to be passed from the beginning all the way to the end. The prediction at time step $t$ depends on all previous predictions at time step $t\textquoteright<t$. However, when a typical RNN is used in a larger context (i.e. a relatively large time steps), the RNN suffers from the issue of vanishing gradient descent 9. LSTMs, a special kind of RNN, can solve this long-term dependency problem. (Christopher Olah. “Understanding LSTM Networks.”) Each cell in a typical LSTMs network contains 3 gates (i.e., forget gate, input gate, and output gate) to decide whether or not information should be maintained in the cell state $C_{t}$. For CountVectorizer and TfidfVectorizer, each sample of text is converted into a 1-d feature vector of size 10000. As a result, the number of time steps (i.e. the maximum amount of word vectors for each sample) for these two can only be set to 1, as the pre-trained representations are done at the sample’s level. By contrast, the number of time steps for Word2Vec can either be 1, if we simply take an average of the word embeddings, or the length of the sentence, where each word has an embedding and thus the pre-trained representations are done at the word’s level. We choose the approach with 1 timestep in our model because it requires less computation power. Meanwhile, we also do the length of the sentence, and 200 time steps are chosen as 200 is close to the mean amount of words in each sample and it is a fairly common choice in practice. However, since we do not have enough computation power to fine-tune (grid search) our model, we leave it out for our model and include it only in the final section. In the LSTM layer, a dropout rate of 0.2, a common choice in practice10 , is used to prevent overfitting. Grid search is performed in order to pick decent values of hyperparameters, including the number of hidden units in the LSTM layer, the number of hidden layers, the activation functions and the number of nodes in the hidden layer, and the optimizer. Relatively small numbers of hidden layers (i.e., {0, 1, 2}) and nodes (i.e., {200, 400, 600}) are selected as the basis for grid search, because this is a simple binary classification task and too many of them would cause overfitting. Due to the limitation of computing resource, the grid search for LSTMs is divided into four sequential steps. Instead of performing grid search on all the hyperparameters all at once, the grid search is first done on the number of hidden layers and all other hyperparameters are randomly selected from the subset. Then, the grid search is done on the number of nodes in the hidden layer(s), using the best number of hidden layer found in step 1. The grid search completes when all four steps are finished. In each step we used K-fold cross validation with $K=3$. #### 2.6.3 Random Forest A random forest is an ensemble classifier that estimates based on the combination of different decision trees. So random forest will fit a number of decision tree classifiers on various subsamples of the dataset. A random best subsets are built by each tree in the forest. In the end, it gives the best subset of features among all the random subsets of features. In our project, 3 random forest algorithms have been applied with models count vectorizer, tfidf and word-to-vector. Random forest algorithm requires 4 hyperparameters to tune, such as the number of trees in the forest (i.e., {200, 400, 800}); the maximum depth of the tree (i.e., {1,5,9}); the minimum number of samples required to be at a lead node (i.e., {2, 4}); The minimum number of samples at each leaf node has the effect of smoothing the model, especially during regression; the minimum number of samples required to be at a leaf node (i.e., {5, 10}). All parameters are applied to grid search and in the end, the best set of parameters can be determined as we used K-fold cross validation with $K=3$. #### 2.6.4 Logistic Regression Logistic regression is a statistical machine learning algorithm that classifies the data by considering outcome variables on extreme ends and this algorithm is providing a discriminatory line between classes. Compared to another simple model, linear regression, which requires hard threshold in classification, logistic regression can overcome threshold values for a large dataset. Logistic regression produces a logistic curve, which is limited to values between 0 to 1, by adding sigmoid function in the end. In regards to our project, three logistic regressions have been applied with models CountVectorizer, TF-IDF and Word2Vec. We did grid search on the solvers, including newton-cg, sag, lbfgs and liblinear. Grid search is also performed on the inverse of regularization parameter with values being {0, 4, 10}. Best parameter sets can be determined as we used K-fold cross validation with $K=3$. #### 2.6.5 Support Vector Machine (SVM) SVM is a supervised machine learning algorithm in which a hyperplane is created in order to separate and categorize features. The optimal hyperplane is usually calculated by creating support vectors on both sides of the hyperplane in which each vector must maximize the distance between each other. In other words, the larger the distance between each vector around the hyperplane, the more accurate the decision boundary will be between the categories of features. In regards to our project, we fit 3 support vector machines on CountVectorizer, TfidfVectorizer, and WordToVectorizer. An SVM requires specific parameters such as a kernel type, $C$, maximum iterations, etc. In our case, we needed to determine the optimal $C$ as well as the optimal kernel for each fit. We used K-fold cross validation with $K=3$. A grid search of kernel types and $C$ was performed in order to give us the most accurate svm model. The parameters we used for each kernel were linear and rbf while the values we used for $C$ were 0.25 ,0.5, and 0.75. Once the grid search was completed for these hyperparameters, the model was evaluated with the most optimal hyperparameters using cross validation of 3 splits. ## 3 Results Grid Search Results CountVectorizer TF-IDF Word2Vec SVM Kernel = Linear Kernel = Linear Kernel = Linear C = 0.25 C = 0.75 C = 0.75 Logistic Regression Solver = sag Solver = sag Solver = newton-cg C = 21.54 C = 7.74 C = 3593.81 Random Forest Max Depth = 9 Max Depth = 9 Max Depth = 9 Min_samples_leaf = 2 Min_samples_leaf = 4 Min_samples_leaf = 2 Min_samples_split = 10 Min_samples_split = 5 Min_samples_split = 10 N_estimators = 200 N_estimators = 400 N_estimators = 400 ANN Activation = relu Activation = sigmoid Activation = relu Optimizer = Adam Optimizer = Adam Optimizer = Adam Hidden_layers = 2 Hidden_layers = 3 Hidden_layers = 1 Num_Neurons = 600 Num_Neurons = 400 Num_Neurons = 600 LSTM Activation = sigmoid Activation = sigmoid Activation = relu Optimizer = Adam Optimizer = Adam Optimizer = Adam Hidden_layers = 2 Hidden_layers = 2 Hidden_layers = 2 Memcells = 200 Memcells = 200 Memcells = 200 Num_Neurons = 200 Num_Neurons = 600 Num_Neurons = 600 Mean Test Scores SVM ANNs LSTMs LOGISTIC RANDOM FOREST CV 93.06% 94.29% 94.88% 94.45% 87.64% TFIDF 94.58% 93.73% 93.89% 94.79% 87.64% Word2Vec 91.17% 93.06% 92.29% 91.30% 88.60% ANN Loss and Accuracy LSTM Loss and Accuracy The model is evaluated using a 3-fold of cross validation. Out of the fifteen models, CountVectorizer with LSTMs performs the best. Word2Vec performs the worst among the three pre-training algorithms. Random forest performs the worst among the five fine-tuning algorithms. ## 4 Discussion Among our three pre-training models, CountVectorizer achieves in general the best performance comparatively and Word2Vec performs relatively poor amongst the three models. The essential idea behind both CountVectorizer and TF-IDF is computing a score which depends on the frequency of the word belonging to the vocabulary. However, comparing to CountVectorizer, the TF-IDF includes an extra inverse document frequency that “penalizes” (apparently masks) the contextual meaning within the words that appear more frequently across documents. They represent the importance of the word within a document. The results may imply that even though the penalization is smoothed by a log function, the punishment may be too high. The results also show that in general neural networks do the best consistently, as neural networks serve as a powerful universal approximator. However, the loss and accuracy plots show that we are using too many epochs and thus have the issue of overfitting. This is because our pre-training model is already very strong so it learns a good contextual representation of text. As a result, the epochs needed for downstream task are not much. In addition, one thing to note is that logistic regression also performs very well. This implies that our data are mostly linearly separable. While neural networks can fit the data very well, but they run the risk of overfitting the data. As a result, neural networks are not as good as SVM and Logistic Regression for TF- IDF. A combination of CountVectorizer and LSTMs is the best among all the models. While LSTMs with one timestep are very similar to ANN in terms of architecture, LSTMs have gates and a tanh activation function inside the module. This different design may let LSTMs perform slightly better than ANN. Word2Vec does not perform well. One reason is that we are simply taking an average of the word embedding vectors to get a generalized vector representation of each sample of paragraph. Taking an average fails to represent the dependencies between words. Another reason is that we do not use pre-trained Word2Vec embeddings available online from huge corpus but instead build our own from the dataset. While we thought that building our own Word2Vec would make the model specific to this task, the results show that Word2Vec may need to be built from larger dataset. ## 5 Conclusion This report provides a fairly simple approach to encode texts and how the presence of words in general impacts the classification of texts as real and fake. We achieved high accuracy results in most of our algorithms and in particular neural networks generally do better than the others. What’s worth noting is that our LSTMs only use a timestep of 1 and are essentially multi-layer perceptrons. Still, as mentioned is the LSTM’s method section, the LSTMs with the real recurrence are performed by using Word2Vec for representations at the word’s level. In this case, each word has its own vector, and a sample will be a collection of vectors and thus a 2-D matrix. As mentioned before, each vectorized word will become a timestep, and a total of 200 timesteps is used (If the paragraph has more than 200 words, only the first 200 words will be selected). We run our model and get the following results. The results seem solid, but this approach is not included in our model because it takes too much time to run and we do not have time to fine-tune the hyperparameters. But in future work, we believe that using LSTMs with real recurrence will give an even better results. While we achieve great performance in this dataset, the question remains as to whether X (to be replaced by the best model) can still perform well in tasks that classify news into more than two categories, such as the Fake News Challenge. In that case, a simple unidirectional LSTMs may not be so well and may need to be replaced by a bidirectional one. In addition, it would be interested to know how well our pre-trained model performs in other downstream tasks, such as Spam Detection. Lastly, in our model, the pre-training is done on the dataset given (will make the model specific to the task), instead of on the big corpus available online, such as Google’s pre-trained Word2Vec model. If the task were a classification of four or eight categories, pre-trained model on large corpus may perform better as the model is pre-trained on more words. We can also try to improve the training by using different word embeddings. While we only chose only 3 different types of embeddings, we could have tried different embeddings such as GloVe and the features used are entirely dependent only on context words. We can use different forms for encoding texts which can be used to be trained using these algorithms to achieve a better model. In another State-of-the-art pre-trained models can be used if the task is no longer a binary classification. Models like Transformer and BERT will be strong candidates as they have learned a very strong representation that takes the context into account when computing an embedding for a word. Unlike LSTMs whose sequential nature prohibits parallelization, the Transformer and the BERT can achieve parallelization by replacing recurrence with the attention mechanism. Thus, they require less computation power and can be easily fine- tuned in downstream tasks. ## 6 Appendix ## Github Repo https://github.com/Sairamvinay/Fake-News-Dataset ## Author Contributions Sairamvinay Vijayaraghavan: Project Planning, Problem Formation, DataSet Search, POS Distribution graph, Code for CountVectorizer, Word2Vec, ANN, Randomforest,To parse csv files (readdata), Code integration for TextVectorizer, Grid Search model running, ROC model running, Code Base Cleanup and management (further cleanup), PowerPoint Checking, Report Analysis for W2V, ANN, Report editing Zhiyuan Guo: Project Planning, DataSet Search, Polarity Graphs, Code for LSTM, RandomForest, Adding Functionality and Readability in each of the scripts, Code Integration, Grid Search model running, ROC model running, PowerPoint Development, Report Analysis for TFIDF and LSTM, Report Analysis for the Abstract, the Discussion, Conclusion, Pipeline Diagram, Report editing Ye Wang: Project Planning, DataSet Search, Code for TFIDF, PCA, Grid Search model running, ROC model running, Report Integration into Latex, Report Analysis of the Results (table creations), Report Analysis for the Outlier Removal, Random Forest, Report editing John Voong: Word2Vec, DataCleanup (StopWord Cleanup), Grid Search model running, ROC model running, PowerPoint Development, Report Analysis for W2V, Pipeline Diagram, Report editing, Paper structure Wenda Xu: Code for PCA, ROC model running, Code Base Cleanup and management, PowerPoint Development, Report Analysis about Count Vectorizer, Report Analysis about Logistic Regression Armand Nasseri: Project Planning, Dataset search, Code for SVM, Data Cleanup (StopWord Cleanup), ROC model running, PowerPoint Development, Report Analysis about SVM Jiaru Cai: Outlier Removal, Accuracy and Loss Plots for Neural Network, PowerPoint Framework Kevin Vuong: DataCleanup (remove punctuations), Code for Logistic Regression, Grid Search model running, PowerPoint Cleanup, Report Analysis about Data Cleanup, Introduction and Abstract Linda Li: Unigram and Bigram analysis, Code for ROC plots, Report Analysis of the Data Cleanup section, Graph analysis Eshan Wadhwa: Related Work, References and Citation (Introduction and Field research), Report Editing, PowerPoint slides, ## References [1] Samir Bajaj, “The Pope Has a New Baby!” Fake News Detection Using Deep Learning”, Winter 2017, https://pdfs.semanticscholar.org/19ed/b6aa318d70cd727b3cdb006a782556ba657a.pdf [2] Arjun Roy, Kingshuk Basak, Asif Ekbal, and Pushpak Bhattacharyya, “A Deep Ensemble Framework for Fake News Detection and Classification”, 12 November 2018, https://arxiv.org/pdf/1811.04670.pdf [3] Niall J. Conroy, Victoria L. Rubin, and Yimin Chen, “Automatic Deception Detection: Methods for Finding Fake News”, November 2015, https://asistdl.onlinelibrary.wiley.com/doi/epdf/10.1002/pra2.2015.145052010082. [4] Liang Wu and Huan Liu, “Tracing Fake-News Footprints: Characterizing Social Media Messages by How They Propagate”, February 2018, http://www.public.asu.edu/~liangwu1/WSDM18_TraceMiner.pdf [5] Adrian Colyer, “Tracing fake news footprints: characterizing social media messages by how they propagate”,the morning paper, February 2018, https://blog.acolyer.org/2018/02/19/tracing-fake-news-footprints- characterizing-social-media-messages-by-how-they-propagate/ [6] Kai Shu, Amy Sliva, Suhang Wang, Jiliang Tang and Huan Liu, “Fake News Detection on Social Media: A Data Mining Perspective”, August 2017, https://arxiv.org/abs/1708.01967 [7] Jiawei Zhang, Bowen Dong and Philip S. Yu, “FAKEDETECTOR: Effective Fake News Detection with Deep Diffusive Neural Network”, August 2019, https://arxiv.org/pdf/1805.08751.pdf [8] Sepp Hochreiter and Jurgen Schmidhuber, “Long short-term memory”, November 1997, http://www.bioinf.jku.at/publications/older/2604.pdf [9] Yoshua Bengio, Patrice Simard, and Paolo Frasconi. “Learning long-term dependencies with gradient descent is difficult”, March 1994, http://www.comp.hkbu.edu.hk/~markus/teaching/comp7650/tnn-94-gradient.pdf [10] Gaofeng Cheng, Vijayaditya Peddinti, Daniel Povey, et al., “An Exploration of Dropout with LSTMs”. August 2017, https://www.danielpovey.com/files/2017_interspeech_dropout.pdf [11] Juan Ramos. “Using tf-idf to determine word relevance in document queries”, December 2003, https://www.cs.rutgers.edu/~mlittman/courses/ml03/iCML03/papers/ramos.pdf [12] Gerard Salton and Christopher Buckley. “Term-weighting approaches in automatic text retrieval”, January 1988, https://www.sciencedirect.com/science/article/abs/pii/0306457388900210 [13] Jason Brownlee. “How to Prepare Text Data for Machine Learning with scikit-learn”, August 2019, https://machinelearningmastery.com/prepare-text-data-machine-learning-scikit- learn/

2024-09-04T02:54:58.923713

2003.04986

arxiv-papers

# Investigating an approach for low resource language dataset creation, curation and classification: Setswana and Sepedi ###### Abstract The recent advances in Natural Language Processing have been a boon for well represented languages in terms of available curated data and research resources. One of the challenges for low-resourced languages is clear guidelines on the collection, curation and preparation of datasets for different use-cases. In this work, we take on the task of creation of two datasets that are focused on news headlines (i.e short text) for Setswana and Sepedi and creation of a news topic classification task. We document our work and also present baselines for classification. We investigate an approach on data augmentation, better suited to low resource languages, to improve the performance of the classifiers. languageresourceLanguage Resources Investigating an approach for low resource language dataset creation, curation and classification: Setswana and Sepedi Vukosi Marivate${}^{1}{}^{,}{}^{2}$, Tshephisho Sefara2, Vongani Chabalala3, Keamogetswe Makhaya4, Tumisho Mokgonyane5, Rethabile Mokoena6, Abiodun Modupe${}^{7}{}^{,}{}^{1}$ --- University of Pretoria1, CSIR2, University of Zululand3, University of Cape Town4, University of Limpopo5, North-West University6, University of the Witwatersrand7 <EMAIL_ADDRESS><EMAIL_ADDRESS> Abstract content ## 1\. Introduction The most pressing issue with low-resource languages is of insufficient language resources. In this study, we introduce an investigation of a low- resource language that provides automatic formulation and customization of new capabilities from existing ones. While there are more than six thousand languages spoken globally, the openness of resources for each is extraordinarily unbalanced [Nettle, 1998]. For example, if we focus on language resources annotated on the public domain, as of November 2019, AG corpus released about $496,835$ news articles only in English languages from more than $200$ sources111http://groups.di.unipi.it/~gulli, Reuters News Dataset [Lewis, 1997] comprise an roughly $10,788$ annotated texts from the Reuters financial newswire. The New York Times Annotated Corpus [Sandhaus, 2008] holds over $1.8$ million articles, and there are no standard annotated tokens in the low-resource language. Google Translate only supports around 100 languages [Johnson et al., 2017]. A significant number of bits of knowledge focus on a small number of languages neglecting $17\%$ out of the world’s language categories label as low-resource [Strassel and Tracey, 2016], which makes it challenging to develop various mechanisms for Natural Language Processing (NLP). In South Africa several of the news websites (private and public) are published in English, even though there are 11 official languages (including English). We list the top premium newspapers by circulation as per first Quarter 2019 [Bureau of Circulations, 2019] in Table 1. We do not have a distinct collection of a diversity of languages with most of the reported datasets as existed in English, Afrikaans and isiZulu. In this work, we aim to provide a general framework that enables us to create an annotated linguistic resource for Setswana and Sepedi news headlines. We apply data sources of the news headlines from the South African Broadcast Corporation (SABC) 222http://www.sabc.co.za/, their social media streams and a few acoustic news. Unfortunately, we do not have any direct access to news reports, so we hope this study will promote collaboration between the national broadcaster and NLP researchers. Table 1: Top newspapers in South Africa with their languages Paper | Language | Circulation ---|---|--- Sunday Times | English | 260132 Soccer Laduma | English | 252041 Daily Sun | English | 141187 Rapport | Afrikaans | 113636 Isolezwe | isiZulu | 86342 Sowetan | English | 70120 Isolezwe ngeSonto | isiZulu | 65489 Isolezwe ngoMgqibelo | isiZulu | 64676 Son | Afrikaans | 62842 The rest of the work is organized as follows. Section 2. discusses prior work that has gone into building local corpora in South Africa and how they have been used. Section 3. presents the proposed approach to build a local news corpora and annotating the corpora with categories. From here, we focus on ways to gather data for vectorization and building word embeddings (needing an expanded corpus). We also release and make pre-trained word embeddings for 2 local languages as part of this work [Marivate and Sefara, 2020a]. Section 4. investigate building classification models for the Setswana and Sepedi news and improve those classifiers using a 2 step augmentation approach inspired by work on hierarchical language models [Yu et al., 2019]. Finally, Section 5. concludes and proposes a path forward for this work. ## 2\. Prior Work Creating sizeable language resources for low resource languages is important in improving available data for study [Zoph et al., 2016] and cultural preservation. If we focus our attention on the African continent, we note that there are few annotated datasets that are openly available for tasks such as classification. In South Africa, the South African Center for Digital Language Resources (SADILAR) 333www.sadilar.org has worked to curate datasets of local South African languages. There remain gaps such as accessing large corpora and data from sources such as broadcasters and news organizations as they have sizeable catalogs that are yet to make it into the public domain. In this work, we work to fill such a gap by collecting, annotating and training classifier models for news headlines in Setswana and Sepedi. As the data that we do find publicly is still small, we also have to deal with the challenges of Machine Learning on small data. Machine learning systems perform poorly in presence of small training sets due to overfitting. To avoid this problem, data augmentation can be used. The technique is well known in the field of image processing [Cubuk et al., 2019]. Data augmentation refers to the augmentation of the training set with artificial, generated, training examples. This technique is used less frequently in NLP but a number of few studies applied data augmentation. ?) use data augmentation to counteract overfitting where recurrent neural network (RNN) Encoder-Decoder is implemented specifically geared toward a low- resource setting. Authors apply data augmentation by finding words that share word stem for example fizzle and fizzling share fizzl. Then authors replace a stem with another string. ?) apply data augmentation by using synonyms as substitute words for the original words. However, ?) states that synonyms are very limited and the synonym-based augmentation cannot produce numerous different patterns from the original texts. Hence, ?) proposes contextual data augmentation by replacing words that are predicted by a language model given the context surrounding the original words to be augmented. As ?) states that these techniques are valid, they are not often used in practice because they have a high cost of implementation relative to performance gain. They propose an easy data augmentation as techniques for boosting performance on text classification tasks. These techniques involve synonym replacement, random insertion, random swap, and random deletion of a word. Authors observed good performance when using fraction of the dataset (%):1, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, as the data size increases, the accuracy also increases for augmented and original data. Original data obtained highest accuracy of 88.3% at 100% data size while augmented data obtained accuracy of 88.6% at 50% data size. In this work, we investigate the development of a 2 step text augmentation method in order to be improve classification models for Setswana and Sepedi. To do this we had to first identify a suitable data source. Collect the data, and then annotate the datasets with news categories. After the data is collected and annotated, we then worked to create classification models from the data as is and then use a word embedding and document embedding augmentation approach. ## 3\. Developing news classification models for Setswana and Sepedi Here we discuss how data was collected as well as the approach we use to build classification models. ### 3.1. Data Collection, Cleaning and Annotation Before we can train classification models, we first have to collect data for 2 distinct processes. First, we present our collected news dataset as well as its annotation. We then discuss how we collected larger datasets for better vectorization. #### 3.1.1. News data collection and annotation The news data we collected is from the SABC444http://www.sabc.co.za/ Facebook pages. The SABC is the public broadcaster for South Africa. Specifically, data was collected from the Thobela FM (An SABC Sepedi radio station)555https://www.facebook.com/thobelafmyaka/ and Motsweding FM (An SABC Setswana radio station)666https://www.facebook.com/MotswedingFM/. We scraped the news headlines that are published as posts on both Facebook pages. We claim no copyright for the content but used the data for research purposes. We summarize the datasets in Table 2. We visualize the token distributions in Sepedi and Setswana in Figures 1 and 2 respectively. Table 2: News Data Sets | Setswana | Sepedi ---|---|--- Corpus Size (headlines) | 219 | 491 Number of Tokens (words) | 1561 | 3018 Figure 1: Setswana Wordcloud Figure 2: Sepedi Wordcloud As can be seen, the datasets are relatively small and as such, we have to look at other ways to build vectorizers that can better generalize as the word token diversity would be very low. We annotated the datasets by categorizing the news headlines into: _Legal_ , _General News_ ,_Sports_ , _Other_ , _Politics_ , _Traffic News_ , _Community Activities_ , _Crime_ , _Business_ and _Foreign Affairs_. Annotation was done after reading the headlines and coming up with categories that fit both datasets. We show the distribution of the labels in both the Setswana and Sepedi data sets in Figures 3 and 4 respectively. For this work, we only explore single label categorization for each article. It remains future work to look at the multi-label case. As such, there might be some noise in the labels. Examples from the Sepedi annotated news corpus are shown next: > _Tsela ya N1 ka Borwa kgauswi le Mantsole Weighbridge ka mo Limpopo ebe e > tswaletswe lebakanyana ka morago ga kotsi yeo e hlagilego._ Traffic > > _Tona ya toka Michael Masutha,ore bahlankedi ba kgoro ya ditirelo tsa > tshokollo ya bagolegwa bao ba tateditswego dithieeletsong tsa khomisene ya > go nyakisisa mabarebare a go gogwa ga mmuso ka nko,ba swanetse go hlalosa > gore ke ka lebaka la eng ba sa swanelwa go fegwa mesomong_ Legal Figure 3: Setswana news title category distribution Figure 4: Sepedi news title category distribution The full dataset is made available online [Marivate and Sefara, 2020b] for further research use and improvements to the annotation777https://zenodo.org/record/3668495. As previously discussed, we used larger corpora to create language vectorizers for downstream NLP tasks. We discuss this next. #### 3.1.2. Vectorizers Before we get into the annotated dataset, we needed to create pre-trained vectorizers in order to be able to build more classifiers that generalize better later on. For this reason we collected different corpora for each language in such as way that we could create Bag of Words, TFIDF, Word2Vec [Mikolov et al., 2013] and FastText [Bojanowski et al., 2017] vectorizers (Table 3). We also make these vectorizers available for other researchers to use. Table 3: Vectorizer Corpora Sizes in number of lines (number of tokens) Source | Setswana | Sepedi ---|---|--- Wikipedia | 478(_21924_)888https://tn.wikipedia.org/ | 300(_10190_)999https://nso.wikipedia.org/ JW300101010http://opus.nlpl.eu/JW300.php | 874464(_70251_) | 618275(_53004_) Bible | 3110(_40497_) | 29723 Constitution111111https://www.justice.gov.za/legislation/constitution/pdf.html | 7077(_3940_) | 6564(_3819_) SADILAR121212https://www.sadilar.org/index.php/en/resources | 33144(_61766_) | 67036(_87838_) Total | 946264(_152027_) | 721977(_149355_) ### 3.2. News Classification Models We explore the use of a few classification algorithms to train news classification models. Specifically we train * • Logistic Regression, * • Support Vector Classification, * • XGBoost, and * • MLP Neural Network. To deal with the challenge of having a small amount of data on short text, we use data augmentation methods, specifically a word embedding based augmentation [Wang and Yang, 2015], approach that has been shown to work well on short text [Marivate and Sefara, 2019]. We use this approach since we are not able to use other augmentation methods such as synonym based (requires developed Wordnet Synsets [Kobayashi, 2018]), language models (larger corpora needed train) and back-translation (not readily available for South African languages). We develop and present the use of both word and document embeddings (as an augmentation quality check) inspired by a hierarchical approach to augmentation [Yu et al., 2019]. ## 4\. Experiments and Results This Section presents the experiments and results. As this is still work in progress, we present some avenues explored in both training classifiers and evaluating them for the task of news headline classification for Setswana and Sepedi. ### 4.1. Experimental Setup For each classification problem, we perform 5 fold cross validation. For the bag-of-words and TFIDF vectorizers, we use a maximum token size of 20,000. For word embeddings and language embeddings we use size 50. All vectorizers were trained on the large corpora presented earlier. #### 4.1.1. Baseline Experiments We run the baseline experiments with the original data using 5-fold cross validation. We show the performance (in terms of weighted F1 score) in the Figures 5 & 6\. We show the baseline results as _orig_. For both the Bag-of- Words (TF) and TFIDF, the MLP performs very well comparatively to the other methods. In general the TFIDF performs better. Figure 5: Baseline classification model performance for Setswana news title categorisation Figure 6: Baseline classification model performance for Sepedi news title categorisation #### 4.1.2. Augmentation We applied augmentation in different ways. First for Sepedi and Setswana word embeddings (word2vec), we use word embedding-based augmentation. We augment each dataset 20 times on the training data while the validation data is left intact so as to be comparable to the earlier baselines. We show the effect of augmentation in Figure 5 and 6 (performance labeled with _aug_ The contextual, word2vec based, word augmentation improves the performance of most of the classifiers. If we now introduce a quality check using doc2vec (Algorithm 1) we also notice the impact on the performance for Sepedi (Figure 6 _aug qual_ ). We were not able to complete experiments with Setswana for the contextual augmentation with a quality check, but will continue working to better under stand the impact of such an algorithm in general. For example, it remains further work to investigate the effects of different similarity thresholds for the algorithm on the overall performance, how such an algorithm works on highly resourced languages vs low resourced languages, how we can make the algorithm efficient etc. Input: $s$: a sentence, $run$: maximum number of attempts at augmentation Output: $\hat{s}$ a sentence with words replaced 1 def _Augment(_s,run_)_: 2 Let $\vv{V}$ be a vocabulary; 3 for _ $i$ in range(_run_) _ : 4 $w_{i}\leftarrow$ randomly select a word from $s$; 5 $\vv{w}\leftarrow$ find similar words of $w_{i}$; 6 $s_{0}\leftarrow$ randomly select a word from $\vv{w}$ given weights as distance; 7 $\hat{s}\leftarrow$replace $w_{i}$ with similar word $s_{0}$; 8 $\vv{s}\leftarrow Doc2vec(s)$; 9 $\vv{\hat{s}}\leftarrow Doc2vec(\hat{s})$; 10 $similarity$ $\leftarrow$ Cosine Similarity($\vv{s}$, $\vv{\hat{s}}$); 11 if _$similarity$ $>$ $threshold$_ : 12 return($\hat{s}$); 13 14 15 16 17 18 Algorithm 1 Contextual (Word2vec-based) augmentation algorithm with a doc2vec quality check Figure 7: Word2Vec feature based performance for news headline classification Figure 8: Confusion Matrix of News headline classification models It also interesting to look at how performance of classifiers that were only trained with word2vec features would fair. Deep neural networks are not used in this current work and as such we did not use recurrent neural networks, but we can create sentence features from - word2vec by either using: the mean of all word vectors in a sentence, the median of all word vectors in a sentence or the concatenated power means [Rücklé et al., 2018]. We show the performance of using this approach with the classifiers used for Bag of Words and TFIDF earlier in Figure 8. The performance for this approach is slightly worse with the best results for Sepedi news headline classification being with XGBoost on the augmented data. We hope to improve this performance using word2vec feature vectors using recurrent neural networks but currently are of the view that increasing the corpora sizes and the diversity of corpora for the pre-trained word embeddings may yield even better results. Finally, we show the confusion matrix of the best model in Sepedi on a test set in Figure 8. The classifier categorises _General News_ , _Politics_ and _Legal_ news headlines best. For the others there there is more error. A larger news headline dataset is required and classification performance will also need to be compared to models trained on full news data (with the article body). For the Setswana classifiers, the confusion matrix shows that the data skew results in models that mostly can categorise between categorises _General News_ and _Other_. We need to look at re-sampling techniques to improve this performance as well as increasing the initial dataset size. ## 5\. Conclusion and Future Work This work introduced the collection and annotation of Setswana and Sepedi news headline data. It remains a challenge that in South Africa, 9 of the 11 official languages have little data such as this that is available to researchers in order to build downstream models that can be used in different applications. Through this work we hope to provide an example of what may be possible even when we have a limited annotated dataset. We exploit the availability of other free text data in Setswana and Sepedi in order to build pre-trained vectorisers for the languages (which are released as part of this work) and then train classification models for news categories. It remains future work to collect more local language news headlines and text to train more models. We have identified other government news sources that can be used. On training embedding models with the data we have collected, further studies are needed to look at how augmentation using the embedding models improve the quality of augmentation. ## 6\. Bibliographical References ## References * Bojanowski et al., 2017 Bojanowski, P., Grave, E., Joulin, A., and Mikolov, T. (2017). Enriching word vectors with subword information. Transactions of the Association for Computational Linguistics, 5:135–146. * Bureau of Circulations, 2019 Bureau of Circulations, A. (2019). Newspaper circulation statistics for the period january–march 2019 (abc q1 2019). * Cubuk et al., 2019 Cubuk, E. D., Zoph, B., Mane, D., Vasudevan, V., and Le, Q. V. (2019). Autoaugment: Learning augmentation strategies from data. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 113–123. * Johnson et al., 2017 Johnson, M., Schuster, M., Le, Q. V., Krikun, M., Wu, Y., Chen, Z., Thorat, N., Viégas, F., Wattenberg, M., Corrado, G., et al. (2017). Google’s multilingual neural machine translation system: Enabling zero-shot translation. Transactions of the Association for Computational Linguistics, 5:339–351. * Kobayashi, 2018 Kobayashi, S. (2018). Contextual augmentation: Data augmentation by words with paradigmatic relations. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers), volume 2, pages 452–457. * Lewis, 1997 Lewis, D. D. (1997). Reuters-21578 text categorization collection data set. * Marivate and Sefara, 2019 Marivate, V. and Sefara, T. (2019). Improving short text classification through global augmentation methods. arXiv preprint arXiv:1907.03752. * Marivate and Sefara, 2020a Marivate, V. and Sefara, T. (2020a). African embeddings [nlp]. https://doi.org/10.5281/zenodo.3668481, February. * Marivate and Sefara, 2020b Marivate, V. and Sefara, T. (2020b). South African news data dataset. https://doi.org/10.5281/zenodo.3668489. * Mikolov et al., 2013 Mikolov, T., Sutskever, I., Chen, K., Corrado, G. S., and Dean, J. (2013). Distributed representations of words and phrases and their compositionality. In Advances in neural information processing systems, pages 3111–3119. * Nettle, 1998 Nettle, D. (1998). Explaining global patterns of language diversity. Journal of anthropological archaeology, 17(4):354–374. * Rücklé et al., 2018 Rücklé, A., Eger, S., Peyrard, M., and Gurevych, I. (2018). Concatenated power mean word embeddings as universal cross-lingual sentence representations. arXiv preprint arXiv:1803.01400. * Sandhaus, 2008 Sandhaus, E. (2008). The new york times annotated corpus. Linguistic Data Consortium, Philadelphia, 6(12):e26752. * Silfverberg et al., 2017 Silfverberg, M., Wiemerslage, A., Liu, L., and Mao, L. J. (2017). Data augmentation for morphological reinflection. In Proceedings of the CoNLL SIGMORPHON 2017 Shared Task: Universal Morphological Reinflection, pages 90–99. * Strassel and Tracey, 2016 Strassel, S. and Tracey, J. (2016). Lorelei language packs: Data, tools, and resources for technology development in low resource languages. In Proceedings of the Tenth International Conference on Language Resources and Evaluation (LREC’16), pages 3273–3280. * Wang and Yang, 2015 Wang, W. Y. and Yang, D. (2015). That’s so annoying!!!: A lexical and frame-semantic embedding based data augmentation approach to automatic categorization of annoying behaviors using# petpeeve tweets. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pages 2557–2563. * Wei and Zou, 2019 Wei, J. and Zou, K. (2019). Eda: Easy data augmentation techniques for boosting performance on text classification tasks. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pages 6383–6389. * Yu et al., 2019 Yu, S., Yang, J., Liu, D., Li, R., Zhang, Y., and Zhao, S. (2019). Hierarchical data augmentation and the application in text classification. IEEE Access, 7:185476–185485. * Zhang et al., 2015 Zhang, X., Zhao, J., and LeCun, Y. (2015). Character-level convolutional networks for text classification. In Advances in neural information processing systems, pages 649–657. * Zoph et al., 2016 Zoph, B., Yuret, D., May, J., and Knight, K. (2016). Transfer learning for low-resource neural machine translation. arXiv preprint arXiv:1604.02201.

2024-09-04T02:54:58.932182

2003.04991

arxiv-papers

# Unsupervised and Interpretable Domain Adaptation to Rapidly Filter Tweets for Emergency Services Jitin Krishnan Department of Computer Science George Mason University Fairfax, VA, USA <EMAIL_ADDRESS>Hemant Purohit Department of Information Sciences & Technology George Mason University Fairfax, VA, USA <EMAIL_ADDRESS>Huzefa Rangwala Department of Computer Science George Mason University Fairfax, VA, USA <EMAIL_ADDRESS> ###### Abstract During the onset of a natural or man-made crisis event, public often share relevant information for emergency services on social web platforms such as Twitter. However, filtering such relevant data in real-time at scale using social media mining is challenging due to the short noisy text, sparse availability of relevant data, and also, practical limitations in collecting large labeled data during an ongoing event. In this paper, we hypothesize that unsupervised domain adaptation through multi-task learning can be a useful framework to leverage data from past crisis events for training efficient information filtering models during the sudden onset of a new crisis. We present a novel method to classify relevant social posts during an ongoing crisis without seeing any new data from this event (fully unsupervised domain adaptation). Specifically, we construct a customized multi-task architecture with a multi-domain discriminator for crisis analytics: multi-task domain adversarial attention network (MT-DAAN). This model consists of dedicated attention layers for each task to provide model interpretability; critical for real-word applications. As deep networks struggle with sparse datasets, we show that this can be improved by sharing a base layer for multi-task learning and domain adversarial training. The framework is validated with the public datasets of TREC incident streams that provide labeled Twitter posts (tweets) with relevant classes (Priority, Factoid, Sentiment) across 10 different crisis events such as floods and earthquakes. Evaluation of domain adaptation for crisis events is performed by choosing one target event as the test set and training on the rest. Our results show that the multi-task model outperformed its single-task counterpart. For the qualitative evaluation of interpretability, we show that the attention layer can be used as a guide to explain the model predictions and empower emergency services for exploring accountability of the model, by showcasing the words in a tweet that are deemed important in the classification process. Finally, we show a practical implication of our work by providing a use-case for the COVID-19 pandemic. ###### Index Terms: Social Media, Crisis Analytics, Text Classification, Unsupervised Domain Adaptation, Interpretability Figure 1: Problem Statement: Interpretably predict labels for tweets collected during an ongoing crisis using only the past crisis data, given a) unavailability of labeled data in the ongoing event, and b) need for interpretability of machine reasoning behind data filtering for emergency managers. ## I Introduction During the sudden onset of a crisis situation, social media platforms such as Twitter provide valuable information to aid crisis response organizations in gaining real-time situational awareness [1]. Effective analysis of important information such as affected individuals, infrastructure damage, medical emergencies, or food and shelter needs can help emergency responders make time-critical decisions and allocate resources in the most effective manner [2, 3, 4]. Several machine learning systems have been deployed to help towards this humanitarian goal of converting real-time social media streams into actionable knowledge. Classification being the most common task, researchers have designed models [5, 6, 7, 3] that classify tweets into various crisis- dependent categories such as priority, affected individuals, type of damage, type of assistance needed, usefulness of the tweet, etc. Social media streams contain short, informal, and abbreviated content; with potential linguistic errors and sometimes contextually ambiguous. These inherently challenging properties of tweets make their classification task and formulation less trivial when compared to traditional text classification tasks. In this paper, we address two practically important and underdeveloped aspects of current research in social media mining for crisis analytics to classify relevant social web posts: a) a fully unsupervised domain adaptation, and b) interpretability of predictions. A fully unsupervised domain adaptation uses no data from the ongoing crisis to train the model. Nguyen et al., 2016 [5] showed that their convolutional neural network (CNN) model does not require feature engineering and performed better than the state-of-the-art methods; one of their models being completely unsupervised [5]. Similarly, Alam et al., 2018 [6] designed a CNN architecture with adversarial training on graph embeddings, but utilizing unlabeled target data. Our goal is to construct an unsupervised model that does not require any unlabeled target data with the capability of being interpretable. We specifically address the problem of data sparsity and limited labels by designing a multi-task classification model with domain adversarial training; which, to the best of our knowledge, is not explored in social media mining for crisis analytics. Another crucial component of our model is interpretability. In prior works, when a top performing model produces an accuracy of $78\%$, for instance, it is unclear how trustworthy it is and what features are deemed important in the model’s decision-making process. An interpretable model like ours can present with a convincing evidence of which words the classifier deems important when making a certain prediction, and helps ensure reliability for domain users, e.g., emergency managers. Contributions: a) To address the problems of data sparsity and limited labels, we construct a customized multi-task learning architecture (MT-DAAN) to filter tweets for crisis analytics by training four different classification tasks (c.f. examples in Fig. 3) across ten different crisis events under domain shift. This multi-task domain adversarial model consists of dedicated attention layers for each task for interpretability and a domain classifier branch to promote the model to be domain-agnostic. b) We demonstrate that the attention layers provide interpretability for the predictions made by the classifiers; with the goal to aid emergency services in a more meaningful way. c) We empirically validate the performance of the underlying single-task attention-based neural network architecture by comparing it to the state-of- the-art methods, for improving generalizability and interpretability for domain adaptation in unsupervised tweet classification tasks in general. d) Additionally, through experiments, we show that deep networks struggle with small datasets, and that this can be improved by sharing the base layer for multi-task learning and domain adversarial training. ## II Related Work and Background ### II-A Domain Adaptation Domain Adaptation in text classification tasks has a long line of fruitful research [8, 9, 10] that try to minimize the difference between the domains so that a model trained solely on one domain is generalizable to unseen test data from a completely different domain. With the introduction of Domain- Adversarial training of Neural Networks (DANN) [11], many state-of-the-art models now utilize unlabeled target data to train classifiers that are indiscriminate toward different domains. The speciality of this architecture is that it consists of an extra branch, which performs domain classification using unlabeled data from different domains. Thus, both task and domain classifiers share some bottom layers but have separate layers towards the top. A negative gradient (gradient reversal) from the domain classifier branch is back-propagated to promote the features at the lower layers of the network incapable of discriminating the domains. Recent works such as Adversarial Memory Network (AMN) [12] utilizes attention, in addition to DANN, to bring interpretability to capture the pivots for sentiment classification. Hierarchical Attention Network (HATN) [13] expands upon AMN by first extracting pivots and then jointly training networks for both pivots and non- pivots. For filtering social web data for crisis analytics, these models do not suffice and need customized expansions due to the following reasons: a) Collecting and using large unlabeled target data from the new/ongoing crisis event may not be practically viable, thus, we aim for a fully unsupervised modeling. b) Having access to unlabeled data from multiple crisis events can alleviate the above problem to an extent by using it to train the domain classifier branch to push the model to be domain independent. c) Due to the low-resource nature of the dataset, binary classifiers may miss important lower level features that can be potentially improved by a multi-task model that shares the lower layers of the network for all the tasks. This is also evident from our results in Table III and IV, which show that deep models that perform much better than simple models on Amazon reviews do not significantly outperform them on TREC tweet dataset for crises. ### II-B Multi-Task Learning Multi-Task Learning (MTL) solves multiple tasks at the same time with a goal to improve the overall generalization capability of the model [14]. Within the context of Deep Learning, MTL is performed by sharing (or constraining) lower level layers and using dedicated upper level layers for various tasks. A rich overview of MTL in Deep Neural Networks is presented by Ruder (2017) [15]. MTL has been a successful strategy over the past few years for many research explorations such as relationship networks [16] in computer vision and Sluice networks [17] in natural language processing. Similar problems in domain adaptation of semantic classification and information retrieval were addressed by jointly learning to leverage large amounts of cross-task data [18]. In low resource datasets such as for crises, the chance of overfitting is very high. Thus, it seems intuitively better for the model to find a shared representation capturing different tasks and not just one, such that feature commonalities across tasks can be exploited. ### II-C Attention Mechanism Attention mechanism [19], originally designed for machine translation problems, has become one of the most successful and widely used methods in deep learning that can look at a part of a sentence at a time like humans. This is particularly useful because of its ability to construct a context vector by weighing on the entire input sequence unlike previous sequence-to- sequence models [20] that used only the last hidden state of the encoder network (typically BiLSTM [21], LSTM [22], or GRU [23]). For example, in a sentence, the context vector is a dot product of the word activations and weights associated with each word; thus leading to an improved contextual memorization, especially for long sentences. Our method incorporates such attention mechanisms to enhance interpretability of the classifier. ## III Methodology ### III-A Problem Statement: Unsupervised Domain Adaptation for Crisis Tweet Classification Using notations in Table I, consider a set $C$ of all crisis events such as Guatemala Earthquake or Typhoon Yolanda. The task of unsupervised domain adaptation for crisis analytics is to train a classifier for a specific target crisis ($c_{t}$) using labeled ($L_{C-c_{t}}$) and unlabeled ($U_{C-c_{t}}$) data from all other crises; where $C-c_{t}$ denotes the set of all crisis events minus the target crisis. We assume that no data record from the target crisis is available for training. Following the traditional domain adaptation terminology, $X_{s}$ = $L_{C-c_{t}}$ represents the labeled data from the source domain $S$ and $Y_{s}$ = $y_{C-c_{t}}$ represents the ground truth labels on which the classifier is trained. And, $X_{t}$ = $L_{c_{t}}$ represents the labeled data from the target domain $T$ and $Y_{t}$ = $y_{c_{t}}$ represents the ground truth labels; both of which are only used for testing the classifier. $X_{d}$ = $U_{C-c_{t}}$ represents the unlabeled data from different domains minus the target. To summarize: Input: $X_{s}$, $Y_{s}$, $X_{d}$ Output: $Y_{t}^{pred}$ $\leftarrow$ $predict(X_{t})$ Notation | Definition ---|--- $C$ | Set of all crisis events $\\{c_{1},c_{2},...\\}$ $L_{c_{k}}$ | Set of labeled data from the event $c_{k}$ $y_{c_{k}}$ | Set of ground truth labels for $L_{c_{k}}$. $m$ | Number of tasks (Number of bits in each label) $U_{c_{k}}$ | Set of unlabeled data from the event $c_{k}$ $T_{x}$ | Number of words in a sentence $x^{<k>}$ | $k$-th word of a sentence $\alpha^{<k>}$ | attention from $k$-th word $a^{<k>}$ | BiLSTM activation from $k$-th word TABLE I: Notations ### III-B Overview In the following sections, we describe three models: Single-Task Attention Network (ST), Single-Task Domain Adversarial Attention Network (ST-DAAN), and Multi-Task Domain Adversarial Attention Network (MT-DAAN). ST is the model we adopt from [24] to build the single-task attention based baseline. ST-DAAN is constructed on top of ST to make the model domain agnostic by performing adversarial training using gradient reversal. Finally, MT-DAAN is constructed on top of ST-DAAN with dedicated attention layers for each task on a shared BiLSTM layer. This is shown in Figure 2. Figure 2: Fully Unsupervised Domain Adaptation Set-up for Multi-Task Crisis Tweet Classification. ### III-C Single-Task Attention Network (ST) We first describe the single-task attention network [24] on top of which we build our models. This model aligns with our goals of interpretability and unsupervised domain adaptation. This BiLSTM based model with Attention gives us three main advantages: 1. 1. Unlike several existing domain adaptation methods that use unlabeled target data to train the domain adversarial component via gradient reversal, this method is a fully unsupervised baseline which also can be customized for multi-task learning. 2. 2. The method uses attention mechanism which in turn weighs each word in a sentence based on its importance. This can be directly utilized for interpretability. 3. 3. The method also runs much faster (only a few minutes), i.e. highly useful in crisis times, as compared to the top performing semi-supervised models such as HATN [13] (hours). This model [24] consists of a BiLSTM layer which produces $T_{x}$ activations, each corresponding to a word in the sentence. These activations are passed through dense and softmax layers and are combined by dot product to produce the context vector $\sum_{k=1}^{T_{x}}\alpha^{<k>}a^{<k>}$, where $a^{<k>}$ is the BiLSTM activation from $k$-th word and $\alpha^{<k>}$ is the attention weight of $k$-th word. Sentences with words greater than $T_{x}$ are stripped and those with words lower than $T_{x}$ are padded. This single-task ($m=1$) attention network is the building block with which rest of the following models are constructed. The single-task binary cross entropy loss function is shown below. $\footnotesize L_{T}=-\frac{1}{N}\sum_{i=1}^{N}[y_{i}\log\hat{y_{i}}+(1-y_{i})\log(1-\hat{y_{i}})]$ (1) where $T$ represents the task, $y$ is the true label, and $\hat{y}$ is the predicted label. ### III-D Single-Task Domain Adversarial Attention Network (ST-DAAN) To study the specific contribution of domain adversarial training, we construct a secondary baseline over the ST architecture by constructing an additional branch with gradient reversal layer which is represented by the green blocks in Figure 2. This is a single-task binary classifier with $m=1$. Domain Adversarial Training of Neural Networks (DANN) [11] was introduced with a goal to confuse the classifier by back-propagating a negative gradient from a separate domain classifier branch (right-most branch, as shown in Figure 2). This makes the classifier agnostic to difference in domains. This back- propagation is implemented using a gradient reversal layer [11] which does nothing during the forward pass but pushes a negative gradient ($-\lambda\frac{\partial L_{d}}{\partial\theta_{f}}$) during the backward (gradient update) pass. $L_{d}$ is the domain classification loss, $\lambda$ is the strength of the reversal, and $f$ represents the lower level layers or features over which the negative gradient update is performed. In our architecture, the goal is to make the BiLSTM layer indiscriminate towards various crisis domains such that the multi-task classification does not depend on the domain from which the tweet/sentence is coming from. The ST-DAAN loss function is shown below. $\footnotesize L_{T}^{\prime}=L_{T}+w_{d}L_{d}$ (2) where $w_{d}$ is the domain adversarial loss weight. $L_{d}$ represents the categorical cross entropy loss for multi-domain discriminator shown below. $\footnotesize L_{d}=-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{|C-c_{t}|}[y_{ij}\log\hat{y_{ij}}]$ (3) where $C-c_{t}$ is the set of all crisis events without the target event. ### III-E Multi-Task Domain Adversarial Attention Network (MT-DAAN) Building on top of ST-DAAN, we construct MT-DAAN, which is intended to classify problems with multiple tasks or labels. For each task, a dedicated attention layer is allocated from which it predicts binary labels. The BiLSTM layer remains exactly the same as in the single-task model but multiple attention blocks are added for each task along with a domain classifier. In the architecture decision process, we first investigated a multi-label classifier where all layers are shared with the final softmax layer making multi-label predictions. In low resource settings, constructing a multi-label classifier using a shared architecture is challenging for two reasons: a) jointly balancing positive and negative samples across all classes is not trivial and potentially challenging to make it extensible when new classes need to be added, and b) attention layer may not always produce class-specific insights as the weights are assigned to train for the combination of labels. On the other hand, in the multi-task architecture with separate attention layers, it is easy to add more classes. If some classes require more training, it is trivial to further tune a model specific to that class. More importantly, $context^{<t_{j}>}$ vector for $j$-th task identifies the influential words from each sentence for that specific task. The complete architecture is shown in Figure 2. MT-DAAN loss function is shown below: $\footnotesize L_{MT-DAAN}=\sum_{k=1}^{m}(w_{k}L_{T_{k}})+w_{d}L_{d}$ (4) where $m$ is the number of tasks, $w_{k}$ is the loss weight and $L_{T_{k}}$ is the loss term for each task, $w_{d}$ is the domain adversarial loss weight, and $L_{d}$ is the domain adversarial loss term. ### III-F Model Interpretability The output ($\alpha$) of the attention layer ($ATT$) of each task, is a $T_{x}$-dimensional vector; $T_{x}$ being the number of words in the sentence. The context vector ($\sum_{k=1}^{T_{x}}\alpha^{<k>}a^{<k>}$) is the product of these attention weights and the $T_{x}$-dimensional activation ($a$) from the $BiLSTM$ layer. $\alpha$ essentially weighs how much each word in the sentence contributes to the classification result. Thus, $\alpha$ is the component that is evaluated for model interpretability. ## IV DATASETS CRISIS EVENTS | Total Tweets | Vocab | Avg #words | P | F | S | I ---|---|---|---|---|---|---|--- 2012 Guatemala Earthquake | 154 | 422 | 18.74 | 104 | 108 | 12 | 15 2013 Typhoon Yolanda | 564 | 1746 | 19.47 | 249 | 46 | 119 | 51 2013 Australia Bushfire | 677 | 2102 | 20.21 | 152 | 213 | 167 | 36 2013 Boston Bombings | 535 | 1755 | 19.30 | 147 | 28 | 234 | 198 2013 Queensland Floods | 713 | 2301 | 19.08 | 293 | 54 | 173 | 215 2014 Chile Earthquake | 311 | 919 | 16.54 | 48 | 26 | 50 | 10 2014 Typhoon Hagupit | 1470 | 2893 | 15.36 | 469 | 375 | 276 | 101 2015 Nepal Earthquake | 2048 | 4026 | 13.77 | 1067 | 377 | 741 | 133 2015 Paris Attacks | 2066 | 4152 | 18.62 | 306 | 183 | 782 | 429 2018 Florida School Shooting | 1118 | 2940 | 21.40 | 329 | 64 | 206 | 70 TABLE II: TREC Dataset Statistics; Showing the number of positive samples for each of the 4 classes. $P$=Priority, $F$=Factoid, $S$=Sentiment, and $I$=Irrelevant. ### IV-A TREC Dataset TREC-IS111http://dcs.gla.ac.uk/~richardm/TREC_IS/ (Text Retrieval Conference - Incident Streams) is a program that encourages research in information retrieval from social media posts with the goal to improve the state-of-the- art social media based crisis analytics solutions. We use the dataset from 2018 track proposal. Statistics of this curated dataset of Twitter downloaded from TREC is shown in Table II. The original dataset consisted of 15 crisis events. However, due to very low data, we trimmed the events and tasks such that there are at least 10 positive samples for each task. The four tasks used in our experiments are shown below: 1. 1. Priority: Different priority levels are assigned for each tweet: low, medium, high, critical. We convert this into a binary classification problem where $low=0$ and $\\{medium$, $high$, $critical\\}=1$. 2. 2. Factoid: ‘Factoid’ is a categorical label that represents if a tweet is stating a fact. Eg: ‘death toll rises ...’ 3. 3. Sentiment: ‘Sentiment’ is a categorical label that represents if a tweet represents a sentiment. Eg: ’Worried.. Thoughts and prayers.’ 4. 4. Irrelevant: ‘Irrelevant’ is a categorical label for tweets that do not provide any relevant information. ### IV-B Amazon Reviews Dataset The standard benchmark dataset222http://www.cs.jhu.edu/~mdredze/datasets/sentiment/ of Amazon reviews [25] is widely used for cross-domain sentiment analysis. We chose four domains: Books (B), Kitchen (K), DVD (D), and Electronics (E). The raw data333https://github.com/hsqmlzno1/HATN/tree/master/raw_data, a part of Blitzer’s original raw dataset, used in this work is from HATN [13]. This dataset consists of $3000$ positive and $3000$ negative samples for each of the $4$ domains. This dataset is used for two purposes: 1) to validate the performance of the state-of-the-art methods including the single-task baseline and 2) to compare and contrast the performance of deep models when trained with rich versus sparse datasets. ### IV-C COVID-19 Tweet Dataset For the COVID-19 use-case, we use Twitter posts collected using CitizenHelper [26] system in March 2020, for the geo-bounding box of the Washington D.C. Metro region. These tweets were annotated by volunteers of regional Community Emergency Response Teams (CERTs), with ‘Relevant’ label denoting how relevant a tweet is for crisis response operations. The label values range on a scale of $1$-$4$. We convert them into binary classes by considering values $1$ and $2$ as $-$ve ($0$) class and values $3$ and $4$ as $+$ve ($1$) class. This dataset consists of $4911$ tweets with $-$ve ($Relevant$=$0$) and $637$ tweets with $+$ve ($Relevant$=$1$) classes. Following unsupervised domain adaptation criteria, the filtering models are trained using only the TREC dataset and evaluated on the COVID-19 tweets. For each independent run of the experiment, a balanced subset of size $637$ for both classes is selected for testing. S $\rightarrow$ T | LR | SVM | CNN | BiLSTM | AMN | HATN | ST ---|---|---|---|---|---|---|--- B $\rightarrow$ K | 76.40 | 75.95 | 81.20 | 84.45 | 81.88 | 87.03 | 87.22 B $\rightarrow$ E | 75.53 | 74.05 | 80.44 | 84.61 | 80.55 | 85.75 | 85.51 B $\rightarrow$ D | 81.08 | 81.43 | 82.94 | 83.52 | 85.62 | 87.07 | 86.32 K $\rightarrow$ B | 76.12 | 75.78 | 78.78 | 80.67 | 79.05 | 84.88 | 81.85 K $\rightarrow$ E | 80.37 | 81.20 | 85.17 | 87.37 | 86.68 | 89.00 | 87.09 K $\rightarrow$ D | 73.32 | 74.98 | 76.41 | 78.49 | 79.50 | 84.72 | 81.13 E $\rightarrow$ B | 74.85 | 74.18 | 78.08 | 81.18 | 77.52 | 84.03 | 81.50 E $\rightarrow$ K | 81.85 | 81.85 | 86.59 | 89.00 | 87.83 | 90.08 | 89.21 E $\rightarrow$ D | 75.82 | 75.83 | 78.35 | 78.46 | 85.03 | 84.32 | 81.37 D $\rightarrow$ B | 81.17 | 82.20 | 82.26 | 84.83 | 84.53 | 87.78 | 87.02 D $\rightarrow$ K | 76.42 | 77.58 | 81.09 | 85.21 | 81.67 | 87.47 | 86.37 D $\rightarrow$ E | 72.47 | 73.68 | 79.56 | 83.66 | 80.42 | 86.32 | 85.63 AVG | 77.12 | 77.39 | 80.91 | 83.45 | 82.52 | 86.54 | 85.02 TABLE III: Performance comparison (accuracy) of various models on the standard benchmark dataset of amazon reviews. Methods in blue do not use any unlabeled target data; hence relevant in our context. Each reported score is an average of 10 independent runs of each experiment. Target | LR | SVM | CNN | BiLSTM | ST ---|---|---|---|---|--- Guatemala Earthquake | 60.14 | 56.76 | 60.47 | 65.54 | 59.97 Typhoon Yolanda | 65.39 | 65.97 | 63.05 | 65.49 | 65.53 Australia Bushfire | 65.61 | 63.23 | 62.10 | 60.10 | 62.44 Boston Bombings | 71.47 | 75.45 | 69.72 | 71.43 | 72.08 Queensland Floods | 65.56 | 64.81 | 64.13 | 66.01 | 66.21 Chile Earthquake | 43.09 | 37.94 | 43.37 | 35.45 | 39.23 Typhoon Hagupit | 49.86 | 46.22 | 49.21 | 54.13 | 52.61 Nepal Earthquake | 57.11 | 55.39 | 58.61 | 60.49 | 61.35 Paris Attacks | 71.43 | 71.72 | 72.50 | 72.14 | 71.31 Florida School Shooting | 58.79 | 63.02 | 58.82 | 59.71 | 60.55 AVG | 60.85 | 60.05 | 60.20 | 61.05 | 61.13 TABLE IV: Performance comparison (accuracy) of unsupervised models on TREC- Priority (tweet) dataset showing that deep models are not strictly superior than simpler models due to data sparsity. Each reported score is an average of 10 independent runs of each experiment. $Source$ = $Everything$ \- $Target$. TARGET | Priority | Factoid ---|---|--- | ST | ST-DAAN | MT-DAAN | ST | ST-DAAN | MT-DAAN | Acc | F1 | Acc | F1 | Acc | F1 | Acc | F1 | Acc | F1 | Acc | F1 Guatemala Earthquake | 59.97 | 62.39 | 69.07 | 69.66 | 69.05 | 69.34 | 68.92 | 68.47 | 79.90 | 80.76 | 84.05 | 97.01 Typhoon Yolanda | 65.53 | 65.47 | 66.07 | 63.73 | 67.42 | 67.30 | 80.50 | 84.42 | 82.71 | 85.61 | 84.36 | 86.93 Australia Bushfire | 62.44 | 66.69 | 61.07 | 63.42 | 61.93 | 64.28 | 64.58 | 60.69 | 65.64 | 60.53 | 65.04 | 60.13 Boston Bombings | 72.08 | 74.29 | 72.34 | 73.37 | 73.80 | 74.74 | 83.10 | 88.51 | 81.42 | 85.90 | 85.82 | 88.82 Queensland Floods | 66.21 | 65.94 | 67.19 | 66.97 | 66.74 | 66.46 | 37.56 | 48.90 | 50.46 | 59.82 | 49.52 | 59.21 Chile Earthquake | 39.23 | 40.92 | 38.91 | 42.37 | 41.80 | 46.33 | 30.38 | 33.97 | 39.87 | 48.68 | 45.28 | 54.58 Typhoon Hagupit | 52.61 | 50.59 | 58.97 | 58.94 | 57.50 | 57.52 | 68.98 | 70.79 | 71.42 | 72.44 | 69.49 | 70.08 Nepal Earthquake | 61.35 | 59.44 | 60.18 | 57.80 | 61.65 | 59.49 | 74.04 | 76.08 | 80.72 | 81.00 | 81.04 | 81.02 Paris Attacks | 71.31 | 76.26 | 70.42 | 74.08 | 74.44 | 77.21 | 75.78 | 80.35 | 82.35 | 84.89 | 82.52 | 85.63 Florida School Shooting | 60.55 | 61.75 | 65.47 | 64.07 | 62.51 | 63.24 | 76.73 | 82.67 | 84.55 | 87.51 | 85.80 | 88.15 AVG | 61.13 | 62.37 | 62.97 | 63.44 | 63.68 | 64.59 | 66.06 | 69.49 | 71.90 | 74.71 | 73.29 | 77.16 TARGET | Sentiment | Irrelevant ---|---|--- | ST | ST-DAAN | MT-DAAN | ST | ST-DAAN | MT-DAAN | Acc | F1 | Acc | F1 | Acc | F1 | Acc | F1 | Acc | F1 | Acc | F1 Guatemala Earthquake | 96.96 | 97.03 | 96.45 | 96.68 | 96.76 | 92.73 | 89.36 | 89.03 | 91.22 | 91.06 | 93.11 | 92.73 Typhoon Yolanda | 75.81 | 77.62 | 77.54 | 79.01 | 76.82 | 78.35 | 76.05 | 79.77 | 78.49 | 80.59 | 80.46 | 82.31 Australia Bushfire | 75.95 | 77.58 | 78.80 | 79.12 | 78.54 | 78.92 | 35.42 | 47.164 | 53.78 | 65.11 | 51.76 | 63.36 Boston Bombings | 81.39 | 81.11 | 80.73 | 80.70 | 82.13 | 82.10 | 58.15 | 55.73 | 58.15 | 57.43 | 61.49 | 61.45 Queensland Floods | 81.69 | 80.39 | 81.05 | 81.39 | 81.53 | 81.32 | 65.68 | 65.36 | 67.26 | 65.72 | 67.88 | 67.27 Chile Earthquake | 92.69 | 92.91 | 93.10 | 93.21 | 93.62 | 93.68 | 75.16 | 84.98 | 80.46 | 86.38 | 80.64 | 86.56 Typhoon Hagupit | 84.98 | 85.86 | 85.15 | 86.14 | 85.43 | 86.38 | 63.21 | 75.04 | 71.50 | 78.25 | 70.22 | 77.27 Nepal Earthquake | 67.75 | 68.42 | 70.20 | 70.51 | 69.96 | 70.31 | 31.79 | 42.10 | 36.97 | 47.41 | 41.49 | 52.87 Paris Attacks | 76.01 | 76.63 | 73.65 | 73.98 | 74.47 | 74.60 | 33.91 | 35.25 | 44.52 | 48.32 | 47.17 | 51.32 Florida School Shooting | 68.77 | 71.77 | 67.06 | 70.03 | 68.14 | 71.05 | 32.66 | 40.90 | 44.22 | 55.27 | 47.64 | 58.65 AVG | 80.20 | 80.93 | 80.37 | 81.08 | 80.74 | 80.94 | 56.14 | 61.53 | 62.66 | 67.55 | 64.19 | 69.38 TABLE V: Unsupervised domain adaptation results on TREC dataset showing performance boost for Priority, Factoid, and Irrelevant tasks. However, Sentiment task did not show a significant improvement. See performance evaluation section for details. Each reported score is an average of 10 independent runs of each experiment. TARGET | Relevant ---|--- | ST | ST-DAAN | MT-DAAN | Acc | F1 | Acc | F1 | Acc | F1 COVID-19 | 73.25 | 77.36 | 74.55 | 77.51 | 77.00 | 78.09 TABLE VI: Unsupervised domain adaptation results for COVID-19 tweets using only the TREC dataset for training. Each reported score is an average of 10 independent runs of each experiment. ## V Results & Discussion We first validate the performance of the adopted unsupervised ST model [24] by comparing it with the following standard neural network architectures and state-of-the-art models used for domain adaption in text. We use the standard benchmark dataset of Amazon reviews. Following the traditional domain adaptation experimental setup, each experiment represented as S $\rightarrow$ T consists of a source domain (S) on which the model is trained and a target domain (T) on which the model is tested. We use Keras deep learning library for our implementations; with $T_{x}$=$200$ for Amazon reviews and $30$ for Tweets. We use Adam optimizer with a dropout of $0.4$, maximum epoch of $50$, early stopping patience of $3$, batch size of $32$, and validation split of $0.15$. 1. 1. Simple Baselines: We construct simple baseline classifiers [27]: Logistic Regression (LR) and Support Vector Machines (SVM). The input to these models are constructed by aggregating the $300$-dimensional word embeddings of words in each review. 2. 2. CNN: A standard Convolutional Neural Network inspired by Kim, 2014 [28] is constructed with the following architecture: $Word\ Embeddings(T_{x},300)\rightarrow Conv1D(128,5)$ $\rightarrow MaxPooling1D(5)$ $\rightarrow Conv1D(128,5)\\\ \rightarrow MaxPooling1D(5)$ $\rightarrow Conv1D(128,5)\\\ \rightarrow GlobalMaxPooling1D()\rightarrow Dense(128)$ $\rightarrow Dense(2)\rightarrow y$. This is combined with dropouts, relu activations, and ending with softmax activation producing labels for binary classification. State-of-the-art deep learning methods for existing social media mining approaches of crisis analytics [6, 5] use a similar architecture. 3. 3. BiLSTM: This is the bottom-most layer in Figure 2 with the activation $a^{<T_{x}>}$ passed through the following: $Dense(10)\rightarrow Dense(2)\rightarrow y$ also including dropouts, relu activation, and ending with softmax. 4. 4. AMN and HATN: AMN [12] and HATN [13] are attention-based methods which use gradient reversal to perform domain adversarial training on the unlabeled data from source and target domains. HATN is an extension to AMN by adding the hierarchical component and jointly training pivot and non-pivot networks. Input to all the models are word vectors444https://code.google.com/archive/p/word2vec/ [29]. The evaluation on amazon reviews shows how well the single-task (ST) model perform when compared to the existing top-performing domain adaptation models on benchmark dataset. Table III shows accuracy scores on the Amazon cross-domain sentiment analysis dataset. HATN uses unlabeled target data, gradient reversal, explicit pivot extraction, and joint training making it a computationally expensive method. As shown in the experimental evaluation, we use the same Amazon dataset and GoogleNews word vectors for our experiments. ST, being unsupervised with no need of unlabeled target data, performed competitively with an overall accuracy of 85.02%; thus establishing a strong fully unsupervised building block for us to build upon. ### V-A Crisis Tweets vs Amazon Reviews Table III and IV show that deep models struggle with small datasets such as TREC-IS tweets. When ST model outperformed Logistic Regression by $\sim 8\%$ on the Amazon reviews dataset, the difference was only less than $1\%$ with no statistical significance on the TREC-Priority dataset. Note that we conduct experiments with various parameter combinations on the deep models when using tweets. For example, $T_{x}=200$ for amazon reviews and $T_{x}=30$ for tweets due to the difference in their average word-length. Books domain of Amazon reviews has $182$ average number of tokens per review with a vocab size of $105920$. On the other hand, the event with highest number of tweets in the TREC dataset (Paris Attacks) has only 18.62 average number of tokens per tweet with a vocab size of $4152$. This difference makes it intuitively challenging to train deep models with several parameters that may lead the model to memorize the entire dataset resulting in poor generalization. Multi-task learning and domain adversarial training try to alleviate this problem by training the shared BiLSTM layer with much more data from different tasks and unlabeled data. ### V-B MT-DAAN Performance Evaluation The primary purpose of the MT-DAAN model is to show that sharing the bottom layer of the model (i.e., shared representation) for different tasks along with domain adversarial training can help improve the generalizability of some of the tasks that are otherwise trained alone in the single-task model. The experiments for MT-DAAN are setup in the same unsupervised way as for single- task. No data from the test crisis is used for training. For example, if we are testing our model for the event ‘Typhoon Yolanda’, no data from this crisis is used for training. Note that the domain classifier component uses unlabeled data only from rest of the crisis; making it a fully unsupervised domain adaptation approach. Performance scores of the four tasks (Priority, Factoid, Sentiment, and Irrelevant) are shown in Table V. The results show clear performance improvement for Priority, Factoid, and Irrelevant tasks. However, Sentiment task did not show significant improvement. We speculate that this is because other tasks do not generalize the bottom layer enough to boost the sentiment classification performance. These results show the usefulness of multi-task learning as well as domain adversarial training where different tasks in multiple domains help each other when the data is sparse and labels are limited. Figure 3: Examples of interpretable results using attention; darker the shade, higher the attention. Recall that no data from the crisis-event for testing is used for training the model. Even then, relevant keywords such as ‘police urging’, ‘death toll rises’, ‘worried’, and ‘thoughts with people’ are correctly picked up by the attention layers of their respective tasks. ### V-C Word Vectors We use fastText[30] as our word embeddings for tweets because of its sub-word usage and the ability to create vectors for arbitrary and out-of-vocabulary words. Although there exists many alternatives, picking the one that works well for a specific dataset is not trivial. We conducted experiments using four choices of word embeddings: fastText [30], GoogleNews [29], Glove [31], and CrisisNLP [32]. Averaging over 10 crises, we received the following accuracy scores (in %) respectively for the above word embeddings: {$80.20$, $81.82$, $81.88$, $80.73$}. Unlike fastText, we fine-tune these pre-trained vectors to create vectors for out-of-vocabulary words. Vectors for words that are already in the vocabulary are locked while tuning for consistency in evaluation. The tweet-based embeddings such as Glove or CrisisNLP did not significantly outperform other models. Glove vectors are 200-dimensional while the rest are 300-dimensional which makes the experiment favoring Glove word vectors. This experiment shows that the problem of finding a strictly superior word vector model for tweets still remains a challenging task. Figure 4: Examples of interpretable results using attention for relevancy prediction of COVID-19 tweets. With $77\%$ accuracy, although the highly attended words in the ‘Relevant’ tweets provide some intuitive sense of interpretability, the highlighted words in the ‘Irrelevant’ tweets are somewhat ambiguous because it is unclear if those words are chosen due to their specific or generic nature. This shows both the benefits and challenges of unsupervised and interpretable domain adaptation. ### V-D Interpretability: Attention Visualization The attention weights used to create the context vector by the dot product operation with word activations represent the interpretable layer in our architecture. These weights represent the importance of each word in the classification process. Some examples are shown in Figures 3 and 4. Stronger the color intensity stronger the word attention. In the first example, ‘boston police urging’ is the reason why the tweet is classified as $+$ve priority. Similarly, ‘death toll rises’ in the Factoid example, ‘worried, prayers’ in the Sentiment example, and ‘thoughts with people’ in the Irrelevant example are clear intuitive indicators of +ve predictions. These examples show the importance of having interpretability as a key criterion in crisis domain adaptation tasks for social media. To the best of our knowledge, in social media mining for crisis analytics, there does not exist a ground truth dataset that highlights the words that explain the labels for tweets. Using our model as a guide, we hope to build a robust evaluation dataset as our immediate next step so that the models can be quantitatively evaluated using robust trust-evaluation methods such as LIME [33]. It is also crucial to note that binary classification tasks such as sentiment analysis of Amazon reviews has a clear class divide that produces intuitive keywords such as ‘good’, ‘excellent’, or ‘great’ for $+$ve reviews and ‘bad’, ‘poor’, or ‘horrible’ for $-$ve reviews. However, for short texts such as tweets shown in Figure 4, ‘relevancy’ can depend on the context and it is unclear which keywords truly represent the examples in the ‘irrelevant’ class. ## VI COVID-19 Use-Case We show a practical implication of our work by applying it to COVID-19 tweets described in Section 4.3. Our goal is to interpretably predict if a COVID-19 tweet is relevant or not; a binary classification task. The models are trained using only the TREC dataset and evaluated on the COVID-19 tweets (a balanced subset of size $637$ for $+$ve and $-$ve labels). We found that a combination of ‘Priority’ and ‘Irrelevant’ labels from TREC performs better to predict COVID-19’s ‘Relevant’ label (this can be trivially verified by constructing two binary classifiers). We augment all three methods (ST, ST-DAAN, and MT- DAAN) with an additional condition before label prediction: $R_{c}=P_{t}\cap\overline{I_{t}}$, which means that a COVID-19 tweet is ‘Relevant’ only if it is predicted both ‘Priority’ = $1$ and ‘Irrelevant’ = $0$. The scores are reported in Table VI and the attention results are shown in Figure 4, demonstrating the effectiveness of our proposed method. ## VII Conclusion We presented a novel approach of unsupervised domain adaptation with multi- task learning to classify relevant information from Twitter streams for crisis management, while addressing the problems of data sparsity and limited labels. We showed that a multi-task learning model that shares the lower layers of the neural network with dedicated attention layers for each task along with a domain classifier branch can help improve generalizability and performance of deep models in the settings of limited data. Furthermore, we showed that using an attention-based architecture can help in interpreting the classifier’s predictions by highlighting the important words that justify the predictions. 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2024-09-04T02:54:58.992673

2003.05106

arxiv-papers

# Self-Sovereign Identity for IoT environments: A Perspective Geovane Fedrecheski1, Jan M. Rabaey4, Laisa C. P. Costa1, Pablo C. Calcina Ccori1, William T. Pereira1, Marcelo K. Zuffo1 {geovane, laisa<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>This research was partially funded by CAPES. 1Interdisciplinary Center on Interactive Technologies, Polytechnic School, University of Sao Paulo, Brazil 4Berkeley Wireless Research Center, Electrical Engineering and Computer Science Department, University California, Berkeley, US ###### Abstract This paper analyses the concept of Self-Sovereign Identity (SSI), an emerging approach for establishing digital identity, in the context of the Internet of Things (IoT). We contrast existing approaches for identity on the Internet, such as cloud-based accounts and digital certificates, with SSI standards such as Decentralized Identifiers (DIDs) and Verifiable Credentials (VCs). To the best of our knowledge, this is the first thorough comparison of these approaches. The benefits and challenges of using DIDs and VCs to identify and authenticate IoT devices and their respective users are discussed. In the end, we establish that SSI, with its owner-centric, privacy-aware and decentrailized approach, provides a viable and attractive option for secure identification of IoT devices and users. ## I Introduction The Internet was developed as a research project to interconnect computers [1]. Protocols like TCP/IP, developed as open standards, allowed computers to connect in a global scale. However, even after the world-changing impacts the Internet had on society over the last decades, it has no pervasive, privacy- preserving, and easy to use mechanism to manage digital identities. Where human activity is involved, a common abstraction is to use accounts, i.e. digital records, often containing personally identifiable information (PII), that are protected by a password and saved on a webserver. Although this method has been working for several decades, it has many security drawbacks, such as the use of weak passwords [2] and the potential for privacy violation. Furthermore, the manual approach of password-protected accounts makes it unsuitable to machine-to-machine interactions, a common scenario in the IoT. More automated solutions can be achieved by using Public Key Certificates (PKCs) that bind names to public keys [3]. Widespread use of PKC, however, is limited to organizations, due to the complexity of current methods. For instance, while websites usually prove their identities to web browsers using certificates, users do not use certificates in the same way, i.e. to prove their identity to the website. Moreover, existing standards were not designed for privacy, as evidenced by the use of real names in known certificate formats such as PGP [4] and X.509 [5]. To aggravate the situation, the assignment of unique names often require centralized architectures, which is inadequate for distributed IoT applications. A recent development towards online identification of users, organizations, and devices has been referred to as “Self-Sovereign Identity” (SSI). The basic premise of SSI is that subjects should own and control their own identity, instead of having it stored and managed by a third party. This approach brings several benefits, including enhanced privacy, control, and decentralization. Two new standards are being proposed to realize SSI, namely, Decentralized Identifiers (DIDs) and Verifiable Credentials (VCs) [6, 7]. While DIDs focus on cryptographic identification, VCs provide a means for privacy-aware and authenticated attribute disclosure. In this paper we analyze existing approaches to identity in the Internet, such as X.509, PGP [4], and SSI. We present a detailed comparison focusing on the data models used to represent identity across different standards. Finally, we discuss what are the benefits of using SSI in the Internet of Things, and identify challenges that must be overcome. ## II Self-Sovereign Identity Self-Sovereign Identity is an approach in which subjects are in full control of their own digital identities [8]. SSI is analogous to offline identifiers, which are carried by the owner (within a physical wallet), but contrasts with current digital identity solutions, which are either based on accounts or digital certificates, and have privacy and centralization issues. While initially proposed by members [8] of online communities, a formal definition of SSI was released recently [9]. Considering an identity to be composed of an identifier associated with a set of name-value attributes, the full self-sovereign identity of an individual is the collection of all identities (i.e. identifiers and attributes) that span a range of decentralized domains, such that the individual is in full control of these identities [9]. As digital privacy concerns have been growing in recent years, interest in SSI has intensified. This led to the definition of a set of technical specifications to implement SSI, which we describe below. ### II-A Decentralized Identifiers Digital identifiers so far have been either centralized or non-resolvable. For example, Uniform Resource Locators (URLs), which can be used to resolve HTML documents, usually depend on domains names assigned by ICANN111Internet Corporation for Assigned Names and Numbers - https://www.icann.org/, a centralized authority. On the other hand, unique, user-generated identifiers such as UUIDs cannot be used to resolve associated metadata. To address this, a new specification for Decentralized Identifiers (DIDs) is being developed with the support of the W3C [6]. The DID has the following syntax: did:btcr:abcdefgh12345678. The did prefix is mandatory, and colons are used to separate a method definition and a method-specific id. A method is a specific set of rules for working with DIDs (the example above uses the Bitcoin method), and the format of the id depends on that method. An open directory of different DID methods is available for public access and open for new submissions222https://w3c-ccg.github.io/did-method-registry/. Each DID is associated with a DID Document (DDo) that contains the DID itself along with public keys, service endpoints, and other metadata. The public key is used to authenticate and encrypt messages, while the endpoint provides a way to message the entity that controls that DID. To control a specific DID, a subject just have to own a private key associated with public keys in the DDo. A common storage mechanism for DDos are Blockchains, from which they can be resolved using the referred DID. On the other hand, in some cases individuals may not want to publish their DIDs, e.g. to avoid identity correlation. In this case, the special peer DID method can be used. Thus, DIDs are unique identifiers that can be resolved to DID Documents, and they allow the establishment of an end-to-end secure channel. What DIDs do not provide, however, is a means for entities to prove claims (attributes) about themselves. ### II-B Verifiable Credentials Verifiable Credentials (VCs) is a W3C recommendation for portable and provable claims about a subject. For instance, a person may claim to have the name Alice, and a device may claim to be of type Camera. The relationship among DIDs and VCs is shown in Figure 1. All VCs refer to the DID of the subject to which they have been assigned (e.g. an IoT device). VCs also contain the DID of its issuer along with a cryptographic proof. This allows a subject to present a VC to a verifier, which can then resolve the DDo of the issuer (and therefore its public key) from a public ledger, e.g. a Blockchain, and check the authenticity of the VC. Figure 2 shows a use case where a user issues a VC to a device. A major incentive for SSI is privacy, therefore VCs are expected to be private and stored in a personal wallet, to be shared only when necessary. To further improve privacy, the VC specification supports zero-knowledge proofs, i.e. a cryptographic technique “where an entity can prove to another entity that they know a certain value without disclosing the actual value” [7]. Figure 1: A DID is the link between a DDo and a set of VCs, much like a primary key can link different tables in a database. This allows a subject associated with a DID to prove its identity. Figure 2: An owner-centric scenario using SSI. Each subject generates its own DDo, while the VC is issued by the device owner. ### II-C Decentralization, privacy, and layered authentication Public key cryptography can be used to derive a shared secret over an insecure channel [10]. However, a known problem is how to trust the origin of the public key. To solve this, a signed certificate that binds a name to a public key was proposed [3]. Two common standards for digital certificates are X.509 [5, 11] and Pretty Good Privacy (PGP) [4]. Although they differ in details, both follow the original definition in which names are tied to public keys and signed by a third party [3]. A crucial challenge faced by certificate-based solutions was ensuring the uniqueness of the names. The most common solution to this was to rely on centralized architectures. For example, the name on the subject field in X.509 must be enforced by a global authority, and the PGP id uses the name of a person plus her email address, which ultimately depends on DNS, which is centralized as well. More recently, the emergence of Blockchain technology allows decentralized consensus for choosing unique names. One problem, however, is that solutions based on certificates put sensitive information in the identifier, which compromises the privacy of certificate holders, and therefore might not be suitable for storage in public, immutable ledgers. An approach to solve this is to limit the exposure of PII on the ledger by only writing anonymous information to it, e.g., public keys. In particular, this approach enables public key storage and lookup, which can be used to create a confidential and non-repudiable channel. Higher-level abstractions can then be used to implement authentication, since the attributes necessary to authenticate users are usually application-specific. This is the solution that results from combining the DID and VC specifications. Containing only pseudonymous information, such as public keys and service endpoints, DID Documents can be used to establish a cryptographically secure channel between two entities. After the confidential channel is created, the entities can exchange VCs, according to the levels of trust necessary to each application. In other words, while DIDs are lower- level and pseudonymous, VCs are application-specific and can be used to authenticate attributes such as name or device type. Finally, it is worth noting that as each DID is usually a high-entropy random string, name collisions actually stop being a concern. ## III Data models for digital identity This section provides analysis and comparisons of existing data models for digital identity. We start by discussing the limitations of password-based accounts, and then proceed to compare data models based on public key cryptography. ### III-A Accounts The most basic method to identify subjects in computer systems is the account: a digital record, usually composed of at least a user name and a password, that identifies a user. Accounts are commonly stored in a server controlled by the service provider. For example, popular IoT vendors require that a device owner have a cloud-based account, so that she can use this virtual identity to configure her devices. While accounts have been used for decades in a variety of systems, they are among the most primitive solutions for digital identities. Among the problems related to account-based authentication are privacy and the use of passwords. With respect to privacy, issues arise because the user is forced to store plaintext PII in a third-party system. Regarding passwords, the literature indicates common problems such as password reuse and difficulty to enforce strong passwords, and points that the most widespread solution is the use of “recommendations” [2], which depends on human factors and are difficult to enforce. TABLE I: Comparison of standardized data models for digital identity. | PGP | X.509 | Self-Sovereign Identity ---|---|---|--- | PGP Key | | Public Key Certificate --- (PKC) | Attribute Certificate --- (AC) | DID Document --- (DDo) | Verifiable Credential --- (VC) Goal | | Prove control of public --- keys and identifier (plus optional attributes) Publish public keys | Prove control of public --- keys and identifier (plus optional attributes) Publish public keys | Prove possession of --- attributes | Prove control of identifier --- Publish public keys and service endpoints | Prove possession of --- attributes Identifier | Name and Email | Qualified Name | Same as PKC | Method-specific DID | Same as DDo | Uniqueness --- of identifier | Global --- authority (DNS) Global authority (CA) | Same as PKC | | Ledger consensus / --- Random number gen. Same as DDo Public Key(s) | 1 primary, N subkeys | 1 | n/a (points to PKC) | N | n/a (points to DDo) Attribute(s) | Attributes field | Extensions field | Attributes field | - | subjectCredential field Endorsement | Signature of many peers | Signature of a CA | Signature of a CA | | Self-signed (optional) --- Indirect through VC Signature of an Issuer | Service --- endpoints - | - | n/a | Yes | n/a | Semantic --- schemas - | - | - | Yes | Yes ### III-B Models based on public key cryptography Pretty Good Privacy (PGP) [4] was created to allow individuals to prove a binding between a public key and an identifier, the latter being composed by a real name and an email address. This binding, along with optional attributes and signatures, is stored in a document called a PGP Key. Conceived as a distributed solution, individuals in the PGP scheme can sign the keys of other individuals, so as to give an endorsement that they are who they say they are, i.e. they are not impersonating someone or using a fake id. This scheme of peer signatures is often referred to as the Web of Trust. X.509 Certificates, created by the X.500 working group, defines a format for Public Key Certificates (PKC) that binds public keys to qualified names [5]. PKCs are widely used in the Internet to authenticate domain names and protect communications. Although technically nothing prevents peer-to-peer signature of X.509 certificates, the vast majority of its usage is under centralized architectures, in which a trusted authority signs the certificate to make it trustworthy. Finally, in certain cases it is useful to have a separate document that, instead of having public key, contains only a name associated with signed attributes. To meet this demand, X.509 proposed a new standard called Attribute Certificate (AC), which contains no public key, but links to a PKC through its subject field [11]. Finally, as previously mentioned, Self-Sovereign Identity is a novel approach that uses Decentralized Identifiers [6] and Verifiable Credentials [7] to prove possession of identifiers and attributes, respectively. ### III-C High-level comparison The following paragraphs compares models used by the PGP, X.509, and SSI standards, according to Table I. #### Goal Both PGP Keys and PKCs are used to publish and prove control of public keys that are tied to identifiers. Also, in these approaches, attributes can be provided either in the same document as the public keys (PGP Key and PKC), or, in the case of X.509, in a separate document (AC). On the other hand, documents in the SSI paradigm have decoupled goals: DDos are be used to prove control of an identifier and to provide a means for establishing a secure communication; and VCs are used to prove possession of attributes. #### Identifier (and uniqueness) While PGP and X.509 use names and other identifiers that depend on centralized entities, in SSI the identifiers are completely decentralized and can be auto- generated, for example by using strong random number generators. Not only this enables easy global uniqueness, but the pseudonymous characteristic of DIDs also enhances privacy, when compared to previous approaches based on real names or email addresses. Pseudonymous identifiers are also more suited for IoT, since devices do not have names or email addresses by default. #### Public Key(s) PKCs are limited to only one public key, while PGP Keys and DDos can have many. PGP still differs from DDos as the former uses a primary key that is tied to an identifier and allows more subkeys to be included, while the latter support multiple public keys without assumptions other than the key type, which usually encodes its purpose, e.g. sign or encrypt. #### Attribute(s) Both PGP Keys and X.509 certificates support arbitrary attributes, either via PKC extensions or dedicated ACs. In self-sovereign identity, a DDo does not support attributes in order to stay anonymous. Instead, all PII is handled only by VCs, which are private by default. #### Endorsement(s) PGP Keys can be signed by one or more peers, but X.509 certificates and VCs can only be signed by a single issuer. DDos are not signed by external entities, and may be self-signed. When a DDo is written to a ledger, however, the transaction will be signed, which can be used to attest the validity of the DDo. Another way of proving endorsement over a DID is to check the signature of a VC associated with that DID. If the VC is signed by a trusted issuer, the DID can be trusted. Furthermore, with respect to who can sign the endorsements, technically it can be anyone, but there are philosophical differences. X.509, for example, was devised to work within a centralized architecture, where only trusted authorities can sign certificates. On the other end of the spectrum, PGP expects peer-to-peer signatures, which ultimately creates a Web of Trust. Finally, VCs does not make strong assumptions on the network structure, although decentralized approaches, especially the ones based on Blockchain, may be favorable. #### Service endpoints a novelty introduced by DDos is the association of a built-in mechanism to reach the owner of a public key. This facilitates the establishment of secure interactions between peers, from web to IoT environments. #### Semantic schemas only SSI-based data models allow extensibility through semantic annotations over JSON documents. The main reason for this is that these technologies only became popular after X.509 and PGP were developed. ### III-D Public key distribution TABLE II: Comparison of data models for key distribution. | Raw Pub Key | PKC | DDo ---|---|---|--- | Associates key material --- to metadata | X | X Privacy: no PII disclosed | X | | X | Key rotation does not --- requires re-signing n/a | | X Serialization formats | | Binary --- Base64 | DER --- PEM | JSON-LD --- JWT Semantic schemas | | | X | Decentralized: user --- generates the artifact X | | X | Decentralized: user --- carries the artifact X | X | X Service endpoint | | | X TABLE III: Comparison of data models for attributes. | PKC | AC | VC ---|---|---|--- | Signed attributes --- about a subject X | X | X | Key rotation does not --- requires re-signing | X | X | Identifier differs from --- key material X | X | X | Attributes decoupled --- from key material | X | X Selective disclosure | | | X Zero-knowledge proofs | | | X Delegation | | X | Revocation | X | X | X Serialization formats | | DER --- PEM | DER --- PEM | JSON-LD --- JWT Semantic schemas | | | X | Decentralized: user --- carries the artifact X | X | X | Decentralized: Verifier --- decoupled from Issuer | | X An important aspect in the design of systems based on asymmetric encryption is the data model used to support key distribution. In the following, we compare three approaches, as shown in Table II: Raw Public Key, Public Key Certificates, and DID Document. #### Raw public key this is the simplest approach, and consists in having a public key shared as a raw array of bytes, often encoded in some ascii-compatible format, such as base64. Although this approach is decentralized and discloses no personal information, it does not allow associated metadata. #### Public Key Certificate as previously discussed, PKCs bind a name and other attributes to a public key, which allows subjects prove their identity. Created before privacy was a major concern, X.509 PKCs always carry PII in the main identifier, and may carry PII in other attributes. Finally, other drawbacks of PKCs include the imposition of specialized serialization formats (DER and PEM), tight coupling of keys and data (which makes key rotation more difficult), and a centralized architecture, i.e. the artifact is not self-generated. #### DID Document DDos associate public keys to pseudonymous metadata, while also allowing key rotation without re-signing any associated metadata. The latter is possible because all signed metadata actually only lives in associated VCs. An important difference to highlight is that DDos are not signed by third parties, thus they cannot authenticate the origin of a public key. If this is necessary, DDos can be composed with VCs to increase security. DDos supports JSON-based serialization formats, which are available in most programming languages and platforms, and can benefit from publicly available semantic schemas. As each user auto-generates their own DIDs and DDos, the management of the identifier is decentralized. Finally, service endpoints in DDos provide a novel way for peers to securely establish secure channels. ### III-E Attribute distribution Four out of the five previously described formats can be used to prove control over attributes: PGP Keys, Public Key Certificates, Attribute Certificates, and Verifiable Credentials. Since PGP Keys are less widely used, we only compare the latter, as shown in Table III. #### Public Key Certificates the encoding of attributes in PKCs leverages the X.509 PKC extension field. Although the reuse of an existing format may seen advantageous in terms of compatibility, the whole certificate must be re-signed when a key is rotated, or when selective disclosure of attributes is necessary. An important drawback not mentioned so far is that it is impossible to disclose only a subset of the attributes in a PKC, without contacting the issuer for a new signature. #### Attribute Certificates differing from PKCs, ACs contain a name and a list of attributes, but no public key, which simplifies key rotation. Finally, while ACs support delegation, in general they have the same drawbacks as PKCs. #### Verifiable Credentials similar to an AC, a VC does not contain public keys, as it focus on binding identifiers to attributes. Among the novelties in the VC standard is the support for selective disclosure without contacting the issuer, which is realized using zero-knowledge cryptography. VCs also leverage JSON, a serialization format that is both human readable and lightweight to parse. VCs and can be further specialized into two formats: JSON Linked Data (JSON- LD)333https://json-ld.org/, a format to serialize linked data; and JSON Web Token (JWT), a widely used format to express security claims444https://jwt.io/. ## IV Benefits and Challenges of SSI for IoT As the IoT continues to evolve, new paradigms that allow spontaneous machine- to-machine interactions started to appear [12, 13]. Necessarily decentralized, the future IoT will require users to be the root of trust of their devices, leading to an owner-centric IoT. As privacy concerns raise in importance, solutions that minimize personal data sharing become paramount. Full realization of these and other features will require novel, open, and secure standards for identity in the IoT. The next paragraphs discuss aspects of self-sovereign identity that are likely to improve decentralized IoT security, while also pointing the factors that will require innovation to bring SSI to IoT, such as support constrained devices. ### IV-A Benefits The benefits of SSI for IoT, such as privacy and decentralization, are discussed below. #### Owner-Centric The user can be the root of trust of her devices. Once a user is the owner and controller of her identity, it is straightforward to create a network of devices that belong to her, for example by provisioning an “owner=Alice” credential to each device. One interesting consequence of this is that no third party is needed to enforce security and administration of devices, as the user herself will be able to do it. Note that in this approach devices can have their own identity as well, and may only use the owner attribute to facilitate the creation of trust relationships, i.e. devices that share the same owner can automatically trust each other. #### Privacy-preserving Personal information is protected. By having the identity of owners and things stored locally, sensitive data that would otherwise be stored in a service provider will now live closer to the owner (usually in a digital wallet). While the user can choose to backup his data for various reasons, she will be able to do so in an encrypted way, as only she will possess the decryption keys. Users and devices will also get to choose with whom they share their credentials, and even be able to do so employing selective disclosure and zero-knowledge proofs techniques, further improving privacy. #### Decentralized No single-point of failure. While identity providers may have been a convenient way to authenticate users and devices so far, it is not clear what happens when a provider stops providing, e.g. when it goes out of business. In the self-sovereign approach, the user decides when her identity starts or stops being valid, and she will have similar controls over her devices. Finally, data breaches, information sharing without user consent, and other issues are minimized when identities are not stored in a high-value data silo that acts as a honeypot for hackers. #### End-to-end security Communications between two endpoints are secure. By exchanging DID Documents and applying asymmetric cryptography, IoT devices can mutually authenticate, derive short-lived symmetric keys, send encrypted messages, and enforce non- repudiation. This approach can also be implemented in a transport-agnostic way, enabling secure communication even among different protocols. #### Layered authentication Separates cryptographic and application-specific authentication. In the former, two devices prove to each other that they are in possession of specific public keys, while in the latter the devices prove different attributes about themselves. This approach allows endpoints to always be cryptographically protected, and leaves higher-level trust requirements to be handled at the application layer. #### Standardized and open approach Fosters interoperability and robustness. Since both DIDs and VCs are being developed as open W3C specifications, companies and researchers are free to build solutions that are interoperable and rely on well-tested data models. #### JSON-based encoding Using JSON enables more applications to handle data extracted from DID Documents and credentials, even if not originally designed to work with SSI. ### IV-B Challenges We now discuss some challenges to apply SSI in IoT environments. #### Constrained devices Fully adopting SSI means that devices need to be able to run asymmetric cryptography and cope with communication overhead of transmitting metadata, such as DID Documents and Verifiable Credentials. #### Asymmetric Cryptography SSI demands execution of encryption algorithms based on asymmetric keys, which can be challenging in devices with limited processing and energy resources. While authors points that constrained processors such as the 32-bits Cortex M0 are well equipped to execute Elliptic Curve Cryptography (ECC) [14], the number of operations still must be controlled to avoid battery draining. A common tactic is to use long lived session keys that are less frequently updated, e.g. once a day. #### Communication overhead Depending on the communication protocol, the size of DDos and VCs may impose a barrier. For example, low energy protocols such as LoRA and BLE have maximum packet sizes of 222 and 244 bytes, respectively, while DDos and VCs easily achieve 500 bytes or more. Therefore, strategies such as compression, fragmentation, and infrequent document transmission, will be necessary. In extreme cases, SSI may not be possible at all, which will require proxy approaches [15]. #### DID Resolution Higlhy constrained devices may not be able to connect to the Internet to download DID Documents at all. A possible solution is to create a local cache of known DIDs, either managed by the device itself or by its gateway. On the other hand, if both devices use peer DIDs, they can simply exchange their DIDs directly, which shifts the problem to securely delivering the DIDs in the first place. #### Software availability The SSI ecosystem is new and there is limited software available for embedded devices. Given the foundational importance of secure cryptographic algorithms and protocols, applications based on SSI should rely on existing libraries that encapsulate complexity and are well tested, which reduces the chances for vulnerabilities. Although reference implementations exist [16], they are focused on cloud and mobile use cases. To fully incorporate SSI into IoT, portable and lightweight libraries tailored for constrained devices must be created and made widely available. ## V Conclusion and perspective As the primary motivation for the development of the Internet was to remotely connect computers, the problem of secure identification of users and devices was left aside. While identity solutions such as accounts and certificates were eventually developed, they feature critical issues such as weak passwords, lack of privacy, and centralization. As it is common for systems to mature over time, as good (and bad) practices are learned, we argue that the Self-Sovereign Identity approach represents an important step forward in the area of digital identity. Particularly in the context of the IoT, this paper showed how SSI can (1) empower owners to have full control over both their identities and their devices, (2) improve privacy by decoupling pseudonymous and sensitive identity records, and (3) allow decentralized identity management by reducing the dependency on third parties. As for the next steps, the realization of SSI in the IoT will demand implementations that are optimized for constrained devices, both for cryptographic operations and low- power communication. Furthermore, wide adoption of SSI will depend on the availability of open software libraries to manipulate DIDs and VCs in IoT devices. To conclude we argue that, if adopted, SSI may significantly benefit security and privacy of IoT applications, and potentially enable new use cases, such as those that involve cross-owner decentralized interactions. ## References * [1] B. M. Leiner, V. G. Cerf, D. D. Clark, R. E. Kahn, L. Kleinrock, D. C. Lynch, J. Postel, L. G. 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2024-09-04T02:54:59.007056

2003.05110

arxiv-papers

# H–He collision-induced satellite in the Lyman-$\alpha$ profile of DBA white dwarf stars Nicole F. Allard 1,2, John F. Kielkopf 3, Siyi Xu 4, Grégoire Guillon 5, Bilel Mehnen 6, Roberto Linguerri 6, Muneerah Mogren Al Mogren 7, Majdi Hochlaf 6, Ivan Hubeny 8 1GEPI, Observatoire de Paris, Université PSL, UMR 8111, CNRS, 61, Avenue de l’Observatoire, F-75014 Paris, France 2Sorbonne Université, CNRS, UMR7095, Institut d’Astrophysique de Paris, 98bis Boulevard Arago, PARIS, France 3Department of Physics and Astronomy, University of Louisville, Louisville, Kentucky 40292 USA 4Gemini Observatory, 670 N. Aóhoku Place, Hilo, HI 96720 HI, USA 5Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR6303, CNRS, Université de Bourgogne Franche Comté, 21078 Dijon Cedex, France 6Université Gustave Eiffel, COSYS/LISIS, 5 Bd Descartes 77454, Champs sur Marne, France 7Chemistry Department, Faculty of Science, King Saud University, PO Box 2455, Riyadh 11451, Kingdom of Saudi Arabia. 8Department of Astronomy, University of Arizona, 933 N Cherry Ave, Tucson, AZ 85719 USA E-mail<EMAIL_ADDRESS> (Accepted XXX. Received YYY; in original form ZZZ) ###### Abstract The spectra of helium-dominated white dwarf stars with hydrogen in their atmosphere present a distinctive broad feature centered around 1160 Å in the blue wing of the Lyman-$\alpha$ line. It is extremely apparent in WD 1425+540 recently observed with HST COS. With new theoretical line profiles based on ab initio atomic interaction potentials we show that this feature is a signature of a collision-induced satellite due to an asymptotically forbidden transition. This quasi-molecular spectral satellite is crucial to understanding the asymmetrical shape of Lyman-$\alpha$ seen in this and other white dwarf spectra. Our previous work predicting this absorption feature was limited by molecular potentials that were not adequate to follow the atomic interactions with spectroscopic precision to the asymptotic limit of large separation. A new set of potential energy curves and electronic dipole transition moments for the lowest electronic states of the H–He system were developed to account accurately for the behaviour of the atomic interactions at all distances, from the chemical regime within 1 Å out to where the radiating H atoms are not significantly perturbed by their neighbors. We use a general unified theory of collision-broadened atomic spectral lines to describe a rigorous treatment of hydrogen Lyman-$\alpha$ with these potentials and present a new study of its broadening by radiative collisions of hydrogen and neutral helium. These results enable ab initio modeling of radiative transport in DBA white dwarf atmospheres. ###### keywords: (stars:) white dwarfs < Stars - stars: atmospheres < Stars - atomic data < Physical Data and Processes - atomic processes < Physical Data and Processes - line: profiles < Physical Data and Processes - molecular data < Physical Data and Processes ††pubyear: 2019††pagerange: H–He collision-induced satellite in the Lyman-$\alpha$ profile of DBA white dwarf stars–7 ## 1 Introduction Theoretical studies of the effects of neutral atom collisions on atomic spectral lines have often been hindered by our ignorance of the atomic potentials. Even for systems as simple as H-H or H-He, the interactions and the electric transition moments are quite difficult to compute with the accuracy which is needed for evaluating a complete line profile. The fundamental theory of calculating the spectral line profile (Allard et al., 1999) requires knowledge of molecular potentials with high accuracy because the shape and strength of the line profile are very sensitive to the details of the molecular potential curves describing the atom-atom collisions. In Allard & Christova (2009) we made an exhaustive study of the red wing of Lyman-$\alpha$ line perturbed by H–He collisions, where we used the potentials and electric dipole transition moments of Theodorakopoulos et al. (1984) and Theodorakopoulos et al. (1987). We considered the high He densities met in cool DZ white dwarfs and examined the range of validity of the one-perturber approximation widely used to calculate the line wings. We have shown there that the extension of the red wing of the Lyman-$\alpha$ line seen in DZ white dwarf spectra depends strongly on the stellar temperature, while it is not dependent on the helium density. We also predicted a blue satellite which only very recently has been observed in Hubble Space Telescope Cosmic Origins Spectrograph (HST COS) observations (Xu et al., 2017). The importance of a correct determination of the blue wing of Lyman-$\alpha$ line to interpret the asymmetrical shape of the Lyman-$\alpha$ line observed with COS is presented in Sect. 2. An accurate prediction of the satellite and consequently the full Lyman-$\alpha$ profile requires exacting new ab initio calculations to obtain the ground and first excited potential energy curves and the corresponding electric dipole transition moments for the H–He system. The new molecular data in Sect. 3 corroborate the prediction of a line satellite in the Lyman-$\alpha$ profile (Allard & Christova, 2009) that is described in Sect. 4. In Allard et al. (1999) we previously derived a classical path expression for a pressure-broadened atomic spectral line shape that includes the effects of a radiative electric dipole transition moment that is dependent on the position of the radiating atom and its dynamic neighbors. Such a comprehensive unified approach employing the precise molecular data is fundamentally necessary to obtain an accurate absorption line profile that is valid over the full breadth of spectral line for the range of densities and temperatures found in stellar atmospheres. Figure 1: COS observation of WD 1425+540. The broad distinctive collision- induced satellite in the blue wing of the Lyman-$\alpha$ line about 1160 Å is clearly visible (Xu et al., 2017). The strong emission at the center of Lyman-$\alpha$ is from Earth’s geocoronal hydrogen above the HST orbit. ## 2 COS observation of WD 1425+540 WD 1425+540 (T=14,490 K, log g=7.95) is the prototype of DBA white dwarfs and it is a helium-dominated white dwarf that also has a large amount of hydrogen in its atmosphere (Bergeron et al., 2011). It was observed with HST COS under program 13453, and the details of observation and data reduction strategy were reported by Xu et al. (2017). Here, we focus on the spectrum of segment B of the G130M grating, which covers 1130-1270 Å, as shown in Fig. 1. As described in Xu et al. (2017), there are two unusual features of the Lyman-$\alpha$ profile in WD 1425+540. First, the line profile is very asymmetric exhibiting an extend blue wing with the satellite feature as noted. Second, previous white dwarf spectral models cannot reproduce the strength of Lyman-$\alpha$ and Balmer-$\alpha$ simultaneously. The derived hydrogen abundance is more than a factor of 10 higher from the Lyman-$\alpha$ measurement than from Balmer-$\alpha$. While WD 1425+540 is the most extreme case so far, these peculiarities have been observed in other DBA white dwarfs as well, e.g. Jura et al. (2012). The asymmetry also could not be produced by white dwarf models of Xu et al. (2017) because the opacity data used for the Lyman-$\alpha$ profile did not take into account the quasi-molecular line satellite predicted in Allard & Christova (2009). Once this feature is included, the observed asymmetry is reproduced (Gänsicke et al., 2018). The need to have both accurate data for Lyman-$\alpha$ and for Balmer-$\alpha$ is essential to determine the hydrogen abundance correctly. The goal of this paper is to develop the foundation of the atomic and molecular physics needed to compute a complete profile without making ad hoc assumptions. We emphasize the importance of accurate potentials and electric dipole transition moment data for this purpose, and here we provide that data for Lyman-$\alpha$. With new potentials of H-He we also compute a model DBA white dwarf spectrum that demonstrates their validity. ## 3 H–He diatomic potentials ### 3.1 Methodology and benchmarks The lowest electronic excited states of hydrogen and helium are at unusually high energies for neutral atoms (> 10 eV) with respect to their ground states, and close to the corresponding ionization thresholds. Hydrogen with $n$= 2 or greater is a Rydberg atom in this sense (Gallagher, 1994). The electronic excited states of H–He diatomic system of interest in the present work correlate adiabatically to those of these atoms. Therefore, for the correct description of the electronic states of the H–He diatomic system consistent with its isolated atomic fragments one needs the inclusion of diffuse functions that can flexibly represent the states. In addition to this, the computation of the possible interactions that may occur between these electronic states and the subsequent mixing of their wavefunctions that results in an apparent change in electric dipole transition moments, require post Hartree-Fock multi-configurational approaches. More specifically, we used the Complete Active Space Self Consistent Field (CASSCF) (Knowles & Werner, 1985; Werner & Knowles, 1985) followed by the internally contracted Multi- Reference Configuration Interaction (MRCI) (Knowles & Werner, 1988; Werner & Knowles, 1988; Shamasundar et al., 2011) methods as implemented in the MOLPRO 2015 package (Werner et al., 2015). In MRCI, the complete CASSCF wave functions are used as a reference. Furthermore, the Davidson correction (MRCI+Q) (Langhoff & Davidson, 1974) has been applied to the resulting energies to account for the lack of size-consistency of the MRCI method. These computations were performed in the $C_{2v}$ point group, where the $B_{1}$ and $B_{2}$ representations were treated on equal footing. Benchmarks on valence-Rydberg electronic states of other molecular systems (Spelsberg & Meyer, 2001; Ndome et al., 2008; Hochlaf et al., 2010) showed the need to use a CASSCF active space larger than the full-valence space. The atomic basis set for the H and He atoms had to be optimized as well. Thus, we performed a series of benchmark computations at different levels of accuracy to find the appropriate states for convergence. Firstly, at the lowest level of accuracy, we adopted a small active space of 3 electrons in 7 molecular orbitals in conjunction with the aug-cc-pV5Z (Dunning, 1989; Kendall et al., 1992) basis set. With this approach, we found inconsistencies in the calculated energies, especially in the asymptotic region. Indeed, with this simplest choice there is a large energy gap of $\sim 0.45$ eV between the two equivalent dissociation limits H($2p\,^{2}P$) + He($1s^{2}\,{}^{1}S$) and H($2s\,^{2}S$) + He($1s^{2}\,{}^{1}S$). Obviously, this gap is unphysical since these two asymptotes should be strictly degenerate because the two H ($n=2$) states have the same energy apart from Lamb shift and negligibly small fine and hyperfine structure. Moreover, we found a spurious second potential well ($D_{e}$ $\sim$ 660 cm-1) in the $C\,\Sigma$ state of H–He at large internuclear separations (for $R_{\mathrm{H-He}}$ $\sim$ 4.2 Å). Thus, at this level of accuracy, a rather poor chemical description of the H–He molecule is obtained in spite of the relatively large size of the MRCI computations with $\sim$ 4.3 x 104 uncontracted configuration state functions (CSFs) per $C_{2v}$ symmetry. This may be linked to some missing correlation energy in the MRCI wavefunctions that can be recovered by means of larger active spaces in the reference CASSCF vector and by adopting more diffuse atomic basis sets. Secondly, we tried an enlarged CASSCF active space of 3 electrons in 14 molecular orbitals in conjunction with the aug-cc-pV6Z (Dunning, 1989; Kendall et al., 1992) basis set. In the subsequent MRCI treatment, the multi- configuration wave functions included $\sim$ 2.1 x 105 uncontracted CSFs per $C_{2v}$ symmetry. With this ansatz, the energy difference between the above mentioned asymptotes was reduced to $\sim$ 0.33 eV but still did not vanish. For modeling based on unified spectral line shape theory an error of this size would be unacceptable. Finally, using the same active space as in the second series of computations, we added a set of diffuse functions to the aug-cc-pV6Z basis set for H and He. Hereafter, this enlarged set will be denoted as aug-cc-pV6Z⋆. The exponents of the added Gaussian primitives, which were left uncontracted, are listed in Table 1 in the Appendix. This approach, compared to the previous ones, solved all the inconsistencies mentioned above. That is, it yielded degenerate H($2p\,^{2}P$) + He($1s^{2}\,{}^{1}S)$ and H($2s\,^{2}S$) + He($1s^{2}\,{}^{1}S)$ dissociation limits and no spurious potential well in the $C\,\Sigma$ state. We note that convergence was reached at this step since a further expansion of the aug-cc- pV6Z⋆ set by inclusion of more diffuse functions led to almost identical results. In these calculations, the MRCI wave functions included more than $7.5\times 10^{5}$ uncontracted CSFs per $C_{2v}$ symmetry species. These relatively large computations for such a small molecular system were necessary to obtain the precision needed to model the Lyman-$\alpha$ profile accurately. Figure 2: Top: short-range part of the potential curves of the H–He molecule: $A$ (red dotted), $B$ (green dashed line) and $C$ (blue solid). Bottom: $X$ (black solid). Note the agreement at short distance with data of Theodorakopoulos et al. (1984) that are overplotted in dotted cyan. Figure 3: Top: long range part of the $C\,\Sigma$ potential curve correlated with $2s$ state. This work (full line), Theodorakopoulos et al. (1984) (dotted line). Bottom: $\Delta V(R)$ (black) and $\tilde{d}(R)$ (blue) at 14500 K for the $C-X$ transition. The atomic separation for the maximum in the $C-X$ difference potential is $R_{\mathrm{max}}\approx 2.2\,$Å as shown in Fig. 4 Note that the $C-X$ transition in this work is forbidden asymptotically as it is a transition between the $2s$ and $1s$ states of the free hydrogen atom at large $R$. ### 3.2 Potential energy curves and transition moments The electronic states investigated in the present contribution correlate, at large internuclear distances, to the H($1s\;^{2}S$) + He($2s^{2}\;{}^{1}S$), H($2s\;^{2}P$) + He($2s^{2}\;{}^{1}S$), and H($2p\;^{2}P$) + He($2s^{2}\;{}^{1}S$) dissociation limits (see Fig. 2 and Table 2 in the Appendix). The MRCI+Q/aug-cc-pV6Z⋆ potential energy curves of the four lowest electronic states of H–He, obtained with the largest active space and basis set as described in the previous section, are represented in Fig. 2 as a function of the internuclear distance, $R_{\mathrm{H-He}}$. This figure shows that the ground state possesses a repulsive potential correlating to the H($1s\,^{2}S$) + He($1s^{2}\,{}^{1}S$) isolated atom asymptote at large distances. The ground $X\,^{2}\Sigma^{+}$ state is repulsive at short range with a shallow well at $4\,$Å. The excited $A\,^{2}\Sigma^{+}$, $B\,^{2}\Pi$ and $C\,^{2}\Sigma^{+}$ states have rather deep potential wells in the molecular region closer than 1 Å, and complex behavior at longer range that can affect transition probabilities and difference potential energies in subtle ways. We refer to these as the $X\,\Sigma$, $A\,\Sigma$, $B\,\Pi$, and $C\,\Sigma$ states, or more succinctly by the letter designations $X$, $A$, $B$, and $C$ in the following. They correlate adiabatically to the H($n=2$) + He($1s^{2}\,{}^{1}S$) dissociation limits at large internuclear separations (see Table 2 in the Appendix). The ordering of the assignments of labels for the states is with $A\,\Sigma$ the lowest and $C\,\Sigma$ the highest inside this close 1 Å region with wells in all the states of the order of $15\,000\;\mathrm{cm}^{-1}$ deep, with minima located at $R_{\mathrm{H-He}}$ = 0.7407, 0.7686, and 0.8095 Å for the $A$, $B$ and $C$ states, respectively (see Table 3 in the Appendix). While the $A$ and $B$ states have potentials with a simple short-range well, the $C$ state also exhibits a potential maximum of $\approx 0.666$ eV at $R_{\mathrm{H-He}}=2.098$ Å. Its presence causes a related maximum in the $C-X$ transition difference potential energy curve which affects the blue wing of Lyman-$\alpha$. Although the $C\,\Sigma$ H-He molecular state shown in Fig. 2 is correlated asymptotically with the $2s$ atomic state, we find that at $R_{\mathrm{H-He}}<7\;$Å the transition probability to the $X\,\Sigma$ ground state is not zero. Detailed electric dipole transition moments between the $X\,\Sigma$ ground state and the $A\,\Sigma$, $B\,\Pi$ and $C\,\Sigma$ excited states as a function of the internuclear distance have been calculated at the MRCI/aug-cc-pV6Z⋆ level. In this calculation almost all the transition moments are rather large, particularly for the $C\,\Sigma$ $\leftarrow$ $A\,\Sigma$ and $B\,\Pi$ $\leftarrow$ $A\,\Sigma$ transitions, where corresponding matrix elements of around -9.2 and -7.5 debye (D or $10^{-18}$ statcoulomb-cm) are calculated, respectively. Fig. 7 in the Appendix offers a detailed view. These transition moments correlate to the correct atomic values at dissociation. In particular, the $\langle X\,\Sigma|DM|C\,\Sigma\rangle$ matrix element of the electric dipole transition moment (DM) vanishes at large $R_{\mathrm{H-He}}$ where the $1s-2s$ transition in the isolated hydrogen atom is forbidden to one-photon electric dipole transitions by parity conservation. ## 4 Lyman-alpha opacity The theory of spectral line shapes, especially the unified approach we developed, determines the contributions of specific spectral lines to stellar opacities and may be incorporated into stellar atmosphere models to make accurate synthesis of stellar spectra possible. The line shape theory accounts for neutral atom broadening and shift in both the centers of spectral lines and their extreme wings with one consistent treatment without ad hoc assumptions about the line shape or potentials. Complete details and the derivation of the theory are provided by Allard et al. (1999). The spectrum, $I(\Delta\omega)$, is the Fourier transform (FT) of a electric dipole transition autocorrelation function, $\Phi(s)$. For a perturber density $n_{p}$, we have $\Phi(s)=e^{-n_{p}g(s)}\;,$ (1) where the decay of the autocorrelation function with time leads to atomic line broadening. (See Eq. (121) of Allard et al. (1999).) Our approach introduces the concept of a modulated electric dipole transition moment $\tilde{d}_{if}(R(t))$ into the line shape calculation. $\tilde{d}_{if}[R(t)]=d_{if}[R(t)]e^{-\frac{V_{i}[R(t)]}{2kT}}\;\;,$ (2) where the potential energy for the initial state is $V_{i}(R)=E_{i}(R)-E_{i}^{\infty}\;\;.$ (3) The difference potential energy $\Delta V(R)$ for a transition $if$ is $\Delta V(R)=V_{if}(R)=V_{f}(R)-V_{i}(R)\;\;.$ (4) The Boltzmann factor $e^{-\frac{V_{i}(R)}{2kT}}$ in Eq. (2) appears because the perturbing atoms or ions are in thermal equilibrium with the radiating atom which affects the probability of finding them initially at a given $R$. This treatment results in Lyman series line wing profiles that exhibit a sensitive dependence on temperature. We had to use electric dipole moments modulated by the Boltzmann factor in the comparison of emission spectra of Lyman-$\alpha$ (Kielkopf & Allard, 1998) and Balmer $\alpha$ (Kielkopf et al., 2002) measured in laboratory. ### 4.1 Study of the characteristics of the line satellite In Allard & Christova (2009) we predicted a line satellite at 1157 Å in spectra computed for the temperature range of cool DZ white dwarfs with potentials published in Theodorakopoulos et al. (1984). However, we noticed an unexpected well of about 150 cm-1 (upper Fig. 3) in the potential energy of the $C\,\Sigma$ state at $R\sim 8$ Å which may be related to the choice of basis states and has no clear physical origin. In this work we use the new ab initio calculations of the potentials over the full range of distances $R$ between the H and He atoms since convergence at large $R$ is now reached. The long range well of the $C\,\Sigma$ state of Theodorakopoulos et al. (1984) and Theodorakopoulos et al. (1987) potentials is not found in these new calculations as we see in Fig. 3. Figure 4: Top: variation with temperature of the line satellite. The He density is $1\times 10^{20}$ cm-3, the temperatures are 14500 K (full black line), $20\,000$ K (blue stars), and $5\,000$ K (red dashed line). Bottom: for the $C-X$ transition, $\Delta V(R)$ (black solid) and $\tilde{d(R)}$ at 5000 K (black solid), $10\,000$ K (red dotted), 14500 K (green dashed), and 20000 K (blue solid). At the highest temperatures the He can reach the inner regions of the lower state $X\,^{2}\Sigma$ potential and enhance the transition probability. The prediction of a satellite in the blue wing of the H–He line profile is related to a potential maximum at $R=2.1$ Å (see Sect. 3.2) of the $C\,\Sigma$ state. This leads to a maximum of the potential energy difference $\Delta V(R)$ in Eq. (4) for this transition shown in Fig. 3. The unified theory predicts that line satellites will be centered periodically at frequencies corresponding to integer multiples of the extrema of $\Delta V(R)$. In the quasi-static limit the first satellite on the line would be at $\Delta\omega=5\,000$ cm-1 corresponding to $\lambda\sim 1150$ Å on the blue side of Lyman-$\alpha$. In this case the maximum in $\Delta V$ occurs at rather small internuclear distance, and is quite sharp. The correspondingly short duration of the close collision leads to a broad satellite centered at $\lambda$ $\sim$ 1160 Å for T=$14\,500$ K (Fig. 4). ### 4.2 Temperature and density dependence For a lower temperature, $T=5\,000$ K (Fig. 4), the duration of the collision is longer, and the line satellite at $\lambda\sim 1153$ Å is sharper and closer to the predicted quasi-static position than at higher temperatures. The oscillations which appear on the red side of the quasi-molecular satellite are due to interference effects described by Royer (1971) and Sando & Wormhoudt (1973). They depend on the relative velocity and therefore on temperature. Consequently velocity averaging would moderate their amplitude in observed spectra. At temperatures below $10\,000$ K the blue wing of Lyman-$\alpha$ shortward of 1150 Å becomes significantly more transparent than at higher temperature, an order of magnitude effect below 1120 Å. Thus this far blue wing is a sensitive indicator of temperature in cool helium-rich WD atmospheres. The satellite amplitude depends on the value of the electric dipole transition moment through the region of the potential extremum responsible for the satellite and on the position of this extremum. The blue line wings shown in Fig. 4 are unchanged in the range $14\,500$ to $20\,000$ K as there is no change with $T$ of $\tilde{d}_{if}[R(t)]$ in the internuclear distance where the potential difference goes through a maximum. $\tilde{d}_{if}[R(t)]$ at $14\,500$ K for the $C-X$ transition is also plotted in Fig. 3. In the former work we used electric dipole transition moments of Theodorakopoulos et al. (1987) where the $C-X$ transition was allowed. Nevertheless the amplitude and position of the line satellite are unchanged as they are due to a range of internuclear distance where the potentials and the dipole moments are almost identical as we see in Fig. 5. The main difference between the two potentials concerns the red wing which is lowered using dipole moments of Theodorakopoulos et al. (1987) where the $A-X$ transition was forbidden. Figure 5: Comparison of the unified line profile using the dipole moments of this work (black line) with the line profile using dipole moments of Theodorakopoulos et al. (1987) (red dashed line) . The He density is $10^{20}$ cm-3 and the temperature is 14500 K. In summary the unified line profile calculation leads to a flat blue wing due to a line satellite. The resulting asymmetry of the Lyman-$\alpha$ line can be easily appreciated in Fig. 5 the blue side of the line is wider than the red side. Measured at the strength of the broad collision-induced 1160 Å satellite, the asymmetry ratio of the width on the blue side to that on the red is as large as 2.2. Consequently, the near wing is clearly far different from a symmetric Lorentzian because the satellite is rather close to the isolated atom line center. This was also the case for the Mg b triplet perturbed by He (Allard et al., 2016). The existence of the asymmetrical shape of these line profiles depends strongly on the maximum value of the potential energy difference $\Delta V(R)$ which predicts the position of the line satellite and on the atomic collision energies at the temperatures of interest. These results enable computing atmosphere models and synthetic spectra which we compare to an HST COS observation of WD 1425+540 in Section 5. ## 5 Model atmosphere and synthetic white dwarf spectrum To demonstrate the importance of a proper treatment of He perturbers on hydrogen lines, synthetic spectra of the white dwarf WD 1425+540 were computed using the stellar atmosphere code TLUSTY (version 207) for computing the atmospheric structure, and a companion program SYNSPEC (version 53) for generating detailed synthetic spectra. For a description of the previous versions (205 and 51) see the works of Hubeny & Lanz (2017) and Hubeny & Lanz (2011a, b). This procedure allows us to study the effect of the H/He ratio on the spectrum, and the development of line wings, though it is not fully self- consistent with the stellar atmosphere model since that would require a treatment of He I optical lines as well. We have computed a number of H-He models, with the basic model parameters, $T_{\rm eff}=14,410$ K and $\log g=7.89$, from Gänsicke et al. (2018), and with varying He/H ratio. For treating the electron and proton broadening of the hydrogen lines we used Tremblay & Bergeron (2009) data. The He/H ratio was adjusted to obtain a reasonable agreement by eye with the observed spectrum, and we found a nominal ratio of $4\times 10^{3}$ ($\log(N_{\mathrm{H}}/N_{\mathrm{He}})\approx-3.6$) fitted the observed profile well. Liebert et al. (1979) found 3.7 from a ground-based H$\beta$ profile, and recently Gänsicke et al. (2018) analyzed the L$\alpha$ profile and adopted a somewhat larger $\log(N_{\mathrm{H}}/N_{\mathrm{He}})\approx-4.0\pm 0.20$. The potential energies for the $n=1$ and $n=2$ electronic states H-He that were used in our models are the ones described in this paper. Stellar opacities were computing using H-He electric dipole moments from the previous work of Theodorakopoulos et al. (1987) in which the $A-X$ transition is forbidden, and also using new dipole transition moments from this work in which the $A-X$ transition is allowed. As shown in Fig. 6, the observed red wing of Lyman-$\alpha$ is consistent with a suppressed $A-X$ transition probability in the region of atomic separation with difference potential energy that would contribute. We conclude that the additional basis states used for the new ab initio potentials improve the calculation of the potential energy curves, but may not capture the dipole transition moments of the real H-He system correctly for the $A-X$ transition. However the combination of this work’s potentials and the dipole moments of Theodorakopoulos et al. (1987) achieve a remarkable fit in Fig. 6 to the HST COS spectrum of WD 1425+540 when incorporated into the unified line shape theory we described here. Figure 6: The observed spectrum of WD 1425+540 (also see Fig. 1) compared with a synthetic white dwarf spectrum in the Lyman-$\alpha$ region. The synthetic spectrum is computed with TLUSTY and SYNSPEC for a temperature of 14 500 K and a He/H ratio of $4\times 10^{3}$ using the unified line profile with the potentials of this work. For the dipole moments of Theodorakopoulos et al. (1987) (red solid line) the $A-X$ transition is forbidden and its contributions to the opacity are suppressed. For the dipole moments of this work (blue dashed line), the $A-X$ transition contributes in the red wing of the model but is absent in the observed spectrum. ## 6 Conclusions The Lyman-$\alpha$ region of the spectrum of a helium-rich white dwarf with hydrogen in its atmosphere is determined by the changes in transition energy and transition probability during the H-He collisions that broaden the atomic spectral line. We developed new H-He potential energies and transition dipole moments for the hydrogen $1s$, $2s$, and $2p$ states as input data for a unified theory calculation of the profile of WD 1425+540 to test the potentials and dipole moments, and to confirm the origin of the short- wavelength “blue” satellite. We found that the spectral line profile from the new molecular data has a satellite feature in the blue wing that agrees with previous work. These results provide a benchmark implementation of ab initio atomic and molecular potentials for the most basic neutral non-resonant atom- atom pair relevant to stellar atmosphere models. The new calculations show how the profile depends on the variation of the electric dipole transition moment and interaction potential energy with atomic separation. A comparison with the observed spectrum of WD 1425+540 was made by using these theoretical opacities in a stellar atmosphere and spectrum synthesis code. While it was not our goal to refine the stellar model based on the new theoretical data, the profiles reproduce the observed spectrum with a reasonable He/H ratio. Further, the absence of an extended red wing of Lyman-$\alpha$ in the observed spectrum suggests that the states of the difference potential that could contribute to that region have the reduced transition dipole moment that was found in previous molecular models. The new work presented here shows clearly that there is an opportunity to use stellar spectra to improve the atomic and molecular physics, ultimately to yield better models for astrophysical applications. For H–He, the $A-X$ transition dipole moment remains uncertain. The blue wing of Lyman-$\alpha$ is sensitive to He density and the structure and temperature of the stellar atmosphere, with a profile that for wavelengths shortward of $1150\,$Å will have reduced opacity from regions with temperatures under $10\,000\,$K. Profiles computed with a unified theory of collision broadening based on accurate data from ab initio molecular physics take into account the strong dependence of the amplitude of the electric dipole transition moment on atom-atom separation ($R$) where the potential energy change $\Delta V(R)$ is an extremum. Incorporated into model atmospheres, this dependence may be used to probe white dwarf or stellar atmospheres for density and temperature. This emphasizes the importance of the accuracy of both the potential energies and the electric dipole transition moments for the line shape calculations that have traditionally assumed electric dipole transition moments are constant (Allard & Kielkopf, 1982; Allard & Koester, 1992; Allard et al., 1994). The effect of collision broadening is central to understanding the opacity of stellar atmospheres, yet there have been only a few definitive comparisons with experimental work for atomic H. (Kielkopf & Allard, 1995, 1998; Kielkopf et al., 2004). This is because of the difficulty of creating an environment in a laboratory experiment simulating a stellar atmosphere with accurate diagnostics. On the theoretical side, the maturing capability of ab initio methods now offers the possibility of accurately computing the interaction of H with H (Drira, 1999; Spielfiedel, 2003; Spielfiedel et al., 2004) and H with He atoms (this work). While an accurate determination of the broadening of Balmer $\alpha$ with high density atomic hydrogen (that is H–H) has been done by Allard et al. (2008), nothing comparable exists for H–He. Our calculations reported in Allard et al. (2008) support the results of Barklem et al. (2000); Barklem et al. (2002) that the Ali & Griem (1966) theory underestimates the actual line width. Recent laboratory measurements show a similar result at high density in environments comparable to white dwarf atmospheres (Kielkopf & Allard, 2014). It would be possible now to similarly improve the calculation of Balmer-$\alpha$ broadening and its contribution to the full white dwarf opacity model. A major improvement to comprehensive theoretical models for DBA white dwarf spectra is within reach that would determine H-He molecular data for $n=3$ excited states, and use those to compute accurate Balmer-$\alpha$ profiles under white dwarf atmosphere conditions. Such results would help understanding the differences in stellar parameters that are found from Balmer and Lyman line profiles. In conclusion, complete unified line profiles based on accurate atomic and molecular physics for both the Lyman-$\alpha$ and Balmer-$\alpha$ lines should be incorporated into the analysis of DBA white dwarf spectra to derive the hydrogen abundance. ## acknowledgements The paper was based on observations made with the NASA/ESA Hubble Space Telescope under program 13453, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. We thank the COST Action CM1405 MOLecules in Motion (MOLIM) of the European Community for support. The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding the research through the Research Group Project No. RGP-333. 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R., Schütz M., et al., 2015, MOLPRO, version 2015.1, a package of ab initio programs * Xu et al. (2017) Xu S., Zuckerman B., Dufour P., Young E. D., Klein B., Jura M., 2017, ApJ, 836, L7 ## appendix Parameters of the H–He molecular potentials are given in Tables 1 and 2. Figure 7 shows the dependence on $R$ of the radiative transition moments between the excited states and the perturbations of those states as the H and He atoms approach from large $R$. Table 1: Exponents of the diffuse uncontracted Gaussian primitives added to the aug-cc-pV6Z basis set to form the presently used aug-cc-pV6Z* basis sets for the H and He atoms. State | 1 | 2 | 3 ---|---|---|--- H(_s_) | 0.00690204 | 0.002520537 | 0.000920468 H(_p_) | 0.026565598 | 0.010533298 | 0.004176468 H(_d_) | 0.055406537 | 0.024364162 | 0.010713761 H(_f_) | 0.106396067 | 0.046204584 | 0.020065249 H(_g_) | 0.168703345 | 0.069928301 | 0.028985598 H(_h_) | 0.175320015 | 0.045069073 | 0.011585793 He(_s_) | 0.017177900 | 0.006596920 | 0.002533450 He(_p_) | 0.050416903 | 0.019858313 | 0.007821833 He(_d_) | 0.094209988 | 0.036827891 | 0.014396494 He(_f_) | 0.151890237 | 0.056684629 | 0.021154402 He(_g_) | 0.232902520 | 0.079072280 | 0.026845675 He(_h_) | 0.248198125 | 0.060632194 | 0.014811808 Table 2: Dissociation fragments, experimental and calculated relative dissociation asymptotic energies, and molecular states for the four lowest electronic states of H–He. Experimental data are from Kramida (2010). Atomic | | Observed | Calculated | Molecular ---|---|---|---|--- H | He | (cm${}^{-}1$) | (cm${}^{-}1$) | $1s\,^{2}S_{g}$ | $1s^{2}\,{}^{1}S_{g}$ | 0a | 0a | $X\,^{2}\Sigma^{+}$ $2p\,^{2}P_{u}$ | $1s^{2}\,{}^{1}S_{g}$ | 82259 | 82308 | $A\,^{2}\Sigma^{+}$, $B\,^{2}\Pi$ $2s\,^{2}S_{g}$ | $1s^{2}\,{}^{1}S_{g}$ | 82259 | 82308 | $C\,^{2}\Sigma^{+}$ aReference | | | | Table 3: Spectroscopic constants and dissociation energies for the three lowest excited electronic states of H–He as deduced from the MRCI+Q /aug-cc-pV6Z* potential energy curves. $R_{e}$ corresponds to the equilibrium distance. $\omega_{e}$ and $\omega_{e}x_{e}$ are the vibrational constants. $\beta_{e}$ and $\alpha_{e}$ are the rotational constants. $D_{e}$ is the dissociation energy. State | $R_{e}$ | $\omega_{e}$ | $\omega_{e}x_{e}$ | $\beta_{e}$ | $\alpha_{e}$ | $D_{e}$ ---|---|---|---|---|---|--- | Å | cm-1 | cm-1 | cm-1 | cm-1 | eV $A\,^{2}\Sigma^{+}$ | 0.74074 | 3697.2 | 149.5 | 38.16 | 2.608 | 2.563 $B\,^{2}\Pi$ | 0.76863 | 3313.4 | 149.8 | 35.44 | 2.629 | 2.218 $C\,^{2}\Sigma^{+}$ | 0.80953 | 2906.3 | 144.0 | 31.95 | 2.551 | 1.638 Figure 7: Potential energy differences in cm-1 and electric dipole transition moments in debye (D or $10^{-18}$ statcoulomb-cm) between the four lowest electronic states of H–He calculated at the MRCI/aug-cc-pV6Z⋆ level. Note that the $C\,\Sigma$ $\leftarrow$ $X\,\Sigma$ is asymptotically forbidden, while transitions between excited states may occur. Upper panel: Energy differences $A\Sigma-B\Sigma$ (blue) and $A\Sigma-C\Pi$ (red). Lower panel: Electric dipole transition moments for H in the presence of He for states contributing to H Lyman-$\alpha$.

2024-09-04T02:54:59.017865

2003.05124

arxiv-papers

# Role of generalized parity in the symmetry of fluorescence spectrum from two-level systems under periodic frequency modulation Yiying Yan<EMAIL_ADDRESS>Department of Physics, School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China Zhiguo Lü<EMAIL_ADDRESS>Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China JunYan Luo Department of Physics, School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China Hang Zheng<EMAIL_ADDRESS>Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China ###### Abstract We study the origin of the symmetry of the fluorescence spectrum from the two- level system subjected to a low-frequency periodic modulation and a near- resonant high-frequency monochromatic excitation by using the analytical and numerical methods based on the Floquet theory. We find that the fundamental origin of symmetry of the spectrum can be attributed to the presence of the generalized parity of the Floquet states, which depends on the driving parameters. The absence of the generalized parity can lead to the asymmetry of the spectrum. Based on the generalized parity, the conditions for the symmetry and asymmetry of the spectrum can be derived, which succeeds in predicting symmetry and asymmetry of the spectrum for the harmonic, biharmonic, and multiharmonic modulations. Moreover, we find that the secular approximation widely used in the analytical calculation may lead to artifact symmetry of the spectrum that vanishes when such approximation is avoided. The present study provides a significant perspective on the origin of the symmetry of the spectrum. ## I Introduction Resonance fluorescence, arising from a quantum emitter driven by an external field and coupled to a radiative reservoir Mollow (1969); Scully and Zubairy (1997); Cohen-Tannoudji _et al._ (1998), is not only an important concept in quantum optics but also has potential application in quantum information technology, for instance, it plays an important role in realizing the single- photon source He _et al._ (2013); Santana _et al._ (2017); Kiršanskė _et al._ (2017). Particularly, the resonance fluorescence of two-level systems has attracted much interest and been studied in various aspects such as spectrum Ficek and Freedhoff (1993); Agarwal _et al._ (1991); Ficek and Freedhoff (1996); Ficek and Rudolph (1999); Peiris _et al._ (2014); Konthasinghe _et al._ (2014); He _et al._ (2015); Toyli _et al._ (2016), squeezing Carmichael (1985); Grünwald and Vogel (2012, 2013), photon statistics Kimble _et al._ (1977); D’Souza _et al._ (1990); Nazir (2008); Pastukhov _et al._ (2014), photon antibunching Itano _et al._ (1988); Ficek _et al._ (1984); Damanet _et al._ (2018), and so on. The line shape of the spectrum is found to depend strongly on the external field that interacts with the quantum emitters as well as the reservoirs to which the quantum emitters are coupled. As is well- known, for a sufficiently strong monochromatic field, the spectrum has a symmetric three-peak structure, known as the Mollow triplet Mollow (1969). More recently, the bi- and multi-chromatically driven quantum systems are of interest Kryuchkyan _et al._ (2017); Antón _et al._ (2017); Yan _et al._ (2018); Saiko _et al._ (2018). In such systems, the spectrum turns out to have a complicated multipeak structure Ficek and Freedhoff (1993); Agarwal _et al._ (1991); Ficek and Freedhoff (1996); Ficek and Rudolph (1999); Peiris _et al._ (2014); Konthasinghe _et al._ (2014); He _et al._ (2015), which can be either symmetric or asymmetric. In principle, the physical origin of the triplet and multipeak structures can be understood in terms of the transitions between the quantum dressed states Cohen-Tannoudji _et al._ (1998) or in terms of the transitions between the semiclassical Floquet states Breuer and Petruccione (1997); Yan _et al._ (2016a). The studies on the resonance fluorescence have enriched the physics concerning the light-matter interaction. The origin of the symmetry of the spectrum has been found in the case of the monochromatic field. Specifically, it is the detailed balance condition that guarantees the symmetry of the Mollow triplet Cohen-Tannoudji _et al._ (1998). As is well-known, the breakdown of such a condition leads to the asymmetry of the spectrum, for instance, in the presence of a pure dephasing reservoir Roy and Hughes (2012); McCutcheon and Nazir (2013) or the counter- rotating terms of the external field under certain conditions Browne and Keitel (2000); Yan _et al._ (2013, 2016a). The dephasing-induced asymmetric Mollow triplet has been experimentally observed in the quantum dots (the pure dephasing arises because of the interaction between the quantum dot and its solid-state environment) Ulrich _et al._ (2011); Ulhaq _et al._ (2013). For the bi- and multi-chromatic fields, the origin of the symmetry of the spectrum is rarely discussed, owing to the fact that the physically transparent spectrum is hardly analytically derived, and has not been comprehensively understood. Recent studies show that the fluorescence spectrum from a driven two-level system with a modulated transition frequency is symmetrically multipeaked for the vanishing detuning while asymmetrically multipeaked for the finite detuning Yan _et al._ (2016b); Kryuchkyan _et al._ (2017); Antón _et al._ (2017); Yan _et al._ (2018). Such an exotic bichromatically driven two-level system with coexistence of the longitudinal and transversal coupling between the system and the applied fields has been experimentally studied in the superconducting qubits Li _et al._ (2013); Pan _et al._ (2017), single molecule Brunel _et al._ (1998), and nitrogen-vacancy spin qubits Rohr _et al._ (2014). The quantum systems under frequency modulation are also of interest in theoretical studies Kibis _et al._ (2009); Macovei and Keitel (2014); Zhao _et al._ (2015); Silveri _et al._ (2013); Macovei _et al._ (2015), the intriguing phenomena of which were reviewed recently Silveri _et al._ (2017). It is worthwhile to note that the bichromatically driven two- level system with frequency modulation differs from those considered in Refs. Agarwal _et al._ (1991); Ficek and Freedhoff (1993), where the two-level systems are transversely driven by a bichromatic field. In such a case, the symmetry of the fluorescence spectrum is found to depend on the average detuning if the strengths of the two components of the bichromatic field are the same; the pronounced asymmetry of the spectrum is revealed when the average detuning is finite and/or the strengths of the two components of the field are unequal Agarwal _et al._ (1991); Ficek and Freedhoff (1993). For a bichromatically amplitude-modulated field, the spectrum is also found to be symmetric and asymmetric for the vanishing and finite detuning, respectively Wilkens and Rza¸ewski (1989). So far the fundamental origin of such a detuning-dependent symmetry remains obscure. In this work, we use both analytical and numerical methods based on the Floquet theory to study the fundamental origin of the symmetry of the fluorescence spectrum from the two-level system under a low-frequency periodic modulation and a near-resonant monochromatic excitation. We address the symmetry and asymmetry of the spectrum by considering the generalized parity of Floquet states rather than the behaviors of the bare-state or dressed-state populations as considered in Refs. Das and Macovei (2013); Macovei _et al._ (2015); Antón _et al._ (2017). The generalized parity is found to guarantee the symmetry of the spectrum while the breaking of such a parity can yield pronouncedly asymmetric spectrum even in the vanishing detuning case. Based on the generalized parity, the conditions for the symmetric and asymmetric spectra are derived, which are not given in the previous works and cannot be derived from the behaviors of the bare or dressed state population. The generalized-parity-induced symmetry of the spectrum is verified and illustrated in the context of the biharmonic modulation by the comparison between the analytical and numerical results. The analytical results are found to be in agreement with the numerically exact results in the regimes where the perturbation theory and secular approximation can be justified. In addition, we find that the spectrum with the secular approximation may have artifact symmetry under certain conditions, i.e., the spectrum with secular approximation is symmetric while the numerically exact calculation shows asymmetric spectra because of the broken parity. The present finding simply interprets the detuning-dependent symmetry in the harmonic modulation case and can also be extended to analyze the symmetry and asymmetry of the spectrum in the multiharmonic modulation cases. Our results suggest that it is feasible to control the symmetry and asymmetry of the spectrum via engineering the generalized parity of the Floquet states. The rest of the paper is organized as follows. In Sec. II, we first discuss the generalized-parity-induced symmetry of the fluorescence spectrum without the secular approximation and further elucidate the symmetry of the spectrum with a physically transparent formal spectrum with the secular approximation. In Sec. III, we analytically and numerically calculate the fluorescence spectrum in the context of the biharmonic modulation to verify the symmetry and asymmetry of the spectrum predicted based on the generalized parity. In the last section, the conclusions are given. ## II Fluorescence spectrum and generalized parity We consider that the transition frequency of the two-level system is modulated periodically via a low-frequency external field $f(t)$ and the two-level system is also excited by a near-resonant monochromatic field, which is described by the following Hamiltonian ($\hbar=1$) $H(t)=\frac{1}{2}[\omega_{0}+f(t)]\sigma_{z}+\frac{\Omega_{x}}{2}(\sigma_{+}e^{-i\omega_{x}t}+\sigma_{-}e^{i\omega_{x}t}),$ (1) where $\sigma_{z(x,y)}$ is the usual Pauli matrix, $\omega_{0}+f(t)$ is the modulated transition frequency, $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$ are the raising and lowering operators, and $\Omega_{x}$ ($\omega_{x}$) is the strength (frequency) of the monochromatic driving. Here we choose $f(t)=f(t+T)$ with $T$ being the fundamental period of the modulation and much greater than $2\pi/\omega_{x}$. This is a generalized model as compared with the previous one considered in Refs. Yan _et al._ (2016b); Kryuchkyan _et al._ (2017); Antón _et al._ (2017). To study the emission processes, we need to take account of the spontaneous decay. Thus, the time evolution of the driven two-level system under study is modeled by the Lindblad master equation. In the frame rotating at the frequency $\omega_{x}$, the Lindblad master equation takes the form $\frac{d}{dt}\tilde{\rho}(t)={\cal L}(t)\tilde{\rho}(t),$ (2) where $\tilde{\rho}(t)$ is the reduced density matrix in the rotating frame and the superoperator ${\cal L}(t)$ is given by ${\cal L}(t)\tilde{\rho}(t)=-i[\tilde{H}(t),\tilde{\rho}(t)]-\kappa/2[\\{\sigma_{+}\sigma_{-},\tilde{\rho}(t)\\}-2\sigma_{-}\tilde{\rho}(t)\sigma_{+}]$ with $\kappa$ being the radiative decay rate. $\tilde{H}(t)$ is the effective Hamiltonian and reads $\tilde{H}(t)=\frac{\Omega_{x}}{2}\sigma_{x}+\frac{1}{2}[\delta+f(t)]\sigma_{z},$ (3) with $\delta=\omega_{0}-\omega_{x}$ being the detuning between the bare transition frequency and monochromatic excitation frequency. This master equation is actually a set of first-order differential equations with periodic coefficients. It can be directly solved by the so-called Floquet-Liouville (FL) approach with a desire accuracy Ho _et al._ (1986); Yan _et al._ (2016b). Although such a Floquet-theory-based numerical method is simple and efficient, it is not physically transparent to analyze the role of generalized parity of Floquet states in the symmetry of the fluorescence spectrum. We use an alternative method which is developed in our previous works Yan _et al._ (2016a, 2018) to solve the master equation and calculate the fluorescence spectrum. We first calculate the Floquet states for $\tilde{H}(t)$ and use them as the bases to reformulate Eq. (2) and derive its analytical formal solutions with the aid of the secular approximation in the Floquet picture. ### II.1 The symmetry of fluorescence spectrum without secular approximation The steady-state fluorescence spectrum is given by the Fourier transform of the time-averaged first-order correlation function Mollow (1969); Ho _et al._ (1986) $S(\Delta)\propto{\rm Re}\frac{1}{T}\int_{0}^{\infty}\int_{0}^{T}\lim_{t^{\prime}\rightarrow\infty}\left\langle\tilde{\sigma}_{+}(t^{\prime}+\tau)\tilde{\sigma}_{-}(t^{\prime})\right\rangle e^{-i\Delta\tau}dt^{\prime}d\tau,$ (4) where $\Delta=\omega-\omega_{x}$ and $\left\langle\tilde{\sigma}_{+}(t^{\prime}+\tau)\tilde{\sigma}_{-}(t^{\prime})\right\rangle$ is the first-order correlation function and the tilde indicates that it is evaluated in the rotating frame. In general, it is difficult to derive an exact analytical spectrum. Nevertheless, we find that it is possible to show that the spectrum is exactly symmetric about $\Delta=0$ when $\delta+f(t)=-[\delta+f(t+T/2)]$ by realizing the fact that the driven two- level system possesses a generalized parity symmetry, i.e., $\sigma_{x}\tilde{H}(t+T/2)\sigma_{x}=\tilde{H}(t).$ (5) Here, the generalized parity transformation consists of an exchange between the up and down states of two-level system ($\sigma_{z}\rightarrow-\sigma_{z}$) and a time shift of half period of the modulation ($t\rightarrow t+T/2$). We state briefly how the generalized parity guarantees the symmetry of the spectrum. Owing to Eq. (5), we can construct a generalized parity transformation in the Liouville space, the details of which can be found in Appendix A. When $\delta+f(t)=-[\delta+f(t+T/2)]$, the superoperator ${\cal L}(t)$ is similarly found to be invariant under the generalized parity transformation. Based on this property, it can be derived from the master equation (2) without the secular approximation that in the steady-state limit, the time-averaged first-order correlation function is a real-valued function in the rotating frame. As a result, the fluorescence spectrum is symmetric about $\Delta=0$. This finding shows that the symmetry of the spectrum occurs when $\delta+f(t)=-[\delta+f(t+T/2)]$ and results from the generalized parity. We will numerically verify the generalized-parity-induced symmetry in Sec. III. ### II.2 The symmetry of fluorescence spectrum with secular approximation To further elucidate the role of the generalized parity in determining the symmetry of the spectrum, we calculate the spectrum in the Floquet picture which allows us to derive a physically transparent formal spectrum with the aid of the secular approximation. According to the Floquet theory Shirley (1965); Sambe (1973), the time- dependent Schrödinger equation governed by $\tilde{H}(t)$ possesses a set of formal solutions $|\tilde{\psi}_{\alpha}(t)\rangle=|\tilde{u}_{\alpha}(t)\rangle e^{-i\tilde{\varepsilon}_{\alpha}t}$, where $|\tilde{u}_{\alpha}(t)\rangle=|\tilde{u}_{\alpha}(t+T)\rangle$ is Floquet state and $\tilde{\varepsilon}_{\alpha}$ is the corresponding real-valued quasienergy. The index $\alpha$ labels independent Floquet states. Substituting the formal solution into the Schrödinger equation, one readily finds that $[\tilde{H}(t)-i\partial_{t}]|\tilde{u}_{\alpha}(t)\rangle=\tilde{\varepsilon}_{\alpha}|\tilde{u}_{\alpha}(t)\rangle.$ (6) On solving this equation, one obtains the Floquet states and quasienergies of the driven two-level system. We use $|\tilde{u}_{\alpha}(t)\rangle$ ($\alpha=\pm$) as the basis to reformulate the master equation (2) and invoke the secular approximation Yan _et al._ (2016a, 2018), yielding $\displaystyle\frac{d}{dt}\tilde{\rho}_{++}(t)$ $\displaystyle=$ $\displaystyle-\Gamma_{{\rm rel}}\tilde{\rho}_{++}(t)+\Gamma_{{\rm s}},$ (7) $\displaystyle\frac{d}{dt}\tilde{\rho}_{+-}(t)$ $\displaystyle=$ $\displaystyle-(i\Delta_{+-}+\Gamma_{{\rm deph}})\tilde{\rho}_{+-}(t),$ (8) where $\tilde{\rho}_{\alpha\beta}(t)=\langle\tilde{u}_{\alpha}(t)|\tilde{\rho}(t)|\tilde{u}_{\beta}(t)\rangle$ is the element of density operator, $\Delta_{+-}=\tilde{\varepsilon}_{+}-\tilde{\varepsilon}_{-}$ is the difference of two quasienergies, and $\Gamma_{{\rm s}}=\kappa\sum_{l}|x_{-+,l}^{(+)}|^{2}$, where $x^{(+)}_{\alpha\beta,l}$ is a time-averaged transition matrix element defined as follows: $x^{(\pm)}_{\alpha\beta,l}=\frac{1}{T}\int^{T}_{0}\langle\tilde{u}_{\alpha}(t)|\sigma_{\pm}|\tilde{u}_{\beta}(t)\rangle e^{-i2\pi lt/T}dt.$ (9) The relaxation rate $\Gamma_{{\rm rel}}$ and dephasing rate $\Gamma_{{\rm deph}}$ are given by $\displaystyle\Gamma_{{\rm rel}}$ $\displaystyle=$ $\displaystyle\kappa\sum_{l}(|x_{+-,l}^{(+)}|^{2}+|x_{-+,l}^{(+)}|^{2}),$ (10) $\displaystyle\Gamma_{{\rm deph}}$ $\displaystyle=$ $\displaystyle\frac{\kappa}{2}\sum_{l}(|x_{+-,l}^{(+)}|^{2}+|x_{-+,l}^{(+)}|^{2}+4|x_{++,l}^{(+)}|^{2}).$ (11) The analytical formal solutions in the Floquet picture can be easily found as follows: $\displaystyle\tilde{\rho}_{++}(t)$ $\displaystyle=$ $\displaystyle\tilde{\rho}_{++}(0)e^{-\Gamma_{{\rm rel}}t}+\tilde{\rho}_{++}^{{\rm ss}}(1-e^{-\Gamma_{{\rm rel}}t}),$ (12) $\displaystyle\tilde{\rho}_{+-}(t)$ $\displaystyle=$ $\displaystyle\tilde{\rho}_{+-}(0)e^{-(\Gamma_{{\rm deph}}+i\Delta_{+-})t},$ (13) where $\tilde{\rho}_{++}^{{\rm ss}}=\frac{\Gamma_{{\rm s}}}{\Gamma_{{\rm rel}}}=\frac{\sum_{l}|x_{-+,l}^{(+)}|^{2}}{\sum_{l}(|x_{+-,l}^{(+)}|^{2}+|x_{-+,l}^{(+)}|^{2})}$ (14) is the steady-state population of the Floquet state. These solutions together with the quantum regression theory enable us to derive a physically transparent spectrum function Yan _et al._ (2016a, 2018) $\displaystyle S(\Delta)$ $\displaystyle\propto$ $\displaystyle\sum_{l}\bigg{\\{}\pi|x_{++,l}^{(+)}|^{2}(\tilde{\rho}_{++}^{{\rm ss}}-\tilde{\rho}_{--}^{{\rm ss}})^{2}\delta(\Delta-l\omega_{z})$ $\displaystyle+4|x_{++,l}^{(+)}|^{2}\tilde{\rho}_{++}^{{\rm ss}}\tilde{\rho}_{--}^{{\rm ss}}\frac{\Gamma_{{\rm rel}}}{\Gamma_{{\rm rel}}^{2}+(\Delta-l\omega_{z})^{2}}$ $\displaystyle+|x_{+-,l}^{(+)}|^{2}\tilde{\rho}_{++}^{{\rm ss}}\frac{\Gamma_{{\rm deph}}}{\Gamma_{{\rm deph}}^{2}+(\Delta-l\omega_{z}-\Delta_{+-})^{2}}$ $\displaystyle+|x_{-+,l}^{(+)}|^{2}\tilde{\rho}_{--}^{{\rm ss}}\frac{\Gamma_{{\rm deph}}}{\Gamma_{{\rm deph}}^{2}+(\Delta-l\omega_{z}+\Delta_{+-})^{2}}\bigg{\\}},$ It is evident that the accuracy of Eq. (LABEL:eq:sfunsa) is limited by the secular approximation when the transition matrix elements $x^{(+)}_{\alpha\beta,l}$ and quasienergies are exactly calculated. As is well-known, the secular approximation can be justified under the strong driving condition, i.e., $\Delta_{+-}\gg\kappa$. In general, we can calculate the quasienergies and transition matrix elements based on both analytical and numerical diagonalization (ND) of the Floquet Hamiltonian $\tilde{H}(t)-i\partial_{t}$ in the Sambe space Shirley (1965); Sambe (1973) , yielding the analytical and semianalytical spectra, respectively. Next, we discuss the parity phenomenon of the Floquet states resulting from Eq. (5). We consider the behavior of the Floquet states under the generalized parity transformation ${\cal P}_{G}$, which is defined as ${\cal P}_{G}|\tilde{u}_{\alpha}(t)\rangle:=\sigma_{x}|\tilde{u}_{\alpha}(t+T/2)\rangle.$ (16) By differentiating $\sigma_{x}|\tilde{u}_{\alpha}(t+T/2)\rangle$ with respect to $t$, we readily obtain $\left[\sigma_{x}\tilde{H}\left(t+T/2\right)\sigma_{x}-i\partial_{t}\right]\sigma_{x}\left|\tilde{u}_{\alpha}\left(t+T/2\right)\right\rangle=\tilde{\varepsilon}_{\alpha}\sigma_{x}\left|\tilde{u}_{\alpha}\left(t+T/2\right)\right\rangle.$ (17) When $\delta+f(t)=-[\delta+f(t+T/2)]$, $\sigma_{x}|\tilde{u}_{\alpha}(t+T/2)\rangle$ satisfies the same differential equation as $|\tilde{u}_{\alpha}(t)\rangle$ because of Eq. (5). Recalling the uniqueness of solutions of the differential equations, in such cases we must have $\sigma_{x}\left|\tilde{u}_{\alpha}\left(t+T/2\right)\right\rangle=\lambda_{\alpha}|\tilde{u}_{\alpha}(t)\rangle,$ (18) where $\lambda_{\alpha}$ is a constant. Furthermore, we have $\lambda_{\alpha}=\pm 1$ because of ${\cal P}_{G}^{2}|\tilde{u}_{\alpha}(t)\rangle=\lambda_{\alpha}^{2}|\tilde{u}_{\alpha}(t)\rangle=|\tilde{u}_{\alpha}(t)\rangle.$ Specifically, when $\delta+f(t)=-[\delta+f(t+T/2)]$, the Floquet states may be even or odd functions under the generalized parity transformation, which is referred to as the generalized parity of the Floquet states. The generalized parity has been previously investigated in other phenomena such as the coherent destruction of tunneling Grossmann _et al._ (1991) and the laser- induced electronic transport Lehmann _et al._ (2003). Clearly, if $\delta+f(t)\neq-\left[\delta+f\left(t+T/2\right)\right]$, Eq. (18) cannot hold as $\sigma_{x}\tilde{H}\left(t+T/2\right)\sigma_{x}\neq\tilde{H}(t)$, i.e., the effective Hamiltonian is no longer invariant under the generalized parity transformation. Consequently, the Floquet states also do not have the generalized parity. Figure 1: The incoherent components of the fluorescence spectrum for $p=3$, $\Omega_{x}=10\kappa$, $\delta=0$, $\Omega_{z}=\omega_{z}=40\kappa$, $r=1$, and various phase. “Ana.” and “Num.” denote the analytical and the FL numerical results, respectively. We show that the symmetry of the spectrum may be a consequence of the generalized parity of the Floquet states. By using Eq. (18) and $x_{\alpha\beta,l}^{(+)}=\left[x_{\beta\alpha,-l}^{(-)}\right]^{\ast}$, it is straightforward to show the following identity for arbitrary integer $l$ from the definition (9) of the transition matrix element: $x_{\alpha\beta,l}^{(+)}=(-1)^{l}\lambda_{\alpha}\lambda_{\beta}\left[x_{\beta\alpha,-l}^{(+)}\right]^{\ast},$ (19) provided $\delta+f(t)=-[\delta+f(t+T/2)]$. It follows that $|x_{\alpha\beta,l}^{(+)}|=|x_{\beta\alpha,-l}^{(+)}|$ (20) also holds for any integer $l$. We emphasize that the relation (20) can be deduced from relation (19), however, the relation (19) cannot be derived from relation (20). With the relation (20), it is straightforward to show that the spectrum (LABEL:eq:sfunsa) is symmetric about $\Delta=0$ Yan _et al._ (2018). Specifically, since $|x_{++,l}^{(+)}|=|x_{++,-l}^{(+)}|$, the emission lines at $\Delta=\pm l\omega_{z}$ (the positions are symmetric about $\Delta=0$) have the equal weights. Moreover, since $|x_{+-,l}^{(+)}|=|x_{-+,-l}^{(+)}|$, we also have $\tilde{\rho}_{++}^{{\rm ss}}=\tilde{\rho}_{--}^{{\rm ss}}$ according to Eq. (14), leading to $|x_{+-,l}^{(+)}|^{2}\tilde{\rho}_{++}^{{\rm ss}}=|x_{-+,-l}^{(+)}|^{2}\tilde{\rho}_{--}^{{\rm ss}}$. That is to say, the emission lines at $\Delta=\pm(l\omega_{z}+\Delta_{+-})$ (the positions are symmetric about $\Delta=0$) have the same weights. It turns out that the symmetry of the spectrum fundamentally originates from the generalized parity of the Floquet states when $\delta+f(t)=-[\delta+f(t+T/2)]$. Conversely, one may expect that the symmetry of the spectrum may break when such a parity is absent. However, it is a formidable task to analytically prove that the spectrum is asymmetric in the absence of the generalized parity. Let us discuss what happens to the formal spectrum if $\delta+f(t)\neq-[\delta+f(t+T/2)]$. Under such a condition, the generalized parity is absent, and thus we cannot have the relation (19). In principle, the absence of the generalized parity will result in two possible situations. One is that the spectrum becomes asymmetric about $\Delta=0$ because the relation $|x^{(+)}_{\alpha\beta,l}|\neq|x^{(+)}_{\beta\alpha,-l}|$ can be derived at least for a certain $l$. The other is that the spectrum is symmetric because the equality $|x^{(+)}_{\alpha\beta,l}|=|x^{(+)}_{\beta\alpha,-l}|$ still holds for any $l$, originating from other kinds of identities between the transition matrix elements rather than the generalized-parity-induced identity (19). Apparently the first situation is more trivial than the second one. Most importantly, the present analysis suggests that the formal spectrum may be symmetric even without the generalized parity. Consequently, we cannot conclude from the formal spectrum (LABEL:eq:sfunsa) that the symmetry of the spectrum breaks as long as the generalized parity is absent. To end this section, we give some remarks on the above findings based on the formal spectrum. First, we find that the symmetry of the spectrum may result from the generalized parity and requires $\delta+f(t)=-[\delta+f(t+T/2)]$. This is consistent with the analysis above without the secular approximation. Moreover, the generalized parity is found to be an important underlying cause of the relation (20), which was numerically found in harmonic modulation case Yan _et al._ (2018). It turns out here that the relation (20) can be established due to the generalized parity in the bi- and multi-harmonic cases. Second, without the generalized parity, namely, when $\delta+f(t)\neq-[\delta+f(t+T/2)]$, the formal spectrum can be either trivially asymmetric or nontrivially symmetric. The symmetry requires the relation (20) in the absence of the generalized parity, namely, Eq. (19). Third, the formal spectrum is derived with the secular approximation and thus the present analysis needs further verification. In what follows we consider a concrete biharmonic modulation to verify whether the generalized parity guarantees the symmetry of the spectrum when the secular approximation is not invoked and we also check whether the relation (20) can be established without the generalized parity and whether such relations lead to the symmetry of the spectrum without the secular approximation. Figure 2: The incoherent components of the fluorescence spectrum for $p=3$, $\delta=5\kappa$, $\Omega_{x}=10\kappa$, $\Omega_{z}=\omega_{z}=40\kappa$, $r=1$, and various phase. ## III Verification of symmetry and asymmetry of the spectrum To calculate fluorescence spectrum, without loss of generality, we mainly consider the biharmonic modulation in this work, namely, the modulation consists of two harmonics $f(t)=\Omega_{z}[\cos(\omega_{z}t)+r\cos(p\omega_{z}t+\phi)],$ (21) where $\Omega_{z}$ and $\omega_{z}=2\pi/T$ are the amplitude and fundamental frequency of the modulation, respectively, $p$ is a positive integer, $r$ is the ratio of the amplitude of the second harmonic to that of the first one, and $\phi$ is a relative phase. Since $\frac{1}{T}\int^{T}_{0}f(t)dt=0$, the condition for the presence of the generalized parity $\delta+f(t)=-[\delta+f(t+T/2)]$ is equivalent to $\delta=0$ and $f(t)=-f(t+T/2)$. The condition for the absence of the generalized parity $\delta+f(t)\neq-[\delta+f(t+T/2)]$ is simply divided into three cases: $\left\\{\begin{array}[]{c}\delta\neq 0\,{\rm and}\,f(t)=-f(t+T/2);\\\ \delta=0\,{\rm and}\,f(t)\neq-f(t+T/2);\\\ \delta\neq 0\,{\rm and}\,f(t)\neq-f(t+T/2).\end{array}\right.$ (22) It is noted that for the biharmonic modulation (21), both $f(t)=-f(t+T/2)$ and $f(t)\neq-f(t+T/2)$ can be realized by setting $p$ odd and even numbers, respectively. To verify above analysis, we calculate the numerically exact fluorescence spectrum from master equation (2) with the FL formalism Ho _et al._ (1986); Yan _et al._ (2016b), which is compared with the analytical and semianalytical results from Eq. (LABEL:eq:sfunsa). The analytical and semianalytical results are obtained by using the transition matrix elements and quasienergies calculated with the Van Vleck perturbation theory and the ND of the Floquet Hamiltonian, respectively. The detailed analytical calculation is presented in Appendix B. In addition, we just focus on the incoherent components of the fluorescence spectrum, which is of interest in the experiments. In principle, similar analysis is applicable to the coherent components. In this work, we mainly consider the parameters regime $\omega_{z}\sim\Omega_{z}\gg\Omega_{x}\gg\kappa$, in which case both the Van Vleck perturbation theory (up to second order in $\Omega_{x}$) and secular approximation can be justified. Importantly, this regime is experimentally accessible in the artificial atoms, e.g., the transmon qubit Li _et al._ (2013). We should emphasize that if the perturbation theory is inapplicable, we can obtain the transition matrix elements and quasienergies by the ND of the Floquet Hamiltonian. We first verify whether the generalized parity guarantees the symmetry of the spectrum. In Fig. 1, we display the incoherent component of fluorescence spectra obtained by the FL numerical method (solid line) and analytical result (dashed line) for $p=3$, $\delta=0$, and various values of $\phi$. Apparently the spectra are symmetric as expected. The analytical results are in agreement with the FL results. These results also show that the spectrum depends weakly on the relative phase $\phi$. In addition, it is straightforward to verify that for other driving parameters, the spectrum is symmetric as well when $p$ is an odd number and $\delta=0$. In Appendix C, we show that when $\delta=0$ and $p$ is odd, the transition matrix elements indeed satisfy Eq. (19), which guarantees the symmetry of the spectrum. The present results suggest that the symmetry of the spectrum appears as long as $\delta=0$ and $f(t)=-f(t+T/2)$ and fundamentally originates from the generalized parity of the Floquet states in such a situation. Figure 3: The incoherent components of the fluorescence spectrum for $p=2$, $\delta=0$, $\Omega_{x}=10\kappa$, $\Omega_{z}=\omega_{z}=40\kappa$, $r=1$, and various phases. We move to examine whether the symmetry of the spectrum breaks when the generalized parity is absent, namely, under the conditions $\delta+f(t)\neq-[\delta+f(t+T/2)]$. We calculate the spectra with the parameters being the same as in Fig. 1 except for the detuning $\delta=5\kappa$, corresponding to the case of $\delta\neq 0$ and $f(t)=-f(T+T/2)$. In Fig. 2, the analytical and FL numerical spectra agree with each other and are found to be asymmetric for the finite detuning, indicating that in spite of $f(t)=-f(t+T/2)$, the asymmetry of spectrum appears when $\delta\neq 0$. Let us consider the case of $\delta=0$ and $f(t)\neq-f(t+T/2)$ by setting $p$ being even. We calculate the spectrum for $p=2$ and the other parameters being the same as in Fig. 1. Figure 3 displays that the analytical and numerical spectra are pronouncedly asymmetric even though $\delta=0$ except for $\phi=\pi/2$ in which case the analytical spectrum is found to be strictly symmetric (see discussion below) while the numerical spectrum is slightly asymmetric [in particular, the intensities of emission lines at $\Delta=\pm\omega_{z}$ are unequal as shown in Fig. 6(a)]. These results confirm that the formal spectrum (LABEL:eq:sfunsa) may be symmetric without the generalized parity of the Floquet states. However, the numerically exact spectrum is asymmetric in the absence of the generalized parity. This shows that the generalized parity plays an important role in determining the symmetry of the exact spectrum. We will further analyze such discrepancy between the analytical and numerical results later. In addition, we find that in contrast with $p=3$, the spectrum is found to depend strongly on relative phase $\phi$ when $p=2$. Finally we calculate the spectra for $\delta\neq 0$ and $f(t)\neq-f(t+T/2)$. Figure 4 shows the spectra obtained for the detuning $\delta=5\kappa$ and the other parameters being the same as in Fig. 3. The spectra are still asymmetric. In general, it is straightforward to verify the asymmetry of the spectrum under the condition that $\delta+f(t)\neq-[\delta+f(t+T/2)]$. All in all, it turns out that the symmetry of the spectrum breaks in the absence of the generalized parity. Conversely, we can say that the symmetry of the spectrum can be fully attributed to the presence of the generalized parity. In contrast to the previous studies, we ascribe the asymmetry to the breaking of the generalized parity rather than the unequal populations of dressed states Antón _et al._ (2017) or the breakdown of relation (20) Yan _et al._ (2018). Figure 4: The incoherent components of fluorescence spectrum for $p=2$, $\delta=5\kappa$, $\Omega_{x}=10\kappa$, $\Omega_{z}=\omega_{z}=40\kappa$, $r=1$, and various phase. Let us explore how the analytical spectrum becomes symmetric in the absence of the generalized parity of the Floquet states. To this end, we show that the relation (20) can originate from the identities different from Eq. (19). Based on the results from the Van Vleck perturbation theory, we analytically derive the identities for the transition matrix elements in the case of vanishing detuning and even $p$. The derivation are given in Appendix C. When $p$ is even, $\delta=0$, and $\phi=\left(1/2+n\right)\pi$ ($n=0,\pm 1,\pm 2,\ldots$), we find that the following relations hold for arbitrary integer $l$: $\displaystyle x^{(+)}_{++,-l}$ $\displaystyle=$ $\displaystyle(-1)^{l}x^{(+)}_{++,l},$ (23) $\displaystyle x^{(+)}_{-+,-l}$ $\displaystyle=$ $\displaystyle-(-1)^{l}e^{-i2\theta_{0}}x^{(+)}_{+-,l},$ (24) where $\theta_{0}$ is a phase defined in Eq. (86). Although the relations (23) and (24) are derived based on the perturbation theory, it is straightforward to show that they hold in the nonperturbative regimes. In Fig. 5, we calculate $x^{(+)}_{++,l}$ $(l=\pm 1,\pm 2)$ with the variation of $\Omega_{x}$ by using the analytical and ND methods. We see that the deviation between the analytical and numerical results becomes larger and larger as $\Omega_{x}$ increases, which is due to the breakdown of the perturbation calculation. Nevertheless, $x^{(+)}_{++,l}$ obtained by the ND method still satisfies Eq. (23). This suggests that the relations (23) and (24) are not limited to the perturbative regimes. More importantly, it follows from the identities (23) and (24) that $|x_{\alpha\beta,l}^{(+)}|=|x_{\beta\alpha,-l}^{(+)}|$, which leads to the symmetry of the formal spectrum (LABEL:eq:sfunsa). That is to say, without the generalized parity of the Floquet states, the relation (20) can also be established from other kinds of the identities for the transition matrix elements instead of the generalized-parity-induced identity (19) under certain conditions. The discrepancy in the symmetry predicted by the analytical and numerical methods shown in Fig. 3(b) indicates that the relations (23) and (24) cannot guarantee the symmetry of the spectrum without the secular approximation. To further verify this, in Fig. 6, we use semianalytical and FL numerical methods to calculate the weights of the emission lines at $\Delta=\pm\omega_{z}$ as the increasing of $\Omega_{x}$ for $p=2$, $\delta=0$, and two values of $\phi$. It is evident that the weights calculated from the semianalytical method (solid and dashed lines) are the same while the weights from the numerical method (dot-dashed and dotted lines) are unequal, indicating that the semianalytical spectrum is symmetric but the numerical spectrum is not symmetric. The present results illustrate that that provided the relation (20) is established in the absence of the generalized parity, the secular approximation can induce artifact symmetry that vanishes if such approximation is not invoked. Figure 5: Transition matrix elements $x^{(+)}_{++,l}$ versus driving strength $\Omega_{x}$, calculated from the analytical method and the numerical method based on the ND of the Floquet Hamiltonian for $p=2$, $\delta=0$, $\Omega_{z}=\omega_{z}=40\kappa$, $\phi=\pi/2$, and $r=1$. Apart from the biharmonic modulation, we find that the conditions for the symmetry and asymmetry of the spectrum, which are derived based on the generalized parity, are applicable to the simple harmonic and multiharmonic modulation cases. For the simple harmonic modulation $f(t)=\Omega_{z}\cos(\omega_{z}t)$, $f(t)=-f(t+T/2)$ is met. Therefore, the symmetry and asymmetry of the spectrum is uniquely controlled by the detuning $\delta$, which simply interprets the detuning-dependent symmetry of the spectrum. Specifically, the spectrum is expected to be symmetric when $\delta=0$ and asymmetric when $\delta\neq 0$. This is consistent with the findings of previous studies Yan _et al._ (2016b); Antón _et al._ (2017); Yan _et al._ (2018). For the multiharmonic modulation $f(t)=\sum_{p=1}^{N}\Omega_{z,p}\cos(p\omega_{z}t+\phi_{p})]$, where $\Omega_{z,p}$ and $\phi_{p}$ are the amplitude and phase of the $p$th harmonic, respectively, either $f(t)=-f(t+T/2)$ or $f(t)\neq-f(t+T/2)$ can be met, similarly to the biharmonic case. We have calculated the spectrum with the FL and semianalytical methods for the cases of $N=3$, $N=4$, and $N=5$. The results (not shown here) further confirm that the symmetry and asymmetry of spectrum fundamentally originate from the presence and absence of the generalized parity of the Floquet states, respectively. Figure 6: Weights of emission lines at $\Delta=\pm\omega_{z}$ versus driving strength $\Omega_{x}$, calculated from the semianalytical method and the FL method, for $p=2$, $\delta=0$, $\Omega_{z}=\omega_{z}=40\kappa$, $r=1$, and two values of $\phi$. “Semiana.” denotes the semianalytical result. ## IV Conclusions In summary, we have studied the fundamental origin of the symmetry of the resonance fluorescence from the two-level system subjected to a periodic frequency modulation and a near-resonant high-frequency monochromatic excitation by using both analytical and numerical methods based on the Floquet theory. In such a driven two-level system, we have found that the generalized parity of Floquet states plays a fundamental role in the symmetry of the spectrum. Specifically, the generalized parity guarantees the symmetry of the spectrum. On the other hand, when the generalized parity is broken, the spectrum becomes asymmetric. This has been illustrated in the context of the biharmonic modulation, the parameters of which can be tuned to induce or break the generalized parity. For the biharmonic modulation, we find that when $\delta=0$ and $f(t)=-f(t+T/2)$, the generalized parity exists and the spectrum is symmetric. When $\delta+f(t)\neq-[\delta+f(t+T/2)]$, the generalized parity is broken and the spectrum is found to be asymmetric. Interestingly, we can obtain pronouncedly asymmetric spectrum by requiring the modulation $f(t)\neq-f(t+T/2)$ even though $\delta=0$. Moreover, these conditions for the symmetry and asymmetry of the spectrum are found to be applicable to the simple harmonic and multiharmonic modulation cases. In addition, we illustrated that the secular approximation may induce artifact symmetry that vanishes if the secular approximation is avoided under certain conditions. The present study gives a deep insight into the origin of the symmetry of the spectrum and reveals a simple relation between the symmetry of the spectrum and the generalized parity of the Floquet states. ###### Acknowledgements. This work was supported by the National Natural Science Foundation of China (Grants No. 11647082, No. 11774311, No. 11774226, and No. 11874260). ## Appendix A Derivation of symmetry of the spectrum without the secular approximation The master equation can be rewritten in a matrix form $\frac{d}{dt}\vec{\tilde{\rho}}(t)={\cal L}(t)\vec{\tilde{\rho}}(t).$ (25) Here the vector is defined as $\vec{\tilde{\rho}}(t)=(\langle\tilde{\sigma}_{+}(t)\rangle,\langle\tilde{\sigma}_{-}(t)\rangle,\langle\tilde{\pi}_{+}(t)\rangle,\langle\tilde{\pi}_{-}(t)\rangle)^{{\rm T}},$ (26) where $\pi_{\pm}=(1\pm\sigma_{z})/2$ and $\langle\tilde{\hat{o}}(t)\rangle\equiv{\rm Tr}[\hat{o}\tilde{\rho}(t)]$. The superoperator ${\cal L}(t)$ in the Liouville space spanned by the matrix bases $\\{\sigma_{\pm},\pi_{\pm}\\}$ is given by ${\cal L}(t)=\left(\begin{array}[]{cccc}i[\delta+f(t)]-\frac{\kappa}{2}&0&-\frac{i\Omega_{x}}{2}&\frac{i\Omega_{x}}{2}\\\ 0&-i[\delta+f(t)]-\frac{\kappa}{2}&\frac{i\Omega_{x}}{2}&\frac{-i\Omega_{x}}{2}\\\ \frac{-i\Omega_{x}}{2}&\frac{i\Omega_{x}}{2}&-\kappa&0\\\ \frac{i\Omega_{x}}{2}&\frac{-i\Omega_{x}}{2}&\kappa&0\end{array}\right).$ (27) If $\delta+f(t)=-[\delta+f(t+T/2)]$, in which case the Hamiltonian is invariant under the generalized parity transformation, one readily finds that ${\cal T}{\cal L}(t+T/2){\cal T}={\cal L}(t),$ (28) where the transformation matrix is given by ${\cal T}=\left(\begin{array}[]{cccc}0&1&0&0\\\ 1&0&0&0\\\ 0&0&-1&0\\\ 0&0&0&-1\end{array}\right),$ (29) and ${\cal T}^{2}=I$ with $I$ being the identity matrix. Similarly to the Hamiltonian, the matrix ${\cal L}(t)$ is invariant under the transformation defined in Eq. (28), which can be regarded as the generalized parity transformation in the Liouville space, similarly to that defined in Eq. (16) of the main text. Let us derive the specific property of the steady state in the long-time limit [as $\det{\cal L}(t)=0$, there exists a nontrivial steady state]. It follows from Eq. (25) that $\frac{d}{dt}\vec{\tilde{\rho}}(t+T/2)={\cal L}(t+T/2)\vec{\tilde{\rho}}(t+T/2),$ (30) which leads to $\displaystyle\frac{d}{dt}{\cal T}\vec{\tilde{\rho}}(t+T/2)$ $\displaystyle=$ $\displaystyle{\cal T}{\cal L}(t+T/2){\cal T}{\cal T}\vec{\tilde{\rho}}(t+T/2)={\cal L}(t){\cal T}\vec{\tilde{\rho}}(t+T/2),$ (31) which means that ${\cal T}\vec{\tilde{\rho}}(t+T/2)=c\vec{\tilde{\rho}}(t)$, owing to the uniqueness of solutions of the differential equation. On using the fact that $\vec{\tilde{\rho}}(t)=\vec{\tilde{\rho}}(t+T)$ as $t\rightarrow\infty$ because of ${\cal L}(t)={\cal L}(t+T)$, we find that $c$ may be either $+1$ or $-1$. It is easy to prove by contradiction that $c=-1$. Suppose that $c=1$, yielding $\langle\tilde{\pi}_{+}(t+T/2)\rangle=-\langle\tilde{\pi}_{+}(t)\rangle$. However, if one considers $\delta+f(t)=0$ in which case ${\cal L}(t)$ is time independent while Eq. (28) still holds, the steady state becomes time independent and one gets $\langle\tilde{\pi}_{+}(t)\rangle=\langle\tilde{\pi}_{+}(t+T/2)\rangle$. By contradiction, one finds that $c=-1$. Consequently, in the steady-state limit, we have ${\cal T}\vec{\tilde{\rho}}(t+T/2)=-\vec{\tilde{\rho}}(t)\quad(t\rightarrow\infty).$ (32) Next, let us derive the property of the principal matrix solution $\Pi(t,t^{\prime})$ of the master equation, which solves the differential equation $\frac{d}{dt}\Pi(t,t^{\prime})={\cal L}(t)\Pi(t,t^{\prime}),$ (33) with the initial condition $\Pi(t^{\prime},t^{\prime})=I$. It is straightforward to show that $\displaystyle\frac{d}{dt}{\cal T}\Pi(t+T/2,t^{\prime}+T/2){\cal T}$ $\displaystyle=$ $\displaystyle{\cal T}{\cal L}(t+T/2){\cal T}{\cal T}\Pi(t+T/2,t^{\prime}+T/2){\cal T}={\cal L}(t){\cal T}\Pi(t+T/2,t^{\prime}+T/2){\cal T},$ (34) namely, ${\cal T}\Pi(t+T/2,t^{\prime}+T/2){\cal T}$ satisfies the same differential equation and the same initial condition as $\Pi(t,t^{\prime})$. As a result, we simply have ${\cal T}\Pi(t+T/2,t^{\prime}+T/2){\cal T}=\Pi(t,t^{\prime}).$ (35) According to the quantum regression theory Mollow (1969), the two-time correlation functions $\vec{\tilde{g}}(t,t^{\prime})=(\langle\tilde{\sigma}_{+}(t)\tilde{\sigma}_{-}(t^{\prime})\rangle,\langle\tilde{\sigma}_{-}(t)\tilde{\sigma}_{-}(t^{\prime})\rangle,\langle\tilde{\pi}_{+}(t)\tilde{\sigma}_{-}(t^{\prime})\rangle,\langle\tilde{\pi}_{-}(t)\tilde{\sigma}_{-}(t^{\prime})\rangle)^{{\rm T}}$ (36) satisfy the same equation as $\vec{\tilde{\rho}}(t)$, however, with a different initial condition $\vec{\tilde{g}}(t^{\prime},t^{\prime})=(\langle\tilde{\pi}_{+}(t^{\prime})\rangle,0,0,\langle\tilde{\sigma}_{-}(t^{\prime})\rangle)^{{\rm T}}.$ (37) Similarly, another set of two-time correlation functions $\vec{\tilde{G}}(t,t^{\prime})=(\langle\tilde{\sigma}_{+}(t^{\prime})\tilde{\sigma}_{+}(t)\rangle,\langle\tilde{\sigma}_{+}(t^{\prime})\tilde{\sigma}_{-}(t)\rangle,\langle\tilde{\sigma}_{+}(t^{\prime})\tilde{\pi}_{+}(t)\rangle,\langle\tilde{\sigma}_{+}(t^{\prime})\tilde{\pi}_{-}(t)\rangle)^{{\rm T}}$ (38) also satisfy the same differential equation as $\vec{\tilde{g}}(t,t^{\prime})$ but with the initial condition $\vec{\tilde{G}}(t^{\prime},t^{\prime})=(0,\langle\tilde{\pi}_{+}(t^{\prime})\rangle,0,\langle\tilde{\sigma}_{+}(t^{\prime})\rangle)^{{\rm T}}.$ (39) Using Eq. (32), we have ${\cal T}\vec{\tilde{g}}(t^{\prime},t^{\prime})=\left(\begin{array}[]{c}0\\\ \langle\tilde{\pi}_{+}(t^{\prime})\rangle\\\ 0\\\ -\langle\tilde{\sigma}_{-}(t^{\prime})\rangle\end{array}\right)=\left(\begin{array}[]{c}0\\\ \langle\tilde{\pi}_{+}(t^{\prime}+T/2)\rangle\\\ 0\\\ \langle\tilde{\sigma}_{+}(t^{\prime}+T/2)\rangle\end{array}\right)=\vec{\tilde{G}}\left(t^{\prime}+\frac{T}{2},t^{\prime}+\frac{T}{2}\right)\quad(t^{\prime}\rightarrow\infty).$ (40) In the steady-state limit, the correlation functions are found to have the following relation $\displaystyle\vec{\tilde{g}}(t,t^{\prime})$ $\displaystyle=$ $\displaystyle\Pi(t,t^{\prime})\vec{\tilde{g}}(t^{\prime},t^{\prime})$ (41) $\displaystyle=$ $\displaystyle{\cal T}\Pi\left(t+\frac{T}{2},t^{\prime}+\frac{T}{2}\right){\cal T}\vec{\tilde{g}}(t^{\prime},t^{\prime})$ $\displaystyle=$ $\displaystyle{\cal T}\Pi\left(t+\frac{T}{2},t^{\prime}+\frac{T}{2}\right)\vec{\tilde{G}}\left(t^{\prime}+\frac{T}{2},t^{\prime}+\frac{T}{2}\right)$ $\displaystyle=$ $\displaystyle{\cal T}\vec{\tilde{G}}\left(t+\frac{T}{2},t^{\prime}+\frac{T}{2}\right)\quad(t^{\prime}\rightarrow\infty).$ It follows that as $t^{\prime}\rightarrow\infty$, $\displaystyle\langle\tilde{\sigma}_{+}(t)\tilde{\sigma}_{-}(t^{\prime})\rangle$ $\displaystyle=$ $\displaystyle\langle\tilde{\sigma}_{+}(t^{\prime}+T/2)\tilde{\sigma}_{-}(t+T/2)\rangle$ (42) $\displaystyle=$ $\displaystyle\langle\tilde{\sigma}_{+}(t+T/2)\tilde{\sigma}_{-}(t^{\prime}+T/2)\rangle^{\ast}.$ In the steady-state limit, the first-order correlation function depends explicitly on time $t^{\prime}$, however, the $t^{\prime}$ dependence can be eliminated by setting $t=\tau+t^{\prime}$ and integrating over $t^{\prime}$ (because the contributions of $t^{\prime}$-dependent terms are negligible to a long-time observation), yielding the $\tau$-dependent first-order correlation function $\displaystyle\bar{\tilde{g}}_{1}(\tau)$ $\displaystyle\equiv$ $\displaystyle\frac{1}{T}\int_{0}^{T}\lim_{t^{\prime}\rightarrow\infty}\langle\tilde{\sigma}_{+}(\tau+t^{\prime})\tilde{\sigma}_{-}(t^{\prime})\rangle dt^{\prime}$ (43) $\displaystyle=$ $\displaystyle\frac{1}{T}\int_{0}^{T}\lim_{t^{\prime}\rightarrow\infty}\langle\tilde{\sigma}_{+}(\tau+t^{\prime}+T/2)\tilde{\sigma}_{-}(t^{\prime}+T/2)\rangle^{\ast}dt^{\prime}$ $\displaystyle=$ $\displaystyle\frac{1}{T}\int_{T/2}^{T+T/2}\lim_{t^{\prime}\rightarrow\infty}\langle\tilde{\sigma}_{+}(\tau+t^{\prime})\tilde{\sigma}_{-}(t^{\prime})\rangle^{\ast}dt^{\prime}$ $\displaystyle=$ $\displaystyle\frac{1}{T}\int_{0}^{T}\lim_{t^{\prime}\rightarrow\infty}\langle\tilde{\sigma}_{+}(\tau+t^{\prime})\tilde{\sigma}_{-}(t^{\prime})\rangle^{\ast}dt^{\prime}$ $\displaystyle=$ $\displaystyle\bar{\tilde{g}}_{1}^{\ast}(\tau),$ where we used relation (42) and the fact that $\langle\tilde{\sigma}_{+}(\tau+t^{\prime}+T)\tilde{\sigma}_{-}(t^{\prime}+T)\rangle^{\ast}=\langle\tilde{\sigma}_{+}(\tau+t^{\prime})\tilde{\sigma}_{-}(t^{\prime})\rangle^{\ast}$ as $t^{\prime}\rightarrow\infty$. This means that the generalized parity guarantees that the correlation function is a real-valued function of $\tau$ in the rotating frame and thus results in the symmetry of the spectrum when $\delta+f(t)=-[\delta+f(t+T/2)]$. This is consistent with the prediction from the spectrum (LABEL:eq:sfunsa). In general, it is a formidable task to show that the spectrum is asymmetric when $\delta+f(t)\neq-[\delta+f(t+T/2)]$ with or without the secular approximation. Nevertheless, from the above derivation, one readily notes that the generalized parity plays an important role in determining the symmetry of the spectrum. Consequently, if such parity breaks, it is not difficult to imagine that the symmetry of the spectrum also breaks trivially if there is no other symmetry-inducing mechanism. ## Appendix B Analytical calculation of quasienergies and transition matrix elements in the biharmonic modulation case We use the Van Vleck perturbation theory Cohen-Tannoudji _et al._ (1998); Hausinger and Grifoni (2010) to analytically calculate the quasienergies and transition matrix elements $x_{\alpha\beta,l}^{(+)}$ for the biharmonic modulation, which leads to the analytical fluorescence spectrum. Since we are interested in the regime of $\Omega_{z},\,\omega_{z}\gg\Omega_{x}$, which is accessible in the experiment Li _et al._ (2013), we use $\Omega_{x}$ as the perturbation parameter. We first transform Eq. (6) with the unitary transformation $e^{S(t)}[\tilde{H}(t)-i\partial_{t}]e^{-S(t)}e^{S(t)}|\tilde{u}_{\alpha}(t)\rangle=\tilde{\varepsilon}_{\alpha}e^{S(t)}|\tilde{u}_{\alpha}(t)\rangle,$ (44) where $S(t)=i\frac{\Omega_{z}}{2\omega_{z}}\left\\{\sin(\omega_{z}t)+\frac{r}{p}[\sin(p\omega_{z}t+\phi)-\sin\phi]\right\\}\sigma_{z}.$ (45) We can define the transformed Floquet states and transformed Hamiltonian as follows: $|u_{\alpha}^{\prime}(t)\rangle=e^{S(t)}|\tilde{u}_{\alpha}(t)\rangle,$ (46) $\displaystyle H^{\prime}(t)$ $\displaystyle=$ $\displaystyle e^{S(t)}[\tilde{H}(t)-i\partial_{t}]e^{-S(t)}$ (47) $\displaystyle=$ $\displaystyle\frac{1}{2}\delta\sigma_{z}+\frac{1}{2}\sum_{l}(f_{l}\sigma_{+}+f_{-l}^{\ast}\sigma_{-})e^{il\omega_{z}t},$ where $f_{l}=\Omega_{x}F_{l},$ (48) and $\displaystyle F_{l}$ $\displaystyle=$ $\displaystyle\frac{1}{T}\int_{0}^{T}e^{i\frac{\Omega_{z}}{\omega_{z}}\left\\{\sin(\omega_{z}t)+\frac{r}{p}[\sin(p\omega_{z}t+\phi)-\sin\phi]\right\\}-il\omega_{z}t}dt$ (49) $\displaystyle=$ $\displaystyle e^{-i\Theta}\sum_{k}J_{k}\left(\frac{r\Omega_{z}}{p\omega_{z}}\right)J_{l-kp}\left(\frac{\Omega_{z}}{\omega_{z}}\right)e^{ik\phi},$ with $\Theta=\frac{r\Omega_{z}}{p\omega_{z}}\sin\phi$ and $J_{k}(z)$ being the Bessel function of the first kind. To proceed, we introduce an extended Hilbert space in which the time-dependent Floquet Hamiltonian $H^{\prime}(t)-i\partial_{t}$ becomes time independent Sambe (1973). One readily introduces the Fourier basis $|l\rangle\equiv\exp(il\omega_{z}t)$ and inner product $\langle l|n\rangle\equiv\frac{1}{T}\int_{0}^{T}\exp[i(n-l)\omega_{z}t]dt=\delta_{l,n}$, where $\delta_{l,n}$ is the Kronecker delta function. Denoting $|\uparrow\rangle$ and $|\downarrow\rangle$ as the eigenstates for $\sigma_{z}$ with the eigenvalues $+1$ and $-1$, respectively, one gets the composite bases $|\uparrow(\downarrow),l\rangle=|\uparrow(\downarrow)\rangle\otimes|l\rangle$. In the extended Hilbert space spanned by such bases, we can obtain the explicit form of the Floquet Hamiltonian, which is written as $\displaystyle H_{{\cal F}}^{\prime}$ $\displaystyle=$ $\displaystyle H^{\prime}(t)-i\partial_{t}$ (50) $\displaystyle=$ $\displaystyle\frac{1}{2}\delta\sigma_{z}+\sum_{n}n\omega_{z}|n\rangle\langle n|+\frac{1}{2}\sum_{n,l}(f_{l}\sigma_{+}+f_{-l}^{\ast}\sigma_{-})$ $\displaystyle\otimes|n+l\rangle\langle n|.$ The Floquet Hamiltonian has an infinite size and is difficult to be diagonalized exactly in analytical calculation. To carry out perturbation calculation, we transform the Floquet Hamiltonian with a further unitary transformation with the Hermitian generator $K$, leading to $\displaystyle H_{{\cal F}}^{\prime\prime}$ $\displaystyle=$ $\displaystyle e^{iK}H_{{\cal F}}^{\prime}e^{-iK}$ (51) $\displaystyle=$ $\displaystyle H_{\cal F}^{\prime}+[iK,H_{{\cal F}}^{\prime}]+\frac{1}{2!}[iK,[iK,H_{{\cal F}}^{\prime}]]+\ldots,$ where the explicit form of $K$ is to be determined by requiring $H_{{\cal F}}^{\prime\prime}$ to be block diagonal. The generator is expanded as $K=K^{(1)}+K^{(2)}+K^{(3)}+\ldots,$ (52) where the superscripts indicate the orders in the perturbation. We use $H_{0}=\frac{1}{2}\delta\sigma_{z}+\sum_{n}n\omega_{z}|n\rangle\langle n|$ and $V=\frac{1}{2}\sum_{n,l}(f_{l}\sigma_{+}+f_{-l}^{\ast}\sigma_{-})\otimes|n+l\rangle\langle n|$ as the dominate and perturbation components, respectively. Up to the second order in $\Omega_{x}$, we have $\displaystyle H_{{\cal F}}^{\prime\prime}$ $\displaystyle\simeq$ $\displaystyle H_{0}+V+[iK^{(1)},H_{0}]+[iK^{(1)},V]+[iK^{(2)},H_{0}]$ (53) $\displaystyle+\frac{1}{2}[iK^{(1)},[iK^{(1)},H_{0}]].$ Next, we discuss under which condition the transformed Hamiltonian may reasonably be block diagonal. For the dominate component $H_{0}$, we simply have $H_{0}|\uparrow(\downarrow),n\rangle=[+(-)\delta/2+n\omega_{z}]|\uparrow(\downarrow),n\rangle\equiv\tilde{\varepsilon}^{(0)}_{+(-),n}|\uparrow(\downarrow),n\rangle$. Provided that $\tilde{\varepsilon}^{(0)}_{+,n}-\tilde{\varepsilon}^{(0)}_{-,n+m}=\delta-m\omega_{z}\approx 0$, we have a subspace spanned by two almost degenerate unperturbed states $|\uparrow,n\rangle$ and $|\downarrow,n+m\rangle$, where $n$ is an arbitrary integer and $m$ is the integer nearest to $\delta/\omega_{z}$. The projection onto such a subspace is realized by the operator: $\Pi_{n}=|\uparrow,n\rangle\langle\uparrow,n|+|\downarrow,n+m\rangle\langle\downarrow,n+m|.$ (54) The eigenvalues of the dominate component $H_{0}$ in the $n$th subspace are well-separated from those in the $(n+l)$th subspace as long as $|l\omega_{z}|\gg|\delta-m\omega_{z}|$ for any $l\neq 0$. Moreover, if we assume that $|\langle\uparrow,n|V|\downarrow,n+l+m\rangle|\ll|\tilde{\varepsilon}^{(0)}_{+,n}-\tilde{\varepsilon}^{(0)}_{-,n+l+m}|,$ (55) which is simply $|f_{-l-m}/2|\ll|l\omega_{z}|$, the transitions between the states in the different subspaces can be neglected up to a certain order in the perturbation Cohen-Tannoudji _et al._ (1998), yielding the following condition $\Pi_{n}H_{{\cal F}}^{\prime\prime}\Pi_{l}=0,$ (56) for $n\neq l$. Therefore, $H_{{\cal F}}^{\prime\prime}$ is block diagonal. The second condition that $K$ cannot have matrix elements inside each subspace of two almost degenerate states is also assumed, i.e., $\Pi_{n}K\Pi_{n}=0.$ (57) The generator can now be fully determined via Eqs. (56) and (57). The nonvanishing elements of $K^{(1)}$ and $K^{(2)}$are given by $\langle\uparrow,n|iK^{(1)}|\downarrow,l\rangle=\frac{1}{2}\frac{f_{n-l}}{\delta+(n-l)\omega_{z}},$ (58) $\langle\downarrow,l|iK^{(1)}|\uparrow,n\rangle=-\frac{1}{2}\frac{f_{n-l}^{\ast}}{\delta+(n-l)\omega_{z}},$ (59) for $n-l\neq-m$, and $\displaystyle\langle\uparrow,n|iK^{(2)}|\uparrow,l\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{4(n-l)\omega_{z}}\left\\{\sum_{k\neq n+m,l+m}\frac{f_{n-k}f_{l-k}^{\ast}}{2}\left[\frac{1}{\delta+(n-k)\omega_{z}}+\frac{1}{\delta+(l-k)\omega_{z}}\right]\right.$ (60) $\displaystyle\left.+\frac{f_{l-n-m}^{\ast}f_{-m}}{\delta+(l-n-m)\omega_{z}}+\frac{f_{n-l-m}f_{-m}^{\ast}}{\delta+(n-l-m)\omega_{z}}\right\\},$ $\displaystyle\langle\downarrow,n|iK^{(2)}|\downarrow,l\rangle$ $\displaystyle=$ $\displaystyle-\frac{1}{4(n-l)\omega_{z}}\left\\{\sum_{k\neq l-m,n-m}\frac{f_{k-n}^{\ast}f_{k-l}}{2}\left[\frac{1}{\delta+(k-n)\omega_{z}}+\frac{1}{\delta+(k-l)\omega_{z}}\right]\right.$ (61) $\displaystyle+\left.\frac{f_{l-n-m}^{\ast}f_{-m}}{\delta+(l-n-m)\omega_{z}}+\frac{f_{n-l-m}f_{-m}^{\ast}}{\delta+(n-l-m)\omega_{z}}\right\\},$ for $n\neq l$. The rest elements of $K^{(1)}$ and $K^{(2)}$ are vanishing. The transformed Hamiltonian have the $2\times 2$ submatrix $H_{{\cal F}}^{\prime\prime(n)}$ in the diagonal, which reads Cohen-Tannoudji _et al._ (1998) $\displaystyle H_{{\cal F}}^{\prime\prime(n)}$ $\displaystyle=$ $\displaystyle H_{0}\Pi_{n}+\Pi_{n}V\Pi_{n}+\frac{1}{2}\Pi_{n}[iK^{(1)},V]\Pi_{n}$ (64) $\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}\frac{\delta}{2}+n\omega_{z}+\sum_{j\neq-m}\frac{|f_{j}|^{2}}{4(\delta+j\omega_{z})}&\frac{f_{-m}}{2}\\\ \frac{f_{-m}^{\ast}}{2}&-\frac{\delta}{2}+(n+m)\omega_{z}-\sum_{j\neq-m}\frac{|f_{j}|^{2}}{4(\delta+j\omega_{z})}\end{array}\right).$ One can diagonalize the submatrix $H_{{\cal F}}^{\prime\prime(n)}$ analytically. Its eigenvalues (quasienergies) are $\tilde{\varepsilon}_{\pm,n}=\frac{1}{2}\left(m\omega_{z}\pm\Omega_{m}\right)+n\omega_{z},$ (65) where $\Omega_{m}=\sqrt{\left[\delta-m\omega_{z}+\sum_{j\neq-m}\frac{|f_{j}|^{2}}{2(\delta+j\omega_{z})}\right]^{2}+|f_{-m}|^{2}}.$ (66) The eigenvectors are given by $\displaystyle|\Psi_{+,n}^{\prime\prime}\rangle$ $\displaystyle=$ $\displaystyle u|\uparrow,n\rangle+v|\downarrow,n+m\rangle,$ (67) $\displaystyle|\Psi_{-,n}^{\prime\prime}\rangle$ $\displaystyle=$ $\displaystyle v|\uparrow,n\rangle-u^{\ast}|\downarrow,n+m\rangle,$ (68) with $\displaystyle u$ $\displaystyle=$ $\displaystyle\frac{f_{-m}}{|f_{-m}|}\sqrt{\frac{1}{2}\left[1+\frac{1}{\Omega_{m}}\left(\delta-m\omega_{z}+\sum_{j\neq-m}\frac{|f_{j}|^{2}}{2(\delta+j\omega_{z})}\right)\right]},$ (69) $\displaystyle v$ $\displaystyle=$ $\displaystyle\sqrt{\frac{1}{2}\left[1-\frac{1}{\Omega_{m}}\left(\delta-m\omega_{z}+\sum_{j\neq-m}\frac{|f_{j}|^{2}}{2(\delta+j\omega_{z})}\right)\right]}.$ (70) The eigenvectors for $H_{{\cal F}}^{\prime}$ can be derived as follows: $|\Psi_{\pm,n}^{\prime}\rangle=e^{-iK}|\Psi_{\pm,n}^{\prime\prime}\rangle\simeq\left(1-iK^{(1)}-iK^{(2)}+\frac{1}{2!}iK^{(1)}iK^{(1)}\right)|\Psi_{\pm,n}^{\prime\prime}\rangle.$ (71) It is straightforward to derive the explicit form of the eigenvectors, which reads $\displaystyle|\Psi_{+,n}^{\prime}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{{\cal{\cal N}}}\left\\{uB|\uparrow,n\rangle-\sum_{j\neq 0}P_{j}|\uparrow,n+j\rangle+vB|\downarrow,n+m\rangle+\sum_{j\neq 0}Q_{j}|\downarrow,n+m+j\rangle\right\\},$ (72) $\displaystyle|\Psi_{-,n}^{\prime}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{{\cal{\cal N}}}\left\\{vB|\uparrow,n\rangle+\sum_{j\neq 0}Q_{-j}^{\ast}|\uparrow,n+j\rangle-u^{\ast}B|\downarrow,n+m\rangle+\sum_{j\neq 0}P_{-j}^{\ast}|\downarrow,n+m+j\rangle\right\\},$ (73) where $B=1-\frac{1}{8}\sum_{l\neq-m}\frac{|f_{l}|^{2}}{(\delta+l\omega_{z})^{2}},$ (74) $\displaystyle P_{j}$ $\displaystyle=$ $\displaystyle\frac{f_{j-m}}{2[\delta+(j-m)\omega_{z}]}\left(v+\frac{uf_{-m}^{\ast}}{2j\omega_{z}}\right)+\frac{u}{4j\omega_{z}}\sum_{k\neq-m}\frac{f_{k+j}f_{k}^{\ast}}{\delta+k\omega_{z}},$ (75) $\displaystyle Q_{j}$ $\displaystyle=$ $\displaystyle\frac{f_{-j-m}^{\ast}}{2[\delta-(j+m)\omega_{z}]}\left(u+\frac{vf_{-m}}{2j\omega_{z}}\right)+\frac{v}{4j\omega_{z}}\sum_{k\neq-m}\frac{f_{k-j}^{\ast}f_{k}}{\delta+k\omega_{z}},$ (76) and ${\cal N}=\sqrt{B^{2}+\sum_{j\neq 0}(|P_{j}|^{2}+|Q_{j}|^{2})}$ is the normalization factor. The Floquet states $|u_{\alpha,n}^{\prime}(t)\rangle$ with the quasienergy $\tilde{\varepsilon}_{\alpha,n}$ can be derived from $|\Psi_{\alpha,n}^{\prime}\rangle$ by replacing $|n\rangle$ with $e^{in\omega_{z}t}$. With above results at hand, we can analytically calculate the transition matrix element $\displaystyle x_{\alpha\beta,l}^{(+)}$ $\displaystyle=$ $\displaystyle\frac{1}{T}\int_{0}^{T}\langle\tilde{u}_{\alpha}(t)|\sigma_{\pm}|\tilde{u}_{\beta}(t)\rangle e^{-il\omega_{z}t}dt=\frac{1}{T}\int_{0}^{T}\langle u_{\alpha}^{\prime}(t)|e^{S(t)}\sigma_{+}e^{-S(t)}|u_{\beta}^{\prime}(t)\rangle e^{-il\omega_{z}t}dt$ (77) $\displaystyle=$ $\displaystyle\sum_{n}\frac{1}{T}\int_{0}^{T}F_{n}\langle u_{\alpha}^{\prime}(t)|\sigma_{+}|u_{\beta}^{\prime}(t)\rangle e^{i(n-l)\omega_{z}t}dt=\sum_{n}F_{n+l}\langle\Psi_{\alpha,0}^{\prime}|\sigma_{+}|\Psi_{\beta,n}^{\prime}\rangle,$ and $\displaystyle\langle\Psi_{+,0}^{\prime}|\sigma_{+}|\Psi_{+,n}^{\prime}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{{\cal N}^{2}}\left\\{u^{\ast}vB^{2}\delta_{n,-m}-\sum_{j\neq 0,n+m}P_{j}^{\ast}Q_{j-n-m}+(u^{\ast}Q_{-n-m}-vP_{n+m}^{\ast})B(1-\delta_{n,-m})\right\\},$ (78) $\displaystyle\langle\Psi_{+,0}^{\prime}|\sigma_{+}|\Psi_{-,n}^{\prime}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{{\cal N}^{2}}\left\\{-(u^{\ast})^{2}B^{2}\delta_{n,-m}-\sum_{j\neq 0,n+m}P_{j}^{\ast}P_{n+m-j}^{\ast}+2u^{\ast}P_{n+m}^{\ast}B(1-\delta_{n,-m})\right\\},$ (79) $\displaystyle\langle\Psi_{-,0}^{\prime}|\sigma_{+}|\Psi_{+,n}^{\prime}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{{\cal N}^{2}}\left\\{v^{2}B^{2}\delta_{n,-m}+\sum_{j\neq 0,n+m}Q_{-j}Q_{j-n-m}+2vQ_{-n-m}B(1-\delta_{n,-m})\right\\},$ (80) $\displaystyle\langle\Psi_{-,0}^{\prime}|\sigma_{+}|\Psi_{-,n}^{\prime}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{{\cal N}^{2}}\left\\{-u^{\ast}vB^{2}\delta_{n,-m}+\sum_{j\neq 0,n+m}P_{j}^{\ast}Q_{j-n-m}+(vP_{n+m}^{\ast}-u^{\ast}Q_{-n-m})B(1-\delta_{n,-m})\right\\},$ (81) where $(1-\delta_{n,-m})$ indicates that the term vanishes for $n=-m$. Clearly, the validity of the perturbation theory is limited to the condition (55). For $\delta\approx 0$, roughly speaking, the above results can be justified when $r\sim 1$ and $\omega_{z}\sim\Omega_{z}\gg\Omega_{x}$. ## Appendix C Equalities for transition matrix elements in the vanishing detuning case For the biharmonic modulation, we show the equalities that the transition matrix elements satisfy under the vanishing detuning condition ($\delta=0$) using the above analytical results, which helps us to understand the symmetry of the spectrum in the main text. It follows from Eq. (49) that $\displaystyle F_{-l}$ $\displaystyle=$ $\displaystyle e^{-i\Theta}\sum_{k}J_{k}\left(\frac{r\Omega_{z}}{p\omega_{z}}\right)J_{-l-kp}\left(\frac{\Omega_{z}}{\omega_{z}}\right)e^{ik\phi}$ (82) $\displaystyle=$ $\displaystyle(-1)^{l}e^{-i\Theta}\sum_{k}J_{k}\left(\frac{r\Omega_{z}}{p\omega_{z}}\right)(-1)^{k(p+1)}$ $\displaystyle\times J_{l-kp}\left(\frac{\Omega_{z}}{\omega_{z}}\right)e^{-ik\phi},$ where we used the relation $J_{-n}(z)=(-1)^{n}J_{n}(z)$. It is evident that when $p$ is an odd number, $p+1$ is even and thus $(-1)^{k(p+1)}=1$, leading to $F_{-l}=(-1)^{l}e^{-i2\Theta}F_{l}^{\ast}.$ (83) When $p$ is an even number, $(-1)^{k(p+1)}=(-1)^{k}$ may be either $+1$ or $-1$. Nevertheless, we can obtain a simple relation between $F_{l}$ and $F_{-l}$ by setting $(-1)^{k}e^{-ik\phi}=e^{ik\phi},$ (84) which yields that $\phi=\left(1/2+n\right)\pi$ $(n=0,\pm 1,\pm 2,\ldots)$. With an even $p$ and such values of phase, we have $F_{l}=(-1)^{l}F_{-l}.$ (85) We should emphasize that Eqs. (83) and (85) hold under different conditions. The former is available when $p$ is odd and regardless of $\phi$ while the latter is established when $p$ is even and $\phi=(1/2+n)\pi$. Provided that $\delta=0$, we get $m=\delta/\omega_{z}=0$. We define the phase of $F_{0}$ via $F_{0}=e^{-i\theta_{0}}|F_{0}|.$ (86) Together with Eqs. (69) and (70), we simply have $v=ue^{i\theta_{0}}$ (87) with the aid of Eq. (83) or (85). Such an equality between $u$ and $v$ is valid only for $\delta=0$ and in the valid regime of Eq. (83) or (85). ### C.1 Odd $p$ We consider that $p$ is an odd number. It follows from Eq. (49) that $\theta_{0}=\Theta$. Using $\delta=0$ and Eqs. (83) and (87), one readily gets from Eqs. (75) and (76) that $\displaystyle Q_{j}$ $\displaystyle=$ $\displaystyle-\frac{f_{-j}^{\ast}}{2j\omega_{z}}\left(u+\frac{vf_{0}}{2j\omega_{z}}\right)+\frac{v}{4j\omega_{z}}\sum_{k\neq 0}\frac{f_{k-j}^{\ast}f_{k}}{k\omega_{z}}$ (88) $\displaystyle=$ $\displaystyle\frac{(-1)^{j+1}e^{i2\Theta}f_{j}}{2j\omega_{z}}\left(u+\frac{vf_{0}^{\ast}e^{-i2\Theta}}{2j\omega_{z}}\right)+\frac{v}{4j\omega_{z}}\sum_{k\neq 0}\frac{f_{-k-j}^{\ast}f_{-k}}{-k\omega_{z}}$ $\displaystyle=$ $\displaystyle\frac{(-1)^{j+1}e^{i\Theta}f_{j}}{2j\omega_{z}}\left(v+\frac{uf_{0}^{\ast}}{2j\omega_{z}}\right)+\frac{e^{i\Theta}u}{4j\omega_{z}}\sum_{k\neq 0}\frac{(-1)^{j+1}f_{k+j}f_{k}^{\ast}}{k\omega_{z}}$ $\displaystyle=$ $\displaystyle(-1)^{j+1}e^{i\Theta}P_{j}.$ From this relation and Eqs. (77)-(80), it is straightforward to show that $\displaystyle\left[x_{-+,-l}^{(+)}\right]^{\ast}$ $\displaystyle=$ $\displaystyle\sum_{n}\frac{F_{n-l}^{\ast}}{{\cal N}^{2}}\left\\{v^{2}B^{2}\delta_{n,0}+\sum_{n\neq 0,n}Q_{-j}^{\ast}Q_{j-n}^{\ast}+2vBQ_{-n}^{\ast}(1-\delta_{n,0})\right\\}$ (89) $\displaystyle=$ $\displaystyle\sum_{n}\frac{F_{-n-l}^{\ast}}{{\cal N}^{2}}\left\\{v^{2}B^{2}\delta_{n,0}+\sum_{j\neq 0,-n}Q_{-j}^{\ast}Q_{j+n}^{\ast}+2vBQ_{n}^{\ast}(1-\delta_{n,0})\right\\}$ $\displaystyle=$ $\displaystyle\sum_{n}\frac{(-1)^{n+l}F_{n+l}e^{i2\Theta}}{{\cal N}^{2}}\left\\{v^{2}B^{2}\delta_{n,0}+\sum_{j\neq 0,n}Q_{j}^{\ast}Q_{n-j}^{\ast}+2vBQ_{n}^{\ast}(1-\delta_{n,0})\right\\}$ $\displaystyle=$ $\displaystyle\sum_{n}\frac{(-1)^{n+l}F_{n+l}e^{i2\Theta}}{{\cal N}^{2}}\left\\{v^{2}B^{2}\delta_{n,0}+\sum_{j\neq 0,n}(-1)^{n}e^{-i2\Theta}P_{j}^{\ast}P_{n-j}^{\ast}+2vB(-1)^{n+1}e^{-i\Theta}P_{n}^{\ast}(1-\delta_{n,0})\right\\}$ $\displaystyle=$ $\displaystyle(-1)^{l}\sum_{n}\frac{F_{n+l}}{{\cal N}^{2}}\left\\{(u^{\ast})^{2}B^{2}\delta_{n,0}+\sum_{j\neq 0,n}P_{j}^{\ast}P_{n-j}^{\ast}-2u^{\ast}BP_{n}^{\ast}(1-\delta_{n,0})\right\\}$ $\displaystyle=$ $\displaystyle-(-1)^{l}x_{+-,l}^{(+)}.$ Similarly, we find that $\left[x^{(+)}_{++,-l}\right]^{\ast}=(-1)^{l}x^{(+)}_{++,l}$. Not surprisingly, due to the generalized parity of the Floquet states, the transition matrix elements satisfy Eq. (19) as long as $\delta+f(t)=-[\delta+f(t+T/2)]$. For the biharmonic modulation, such equalities are established when $p$ is odd and $\delta=0$. ### C.2 Even $p$ We move to consider that $p$ is an even number. In such a case, the generalized parity of the Floquet states is broken even if $\delta=0$. Thus, we cannot expect that the transition matrix elements satisfy Eq. (19). However, we have another type of equality. With Eqs. (85) and (87), one gets $\displaystyle Q_{j}$ $\displaystyle=$ $\displaystyle\frac{f_{-j}^{\ast}}{-2j\omega_{z}}\left(u+\frac{vf_{0}}{2j\omega_{z}}\right)+\frac{v}{4j\omega_{z}}\sum_{k\neq 0}\frac{f_{k-j}^{\ast}f_{k}}{k\omega_{z}}$ (90) $\displaystyle=$ $\displaystyle\frac{(-1)^{j+1}f_{j}^{\ast}}{2j\omega_{z}}\left(u+\frac{vf_{0}}{2j\omega_{z}}\right)+\frac{v}{4j\omega_{z}}\sum_{k\neq 0}\frac{(-1)^{j+1}f_{j-k}^{\ast}f_{-k}}{-k\omega_{z}}$ $\displaystyle=$ $\displaystyle\frac{(-1)^{j+1}e^{-i\theta_{0}}f_{j}^{\ast}}{2j\omega_{z}}\left(v+\frac{u^{\ast}f_{0}}{2j\omega_{z}}\right)+\frac{e^{-i\theta_{0}}u^{\ast}}{4j\omega_{z}}\sum_{k\neq 0}\frac{(-1)^{j+1}f_{j+k}^{\ast}f_{k}}{k\omega_{z}}$ $\displaystyle=$ $\displaystyle(-1)^{j+1}e^{-i\theta_{0}}P_{j}^{\ast}.$ It is straightforward to derive Eqs. (23) and (24) in the main text via Eqs. (77)-(80) and (90). 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2024-09-04T02:54:59.030750

2003.05126

arxiv-papers

# A variability measure for estimates of parameters in interval data fitting Sergey P. Shary Institute of Computational Technologies SB RAS and Novosibirsk State University, Novosibirk, Russia E-mail<EMAIL_ADDRESS> ###### Abstract The paper presents a construction of a quantitative measure of variability for parameter estimates in the data fitting problem under interval uncertainty. It shows the degree of variability and ambiguity of the estimate, and the need for its introduction is dictated by non-uniqueness of answers to the problems with interval data. A substantiation of the new variability measure is given, its application and motivations are discussed. Several examples and a series of numerical tests are considered, showing the features of the new characteristic and the specifics of its use. Keywords: data fitting problem, linear regression, interval data uncertainty, maximum compatibility method, strong compatibility, variability measure. MSC 2010: 65G40, 62J10, 90C90 ## 1 Introduction and problem statement The purpose of this work is to present a quantitative variability measure for estimates of parameters of functional dependencies in the statistics of interval data. This is a relatively young branch of modern data science that does not rely on the probability theory, but makes extensive use of interval analysis methods (see, e. g., the surveys in [4, 7, 10]). Fig. 1: A variability measure can be an estimate of the size of the set of possible solutions. By the term “variability”, we understand the degree of variation and ambiguity of the estimate, and the need for its introduction is dictated by the fact that, in processing interval data, the answer is typically not unique. Usually, we get a whole set of different estimates that are equally consistent (compatible) with the source data and, thus, suitable as solutions to the problem. The extent to which this set is large or small is, partly, characterized by the term “variability”. In traditional probabilistic statistics, estimates of parameters are known to be random variables themselves, and the measure of their variability can be the variance of the estimates, mean absolute difference, median absolute deviation, average absolute deviation, and such like. What could be their analogues in the statistics of interval data? At first glance, the answer to this question seems quite obvious: it can be any value that characterizes the size of the set of solutions to the problem, if it is non-empty. We can even take an enclosure of the solution set obtained by an interval method. A certain disadvantage of this variant is the excessive detailing of the answer given as a box in $\mathbb{R}^{n}$, a large amount of information that still needs to be “digested” and reduced to a compact and expressive form. Sometimes, an interval estimate in the form of an axes- aligned box may inadequately represent the solution set. Another disadvantage is the complexity of finding such an estimate. It is desirable to have a relatively simple and efficiently computable quantity, expressed in a single number, because it would give a general aggregate view of the subject of interest. Similarly to variance and other probabilistic measures, it can serve as an approximate characteristic of the quality of parameter estimation. The greater the variability of an estimate, the less its certainty and the worse its quality, and this can serve as a basis for conclusions about the quality of the estimate. At the same time, the introduced variability measure should not be simply the “size of the solution set”. If this solution set, for example, is unstable and changes abruptly with arbitrarily small changes in the data, then its size is, to some extent, misleading and disorienting (see example in Section 4). A practically useful variability measure should take into account this possible instability of the solution set to the problem and give us a robust value. Fig. 2: An illustration for the data fitting problem under interval uncertainty. In our article, we are within the framework of the data fitting problem (often called regression analysis problem): given results of measurements or observations, it is required to construct a functional dependence of a fixed type that “best fit” these data. Specifically, we need to determine the parameters $x_{1}$, $x_{2}$, …, $x_{n}$ of a linear function of the form $b=x_{1}a_{1}+\ldots+x_{n}a_{n}$ (1) from a number of values of the independent variables $a_{1}$, $a_{2}$, …, $a_{n}$ (also called _exogenous_ , _explanatory_ , _predictor_ or _input_ variables), and the corresponding values of the dependent variable $b$ (also called _endogenous_ , _response_ , _criterion_ or _output_ variable). Both $a_{1}$, $a_{2}$, …, $a_{n}$ and $b$ are not known precisely, and we only have intervals of their possible values (see Fig. 2). To find estimates of the coefficients $x_{1}$, $x_{2}$, …, $x_{n}$, we use the so-called maximum compatibility method (previously called “maximum consistency method”), which was proposed and developed in the works [6, 16, 17, 19] and others. After the estimates for $x_{1}$, $x_{2}$, …, $x_{n}$ are found, we need to somehow evaluate their variability. Our article presents a construction of the variability measure in the above data fitting problem. Note that traditional methods of data fitting and regression analysis, such as the least squares method and its modifications, the least modulus method, etc., cannot be applied to the solution of our problem, since they are unsuitable for situations where the source data are intervals rather than points. ## 2 Formulation of the main results ### 2.1 Maximum compatibility method and tolerable solution set The initial data for our problem is a set of values of independent and dependent variables for function (1), which are obtained as a result of $m$ measurements (observations): $\begin{array}[]{ccccc}\text{\boldmath$a$}_{11},&\text{\boldmath$a$}_{12},&\ldots&\text{\boldmath$a$}_{1n},&\text{\boldmath$b$}_{1},\\\ \text{\boldmath$a$}_{21},&\text{\boldmath$a$}_{22},&\ldots&\text{\boldmath$a$}_{2n},&\text{\boldmath$b$}_{2},\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\ \text{\boldmath$a$}_{m1},&\text{\boldmath$a$}_{m2},&\ldots&\text{\boldmath$a$}_{mn},&\text{\boldmath$b$}_{m}.\end{array}$ (2) These are intervals as we assume that these data are inaccurate and have interval uncertainty due to measurement errors, etc. Both the data (2) and other interval values throughout the text are highlighted in bold mathematical font according to the informal international standard [5]. The first index of the interval values from (2) means the measurement number, and the second one, at $\text{\boldmath$a$}_{ij}$’s, is the number of the independent variable that takes the corresponding value in this measurement. To find an estimate $(\hat{x}_{1},\hat{x}_{2},\ldots,\hat{x}_{n})$ of the parameters of the linear function (1), we “substitute” data (2) into equality (1), thus getting an interval system of linear algebraic equations $\left\\{\ \begin{array}[]{ccccccccc}\text{\boldmath$a$}_{11}x_{1}&+&\text{\boldmath$a$}_{12}x_{2}&+&\ldots&+&\text{\boldmath$a$}_{1n}x_{n}&=&\text{\boldmath$b$}_{1},\\\\[1.0pt] \text{\boldmath$a$}_{21}x_{1}&+&\text{\boldmath$a$}_{22}x_{2}&+&\ldots&+&\text{\boldmath$a$}_{2n}x_{n}&=&\text{\boldmath$b$}_{2},\\\\[1.0pt] \vdots&&\vdots&&\ddots&&\vdots&&\vdots\\\\[1.0pt] \text{\boldmath$a$}_{m1}x_{1}&+&\text{\boldmath$a$}_{m2}x_{2}&+&\ldots&+&\text{\boldmath$a$}_{mn}x_{n}&=&\text{\boldmath$b$}_{m},\end{array}\right.$ (3) or, briefly, $\text{\boldmath$A$}x=\text{\boldmath$b$}$ (4) with an interval $m\times n$-matrix $\text{\boldmath$A$}=(\text{\boldmath$a$}_{ij})$ and interval $m$-vector $\text{\boldmath$b$}=(\text{\boldmath$b$}_{i})$ in the right-hand side. The sets of parameters which are compatible, in this or that sense, with the measurement data (2) form various solution sets for the equations system (3). The most popular of them are the _united solution set_ and _tolerable solution set_. The united solution set, defined as $\varXi_{uni}(\text{\boldmath$A$},\text{\boldmath$b$})=\bigl{\\{}\,x\in\mathbb{R}^{n}\mid\text{ $Ax=b\,$ for some $A\in\text{\boldmath$A$}$ and $b\in\text{\boldmath$b$}$}\,\bigr{\\}},$ corresponds to the so-called weak compatibility between the parameters of function (1) and data (2) (see [6, 16, 17]). The tolerable solution set, defined as $\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})=\bigl{\\{}\,x\in\mathbb{R}^{n}\mid\text{ $Ax\in\text{\boldmath$b$}\,$ for each matrix $A\in\text{\boldmath$A$}$}\,\bigr{\\}},$ corresponds to the so-called strong compatibility between the parameters of function (1) and data (2) (see [19]). Fig. 3: An illustration of the strong compatibility between interval data and a linear function. Further, we assume that the solution to the data fitting problem for function (1) is found by the maximum compatibility method (see [16, 17, 19]). As an estimate of the parameters of function (1), it takes the maximum point of the _recognizing functional_ , a special function that gives a quantitative “compatibility measure” of this estimate with empirical data (2). The maximum compatibility method has two versions, “weak” and “strong”, that differ in understanding how exactly the interval data should be “compatible” with the function that we construct on them. Weak and strong compatibility reflect two different situations that may occur in data processing. In the weak version, it is required that the graph of the constructed function just intersects the measurement uncertainty boxes (see [16, 17]). The strong version implies more stringent condition: it requires that the function graph passes within the “corridors” specified by the intervals $\text{\boldmath$b$}_{i}$, $i=1,2,\ldots,m$, for _any_ values of the independent variables $a_{1}$, $a_{2}$, …, $a_{n}$ from the respective intervals $\text{\boldmath$a$}_{i1}$, $\text{\boldmath$a$}_{i2}$, …, $\text{\boldmath$a$}_{in}$ obtained in the $i$-th measurement (see [19]). This is illustrated in Fig. 3, where the straight line of the function graph goes through the vertical faces of the measurement uncertainty boxes. The weak compatibility is shown in Fig. 2 by two upper straight lines. The lower line in Fig. 2 does not satisfy compatibility condition at all, neither weak nor strong, since it does not intersect some boxes. The “strong version” of the maximum compatibility method has a number of theoretical and practical advantages over the “weak version”. These are polynomial complexity, robustness of estimates and their finite variability, the fact that the strong compatibility partially overcomes the so-called Demidenko paradox, etc. (see details in [19]). Hence, we consider below a strong version of the maximum compatibility method, which corresponds to the tolerable solution set $\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})$ for the interval system of equations (4). Its recognizing functional is usually denoted by “Tol”, $\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$})\;=\,\min_{1\leq i\leq m}\left\\{\,\mathrm{rad}\,\text{\boldmath$b$}_{i}-\left|\;\mathrm{mid}\,\text{\boldmath$b$}_{i}-\sum_{j=1}^{n}\,\text{\boldmath$a$}_{ij}x_{j}\,\right|\,\right\\},$ (5) where $\mathrm{rad}\,\text{\boldmath$b$}_{i}=\tfrac{1}{2}(\overline{\text{\boldmath$b$}}_{i}-\underline{\text{\boldmath$b$}}_{i}),\hskip 65.44133pt\mathrm{mid}\,\text{\boldmath$b$}_{i}=\tfrac{1}{2}(\overline{\text{\boldmath$b$}}_{i}+\underline{\text{\boldmath$b$}}_{i})$ are radii and midpoints of the components of the right-hand side $b$, the arithmetic operations inside the modulus in (5) are those of the classical interval arithmetic (see, e. g., [4, 8, 9]), and the modulus is understood as the maximum absolute value of the points from the interval, $|\text{\boldmath$a$}|=\max\,\\{\,|a|\mid a\in\text{\boldmath$a$}\,\\}=\max\,\bigl{\\{}\,|\underline{\text{\boldmath$a$}}|,|\overline{\text{\boldmath$a$}}|\,\bigr{\\}}.$ Typical graphs of the functional Tol for the one-dimensional case are shown in Fig. 4 and Fig. 5. To solve the data fitting problem for the linear function (1) and data set (2), it is necessary to find the unconstrained maximum, over all $x\in\mathbb{R}^{n}$, of the functional $\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$})$, $\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$})\rightarrow\max,$ and the vector $\hat{x}=\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$})$ at which this maximum is attained provides an estimate of the parameters of function (1). If $\max\,\mathrm{Tol}\,\geq 0$, then the solution set $\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})$, i. e., the set of parameters strongly compatible with the data is non-empty, and $\hat{x}\in\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})$. If $\max\,\mathrm{Tol}\,<0$, then the solution set $\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})$ is empty and there do not exist parameters that are strongly compatible with data (2). However, the argument $\hat{x}$ of $\max\,\mathrm{Tol}\,$ still provides the best compatibility of the constructed linear function with data (2) (more precisely, the least incompatibility). To conclude this subsection, we give a useful result on the tolerable solution set that allows us to investigate whether it is bounded or unbounded, i. e., whether the tolerable solution sets is finite in size or extends infinitely. Irene Sharaya’s boundedness criterion [13] Let the tolerable solution set to an interval linear system $\text{\boldmath$A$}x=\text{\boldmath$b$}$ be nonempty. It is unbounded if and only if the matrix $A$ has linearly dependent noninterval columns. The criterion of boundedness shows that the tolerable solution set is unbounded, in fact, under exceptional circumstances, which are almost never fulfilled in practice, when working with actual interval data. That is, the tolerable solution set is mostly bounded, and the estimates obtained by the strong version of the maximum compatibility method almost always has finite variability. ### 2.2 Variability measures As a quantity characterizing the variability of the estimate of the parameter vector $\hat{x}=(\hat{x}_{1},\hat{x}_{2},\ldots,\hat{x}_{n})$ in the linear function (1), which is obtained by the maximum compatibility method from data (2), we propose $\mathrm{IVE}\,(\text{\boldmath$A$},\text{\boldmath$b$})\;=\;\sqrt{n}\;\max_{\mathbb{R}^{n}}\,\mathrm{Tol}\,\cdot\Bigl{(}\;\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}\,A\,\Bigr{)}\cdot\frac{\displaystyle\bigl{\|}\,\arg\max_{\mathbb{R}^{n}}\,\mathrm{Tol}\,\bigr{\|}_{2}}{\|\hat{\text{\boldmath$b$}}\|_{2}}.$ (6) In this formula, $n$ is the dimension of the parameter vector of function (1) under construction, $\|\cdot\|_{2}$ is the Euclidean norm (2-norm) of vectors from $\mathbb{R}^{n}$, defined as $\|x\|_{2}\;=\;\left(\;\sum_{i=1}^{n}|x_{i}|^{2}\,\right)^{1/2},$ $\mathrm{cond}_{2}\,A$ is the spectral condition number of the matrix $A$, defined as $\mathrm{cond}_{2}\,A\;=\;\frac{\sigma_{\max}(A)}{\sigma_{\min}(A)},$ i. e., the ratio of the maximal $\sigma_{\max}(A)$ and minimal $\sigma_{\min}(A)$ singular values of $A$; it is an extension, to the rectangular case, of the concept of the condition number from computational linear algebra (see e. g. [2, 24]); $\hat{\text{\boldmath$b$}}$ is a certain “most representative” point from the interval vector $b$, which is taken as $\hat{\text{\boldmath$b$}}\;=\;\tfrac{1}{2}(|\mathrm{mid}\,\text{\boldmath$b$}+\mathrm{rad}\,\text{\boldmath$b$}|+|\mathrm{mid}\,\text{\boldmath$b$}-\mathrm{rad}\,\text{\boldmath$b$}|),$ (7) where the operations “mid” and “rad” are applied in componentwise manner. $\varXi_{\mathit{tol}}$ Fig. 4: The maximum value of the recognizing functional gives an idea of the size of the tolerable solution set $\varXi_{tol}$. Despite the definite formula (7) for $\hat{\text{\boldmath$b$}}$, it should be noted that the introduction of this point is, to a large extent, a matter of common sense. The general approach to the definition of $\hat{\text{\boldmath$b$}}$ is that it must be a kind of “most representative” point from the right-hand side vector $b$, and in some situations this choice may be different from formula (7). For example, $\hat{\text{\boldmath$b$}}$ can be a point result of the measurement, around which the uncertainty interval is built later, based on information about the accuracy of the measuring device. Apart from (6), as a measure of relative variability of the parameter estimate, the value $n\;\Bigl{(}\,\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}A\,\Bigr{)}\cdot\frac{\max_{\mathbb{R}^{n}}\,\mathrm{Tol}\,}{\|\hat{\text{\boldmath$b$}}\|_{2}},$ (8) can have a certain significance. Both IVE and value (8) are defined for interval linear systems (4) with nonzero right-hand sides. They can take either positive real values or be infinite. The latter occurs in the only case of $\,\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}A=\infty$, when all the point matrices $A\in\text{\boldmath$A$}$ have incomplete rank, i. e., when $\sigma_{\min}(A)=0$ for every $A\in\text{\boldmath$A$}$. Then the variability measures are set to be infinite. The symbol IVE is built as an abbreviation of the phrase “interval variability of the estimate”. Below, we show that the value IVE adequately characterizes the size of non-empty tolerable solution set for a large class of practically important situations. But it is useful to discuss informal motivations that lead to the estimate IVE and to demonstrate that IVE has an intuitive, clear and even visual meaning. $\varXi_{\mathit{tol}}$$\varXi_{\mathit{tol}}$ Fig. 5: In addition to the maximum of the recognizing functional, the size of the tolerable solution set is also affected by “steepness” of the graph. The tolerable solution set of an interval system of linear algebraic equations is the set of zero level of the recognizing functional Tol (see details in [15]), or, in other words, the intersection of the hypograph of this functional with the coordinate plane $\mathrm{Tol}\,=0$ (this is illustrated in Fig. 4). As a consequence, the magnitude of the maximum of the recognizing functional can, with other things being equal, be a measure of how extensive or narrow the tolerable solution set is. The more $\max\,\mathrm{Tol}\,$, the larger the size of the tolerable solution set, and vice versa. An additional factor that provides “other things being equal” is the slope (steepness) of pieces of hyperplanes of which the polyhedral graph of the functional Tol is compiled (these are straight lines in the 1D case in Fig. 4 and Fig. 5). The slope of the hyperplanes is determined by the coefficients of the equations that define them, which are the endpoints of the data intervals (2). The value of this slope is summarized in terms of the condition number of point matrices from the interval data matrix $A$. Finally, the multiplier $\frac{\|\arg\max\,\mathrm{Tol}\,\|_{2}}{\|\hat{\text{\boldmath$b$}}\|_{2}}\ =\ \frac{\|\hat{x}\|_{2}}{\|\hat{\text{\boldmath$b$}}\|_{2}}$ is a scaling coefficient that helps to provide the commensurability of the final value with magnitudes of the solution, $\arg\max\,\mathrm{Tol}\,$, and the right-hand side vector of the equations system. Thus, formula (6) is obtained. ## 3 A justification of the variability measure Considering the most general case, we should assume that the number of measurements $m$ may not coincide with the number $n$ of unknown parameters of the linear function (1). In this section, we consider only the case $m\geq n$. In other words, the number of measurements (observations) made is not less than the number of function parameters. Then the interval system of linear equations (4) is either square or tall (overdetermined). Of course, the data fitting problem makes sense for $m<n$ too, the maximum compatibility method also works for this case, and the variability measure IVE is then also applicable (see Section 4), but the latter still needs a separate substantiation. ### 3.1 Estimates of perturbations of the solution to rectangular linear systems The starting point of our constructions justifying the choice of (6) exactly in the form described above is the well-known inequality that estimates perturbation $\Delta x$ of a nonzero solution $x$ to the system of linear algebraic equations $Ax=b$ depending on the change $\Delta b$ of the right- hand side $b$ (see, e. g., [2, 24]): $\frac{\|\Delta x\|_{2}}{\|x\|_{2}}\ \leq\ \mathrm{cond}_{2}\,A\,\cdot\frac{\|\Delta b\|_{2}}{\|b\|_{2}}.$ (9) It is usually considered for square systems of linear equations, when $m=n$, but in the case of the Euclidean vector norm and the spectral condition number of matrices, this inequality holds true in the more general case with $m\geq n$. Naturally, estimate (9) makes sense only for $\sigma_{\min}(A)\neq 0$, when $\mathrm{cond}_{2}A<\infty$, i. e., when the matrix $A$ has full column rank. Let us briefly recall its derivation for this case. Given $Ax=b\quad\text{ и }\quad A(x+\Delta x)=b+\Delta b,$ we have $A\Delta x=\Delta b.$ Further, $\displaystyle\displaystyle\frac{\displaystyle\phantom{M}\frac{\|\Delta x\|_{2}}{\|x\|_{2}}\phantom{M}}{\displaystyle\frac{\|\Delta b\|_{2}}{\|b\|_{2}}}\ $ $\displaystyle=\ \frac{\|\Delta x\|_{2}\,\|b\|_{2}\phantom{I}}{\phantom{I}\|x\|_{2}\,\|\Delta b\|_{2}}=\ \frac{\|\Delta x\|_{2}\,\|Ax\|_{2}\phantom{I}}{\phantom{I}\|x\|_{2}\,\|A\Delta x\|_{2}}\ =\ \frac{\|\Delta x\|_{2}}{\|A\Delta x\|_{2}}\;\frac{\|Ax\|_{2}}{\|x\|_{2}}$ $\displaystyle\leq\;\max_{\Delta x\neq 0}\frac{\|\Delta x\|_{2}}{\|A\Delta x\|_{2}}\ \max_{x\neq 0}\frac{\|Ax\|_{2}}{\|x\|_{2}}\ =\ \left(\min_{\Delta x\neq 0}\frac{\|A\Delta x\|_{2}}{\|\Delta x\|_{2}}\right)^{-1}\ \max_{x\neq 0}\frac{\|Ax\|_{2}}{\|x\|_{2}}$ $\displaystyle=\ \bigl{(}\sigma_{\min}(A)\bigr{)}^{-1}\,\sigma_{\max}(A)\,=\ \mathrm{cond}_{2}(A)$ by virtue of the properties of the singular values (see e. g. [3, 24]). A comparison of the beginning and the end of this calculation leads to the inequality (9), which, as is easy to understand, is attainable for some $x$ and $\Delta x$, or, equivalently, for some right-hand sides of $b$ and their perturbations $\Delta b$. Naturally, the above calculations and the resulting estimate make sense only for $\sigma_{\min}(A)\neq 0$. ### 3.2 Interval systems with point matrices Let us consider an interval system of linear algebraic equations $Ax=\text{\boldmath$b$}$ (10) with a point (noninterval) $m\times n$-matrix $A$, $m\geq n$, and an interval $m$-vector $b$ in the right-hand side. We assume that $A$ has full column rank and, therefore, $\mathrm{cond}_{2}\,A<\infty$. Suppose also that the tolerable solution set for system (10) is non-empty, i. e. $\varXi_{tol}(A,\text{\boldmath$b$})=\bigl{\\{}\,x\in\mathbb{R}^{n}\mid Ax\in\text{\boldmath$b$}\,\bigr{\\}}\neq\varnothing$. We need to quickly and with little effort estimate the size of this solution set, and our answer will be a “radius type” estimate for $\varXi_{tol}(A,\text{\boldmath$b$})$. More precisely, we are going to evaluate $\max\|x^{\prime}-\hat{x}\|_{2}$ over all $x^{\prime}\in\varXi_{tol}(A,\text{\boldmath$b$})$ and for a special fixed point $\hat{x}\in\varXi_{tol}(A,\text{\boldmath$b$})$, which is taken as $\hat{x}\ =\ \arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,(x,A,\text{\boldmath$b$}).$ Recall that the argument $\hat{x}$ of the maximum of the recognizing functional for system (10) is an estimate of parameters of linear function (1) from empirical data. Strictly speaking, this point can be determined non- uniquely, but then let $\hat{x}$ be any one of the points at which the maximum is reached. Let $x^{\prime}$ be a point in the tolerable solution set $\varXi_{tol}(A,\text{\boldmath$b$})$. How to evaluate $\|x^{\prime}-\hat{x}\|_{2}$? It is clear that $x^{\prime}$ and $\hat{x}$ are solutions of systems of linear algebraic equations with the matrix $A$ and some right-hand sides $b^{\prime}$ and $\hat{b}$, respectively, from the interval vector $b$. If $\hat{x}\neq 0$ and $\hat{b}\neq 0$, then we can apply inequality (9), considering a perturbation of the solution $\hat{x}$ to the system of linear algebraic equations $Ax=\hat{b}$. Then $\Delta x=x^{\prime}-\hat{x}$, $\Delta b=b^{\prime}-\hat{b}$, and we get $\frac{\|x^{\prime}-\hat{x}\|_{2}}{\|\hat{x}\|_{2}}\ \leq\;\mathrm{cond}_{2}\,A\cdot\frac{\|b^{\prime}-\hat{b}\|_{2}}{\|\hat{b}\|_{2}},$ from where the absolute estimate is obtained $\|x^{\prime}-\hat{x}\|_{2}\ \leq\;\mathrm{cond}_{2}\,A\cdot\|\hat{x}\|_{2}\cdot\frac{\|b^{\prime}-\hat{b}\|_{2}}{\|\hat{b}\|_{2}}.$ (11) The point $\hat{x}$ is found as the result of maximization of the recognizing functional Tol, the point $\hat{b}$ coincides with $A\hat{x}$, the condition number $\mathrm{cond}_{2}\,A$ can be computed by well-developed standard procedures. Therefore, for practical work with inequality (11), one need somehow evaluate $\|b^{\prime}-\hat{b}\|_{2}$. But first, bearing in mind the further application of the deduced estimate in a situation where the matrix $A$ may vary, we somewhat roughen (11) by taking approximately $\|\hat{b}\|_{2}\approx\|\hat{\text{\boldmath$b$}}\|_{2}$, that is, as the norm of the “most representative” point $\hat{\text{\boldmath$b$}}$ of the interval vector $b$, which we defined in Section 2.2: $\|\hat{b}\|_{2}\,\approx\,\|\hat{\text{\boldmath$b$}}\|_{2},\qquad\text{ where }\ \hat{\text{\boldmath$b$}}\,=\,\tfrac{1}{2}\,\bigl{(}\,|\mathrm{mid}\,\text{\boldmath$b$}+\mathrm{rad}\,\text{\boldmath$b$}|+|\mathrm{mid}\,\text{\boldmath$b$}-\mathrm{rad}\,\text{\boldmath$b$}|\,\bigr{)}.$ In doing this, some coarsening is allowed, so instead of (11) we write $\|x^{\prime}-\hat{x}\|_{2}\ \lessapprox\;\mathrm{cond}_{2}\,A\cdot\|\hat{x}\|_{2}\cdot\frac{\|\Delta b\|_{2}}{\|\hat{\text{\boldmath$b$}}\|_{2}}.$ (12) Now it is necessary to determine the increment of the right-hand side $\Delta b=b^{\prime}-\hat{b}$. Its obvious upper bound is $2\,\mathrm{rad}\,\text{\boldmath$b$}$, but it is too crude. To get a more accurate estimate of $\Delta b$, we also consider, along with system (10), a system of linear algebraic equations $Ax=\tilde{\text{\boldmath$b$}},$ (13) for which the right-hand side is obtained by uniform “compressing” the interval vector $b$: $\tilde{\text{\boldmath$b$}}\,:=\,\bigl{[}\,\underline{\text{\boldmath$b$}}+M,\overline{\text{\boldmath$b$}}-M\,\bigr{]},$ (14) where $M\;:=\;\max_{x\in\mathbb{R}^{n}}\;\mathrm{Tol}\,(x,A,\text{\boldmath$b$})\ \geq\ 0.$ Since the maximum $M$ is reached for a certain value of the argument, $\hat{x}$, then $M=\,\min_{1\leq i\leq m}\left\\{\,\mathrm{rad}\,\text{\boldmath$b$}_{i}-\left|\;\mathrm{mid}\,\text{\boldmath$b$}_{i}-\sum_{j=1}^{n}\,\text{\boldmath$a$}_{ij}\hat{x}_{j}\,\right|\,\right\\}\ \leq\,\min_{1\leq i\leq m}\,\mathrm{rad}\,\text{\boldmath$b$}_{i}.$ As a result, $\underline{\text{\boldmath$b$}}+M\leq\overline{\text{\boldmath$b$}}-M$ in componentwise sense, and the endpoints in the interval vector (14) do not “overlap” each other. But the properties of the recognizing functional imply that, for the interval system of linear algebraic equations (13) with the right-hand side (14), the maximum of the recognizing functional is zero: $\max_{x\in\mathbb{R}^{n}}\;\mathrm{Tol}\,(x,A,\tilde{\text{\boldmath$b$}})\ =\ 0.$ Indeed, the values of $\mathrm{rad}\,\text{\boldmath$b$}_{i}$ are summands in all expressions in (5), for which we take the minimum over $i=1,2,\ldots,m$. Hence, if we simultaneously increase or decrease all $\mathrm{rad}\,\text{\boldmath$b$}_{i}$ by the same value, keeping the midpoints $\mathrm{mid}\,\text{\boldmath$b$}_{i}$ unchanged, then the total value of the recognizing functional will increase or decrease by exactly same value. In other words, if we take a constant $C\geq 0$ and the interval $m$-vector $\text{\boldmath$e$}=([-1,1],\ldots,[-1,1])^{\top}$, then, for the system $Ax=\text{\boldmath$b$}+C\text{\boldmath$e$}\,$ with all the right-hand sides expanded by $[-C,C]$, we have $\mathrm{Tol}\,(x,A,\text{\boldmath$b$}+C\text{\boldmath$e$})\ =\ \mathrm{Tol}\,(x,A,\text{\boldmath$b$})+C.$ (15) Therefore, $\max_{x\in\mathbb{R}^{n}}\;\mathrm{Tol}\,(x,A,\text{\boldmath$b$}+C\text{\boldmath$e$})\ =\ \max_{x\in\mathbb{R}^{n}}\;\mathrm{Tol}\,(x,A,\text{\boldmath$b$})+C.$ (16) The uniform narrowing of the right-hand side vector acts on the tolerable solution set and the recognizing functional in a completely similar way. If we narrow down all the components by the same value $M$, then the maximum of the recognizing functional of the new interval system also decreases by $M$. By virtue of the properties of the recognizing functional, the tolerable solution set $\varXi_{tol}(A,\tilde{\text{\boldmath$b$}})$ for system (13) has empty interior (such sets are often called “non-solid” or “meager”), which we will consider equivalent to “having zero size”. Naturally, this is a simplifying assumption, since in reality the tolerable solution set corresponding to the zero maximum of the recognizing functional may be not a single-point set. But we still accept that. This implication is also supported by the fact that the situation with the zero maximum of the recognizing functional is unstable: the corresponding tolerable solution set can become empty with an arbitrarily small data perturbation (see Section 4). Another fact concerning the auxiliary system (13) with the narrowed right-hand side, which follows from (15)–(16), is that the point $\hat{x}$ remains to be the argument of the maximum of the recognizing functional: $\hat{x}\ =\ \arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,(x,A,\tilde{\text{\boldmath$b$}}).$ For this reason, the point $\hat{b}=A\hat{x}$ lies in the interval vector $\tilde{\text{\boldmath$b$}}$ defined by (14). From what has been said, it follows that the solution set for the system $Ax=\text{\boldmath$b$}$ is obtained from the solution set of the system $Ax=\tilde{\text{\boldmath$b$}}$, which has “negligible size” and for which $\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,(x,\text{\boldmath$A$},\tilde{\text{\boldmath$b$}})=0$, through expanding the right-hand side vector $\tilde{\text{\boldmath$b$}}$ in each component simultaneously by $[-M,M]$, where $M=\max_{x\in\mathbb{R}^{n}}\;\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$}).$ The interval vector $\tilde{\text{\boldmath$b$}}\ni b$ may have non-zero size, but we put $[-\Delta b,\Delta b]=([-M,M],\ldots,[-M,M])^{\top}$ in order to make our estimate (12) attainable. Accordingly, in inequality (12) we take $\|\Delta b\|=\max_{x\in\mathbb{R}^{n}}\;\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$}),$ if the Chebyshev norm ($\infty$-norm) is considered, or a value that differs from it by a corrective factor from the equivalence inequality for vector norms, if we take any other norm. As is known, for any vector $y\in\mathbb{R}^{n}$ (see [2]) $\|y\|_{\infty}\leq\|y\|_{2}\leq\sqrt{n}\;\|y\|_{\infty}.$ (17) Then $\|x^{\prime}-\hat{x}\|_{2}\ \lessapprox\,\sqrt{n}\ \,\mathrm{cond}_{2}A\cdot\bigl{\|}\,\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,\bigr{\|}_{2}\cdot\frac{\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,}{\|\hat{\text{\boldmath$b$}}\|_{2}}.$ (18) What happens if the matrix $A$ does not have a full column rank? Then, by virtue of the Irene Sharaya criterion, the nonempty tolerable solution set to the system (10) is unbounded. This is completely consistent with the fact that then $\mathrm{cond}_{2}A=\infty$ and the value of the variability measure IVE is infinite too. ### 3.3 General interval systems Finally, we consider a general interval system of linear equations $\text{\boldmath$A$}x=\text{\boldmath$b$}$, with an essentially interval matrix, i. e., when $\mathrm{rad}\,\text{\boldmath$A$}\neq 0$. In view of the properties of the tolerable solution set (see, e. g., [15]), it can be represented as $\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})\ =\ \bigcap_{A\in\text{\boldmath$A$}}\;\bigl{\\{}\,x\in\mathbb{R}^{n}\mid Ax\in\text{\boldmath$b$}\,\bigr{\\}}\ =\ \bigcap_{A\in\text{\boldmath$A$}}\varXi_{tol}(A,\text{\boldmath$b$}),$ (19) i. e., as the intersection of the solution sets to the individual systems $Ax=b$ with point matrices $A\in\text{\boldmath$A$}$. For each interval linear system $Ax=\text{\boldmath$b$}$ with $A\in\text{\boldmath$A$}$, we have estimate (18), if $A$ has full column rank. Otherwise, if the point matrix $A$ has incomplete column rank and the corresponding solution set $\varXi_{tol}(A,\text{\boldmath$b$})$ is unbounded, then we do not take it into account. Consequently, for the tolerable solution set of the system $\text{\boldmath$A$}x=\text{\boldmath$b$}$, which is the intersection of the solution sets $\varXi_{tol}(A,\text{\boldmath$b$})$ for all $A\in\text{\boldmath$A$}$, the following should be true: $\|x^{\prime}-\hat{x}\|_{2}\ \lessapprox\ \min_{A\in\text{\boldmath$A$}}\,\left\\{\;\sqrt{n}\;\,\mathrm{cond}_{2}A\cdot\bigl{\|}\,\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,\bigr{\|}_{2}\cdot\frac{\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,}{\|\hat{\text{\boldmath$b$}}\|_{2}}\,\right\\}.$ (20) The transition from representation (19) to inequality (20) can be both very accurate and rather crude (as can be seen from considering the intersection of two 1D intervals). It all depends on the size of the intersection of the solution sets of individual systems $Ax=\text{\boldmath$b$}$. On the other hand, the amount of this intersection is indirectly characterized by the magnitude of $\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,$. Taking the above facts into account, we perform approximate estimation of the right-hand side of inequality (20) by moving the minimum over $A\in\text{\boldmath$A$}$ through the curly brackets. First of all, we evaluate the factor $\|\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,\|_{2}$, which changes to the smallest extent, by the constant available to us after the numerical solution of the data fitting problem: $\bigl{\|}\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,(x,A,\text{\boldmath$b$})\bigr{\|}_{2}\,\approx\;\mathrm{const}\;=\;\bigl{\|}\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$})\bigr{\|}_{2}.$ (21) Next, the minimum of $\mathrm{cond}_{2}A$ naturally turns to $\min\mathrm{cond}_{2}A$, and the most important factor $\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,(x,A,\text{\boldmath$b$})$ will be changed to $\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,(x,\text{\boldmath$A$},\text{\boldmath$b$})$. This choice (as well as (21)) is rather rigidly determined by the following reasons. The expression for our variability measure should preserve its simplicity and be uniform for all cases and situations. In particular, if the interval matrix $A$ squeezes to a point matrix $A$, then our measure should turn to the estimate (18) for the point case. Finally, if $\max\,\mathrm{Tol}\,=0$, then our measure must be zero too, since the size of the (stable) tolerable solution set is also zero, and our variability measure should reliably detect such situations. All this taken together leads to the estimate $\|x^{\prime}-\hat{x}\|_{2}\;\,\lessapprox\ \sqrt{n}\ \Bigl{(}\;\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}A\,\Bigr{)}\cdot\bigl{\|}\,\arg\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,\bigr{\|}_{2}\cdot\frac{\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,}{\|\hat{\text{\boldmath$b$}}\|_{2}}.$ (22) The same estimate as (22), by virtue of the equivalence inequality (17), is also true for the Chebyshev norm: $\max_{x^{\prime}\in\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})}\|x^{\prime}-\hat{x}\|_{\infty}\ \lessapprox\ \sqrt{n}\;\,\Bigl{(}\;\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}\,A\,\Bigr{)}\cdot\bigl{\|}\,\arg\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,\bigr{\|}_{2}\cdot\frac{\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,}{\|\hat{\text{\boldmath$b$}}\|_{2}}.$ This completes the rationale for (6). If we want to evaluate the relative size of the tolerable solution set, expressing it in ratio to the norm of its points, then it is reasonable to take $\hat{x}=\arg\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,$ as the “most typical” point from the tolerable solution set $\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})$. Using (17) again, we obtain $\frac{\max_{x^{\prime}\in\varXi_{tol}(\text{\boldmath$A$},\text{\boldmath$b$})}\|x^{\prime}-x^{\prime\prime}\|_{\infty}}{\|\hat{x}\|_{\infty}}\ \lessapprox\ n\;\Bigl{(}\;\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}A\,\Bigr{)}\cdot\frac{\max_{x\in\mathbb{R}^{n}}\,\mathrm{Tol}\,}{\|\hat{\text{\boldmath$b$}}\|_{2}}.$ This gives value (8). ## 4 Numerical examples and some tests First of all, we consider an example of unstable tolerable solution set that changes abruptly with small perturbations in the system of equations. For all interval $2\times 2$-systems of linear algebraic equations of the form $\begin{pmatrix}[-1,1]&[-1,1]\\\\[2.0pt] 1&-1\\\\[2.0pt] \end{pmatrix}\begin{pmatrix}x_{1}\\\\[2.0pt] x_{2}\end{pmatrix}=\begin{pmatrix}{[-1,1]}\\\\[2.0pt] {[1,1+\eta]}\end{pmatrix},\qquad\eta\geq 0,$ (23) the tolerable solution sets are the same: this is the straight line segment joining the points $(0,-1)$ and $(1,0)$ and depicted in Fig. 6. The diameter of the solution set is essentially non-zero (namely, $\sqrt{2}$), but the unconstrained maximum of the recognizing functional Tol for all such systems is zero, and it is attained at the point $(0.5,-0.5)$. $\varXi_{\mathit{tol}}$ Fig. 6: The tolerable solution set for the interval equations systems (23). At the same time, any arbitrarily small increase in the lower endpoint of the interval $[1,1+\eta]$ in the right-hand side of the second equation makes the tolerable solution set empty. An arbitrarily small reduction of the upper endpoint of the interval $[-1,1]$, located in the first component of the right-hand side vector, produces a similar effect. It turns out that the maximum value of the recognizing functional Tol characterizes very precisely the instability of the original solution set and the zero size of the solution sets of perturbed systems. As the second example, we consider the problem of constructing a linear function of two variables $a_{1}$ and $a_{2}$, $b=x_{1}a_{1}+x_{2}a_{2},$ (24) from the interval data obtained in 3 measurements: $\begin{array}[]{c|ccc}&\text{\boldmath$a$}_{1}&\text{\boldmath$a$}_{2}&\text{\boldmath$b$}\\\ \hline\cr\\\\[-8.53581pt] 1&[98,100]&[99,101]&[190,210]\\\\[3.0pt] 2&[97,99]&[98,100]&[200,220]\\\\[3.0pt] 3&[96,98]&[97,99]&[190,210]\end{array}$ Note that in these data the three-dimensional uncertainty boxes of measurements 1 and 2, as well as 2 and 3, substantially “overlap” each other: their intersections are boxes with non-empty interiors, the sizes of which are comparable to the sizes of the original data boxes. $x_{1}$$x_{2}$$\hat{\text{\boldmath$x$}}$ Fig. 7: The tolerable solution set of the system of equations (25) in comparison with the box constructed by using the estimate IVE. To determine the coefficients $x_{1}$ and $x_{2}$, we compose an interval $3\times 2$-system of linear algebraic equations $\begin{pmatrix}[98,100]&[99,101]\\\\[2.0pt] [97,99]&[98,100]\\\\[2.0pt] [96,98]&[97,99]\end{pmatrix}\begin{pmatrix}x_{1}\\\\[2.0pt] x_{2}\end{pmatrix}=\begin{pmatrix}{[190,210]}\\\\[2.0pt] {[200,220]}\\\\[2.0pt] {[190,210]}\end{pmatrix}.$ (25) Its matrix has incomplete rank, since its member is a point matrix with the rank 1: $\begin{pmatrix}98&99\\\ 98&99\\\ 98&99\end{pmatrix}.$ (26) The united solution set for system (25) is unbounded, therefore it is hardly possible to determine, with certainty, the coefficients of the linear function (24) satisfying the weak compatibility between parameters and data (see Section 2). However, the interval matrix of system (25) does not contain linearly dependent point columns, and therefore, according to the Irene Sharaya criterion [13] (see Section 2.1), the tolerable solution set is bounded. It is depicted in Fig. 7, which is drawn by the procedure EqnTol2D from the package IntLinInc2D [14]. The minimum spectral condition number of the point matrices contained in the interval matrix of (25) is $103.83$, and it is reached on the matrix $\begin{pmatrix}100&99\\\ 97&100\\\ 96&99\end{pmatrix}.$ This result can be obtained, for example, using the simulated annealing algorithm on the set of point matrices contained in the interval matrix of (25). Numerical solution of the maximization problem for the recognizing functional Tol can be carried out within MATLAB environment, using the free program tolsolvty2.m (available from the author’s page at ResearchGate [21]). It implements a modified version of the so-called $r$-algorithms for non- differentiable optimization proposed and developed by N.Z. Shor and N.G. Zhurbenko [20]. Using the precision settings specified in it “by default”, we get $\max\,\mathrm{Tol}\,=1.9095$, which is reached at $\hat{x}=(5.1857\cdot 10^{-7},2.0603)^{\top}$. Then, $\mathrm{IVE}\,=\sqrt{2}\cdot 1.9095\cdot 103.83\cdot\frac{\|\hat{x}\|_{2}}{\sqrt{200^{2}+210^{2}+200^{2}}}=1.6399.$ Interval hull of the tolerable solution set for system (25) (that is, its optimal interval enclosure) is the box $\begin{pmatrix}[-0.9620,3.0227]\\\\[2.0pt] [-0.9320,3.0127]\end{pmatrix},$ which can also be found by the procedure EqnTol2D. We see that the value of IVE is in satisfactory agreement with the radii of the components of the optimal estimate of the solution set, equal to $1.9924$ and $1.9724$ respectively. In the maximum compatibility method, the argument $\hat{x}=\arg\max_{x\in\mathbb{R}^{n}}\mathrm{Tol}\,$ of the unconstrained maximum of the recognizing functional plays a crucial role, and, in fact, our variability estimate IVE relies heavily on it. This is why it makes sense to look at the box $\hat{\text{\boldmath$x$}}$ with the components $[\hat{x}_{i}-\mathrm{IVE}\,,\hat{x}_{i}+\mathrm{IVE}\,]$, $i=1,2$. It is also depicted in Fig. 7, and the substantial asymmetry of its location relative to the solution set is, of course, explained by the specific position of the center, the point $\hat{x}$, as well as the ill-conditioning of the point systems from (25). With other data, the box $\hat{\text{\boldmath$x$}}$ estimates the tolerable solution sets significantly better (see further). Next, we give an example of the opposite type (in a sense, dual to the previous example), where a linear function of three variables $b=x_{1}a_{1}+x_{2}a_{2}+x_{2}a_{3}$ (27) is to be constructed from the data of two experiments summarized below: $\begin{array}[]{c|cccc}&\text{\boldmath$a$}_{1}&\text{\boldmath$a$}_{2}&\text{\boldmath$a$}_{3}&\text{\boldmath$b$}\\\ \hline\cr\\\\[-8.53581pt] 1&[98,100]&[97,99]&[96,98]&[190,210]\\\\[3.0pt] 2&[99,101]&[98,100]&[97,99]&[200,220]\end{array}$ To find the parameters of function (27), we come to an underdetermined interval system of linear algebraic equations $\begin{pmatrix}[98,100]&[97,99]&[96,98]\\\\[2.0pt] [99,101]&[98,100]&[97,99]\end{pmatrix}\begin{pmatrix}x_{1}\\\\[1.0pt] x_{2}\\\\[1.0pt] x_{3}\end{pmatrix}=\begin{pmatrix}{[190,210]}\\\\[2.0pt] {[200,220]}\end{pmatrix}.$ (28) Its matrix is the transposed matrix of system (25), so $\min_{A\in\text{\boldmath$A$}}\mathrm{cond}_{2}\,A$ is the same for it. Also, the matrix of system (28) contains a point matrix of the incomplete rank 1, which is transposed for (26) (and many more such matrices). $x_{1}$$x_{2}$$x_{3}$ Fig. 8: The tolerable solution set for the interval equations system (28). Again, the united solution set for system (28) is unbounded, and it is difficult (if at all possible) to determine the coefficients of the linear function (27), relying on the weak compatibility between parameters and data, due to “infinite variability” of the resulting estimate. Nevertheless, in these adverse conditions, the nonempty tolerable solution set to the interval system of equations (28) is bounded by virtue of the Irene Sharaya criterion [13] (see Section 2.1). In Fig. 8, the tolerable solution set is depicted as a thin hexagonal plate. Computation of the maximum of the recognizing functional for this system using the code tolsolvty2.m gives the value $\max\mathrm{Tol}\,=3.9698$, which is reached at the point $\hat{x}=\arg\max\mathrm{Tol}\,=\,\bigl{(}\,2.0603,3\cdot 10^{-6},2.1\cdot 10^{-6}\,\bigr{)}^{\top}.$ It can be taken as an estimate of the coefficients in (27). Then the varibility measure of the above estimate is $\mathrm{IVE}\,=\sqrt{2}\cdot 3.9698\cdot 103.83\cdot\frac{\|\hat{x}\|_{2}}{\sqrt{200^{2}+210^{2}}}=4.1413.$ Interval hull (optimal interval enclosure) of the tolerable solution set for system (28) is the box $\begin{pmatrix}[-1.9747,4.0302]\\\\[2.0pt] [-1.9899,4.0759]\\\\[2.0pt] [-1.9949,4.1071]\end{pmatrix},$ which can also be computed by the procedure EqnTolR3. The radii of the components of this interval vector are $3.0024$, $3.0329$, $3.0510$ respectively, which is also not very different from the value of IVE. The example shows that the value IVE works even in the case of $m<n$, when the number of measurements is less than the number of parameters to be determined. But a rigorous justification of this fact is waiting for its study. To conclude the section, we present, in Table 1, the results of numerical tests for the interval linear $n\times n$-system $\left(\begin{array}[]{cccc}\theta&{[0,2]}&\cdots&{[0,2]}\\\\[1.0pt] {[0,2]}&\theta&\cdots&{[0,2]}\\\\[1.0pt] \vdots&\vdots&\ddots&\vdots\\\\[1.0pt] {[0,2]}&{[0,2]}&\cdots&\theta\end{array}\right)\;\left(\begin{array}[]{@{\,}c@{\,}}x_{1}\\\\[1.0pt] x_{2}\\\\[1.0pt] \vdots\\\\[1.0pt] x_{n}\end{array}\right)=\left(\begin{array}[]{@{\;}c@{\;}}{[1,K]}\\\\[1.0pt] {[1,K]}\\\\[1.0pt] \vdots\\\\[1.0pt] {[1,K]}\end{array}\right),$ (29) with various $n$ and $K$. System (29) resembles that proposed in [9], having exactly same matrix. But the right-hand sides were taken as positive intervals $[1,K]$, since the original balanced intervals $[-1,1]$ in the system from [9] make the tolerable solution set “too symmetric”. Table 1: Results of the computational tests with system (29) $\theta\rule[-8.53581pt]{0.0pt}{22.76219pt}$ | $\mathrm{IVE}\,$ | $\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol})\|_{\infty}$ | $\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol})\|_{2}$ | $\theta\rule[-8.53581pt]{0.0pt}{22.76219pt}$ | $\mathrm{IVE}\,$ | $\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol})\|_{\infty}$ | $\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol})\|_{2}$ ---|---|---|---|---|---|---|--- $n=5$, $K=10$ | $n=10$, $K=10$ 2.0 | 1.019 | 1.25 | 2.795 | 6.0 | 0.894 | 0.5 | 1.581 4.0 | 1.081 | 0.875 | 1.957 | 9.0 | 1.491 | 0.389 | 1.230 6.0 | 0.786 | 0.639 | 1.429 | 12.0 | 0.582 | 0.313 | 0.988 8.0 | 0.681 | 0.5 | 1.118 | 15.0 | 0.495 | 0.26 | 0.822 10.0 | 0.534 | 0.41 | 0.917 | 20.0 | 0.396 | 0.203 | 0.640 $n=5$, $K=20$ | $n=10$, $K=20$ 2.0 | 2.953 | 3.75 | 8.385 | 6.0 | 2.489 | 1.333 | 4.216 4.0 | 2.698 | 2.125 | 4.752 | 9.0 | 1.831 | 0.944 | 2.987 6.0 | 2.015 | 1.472 | 3.292 | 12.0 | 1.432 | 0.729 | 2.306 8.0 | 1.591 | 1.125 | 2.516 | 15.0 | 1.255 | 0.593 | 1.876 10.0 | 1.378 | 0.91 | 2.035 | 20.0 | 0.985 | 0.453 | 1.431 The interval matrix of system (29) is known to be regular if and only if $\theta>n$ for even $n$ and $\theta>\sqrt{n^{2}-1}$ for odd $n$ [9]. Consequently, in Table 1, the first two rows that correspond to each separate case of $n$ and $K$ refer to systems with singular matrices. As the parameter $\theta$ grows, the matrix of the system becomes regular and better conditioned. The values of IVE are compared with $\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol})\|_{\infty}$ and $\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol})\|_{2}$, that is, with the Chebyshev norm (max-norm) and the Euclidean norm of the radius of the interval hull of the tolerable solution set (denoted as $\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol}$). We can see that, with the exception of two cases corresponding to $n=5$ and $K=10,20$, the values of IVE are always within the two-sided bounds given by $\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol})\|_{\infty}$ (lower bound) and $\|\,\mathrm{rad}\,(\,\scalebox{0.67}[0.87]{$\Box$\hskip 1.0pt}\varXi_{tol})\|_{2}$ (upper bound). And that is reasonable. Overall, our numerical experiments confirm the adequacy of the new measure of variability, which gives quite satisfactory approximate estimates of the size of the tolerable solution sets in various situations. ## 5 Discussion IVE is the first measure of variability proposed in the statistics of interval data, for estimation using the maximum compatibility method, and we can not compare IVE with similar other measures, since they simply do not exist. However, it is useful to correlate the estimate IVE with the ideal mathematical characteristics of the solution set, such as its diameter, in terms of computational convenience and laboriousness. First of all, IVE reflects instabilities in the solution set better than the diameter (see the first example in Section 4). An instability of the tolerable solution set for an interval linear system arises in the case when the maximum value of the recognizing functional is zero, $\max\mathrm{Tol}\,=0$. Then the tolerable solution set can be either a single-point or an extended set with non-zero diameter and empty interior [15]. After an arbitrarily small perturbation of data, the latter situation can abruptly turn into the empty solution set. In any case, this phenomenon is signaled by the zero value of the maximum of the recognizing functional. The corresponding variability measure IVE is also zero, which is quite natural: it makes sense to show only “stable size” of the solution set. The equality of IVE to zero or “almost zero” thus allows us to diagnose unstable cases. Next, the problem of computing the diameter, in 2-norm, of the tolerable solution set to an interval linear system of equations is NP-hard in general. This follows from its reducibility to the quadratic programming problem with linear constraints (see [22]). Indeed, the membership of a point in the tolerable solution set to an interval $m\times n$-system of equations is determined by a system of linear inequalities of the size $2m\times 2n$ (the Rohn theorem [12]). To compute the diameter of the tolerable solution set in 2-norm, we have to maximize the quadratic objective function $\|x^{\prime}-x^{\prime\prime}\|^{2}_{2}$ over all pairs of points $x^{\prime}$, $x^{\prime\prime}$ from the tolerable solution set, i. e. satisfying $2m\times 2n$-systems of linear inequalities. So, computing the diameter of the tolerable solution set is not easy. The diameter of the interval hull of the tolerable solution set can be computed more simply, but it is not better than IVE in any case, since the interval hull is not the solution set itself, and the coarsening resulted from such a replacement may be large. Calculation of IVE by formula (6) requires solving the data fitting problem, that is, finding $\max\,\mathrm{Tol}\,$ and $\arg\max\,\mathrm{Tol}\,$, and then we need to evaluate the minimum of the condition number of the matrices from the interval data matrix. In turn, the recognizing functional Tol is a concave piecewise linear function [15], so computing its maximum is polynomially complex. The author efficiently solves it by nonsmooth optimization methods developed in recent decades, in particular, using $r$-algorithms proposed by N.Z. Shor [20], or using separating plane algorithms (see, e. g., [11, 23]). The most difficult part in calculating IVE is thus evaluating the minimum condition number of point matrices from a given interval matrix. Computing the exact minimum of the condition number is not simple, but to address practical problems which will apply the value IVE, it is sufficient to have an approximate estimate for $\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}\,A$ from above. This follows from our considerations in Section 3.3. Sometimes, it is not necessary to compute $\min\,\mathrm{cond}_{2}\,A$ at all, if we have to compare, with each other, the variability of the estimates obtained for the same data matrix $A$. If the interval matrix is “sufficiently narrow”, being not very different from a point matrix, then we can approximate $\min_{A\in\text{\boldmath$A$}}\,\mathrm{cond}_{2}\,A\;\approx\;\mathrm{cond}_{2}(\mathrm{mid}\,\text{\boldmath$A$}).$ (30) But in general, this recipe may work poorly, since the left and right sides of the approximate equality (30) can be quite different. In the examples with systems (25) and (28) from Section 4, the condition number of the midpoint matrix is $2.38\cdot 10^{4}$, and, using the simplified formula (30), we are mistaken in evaluating the variability measure IVE by more than 20 times. In the general case, for a more accurate evaluation of $\min\,\mathrm{cond}_{2}\,A$, we can use popular evolutionary optimization methods, such as a genetic algorithm, simulated annealing, particle swarm optimization, etc., within the interval matrix $A$. In the numerical experiments from Section 4, the minimum of the condition number was found using the standard program of simulated annealing from free computer math system Scilab. Note that there is a fundamental difference between computing the variability measure IVE and computing the diameter of the tolerable solution set: in the first case, we calculate a minimum, while in the second we have to find a maximum. Using traditional optimization methods and various heuristics, in the first case we compute an approximation to the minimum from above, and in the second case we find an approximation to the maximum from below. If we want to get, with our variability measure, a guaranteed outer estimation of the solution set, then the upper estimate, which is obtained by calculating the minimum in IVE, is more preferable. There exists one more viewpoint at the variability measure IVE. In traditional probabilistic statistics, the phenomenon of collinearity of data (also called “multicollinearity”) plays a large role. It is the presence of a linear dependence between the input (predictor) variables of the regression model. The $k$ variables of the model in question are usually called _collinear_ if the vectors representing them lie in a linear space of dimension less than $k$ [1], so that one of these vectors is a linear combination of the others. In practice, such exact collinearity of data is rare, but real computational problems in data fitting often starts when the data vectors are “almost linearly dependent”. Then the parameter estimates are unstable, which leads to increased statistical uncertainty, i. e., an increase in the variance of the estimates. According to modern views, the collinearity of data is most adequately described by the condition number of the matrix made up of these data (see, e. g., [1], Chapter 3). In this sense, our IVE is, in fact, a measure of the collinearity of the data, corrected with the help of the actual value of the estimate and compatibility of this estimate with the data (which is indicated by the maximal value of the recognizing functional). The minimum over all $\mathrm{cond}_{2}A$ for $A\in\text{\boldmath$A$}$ is taken due to the specifics of the strong compatibility of parameters and data, and it agrees well with the regularizing role of the tolerable solution set (see [18]). With this interpretation, IVE makes sense even with a negative maximum of the recognizing functional, max Tol, when the tolerable solution set is empty and the parameters of the linear function (1), which are strongly compatible with the data, do not exist. The absolute value of IVE still shows, up to a certain scaling coefficient, a measure of the collinearity of the data (a measure of their ill-conditioning), and the negative sign indicates the status of the solution to the problem, i. e., that the parameter vector computed is not strongly compatible with the data, but only provides the best possible approximation for the input data of the problem. The immediate goal of further research is to justify the use of IVE for underdetermined situations, where the number $m$ of observations is less than the number $n$ of parameters to be determined. The maximum compatibility method works well in this case too, we get parameter estimates and we can calculate their values of IVE, but its application needs to be justified, at least at the same level of rigor as was done in this work for $m\geq n$. #### Aknowledgements The author is grateful to Alexander Bazhenov, Sergey Kumkov, and Sergei Zhilin for stimulating discussions and valuable comments on the work. Also, the author thanks the anonymous reviewers for their constructive criticism and good suggestions. ## References * [1] D.A. Belsley, E. Kuh, R.E. Welsch, Regerssion Diagnostics, Wiley-Interscience, Hoboken, N.J., 1980, 2004. * [2] G.H. Golub, Ch.F. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, 1996, 2013. * [3] R.A. Horn, Ch.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1994. * [4] L. Jaulin, M. Kieffer, O. Didrit, E. Walter, Applied Interval Analysis, Springer, London, 2001. * [5] R.B. Kearfott, M. Nakao, A. Neumaier, S. Rump, S.P. Shary, P. van Hentenryck, Standardized notation in interval analysis, Computational Technologies 15 (2010), No. 1, pp. 7–13. * [6] V. Kreinovich, S.P. Shary, Interval methods for data fitting under uncertainty: a probabilistic treatment, Reliable Computing 23 (2016), pp. 105–140. URL: http://interval.louisiana.edu/reliable-computing-journal/volume-23/reliable-computing-23-pp-105-140.pdf (accessed 10 March 2020). * [7] M. Milanese, J. Norton, H. Piet-Lahanier, E. Walter (Eds.), Bounding Approaches to System Identification, Plenum Press, New York, 1996. DOI: 10.1007/978-1-4757-9545-5 * [8] R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, 2009. * [9] A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990. * [10] H.T. Nguyen, V. Kreinovich, B. Wu, G. Xiang, Computing Statistics under Interval and Fuzzy Uncertainty. Applications to Computer Science and Engineering, Springer, Berlin-Heidelberg, 2012. * [11] E.A. Nurminski, Separating plane algorithms for convex optimization, Mathematical Programming 76 (1997), pp. 373–391. DOI: 10.1007/BF02614389 * [12] J. Rohn, Inner solutions of linear interval systems, in: Nickel K. (Ed.), Interval Mathematics 1985, Lecture Notes on Computer Science 212, Springer, New York, 1986, pp. 157–158. * [13] I.A. Sharaya, On unbounded tolerable solution sets, Reliable Computing 11 (2005), pp. 425–432. DOI: 10.1007/s11155-005-0049-9 * [14] I.A. Sharaya, IntLinInc2D, a MATLAB software package for visualization of solution sets to interval linear 2D systems. Novosibirsk, 2014. URL: http://www.nsc.ru/interval/sharaya * [15] S.P. Shary, Solving the linear interval tolerance problem, Mathematics and Computers in Simulation 39 (1995), pp. 53–85. DOI: 10.1016/0378-4754(95)00135-K * [16] S.P. Shary, Solvability of interval linear equations and data analysis under uncertainty, Automation and Remote Control 73 (2012), pp. 310–322. DOI: 10.1134/S0005117912020099 * [17] S.P. Shary, Maximum consistency method for data fitting under interval uncertainty, Journal of Global Optimization 66 (2016), pp. 111–126. DOI: 10.1007/s10898-015-0340-1 * [18] S.P. Shary, Interval regularization for imprecise linear algebraic equations. Deposited in arXiv.org on 27 Sep 2018, No. arXiv:1810.01481. * [19] S.P. Shary, Weak and strong compatibility in data fitting under interval uncertainty, Advances in Data Science and Adaptive Analysis 12 (2020) (in press). * [20] N.Z. Shor, N.G. Zhurbenko, A minimization method using space dilation in the direction of difference of two successive gradients, Cybernetics 7(3) (1971), pp. 450–459. DOI: 10.1007/BF01070454 * [21] TOLSOLVTY2, a MATLAB code for examining solvability of the interval linear tolerance problem. URL: https://www.researchgate.net/publication/294889566_TOLSOLVTY2 * [22] S.A. Vavasis, Complexity theory: Quadratic programming, in: C.A. Floudas and P.M. Pardalos (Eds.), Encyclopedia of Optimization. Second Edition, New York, Springer, 2009, pp. 451–453. * [23] E. Vorontsova, Extended separating plane algorithm and NSO-solutions of PageRank problem, in: Y. Kochetov, M. Khachay, V. Beresnev, E. Nurminski, P. Pardalos (Eds.), Discrete Optimization and Operations Research. Proceedings of 9th International Conference DOOR 2016, Vladivostok, Russia, September 19-23, 2016, Lecture Notes in Computer Science 9869, Cham, Switzerland, Springer International, 2016, pp. 547–560. DOI: 10.1007/978-3-319-44914-2`_`43 * [24] D.S. Watkins, Fundamentals of Matrix Computations, Wiley-Interscience, New York, 2002.

2024-09-04T02:54:59.042857

2003.05133

arxiv-papers

# Model Order Reduction in Neuroscience Bülent Karasözen Institute of Applied Mathematics & Department of Mathematics, Middle East Technical University, Ankara-Turkey<EMAIL_ADDRESS> ###### Abstract Human brain contains approximately $10^{9}$ neurons, each with approximately $10^{3}$ connections, synapses, with other neurons. Most sensory, cognitive and motor functions of our brains depend on the interaction of a large population of neurons. In recent years, many technologies are developed for recording large numbers of neurons either sequentially or simultaneously. Increase in computational power and algorithmic developments have enabled advanced analyses of neuronal population parallel to the rapid growth of quantity and complexity of the recorded neuronal activity. Recent studies made use of dimensionality and model order reduction techniques to extract coherent features which are not apparent at the level of individual neurons. It has been observed that the neuronal activity evolves on low-dimensional subspaces. The aim of model reduction of large-scale neuronal networks is accurate and fast prediction of patterns and their propagation in different areas of brain. Spatiotemporal features of the brain activity are identified on low dimensional subspaces with methods such as dynamic mode decomposition (DMD), proper orthogonal decomposition (POD), discrete empirical interpolation (DEIM) and combined parameter and state reduction. In this paper, we give an overview about the currently used dimensionality reduction and model order reduction techniques in neuroscience. Keywords:neuroscience, dimensionality reduction, proper orthogonal decomposition, discrete empirical interpolation, dynamic mode decomposition, state and parameter estimation. Classification[MSC 2010]: 93A15,92C55, 37M10,37M99,37N40,65R32. ## 1 Introduction Due to the advances in recording and imaging technologies, the number of recorded signals from brain cells increased significantly in the last few years. The recorded spatio-temporal neural activity give rise to networks with complex dynamics. In neuroscience, molecular and cellular level details are incorporated in large-scale models of the brain in order to reproduce phenomena such as learning and behavior. The rapid growth of simultaneous neuronal recordings in scale and resolution brings challenges with the analysis of the neuronal population activity. New computational approaches have to be developed to analyze, visualize, and understand large-scale recordings of neural activity. While algorithmic developments and the availability of significantly more computing power have enabled analysis of larger neuronal networks, these advances cannot keep pace with increasing size and complexity of recorded activity. The activity of complex networks of neurons can often be described by relatively few distinct patterns. Model order reduction techniques enable us to identify the coherent spatial–temporal patterns. The presence or absence of a neural mechanism can be analyzed for neuronal populations. Dimensionality reduction methods [6] are data-driven statistical techniques for forming and evaluating hypotheses about population activity structure, which are summarized in Section 2. One of the goals of neurosciences is fast and accurate predictions of the potential propagation in neurons. The differential equations describing the propagation of potential in neurons were developed and are described by Hodgkin and Huxley equations [12]. They consists of a coupled system of ordinary and partial differential equations (ODEs and PDEs). The dimension of the associated discretized systems is very large for accurately simulating neurons with realistic morphological structure and synaptic inputs. In Section 3 we present two model order reduction approaches based on POD and DEIM [5] which can predict accurately the potential propagation in large scale neuronal networks leading to important speedups [17, 16, 2]. Using the functional neuroimagining data from electroencephalography (EEG) or functional magnetic resonance imaging (fMRI), different regions of the brain can be inferred by dynamic causal modeling (DCM) [7]. Effective connectivity is parameterised in terms of coupling among unobserved brain states and neuronal activity in different regions. In Section 4 we describe a combined state and parameter reduction for parameter estimation and identification [10] to extract effective connectivity in neuronal networks from measured data, such as data from electroencephalography (EEG) or functional magnetic resonance imaging (fMRI). In Section 5 the data- driven, equation free, model order reduction method dynamic mode decomposition (DMD) is described for identifying sleep spindle networks [3]. Reduced order models with POD and DEIM and four variants of them are presented for neuronal synaptic plasticity and neuronal spiking networks in Section 6. ## 2 Dimensionality reduction methods Coordination of responses across neurons exists only at the level of the population and not at the level of single neurons. The presence or absence of a neural mechanism can be analyzed for neuronal populations. Dimensionality reduction methods are data-driven statistical techniques for forming and evaluating hypotheses about population activity structure. They produce low- dimensional representations of high-dimensional data with the aim to extract coherent patterns which preserve or highlight some feature of interest in the data [6]. The recorded neurons of dimension $D$ are likely not independent of each other, because they belong to a common network of neuronal populations. From the high-dimensional data of neuronal recordings, a smaller number of explanatory variables $K$ ( $K<D$) are extracted with the help of dimensionality reduction methods. The explanatory variables are not directly observed, therefore they are referred to as latent variables. The latent variables define a $K$-dimensional space representing coherent patterns of the noisy neural activity of $D$ neurons. There exists several dimensionality reduction methods which differ in the statistical interpretation of the preserved and discarded features of the neuronal populations. We summarize the commonly used statistical methods of dimensionality reduction methods in [6], where further references about the methods can be found. Principal component and factor analysis; The most widely known technique to extract coherent patterns from high dimensional data is the modal decomposition. A particularly popular modal decomposition technique is principal component analysis (PCA), which derives modes ordered by their ability to account for energy or variance in the data. In particular, PCA is a static technique and does not model temporal dynamics of time-series data explicitly, so it often performs poorly in reproducing dynamic data, such as recordings of neural activity. The low-dimensional space identified by PCA captures variance of all types, including firing rate variability and spiking variability, whereas factor analysis (FA) discards the independent variance for each neuron. and preserves variance that is shared across neurons. Time series methods: The temporal dynamics of the population activity can be identified if the data comes from a time series. The most commonly used time series methods for dimensionality reduction neural recordings are: hidden Markov models (HMM) [18], kernel smoothing followed by a static dimensionality reduction method, Gaussian process factor analysis (GPFA) [35], latent linear dynamical systems (LDS) [4] and latent nonlinear dynamical systems (NLDS) [26]. They produce latent neural trajectories that capture the shared variability across neurons. The HMM is applied when a jump between discrete states of neurons exists, other methods identify smooth changes in firing rates over time. Methods with dependent variables: On many neuronal recordings the high- dimensional firing rate space is associated with labels of one or more dependent variables, like stimulus identity, decision identity or a time index. The dimensionality reduction aims in this case to project the data such that differences in these dependent variables are preserved. The linear discriminant analysis (LDA) can be used to find a low-dimensional projection in which the $G$ groups to which the data points belong are well separated. Nonlinear dimensionality reduction methods: All the previous methods assume a linear relationship between the latent and observed variables. When the data lies on a low-dimensional, nonlinear manifold in the high-dimensional space, a linear method may require more latent variables than the number of true dimensions of the data. The most frequently used methods to identify nonlinear manifolds are Isomap79 [31] and locally linear embedding (LLE) [28]. Because the nonlinear methods use local neighborhoods to estimate the structure of the manifold, population responses may not evenly explore the high-dimensional space. Therefore theses methods should be used with care. ## 3 Proper orthogonal decomposition (POD) and discrete empirical interpolation (DEIM) for Hodgin-Huxley model One of the goals of neurosciences is fast and accurate predictions of the potential propagation in neurons. The differential equations describing propagation of potential in neurons are described by Hodgkin and Huxley (HH) cable equations [12]. They consists of a coupled system of ordinary (ODEs) and partial differential equations PDEs. Accurate simulation of morphology, kinetics and synaptic inputs of neurons requires solution of large systems of nonlinear ODEs. The complexity of the models are determined by the synapse density of the dentritic length ($1\mu$ one micron). In simulations, for one synapse per micron on a cell $5$ mm dendrite at $5,000$ compartments each with $10$ variables are needed, which results in $50,000$ coupled nonlinear system of ODEs [17, 16]. To recover complex dynamics, efficient reduced order neuronal methods are developed using the POD and DEIM from the snapshots of the in space and time discretized coupled PDEs and ODEs [17, 16, 2]. In this section we describe two of them. They differ in the formulation of the HH cable equation and of the equations for the gating variables. ### 3.1 Morphologically accurate reduced order modeling The neuronal full order models (FOMs) in [17, 16] consists of $D$ branched dendritic neurons $B=\sum_{d=1}^{D}B_{d}$ meeting at the soma, where the $d^{th}$ has $B_{d}$ branches. It is assumed that the branch $b$ carries $C$ distinct ionic currents with associated densities and $G_{bc}(x)$ and reversal potentials $E_{c},c=1,\ldots,C$. The kinetics of current $c$ on branch $b$ is governed by the $F_{c}$ gating variables, $w_{bcf},f=1,\ldots,F_{c}$. When subjected to input at $S_{b}$ synapses, the nonlinear HH cable equation for the transmembrane potential $v_{b}(x,t)$ with the equation for the gating variables $w_{bcf}$ is given by (see [2] Fig.1, model network with three cables) $\displaystyle a_{b}C_{m}\frac{\partial v_{b}}{\partial t}=$ $\displaystyle\frac{1}{2R_{i}}\frac{\partial}{x}\left(a_{b}^{2}\frac{\partial v_{b}}{\partial x}\right)$ (1) $\displaystyle- a_{b}\sum_{c=1}^{C}G_{bc}(x)(v_{b}-E_{c})\prod_{f=1}^{F_{c}}w_{bcf}^{q_{cf}}$ $\displaystyle\frac{1}{2\pi}\sum_{s=1}^{S_{b}}g_{bs}(t)\delta(x-x_{bs})(v_{b}-E_{bs})$ $\displaystyle\frac{\partial w_{bcf}}{\partial t}$ $\displaystyle=$ $\displaystyle\frac{w_{cf,\infty}(v_{b})-w_{bcf}}{\tau_{cf}(v_{b})},\quad 0<x<l_{b},\;t>0,$ (2) where $g_{bs}(nS)$ is the time course, $x_{bs}$ is the spatial location and $E_{bs}$ is the reversal potential of the $s$th synapse on branch $b$. The variables and parameters in (1) are described in [17, 16]. These branch potentials interact at $J$ junction points, where junction $J$ denotes the soma. The $D$ dendrites join at soma. Continuity of the potential at the soma leads to a common value at current balance denoted by $v_{\sigma}(t)$. Then the networked form of (1) becomes $\displaystyle a_{b}C_{m}\frac{\partial v_{\sigma}}{\partial t}=$ $\displaystyle\frac{\pi}{A_{\sigma}R_{i}}\sum_{d=1}^{D}\frac{\partial}{\partial x}\left(a_{b_{J}^{d}}^{2}(l_{b_{J}^{d}})\frac{\partial v_{b_{J^{d}}}(l_{b_{J^{d}}},t)}{\partial x}\right)$ (3) $\displaystyle- a_{b}\sum_{c=1}^{C}G_{\sigma c}(x)(v_{\sigma}-E_{c})\prod_{f=1}^{F_{c}}w_{\sigma cf}^{q_{cf}}(t)$ $\displaystyle\frac{1}{A_{\sigma}}\sum_{s=1}^{S_{b}}g_{\sigma s}(t)(v_{\sigma}(t)-E_{\sigma s})$ $\displaystyle\frac{\partial w_{\sigma cf}(t)}{\partial t}$ $\displaystyle=$ $\displaystyle\frac{w_{cf,\infty}(v_{\sigma}(t))-w_{\sigma cf}(t)}{\tau_{cf}(v_{\sigma})(t)},\quad 0<x<l_{b},\;t>0.$ (4) When the cell is partitioned into $N$ compartments, with $C$ distinct ionic currents per compartment and with $F$ gating variables per current, the following nonlinear ODEs are obtained $\displaystyle v^{\prime}(t)=$ $\displaystyle Hv(t)-(\Phi(w(t)e).v(t)+\Phi(w(t))E_{i}$ (5) $\displaystyle+G(t).(v(t)-E_{s}),\quad v(t)\in\mathbb{R}^{N}$ $\displaystyle w^{\prime}(t)=$ $\displaystyle(A(v(t))-w(t))./B(v(t)),\quad w(t)\in\mathbb{R}^{N\times C\times F}$ (6) where $H\in\mathbb{R}^{N\times N}$ is the Hines matrix [11], $e=[1\;1\cdots 1]^{T}\in\mathbb{R}^{C}$ and the ‘dot’ operator, $a.b$, denotes element-wise multiplication. $E_{i}$ and $E_{s}$ are respectively the vector of channel reversal potentials and the vector of synaptic reversal potentials, respectively Eq. (5) is discretized in time by the second order discretized Euler scheme [11]. In [16] using the snapshots of $v(t)$ and of the nonlinear term $N(v(t),w(t))\equiv(\Phi)w(t))e).v(t)-\Phi(w(t)))E_{i}$ at times $t_{1},t_{2},\ldots,t_{n}$ the POD and DEIM modes are constructed. The reduced membrane potential $v_{r}$ are constructed using the POD basis, the reduced gating variables $w_{r}$ are obtained after applying the DEIM to the nonlinear terms. The reduced order model in [16] preserves the spatial precision of synaptic input, captures accurately the subthreshold and spiking behaviors. In [17] a linearized quasi active reduced neuronal model is constructed using balanced truncation and ${\mathcal{H}}_{2}$ approximation of transfer functions in time. ROMs preserve the input-output relationship and reproduce only subthreshold dynamics. ### 3.2 Energy stable neuronal reduced order modeling In [1, 2] a different form of the HH cable equation and ODEs for gating variables is considered. The intracellular potential $v(x,t)$ and three gating variables $m(x,t),\;h(x,t)$, and $n(x,t)$ describe the activation and decativation of the ion channels, of the sodium channels and of the potassium channels, respectively. A single cable in the computational domain $(x,t)\in[0,L]\times(0,T]$ describing the distribution of the potential $u(x,t)$ is given by [1, 2] $\frac{\partial u}{\partial t}=\frac{\mu}{a(x)}\left(a(x)^{2}u_{x}\right)_{x}-\frac{1}{C_{m}}g(m,h,n)u+\frac{1}{C_{m}}f(m,h,n,x,t),$ (7) where $a(x)$ the radius of the neurons and $C_{m}$ is specific membrane capacitance, $\mu=\frac{1}{2C_{m}R_{i}}>0$ the ratio with $R_{i}$ the axial resistivity. The conductance $g(x,t)$ is a polynomial of the gating variables $g(x,t)=g_{1}m^{3}h+g_{2}n^{4}+g_{3}>0,$ (8) with the source term $f(m,h,n,x,t)=g_{1}E_{1}m^{2}h+g_{2}E_{2}n^{4}+g_{3}E_{3}-i(x,t),$ (9) where $E_{l},\;l=1,2,3$ are equilibrium potentials and $i(x,t)$ input current at $x$ $i(x,t)=\sum_{s=1}^{N_{s}}i_{s}(x,t),\quad x\in[0,L].$ (10) The nonlinear ODEs for the gating variables are given by $\displaystyle\frac{\partial m}{\partial t}$ $\displaystyle=$ $\displaystyle\alpha_{m}(v(x,t)(1-m(x,t))-\beta_{m}v(x,t))m(x,t),$ $\displaystyle\frac{\partial h}{\partial t}$ $\displaystyle=$ $\displaystyle\alpha_{h}(v(x,t))(1-h(x,t))-\beta_{h}v(x,t))h(x,t),$ (11) $\displaystyle\frac{\partial n}{\partial t}$ $\displaystyle=$ $\displaystyle\alpha_{n}(v(x,t))(1-n(x,t))-\beta_{n}v(x,t))n(x,t),$ Expression for $\alpha_{m},\;\alpha_{h},\;\alpha_{n},\;\beta_{m},\;\beta_{h},\;\beta_{n}$ and boundary conditions can be found in [2]. In [1, 2], a model network with three cables connected to a soma is used. The equations governing the potential propagation in the network $N_{c}$ neuron cables-dentrites and /or axons with the superscript ${}^{(c)},\;c=1,\ldots N_{c}$, are given as $\displaystyle\frac{\partial v^{(c)}}{\partial t}=$ $\displaystyle\frac{\mu}{a^{(c)}(x^{(c)})}\left(\left(a^{(c)}\left(x^{(c)}\right)^{2}\right)v^{(c)}_{x}\right)_{x}-\frac{1}{C_{m}}g\left(m^{(c)},h^{(c)},n^{(c)}\right)u^{(c)}$ $\displaystyle+$ $\displaystyle\frac{1}{C_{m}}f\left(m^{(c)},h^{(c)},n^{(c)},x^{(c)},t\right)$ (12) $\displaystyle\frac{\partial m^{(c)}}{\partial t}$ $\displaystyle=$ $\displaystyle\alpha_{m}(v^{(c)}(1-m^{(c)})-\beta_{m}v^{(c)})m^{(c)},$ $\displaystyle\frac{\partial h^{(c)}}{\partial t}$ $\displaystyle=$ $\displaystyle\alpha_{h}(v^{(c)})(1-h^{(c)})-\beta_{h}v{(c)})h^{(c)},$ (13) $\displaystyle\frac{\partial n^{(c)}}{\partial t}$ $\displaystyle=$ $\displaystyle\alpha_{n}(v^{(c)}))(1-n^{(c)})-\beta_{n}v^{(c)})n^{(c)},$ for $x^{(c)}\in\Omega^{(c)}=[0,L^{(c)}]$ together with the boundary conditions. The semi-discrete form of these equations are approximated using energy stable summation by parts [1, 2] for the model network. The reduced order bases (ROB) for multiple cables of identical lengths are assembled into a network in block form. The block structure of the ROB allows a flexible structure-preserving model reduction approach with an independent approximation in each cable and energy stability and accuracy properties follow from this block structure. Computation of the time varying reduced variables in the gating equations at every time $t$ is costly because they scale with dimension of FOM. A nonnegative variant of the discrete empirical interpolation method (NNDEIM) is developed in [2] to preserve the structure and energy stability properties of the equations. The capability of the greedy-based approach to generate accurate predictions in large-scale neuronal networks is demonstrated for system with more than $15,000$ degree of freedoms (dofs). The state variable ROB of dimension $l=15$ with POD modes together with the nonnegative ROBs of dimension $p=60$ with NNDEIM modes are constructed using a greedy approach to predict the potential variation at the soma. The speedup of simulations is about $20$ larger than Galerkin projection only is $1.3$ without using the NDEIM. ## 4 Combined state and parameter reduction for dynamic causal modelling In neuroscience different regions of the brain are inferred using neuroimagining data from EEG or fMRI recordings using the method od dynamic causal modeling (DCM) [7]. Effective connectivity is parameterised in terms of coupling among unobserved brain states and neuronal activity in different regions. In DCM the neuronal activity is of the observed brain region is represented as a SISO (single input single output) linear state-space system $\dot{x}=A_{\mathrm{d}yn}(\mu)x+B_{\mathrm{d}yn}u$ (14) with the parameterized connectivity $A_{\mathrm{d}yn}(\mu)$ and external input matrices $B_{\mathrm{d}yn}$. Linearization of the nonlinear DCM hemodynamic forward sub-model (balloon model) [7] transforms the neuronal activity to the measured BOLD (blood oxygen level dependent) response. Linearization around the equilibrium results in the following single input, single output (SISO) system: $\displaystyle B_{obs}$ $\displaystyle:=$ $\displaystyle(1\;0\;0\;0)^{T},\quad C_{obs}=(0\;0\;V_{0}k_{1}\;V_{0}k_{2}),$ (15) $\displaystyle\dot{z}_{i}$ $\displaystyle=$ $\displaystyle A_{obs}z_{i}+B_{obs}x_{i},$ (16) $\displaystyle y_{i}$ $\displaystyle=$ $\displaystyle C_{obs}z_{i},$ (17) $\displaystyle z_{0}$ $\displaystyle=$ $\displaystyle(0\;0\;0\;0)^{T},$ (18) $A_{\mathrm{o}bs}:=\left(\begin{array}[]{cccc}\frac{1}{\tau_{s}}&\frac{1}{\tau_{f}}&0&0\\\ 1&0&0&0\\\ 0&\frac{1}{\tau_{0}E_{0}}(1-(1-E_{0})(1-\ln(1-E_{0})))&\frac{1}{\tau_{0}}&\frac{1-\alpha}{\tau_{0}\alpha}\\\ 0&\frac{1}{\tau_{0}}&0&\frac{1}{\tau_{0}\alpha}\end{array}\right).$ (19) The fMRI measurements at the $i^{th}$ brain region are reflected by the output variables $y_{i}$. For the meaning of the variables and parameters in (15) and (19) we refer to [10, 9]. The linearized forward sub-models are embedded into the fMRI connectivity model $\left(\begin{array}[]{c}\dot{x}\\\ \dot{z}_{1}\\\ \dot{z}_{2}\\\ \vdots\\\ z_{N_{dyn}}\end{array}\right)=\left(\begin{array}[]{ccccc}A_{dyn}(\mu)&0&0&\cdots&0\\\ \delta_{1,1}&A_{obs}&0&&\\\ \delta_{2,1}&0&A_{obs}&&\\\ \vdots&&\ddots&\\\ \delta_{1,N_{dyn}}&&&A_{obs}\end{array}\right)\left(\begin{array}[]{c}x\\\ z_{1}\\\ z_{2}\\\ \vdots\\\ z_{N_{dyn}}\end{array}\right)+\left(\begin{array}[]{c}B_{dyn}\\\ 0\\\ 0\\\ \vdots\\\ 0\end{array}\right)v,$ (20) $y=\left(0\left(\begin{array}[]{ccc}C_{obs}&&\\\ &\ddots&\\\ &&C_{obs}\end{array}\right)\right)\left(\begin{array}[]{c}x\\\ z_{1}\\\ z_{2}\\\ \vdots\\\ z_{N_{dyn}}\end{array}\right),$ (21) where $\delta_{ij}\in\mathbb{R}^{4\times N_{\mathrm{d}yn}}$ denotes the Kronecker matrix with non-zero elements located at the $(i,j)^{th}$ component. The linearized state-space forward model (20) and (21) corresponds to a multiple input, multiple output (MIMO) system $\dot{x}(t)=A(\mu)x(t)+Bu(t),\qquad y(t)=Cx(t),$ (22) where $x\in\mathbb{R}^{N}$ is the internal state, $u\in\mathbb{R}^{J}$ the external input, $y\in\mathbb{R}^{O}$ the observed output and $\mu$ are the parameters describing different conditions. For large number of $M:=N^{2}$ parameters, the computational cost for inferring the parameters and states is very high. In [10, 8] a combined state and parameter model order reduction is developed for parameter estimation and identification to extract effective connectivity. The inversion procedure consists of two phases, the offline and online phases. In the offline phase, the underlying parameterized model is reduced jointly in states and parameters. In online phase, the reduced order model’s parameters are estimated to fit the observed experimental data. Using state and parameter reduction, the computational cost is reduced in the offline phase. The simultaneous reduction of state and parameter space is based on Galerkin projections with the orthogonal matrices for the state $V\in\mathbb{R}^{N\times n}$ and for the parameters $P\in\mathbb{R}^{M\times m}$. The reduced model is of lower order $n<<N,\;m<<M$ than the original full order model. The reduced states $x_{r}(t)\in\mathbb{R}^{n}$ and the reduced parameters $\mu\in\mathbb{R}^{m}$ are computed as $\dot{x}_{r}(t)=A_{r}(\mu_{r})x_{r}(t)+B_{r}u(t),\qquad y_{r}(t)=C_{r}x(t)$ (23) with a reduced initial condition $x_{r,0}=V^{T}x_{0}$ and the reduced components $\displaystyle\mu_{r}$ $\displaystyle=$ $\displaystyle P^{T}\mu\in\mathbb{R}^{m},$ $\displaystyle A_{r}(\mu_{r})$ $\displaystyle=$ $\displaystyle V^{T}A(P\mu_{r})V\in\mathbb{R}^{n\times n},$ $\displaystyle B_{r}$ $\displaystyle=$ $\displaystyle V^{T}B\in\mathbb{R}^{n\times J},$ $\displaystyle C_{r}$ $\displaystyle=$ $\displaystyle CV\in\mathbb{R}^{O\times m}.$ In the online phase, the optimization based inverse problem is combined with the reduction of state and parameter space. The inversion is based on generalized data-driven optimization approach to construct the ROMs in [23] and enhanced with the Monte-Carlo method to speed up the computations. The state projection $V\in\mathbb{R}^{N\times n}$ and parameter projection $P\in\mathbb{R}^{m\times m}$ are determined iteratively based on a greedy algorithm by maximizing the error between the high-fidelity original and the low-dimensional reduced model in the Bayesian setting. Numerical experiments with the DCM model in [23] show highly dimensional neuronal network system is efficiently inverted due to the short offline durations. In the offline phase, Monte-Carlo enhanced methods require more than an order of magnitude less offline time compared to the original and data-misfit enhanced methods. In the online phase the reduced order model has a speedup factor about an order of magnitude more compared to the full-order inversion. The output error of the data-misfit enhanced method is close to full order method. The output-errors of Monte-Carlo decrease with increasing number of simulation but does not reach the output error of the full order system. The source code is available in MATLAB [8]. ## 5 Dynamic mode decomposition Dynamic mode decomposition (DMD) is a data-driven, equation free ROM technique [20]. It was initially developed to reduce the high dimensional dynamic data obtained from experiments and simulations in the fluid mechanics into a small number of coupled spatial–temporal modes [29, 30]. DMD was applied to explore spatial–temporal patterns in large-scale neuronal recordings in [3]. DMD can be interpreted as combination of discrete Fourier transform (DFT) in time and principal component analysis (PCA) [15] in space. Both PCA and independent component analyses (ICA) [13] are static techniques, which perform poorly in reproducing dynamic data, such as recordings of neural activity. The data is taken from electrocorticography (ECoG) recordings. Voltages from $n$ channels of an electrode array sampled every $\Delta t$. These measurements are arranged at snapshot $k$ to the column vectors ${\mathbf{x}}_{k}$. The $m$ snapshots in time construct to data matrices shifted in time with $\Delta t$ ${\mathbf{X}}=\left[\begin{array}[]{cccc}|&|&&|\\\ {\mathbf{x}}_{1}&{\mathbf{x}}_{2}&\cdots&{\mathbf{x}}_{m-1}\\\ |&|&&|\end{array}\right],\quad{\mathbf{X}}^{\prime}=\left[\begin{array}[]{cccc}|&|&&|\\\ {\mathbf{x}}_{2}&{\mathbf{x}}_{3}&\cdots&{\mathbf{x}}_{m}\\\ |&|&&|\end{array}\right]$ (24) These matrices are assumed to be related linearly in time ${\mathbf{X}}^{\prime}={\mathbf{A}}{\mathbf{X}}.$ (25) The DMD of the data matrix pair ${\mathbf{X}}$ and ${\mathbf{X}}^{\prime}$ is given by the eigendecomposition of ${\mathbf{A}}$ using the singular value decomposition (SVD) of the data matrix ${\mathbf{X}}=U\Sigma V^{*}$ by computing the pseudoinverse ${\mathbf{A}}\approx{\mathbf{X}}^{\prime}{\mathbf{X}}^{\dagger}\equiv{\mathbf{X}}^{\prime}{\mathbf{V}}{\mathbf{\Sigma}}^{-1}{\mathbf{U}}^{*}.$ The spatio-temporal modes are computed by the exact DMD algorithm [32]. Because DMD does not contain explicit spatial relationship between neighboring measurements, traveling waves occurring in neuronal networks can not be captured well with a few coherent modes. DMD was also used as a windowed technique with a temporal window size constrained by lower bound as for the discrete Fourier transformation (DFT). In contrast to the fluid dynamics where $n>>m$, in neuroscience the electrode arrays that have tens of channels $n$ in the recordings with $m$ number of snapshots in the windows data per second, so that $n<m$. The number of singular values $v$ of ${\mathbf{X}}$ are less than $n$ and $m-1$, which restricts the maximum number of DMD modes and eigenvalues to $n$. Because of this the dynamics can be captured over $m$ snapshots. The rank mismatch is resolved by appending to the snapshot measurements with $h-1$ time-shifted versions of the data matrices. The augmented data matrix ${\mathbf{X}}_{\mathrm{a}ug}$ is given as ${\mathbf{X}}_{\mathrm{a}ug}=\left[\begin{array}[]{cccc}|&|&&|\\\ {\mathbf{x}}_{1}&{\mathbf{x}}_{2}&\cdots&{\mathbf{x}}_{m-h}\\\ |&|&&|\\\ |&|&&|\\\ {\mathbf{x}}_{2}&{\mathbf{x}}_{3}&\cdots&{\mathbf{x}}_{m-h-1}\\\ |&|&&|\\\ &&\cdots&\\\ |&|&&|\\\ {\mathbf{x}}_{h}&{\mathbf{x}}_{h+1}&\cdots&{\mathbf{x}}_{m-1}\\\ |&|&&|\\\ \end{array}\right].$ (26) The augmented matrices ${\mathbf{X}}_{{\mathrm{a}ug}}$ and ${\mathbf{X}}^{\prime}_{{\mathrm{a}ug}}$ are Hankel matrices, which are constant along the skew diagonal, as in the Eigenvalue Realization Algorithm (ERA) [14]. The number of the stacks $h$ is chosen such that $hn>2m$. A measure to determined the optimal number of stacks $h$ is the approximation error $E=\frac{||{\mathbf{X}}-\hat{\mathbf{X}}||_{F}}{||{\mathbf{X}}||_{F}}$ where $||\cdot||_{F}$ is the Frobenius norm. The approximation error $E$ is decreasing with increasing number of stacks $h$ and reaches a plateau, so that the DMD accuracy does not significantly increases. DMD is applied in [3] as an automated approach to detect and analyze reliably spatial localization and frequencies of sleep spindle networks from human ECoG recordings. A MATLAB implementation is available at github.com/bwbrunton/dmd- neuro/. ## 6 Reduced order modeling of biophysical neuronal networks Recently reduced order models for ODEs $\dot{x}(t)=A(t)x(t)+f(x(t))+Bu(t)$ (27) are constructed using POD and DEIM to investigate input-output behavior of the neuronal networks in the brain [22, 21], where $x(t)$ are state, and $u(t)$ are input variables, respectively. The model in [22] is based on the chemical reactions of molecules in synapses, that are the intercellular information transfer points of neurons. The signaling pathways in striatal synaptic plasticity are modeled in [19]. This model describes how certain molecules, which are a prerequisite for learning in the brain, act in synapses. The stoichiometric equations obey the law of mass action, which leads to a deterministic system of $44$ ODEs of the form (27) . The state $x(t)$ of the control system (27) is a collection of ions, molecules, and proteins that act in neuronal synapses. The linear part of (27) is sparse, the nonlinearities are quadratic. The time dependent stimulus $u(t)$ consists of molecules that are important for neuronal excitability and plasticity, calcium and glutamate. In [21], a nonlinear biophysical network model is considered, describing synchronized population bursting behavior of heterogeneous pyramidal neurons in the brain [27]. Neurons communicate by changing their membrane voltage to create action potentials (spikes), propagating from cell to cell. Spiking is the fundamental method of sensory information processing in the brain, and synchronized spiking is an emergent property of biological neuronal networks. The ODE system (27) in [21] consists of the states $x(t)$ as a collection of $50$ neurons, each modeled with $10$ ODEs, leading totally to a system of ODEs of dimension $500$. Each cell is modeled with Hodgkin-Huxley equations, where each cell has only two compartments (dendrites and soma) and these compartments have different ion channels. The state variables $x(t)$ include the voltages of somatic and dendritic compartments, dendritic calcium concentration, synaptic and ion channel gating variables and the input $u(t)$ is an injected current. Additionally, the soma compartment voltages are coupled to dentritic compartments of randomly chosen cells. This networking of the output of cells as input to other cells is key for producing synchronized population behavior. The nonlinearities are Hodgkin-Huxley type, i.e. exponential functions as well as cubic and quartic polynomials. In [22], POD+DEIM was applied to a data-driven biological model of plasticity in the brain (27). The ROMs with POD-DEIM reduce significantly the simulation time and error between the original model and reduced order solutions can be tuned by adjusting the number POD and DEIM bases independently. When the ROMs are trained in a matching time interval of $10000$ seconds, accurate results are obtained. However, generalizing the reduced model to longer time intervals is challenging, which is characteristic for all nonlinear models. In [21], the network model (27) is reduced with localized DEIM (LDEIM) [24], discrete adaptive POD (DAPOD) [33, 34], and adaptive DEIM [25]. DEIM and the variations are used here in combination with POD. 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2024-09-04T02:54:59.054047

2003.05150

arxiv-papers

# Illuminating the dark ages: Cosmic backgrounds from accretion onto primordial black hole dark matter G. Hasinger ###### Abstract The recent interpretation of cold dark matter as the sum of contributions of different mass Primordial Black Hole (PBH) families [1] could explain a number of so far unsolved astrophysical mysteries. Here I assume a realistic $10^{-8}$–$10^{10}$ M⊙ PBH mass distribution providing the bulk of the dark matter, consistent with all observational constraints. I estimate the contribution of baryon accretion onto this PBH population to various cosmic background radiations, concentrating first on the cross-correlation signal between the Cosmic X–ray and the Cosmic infrared background fluctuations discovered in deep Chandra and Spitzer surveys. I assume Bondi capture and advection dominated disk accretion with reasonable parameters like baryon density and effective relative velocity between baryons and PBH, as well as appropriate accretion and radiation efficiencies, and integrate these over the PBH mass spectrum and cosmic time. The prediction of the PBH contribution to the X–ray background is indeed consistent with the residual X–ray background signal and the X–ray/infrared fluctuations. The predicted flux peaks at redshifts z$\approx$17–30, consistent with other constraints requiring the signal to come from such high redshifts. The PBH contribution to the 2–5 $\mu$m cosmic infrared background fluctuations is only about 1%, so that these likely come from star formation processes in regions associated with the PBH. I discuss a number of other phenomena, which could be significantly affected by the PBH accretion. Magnetic fields are an essential ingredient in the Bondi capture process, and I argue that the PBH can play an important role in amplifying magnetic seed fields in the early universe and maintaining them until the galactic dynamo processes set in. Next I study the contribution of the assumed PBH population to the re-ionization history of the universe and find that they do not conflict with the stringent ionization limits set by the most recent Planck measurements. X–ray heating from the PBH population can provide a contribution to the entropy floor observed in groups of galaxies. The tantalizing redshifted 21-cm absorption line feature observed by EDGES could well be connected to the radio emission contributed by PBH to the cosmic background radiation. Finally, the number of intermediate-mass black holes and the diffuse X–ray emission in the Galactic Center region are not violated by the assumed PBH dark matter, on the contrary, some of the discrete sources resolved in the deepest Chandra observations of the Galactic Ridge could indeed be accreting PBH. ## 1 Introduction Recent years saw a revival of the idea originally put forward by S. Hawking [2], that Primordial Black Holes (PBH) could make up the so far elusive Dark Matter. LIGOs first detection of gravitational waves from merging binary black holes of approximately equal masses in the range 10–30 M⊙ [3, 4] led to the suggestion that these could be a signature of dark matter stellar mass PBH [5, 6, 7] in a mass window not yet excluded by other astrophysical constraints. A recent review about the rich literature constraining the possible contributions of PBH to the dark matter is e.g. given in [8]. In a recently published theoretical prediction [1, 9] PBH are created in the QCD phase transitions (around 100 MeV) of different particle families freezing out of the primordial Quark-gluon plasma within the first two seconds after the inflationary phase. When W+/-, Z bosons, baryons, pions are created, and e+e- pairs annihilate, they leave an imprint in form of a significant reduction of the sound speed at the corresponding phase transitions, and allow regions of high curvature to collapse and form PBH [see also 10]. The typical mass scale of these PBH is defined by the size of the horizon at the time of the corresponding phase transition. In this model four distinct populations of PBH in a wide mass range are formed: planetary mass black holes at the W+/-, Z transition, PBH of around the Chandrasekhar mass when the baryons (protons and neutrons) are formed from 3 quarks, PBH of masses of order 30 M⊙ (these correspond to the LIGO black holes), when pions are formed from two quarks, and finally supermassive black holes (SMBH) at the e+e- annihilation [see also 11]. Another remarkable aspect of this theory is, that the gravitational energy released at at the PBH collapse locally reheats regions (hot spots) around the black holes to the electroweak transition scale (around 100 GeV), where chiral sphaleron selection effects can introduce the matter/antimatter asymmetry. The PBH in this picture would therefore also be responsible for the baryogenesis and fix the ratio of dark matter to baryons. Clustering of the PBH in a very wide mass distribution could alleviate some of the more stringent observational constraints on the allowed contribution of PBH to the dark matter [7, 12]. The interpretation of cold dark matter as the sum of contributions of different mass PBH families could explain a number of so far unsolved mysteries, like e.g. the massive seed black holes required to create the supermassive black holes in the earliest QSOs [13], the ubiquitous massive LIGO/VIRGO massive binary black holes [e.g. 6], or even the putative "Planet X" PBH in our own Solar System [14]. The most abundant family of PBH should be around the Chandrasekhar mass (1.4 M⊙). This prediction may already have been vindicated by the recent OGLE/GAIA discovery of a sizeable population of putative black holes in the mass range 1–10 M⊙ [15]. The microlensing survey OGLE has detected $\sim$60 long-duration microlensing events. About 20 of these have GAIA DR2 parallax distances of a few kpc, which break the microlensing mass–distance degeneracy and allow the determination of masses in the few solar mass range, implying that these objects are most likely black holes, since stars at those distances would be directly visible by OGLE. Important fingerprints of a population of PBH may be hidden in the Cosmic infrared and X–ray background radiation (see [16] for a comprehensive review). Indeed, [6] argues, that the near-infrared Cosmic background (CIB) anisotropies detected in deep Spitzer [17, 18, 19, 20] and Akari [21] images, which cannot be accounted for by known galaxy populations [22], could be connected to PBH. Similar fluctuations were discovered in the Cosmic X–ray background (CXB) observed in a deep Chandra survey, which are correlated with the CIB anisotropies in the same field [23]. Later studies of wider/deeper fields covered by both Chandra and Spitzer [24, 25, 26] have substantially improved the detection significance of the observed signal. The X–ray fluctuations contribute about 20% to the CIB signal, indicating that black hole accretion should be responsible for such highly efficient X–ray emission. Similar studies of deep fields observed with the Hubble Space Telescope in the optical range do not show such a cross-correlation signal down to mAB$\sim$28 [see 16]. The angular scale of the fluctuation power spectra of the CIB and CXB reach values >1000", much larger than expected for the known galaxy populations [27]. All of these findings can be understood, if the fluctuation signal comes from a high-redshift (z$\gtrsim$12) population of black holes. The spectral shape of the CXB fluctuations determined from a combination of the deepest/widest fields [26] can be fit either with a very high redshift population of obscured black holes, or with completely unobscured black hole accretion. Original models [28] invoked highly obscured Direct Collapse Black Holes formed in metal-free halos at z>12 to explain the observed CIB and CXB signal. However, accreting massive black holes have recently been firmly ruled out as the source of these fluctuations [29], because they would require an unfeasible amount of black hole accretion at z>6, locking up a larger amount of mass in massive black holes at high redshift, than the known black hole mass function at z=0. These authors also ruled out local diffuse emission as the source of the X–ray fluctuations. The CXB has been largely resolved into discrete sources in deep X–ray images, either directly [see 30, 31], or by crosscorrelating with the deepest Hubble galaxy catalogues [32, 33]. However, [32] show that some marginally significant diffuse CXB still remains after accounting for all discrete contributions. This is consistent with the independent determination of [34]. The residual unresolved flux is about 3 times larger than the X-ray flux associated with the above CXB/CIB fluctuations. Given the difficulties in explaining the CIB/CXB correlation with known classes of sources, and motivated by the notion that the dark matter could be dominated by an extended mass distribution of PBH, I constructed a toy model to explore the potential contribution to the cosmic backgrounds by the accretion of baryons throughout cosmic history onto such a population of early black holes. Assuming a combination of Bondi-Hoyle-Lyttleton quasi-spherical capture at large distances from the PBH, and advection-dominated disk accretion flows (ADAF) in the vicinity of the central object, I can explain the observed residual CXB flux and the CXB/CIB crosscorrelation with minimal tuning of the input parameters, and find a maximum contribution to the extragalactic background light in the redshift range 15<z<30\. I further estimate that this accretion onto PBH can produce enough flux to significantly contribute to the pre-ionization of the intergalactic medium with UV photons by a redshift z$\gtrsim$15 and to the pre-heating of the baryons with X–ray photons, observed as an "entropy floor" in the X–ray emission of galaxy groups. In section 2 the assumed PBH mass distribution is introduced and contrasted with recent observational limits on the PBH contribution to the dark matter. The basic ingredients of the toy model for the accretion onto PBH are presented in section 3. The assumed radiation mechanism and efficiency is discussed in section 4. The contribution of the PBH emission to the different bands is compared with the observational constraints in section 5. Other potential diagnostics of this putative dark matter black hole population are discussed in section 6, and conclusions are presented in section 7. Throughout this work a $\Lambda$CDM cosmology with $\Omega_{M}$=0.315, $\Omega_{\Lambda}$=0.685, and H0=67.4 km s-1 Mpc-1 [35] is used. These parameters define the baryon density $\Omega_{bar}$=0.049, the dark matter density $\Omega_{DM}$=0.264, and the critical mass density of the universe $\rho_{crit}$=1.26$\times 10^{20}M_{\odot}~{}{\rm Gpc}^{-3}$. All logarithms in this paper are taken to the base 10. ## 2 The assumed PBH mass distribution The theoretical predictions in [1, 9, 11, 36] yield a broad distribution of PBH masses with a number of peaks corresponding to the particle families freezing out from the Big Bang. Depending on the spectral index $n_{s}$ of the primordial curvature fluctuation power spectrum, the PBH mass distribution has a different overall slope. [36] find consistency of these predictions with a number of recent observational limits on the PBH contribution to the dark matter, but there is a tension of their models with the Cosmic Microwave Background (CMB) constraints from accretion at large PBH masses [37, 38]. Recent limits from gravitational lensing of type Ia supernovae on a maximum contribution of stellar-mass compact objects to the dark matter of around 35% [39], and from the LIGO OI gravitational wave merger rate of black holes in the mass range 10–300 M⊙ [40] are also in tension with these models. An additional important constraint comes from a comparison of the predicted PBH fraction with the measured local mass function of supermassive black holes (SMBH) in the centers of nearby galaxies. Integrating the local SMBH mass function of [41] (see figure 1) in the range $10^{6}$–$10^{10}$ M⊙ yields a local SMBH mass density of $\rho_{SMBH}$=6.3$\times$105 M⊙ Mpc-3, corresponding to a dark matter fraction of fSMBH=1.89$\times$10-5, which is about a factor of 10–100 lower than the fPBH predictions in [1, 36]. Figure 1: The PBH mass spectrum (thick red line) assumed for this work (García-Bellido, 2020, priv. comm.), compared to a number of observational constraints. Microlensing limits from SNe [39], EROS [42], and the Subaru M31 survey [43] are shown as solid, dashed and dotted green lines, respectively. LIGO limits from gravitational merger event rates are shown as blue solid line for subsolar masses [44], and as blue dashed line for 10-300 M⊙ [40]. The CMB accretion limits from [37] are shown as orange dashed line. Multiwavelength limits from the Galactic Center [45] are shown in magenta for X-ray (solid) and radio (dashed) observations. Finally, the local SMBH mass function [41] is shown as black line at 106-10 M⊙. For these reasons, García-Bellido et al. (2020 in prep.) are revising their model parameters in order to predict a steeper PBH mass function at large MPBH and shared one of their new models, shown as red curve in figure 1. Here a value of ns=0.987 is assumed for the spectral index of the primordial fluctuation power spectrum, as well as a running curvature of dns=$-$0.0006. The integral of this PBH distribution over the whole mass range yields fPBH=1. On the other hand, the distribution yields only $\sim$40% of the dark matter in the peak mass range [0.1,10] M⊙, and is thus fully consistent with the microlensing constraints in figure 1. In the mass range of the LIGO black hole binaries it predicts just the right amount of dark matter to explain the gravitational wave merger rates, and in the SMBH range it is consistent with the local black hole mass function (taking into account the accretion onto supermassive PBH over cosmic time producing the bulk of the X-ray background [46]). Apart from small sections, the new PBH mass function is thus fully consistent with the most recent observational constraints. ## 3 Baryon accretion onto the PBH In the following I use the PBH mass spectrum presented in section 2 to calculate the accretion of baryons onto PBH over cosmic time, and to predict the electromagnetic emission from this process. As we will see, for most of the cosmic history these black holes move at supersonic speeds among the baryons and will therefore undergo Bondi-Hoyle-Lyttleton quasi-spherical capture [47, 48, 49, 50]. In the Bondi-Hoyle picture of a black hole moving supersonically through a homogeneous gas, the capture happens in the wake of the moving object. Behind the object, material moves in from a wide cone, and needs to lose angular momentum before it can fall towards the black hole. The gas is in principle collisionless, so that only the magnetic field trapped in the plasma allows particles to lose angular momentum and start to behave like a fluid. This gas forms the accretion flow, in which it is adiabatically heated. The accreting gas is ionized and embedded in the magnetic field. Any plasma drawn in by the gravitational field will carry along the magnetic field. Shvartsman [51] argues that in the black hole tail, where the matter flow stops, the gravitational and magnetic energy densities become nearly equal. This equipartition is preserved in the infalling flow and thus the magnetic field grows towards the black hole. Like the heat has to be ultimately radiated away, the magnetic field needs a way to dissipate energy on its way inward. [52] discuss that the most likely dissipation mechanism for the magnetic field is reconnection of field lines in narrow current sheets, similar to the processes we observe in solar flares and active galactic nuclei. Magnetic reconnection causes the acceleration and non-thermal heating of a small fraction of the infalling electrons. In parallel, decoupled magnetic field lines can carry some of the amplified magnetic field outward and eject plasma [52]. An important question is, whether the accretion flow is spherically symmetric close to the black hole, or whether an accretion disk is formed. Originally most researchers assumed spherical accretion for PBH [e.g. 53, 54, 38]. However, [37] argues, that the accreted angular momentum is large enough, that an accretion disk is formed, at least close to the black hole. According to these authors, the granularity of the PBH distribution and the formation of PBH binaries at the scale of the Bondi radius will imprint density and velocity gradients into the relative distribution of baryons and PBH, such that ultimately an accretion disk and an advection-dominated accretion flow (ADAF) will form [55]. The formation of an ADAF disk significantly reduces the accretion rate and the radiative efficiency [56], compared to spherical accretion. But to first order the Bondi-Hoyle-Lyttleton mechanism can be used to estimate the accretion rate $\dot{M}$ onto the PBH [37, 8]. Bondi [49] discusses two different approximations to the spherical gas accretion problem, (i) the velocity-limited case, where the motion of the accreting object through the gas is dominant and an accretion column is formed in the wake of the moving object, and (ii) the temperature-limited case, where the sound speed of the gas is dominant and a spherical accretion flow forms. In the velocity-limited case (i) the mass accretion rate is given as $\dot{M}=2.5\pi\rho(GM)^{2}v_{rel}^{-3},$ (3.1) where $\rho$ is the gas density, $M$ is the PBH mass, and $v_{rel}$ is the relative velocity between object and gas. In the temperature-limited case (ii) with negligible relative velocity, the thermal velocity of the gas particles is dominant and the corresponding accretion rate is given by $\dot{M}=2.5\pi\rho(GM)^{2}c_{s}^{-3},$ (3.2) where $c_{s}$ is the sound speed. For intermediate cases, [49] introduces an effective velocity $v_{eff}=\sqrt{v_{rel}^{2}+c_{s}^{2}}$ (3.3) and the corresponding mass accretion rates becomes $\dot{M}=2\lambda\pi\rho(GM)^{2}v_{eff}^{-3},$ (3.4) where the so called accretion eigenvalue $\lambda$ is is a fudge factor of order unity, dependent on non-gravitational aspects of the problem, like e.g. the gas equation of state or outflows from feedback effects. Different authors have discussed this parameter for the particular application of gas accretion onto PBH in the early universe. [53] find values of $1.12>\lambda>10^{-3}$, depending e.g. on the PBH mass. For masses of order 1 M⊙ they find $\lambda=1.12$. [5] discriminate between isothermal and adiabatic gas with accretion eigenvalues of $\lambda$=1.12, and 0.12, respectively. In this paper I assume an eigenvalue $\lambda$=0.05. The motivation for this choice is discussed in section 4, while section 5 and 6 show that this choice fits the observational constraints quite well. Figure 2: Left: Baryon temperature as a function of redshift. Right: Mean relative velocity $\langle v_{rel}\rangle$ between dark matter and baryons, sound speed $c_{s}$ and the effective velocity $v_{eff}$ (eq. 3.8) as a function of redshift. Let us first look at the thermal history and thus the sound speed of the gas over cosmic history. A nice summary is given in figure 15 of [57]. Despite having decoupled from the CMB at z$\approx$1089, the gas temperature continues to follow the temperature evolution T$\propto$(1+z) of the background photons due to Compton scattering off residual ionized electrons from the recombination era. Below redshifts z$\approx$200 the residual ionization in the gas is low enough, that it decouples from the background radiation and cools adiabatically following the relation T$\propto$(1+z)2. When the first objects form and reionization starts around z$\lesssim$20, the gas is heated up to temperatures $\sim$104 K. The details of re-ionization are still uncertain and will be discussed below. I have deliberately chosen a redshift of z$\approx$20 for re-ionization to become dominant, with full ionization occurring around z$\approx$7\. Finally, at z<3, when the bulk of the cosmic baryons are falling into increasingly larger dark matter halos and become virialized, they are further heated up to form the warm/hot intergalactic medium at temperatures $10^{5-7}$K [58]. Using figure 2b in this paper I estimate average temperatures for the IGM of 5$\times 10^{4}$, 1.5$\times 10^{5}$, and 8$\times 10^{5}$ K at z=2, 1, 0, respectively. The baryon temperature as a function of redshift assumed in this work is shown in figure 2 (left). The sound speed of the gas is given by $c_{s}=\sqrt{\frac{\gamma kT}{\mu m_{H}}},$ (3.5) where $\gamma$=5/3 for an ideal monoatomic gas, $\mu$=1.22 is the mean molecular weight including a helium mass fraction of 0.24, $m_{H}$ is the mass of the hydrogen atom, and $T$ is the temperature of the baryons as a function of cosmic history discussed above [59]. The sound speed as a function of redshift is the dotted curve shown in figure 2 (right). I now discuss the relative velocity $v_{rel}$ between the dark matter PBH and the baryons throughout cosmic history. In the radiation-dominated phase of the universe at z>1089, the dark matter is already hierarchically clustering under the influence of its own gravity. The sound speed of the photon-baryon fluid is very high, of order one third of the velocity of light, and thus the normal matter undergoes baryonic acoustic oscillations [60, 61]. This leads to a spatial separation between baryons and dark matter and thus to a Gaussian distribution of relative velocities with an average around $\langle v_{rel}\rangle$$\approx$30 km/s [see 59, 62]. At z$\approx$1089, when electrons and protons combine and the universe becomes transparent, the sound speed of the gas dramatically drops to $\sim$6 km/s. The dark matter PBH kinematically decouple from the baryons and their relative velocities become highly supersonic. In the linear growth phase of the universe, at scales larger than the gas Jeans-length, the dark matter and the baryons fall in the same gravitational potentials of the cosmic web and thus their relative velocity decreases with the cosmic expansion: $\langle v_{rel}\rangle_{linear}\approx 30~{}{\frac{1+z}{1000}}~{}{\rm km~{}s}^{-1}.$ (3.6) This relation is shown as the right branch of the dashed line in figure 2 (right), above redshifts $z\gtrsim 20$. From this figure it becomes apparent, that between recombination and re-ionization the PBH move with highly supersonic, but decreasing velocities through the gas, due to the decaying sound waves. As we will see below, in this branch of the velocity curve the contribution of PBH to the cosmic backgrounds has a maximum. At lower redshifts, at scales smaller than the gas Jeans-length, the hierarchical clustering becomes non-linear and baryons falling into growing dark matter halos are virialized. As a consequence, the velocity dispersion between dark matter and gas increases again towards lower redshifts, scaling as $M_{Halo}^{1/3}$, where $M_{Halo}$ is the mass of the dark matter halo becoming non-linear. I used two different methods to estimate the average virial velocity for redshifts z$\lesssim$20\. First, the Millenium Simulation run described in [63] gives the mass function of dark matter halos with halo masses $M_{Halo}$>$10^{10}M_{\odot}$ for five different redshifts between z=10 and z=0. I extrapolated these simulated mass functions to lower masses ($M_{Halo}$>10${}^{3}M_{\odot}$) using the empirical universal halo mass function shape found through simulations by [64]. For every mass bin I determined the virial velocity according to the calibration of the velocity dispersion as a function of halo mass described in [65], and then calculated the average for each epoch. These virial velocities are shown as crosses in figure 2 (right). The extrapolation to halo masses as small as $M_{Halo}>10^{3}M_{\odot}$ is rather uncertain, both for the mass function and the velocity dispersion, because the cosmological simulations do not have a fine enough mass resolution at this scale. Also, the velocity dispersion relevant for Bondi capture onto PBH is determined by the smallest mass scales becoming non-linear at any redshift. A second possibility to calculate the relative velocities in the non-linear phase is therefore to determine the velocity dispersion directly from the dark matter power spectrum and integrate this over the smallest non-linear scales. This calculation has been performed by M. Bartelmann (2020, priv. comm.), adopting the normalization of the primordial power spectrum of $\sigma_{8}$=0.8. The relative velocity in the non-linear regime can be approximated by $\langle v_{rel}\rangle_{nonlinear}\approx 620~{}(1+z)^{-2.3}~{}{\rm km~{}s}^{-1},$ (3.7) and is shown as the left branch ($z\lesssim 20$) of the dashed line in figure 2 (right). At z=2 the cluster velocity dispersion agrees with this estimate, but it systematically overestimates the small-scale velocity dispersion at larger redshifts. Since we are interested in the total contribution of PBH to the electromagnetic radiation of the universe, we have to average over the whole Gaussian distribution of relative velocities. The Bondi accretion rate is proportional to $v_{rel}^{-3}$ (see above), and therefore smaller velocities dominate. For this particular case [38] propose to replace the quadratic average of relative velocity and sound speed in Bondi’s formula (3.3) above with their harmonic mean: $v_{eff}=\sqrt{\langle v_{rel}\rangle~{}c_{s}.}$ (3.8) This is the assumption I adopt here, and the resulting effective velocity $v_{eff}$ is shown as solid red curve in figure 2 (right). With equation (3.8) the accretion rate becomes $\dot{M}=2\lambda\pi\rho(GM)^{2}~{}(\langle v_{rel}\rangle~{}c_{s})^{-3/2}$ (3.9) It is interesting to note that in the range 200<z<20 both relative velocity and sound speed decrease linearly with (1+z). Therefore the mass accretion rate is expected to be constant in this era. It is important to understand that the redshift, at which both the sound speed and the relative velocity of the gas turn around due to re-ionization and virialization, respectively, and rapidly increase towards lower redshift, is crucial for our analysis. The redshift, where the minimum velocity occurs, ultimately determines the maximum flux contribution of PBH accretion to the cosmic backgrounds. The calculation of the Bondi accretion rate in equation (3.9) requires the density $\rho$ as a function of redshift. With $\Omega_{bar}$=0.049 and $\rho$=n$\cdot$mH, where $n$ is the number density of particles, I find $n=250~{}\left(\frac{1+z}{1000}\right)^{3}~{}{\rm cm}^{-3}.$ (3.10) I define $\dot{m}$ as the normalized mass accretion rate $\dot{m}=\dot{M}/\dot{M}_{Edd}$, with the Eddington accretion rate $\dot{M}_{Edd}$=1.44$\times 10^{17}M/M_{\odot}$ g s-1. Then I can rewrite equation (3.9) into normalized quantities $\dot{m}=\lambda\cdot 0.321\left(\frac{1+z}{1000}\right)^{3}~{}\left(\frac{M}{M_{\odot}}\right)\left(\frac{v_{eff}}{1~{}{\rm km~{}s}^{-1}}\right)^{-3}$ (3.11) With a very broad PBH mass spectrum, including intermediate-mass and supermassive black holes (MPBH>1000 M⊙), it is important to include the effective viscosity due to the Hubble expansion in the early universe [53]. The Bondi radius determines the amount mass captured by the PBH: $r_{B}={\frac{G~{}M}{v_{eff}^{2}}}\approx 1.34\cdot 10^{16}\left(\frac{M}{M_{\odot}}\right)\left(\frac{v_{eff}}{1~{}{\rm km~{}s}^{-1}}\right)^{-2}cm.$ (3.12) This is shown for two different PBH masses in figure 8 (left). The characteristic time scale for accretion is the Bondi crossing time $t_{cr}=r_{B}/v_{eff}$, which can be compared to the Hubble time $t_{H}$ at the corresponding redshift. If $t_{cr}<t_{H}$ there will be stationary accretion, while for Bondi crossing times larger than the Hubble time the accretion is suppressed. For every redshift we can calculate a critical PBH mass Mcr, below which the steady-state Bondi assumption can be applied. For redshifts z=1000, 200, 20 this critical mass corresponds to $log(M_{cr}/M_{\odot})$=5.3, 4.8, 3.4, respectively. At redshifts below z=20 Mcr rapidly increases to values above 106 M⊙. For PBH masses close to and above Mcr the Bondi accretion rate can be scaled by the Hubble viscosity loss given in the dashed curve in figure 3 (left) of [53]. Inserting $v_{eff}$ from equation (3.8) and figure 2 (right) into equation (3.11), assuming an accretion eigenvalue $\lambda$=0.05 and applying the above Hubble viscosity correction, I can finally calculate the normalized accretion rate as a function of redshift and PBH mass. The results are shown in figure 3 (left). For PBH masses smaller than $\sim$1000 M⊙ the normalized accretion rate is roughly constant in the redshift range 20<z<200 due to the fact that the density and velocity dependence on redshift in equation (3.9) roughly cancel out (see also the lower panel of figure 4 in [38]). At z<20 $\dot{m}$ drops dramatically because of the effective velocity increase. PBH masses larger than $\sim 10^{4}$ M⊙ reach accretion rates close to the Eddingon limit at z$\gtrsim$100, but are significantly affected by the Hubble viscosity at z$\gtrsim$20\. For all PBH masses the accretion rate is small enough, that the growth of the PBH population can be neglected over cosmic time (PBH with masses in the range 105-7 M⊙ accrete about 0.5–2% of their mass until z>20). Figure 3: Left: Normalized accretion rate onto PBH with masses in the range 0.1–107 M⊙ as a function of redshift. Right: Radiative efficiencies derived from the accretion rates, assuming the hot accretion flow model of [56] with a viscous heating parameter $\delta$=0.5. Figure 4: Spectra of the hot disk accretion flow (ADAF) from [55] with a viscous heating parameter $\delta$=0.5, divided by the normalized accretion rate. Left: accretion onto a 10 M⊙ black hole for different accretion rates, as indicated. Right: same for an accretion rate of $log(\dot{m})$=-1.6 but different black hole masses (as indicated) . ## 4 Accretion spectrum and radiative efficiency For the accretion flow and the electromagnetic emission mechanism I follow [37, 8] and assume the formation of an accretion disk. Accretion rates close to the Eddington limit will lead to the formation of a standard Shakura- Sunyaev disk [66], which has a canonical radiative efficiency $\eta\approx 0.1$. For much lower accretion rates $\dot{m}$$\ll$1 an advection-dominated hot accretion flow [55] is expected, with a significantly lower radiation efficiency [56], roughly scaling according to $\eta\propto\dot{m}$. Figure 4 shows hot accretion flow spectra from [55] with a viscous heating parameter $\delta$=0.5 for black holes, normalized by Eddington luminosity and mass accretion rate. The left graph shows radiation from a 10 M⊙ black hole at different mass accretion rates. The right graph shows the spectrum from black holes with different masses in the range 10-109 M⊙ and a mass accretion rate $log(\dot{m})$=–1.6. It is important to understand, that for advection dominated accretion flows not all the matter entering the Bondi radius will actually reach the black hole. This is due to feed-back mechanisms acting on the accreted gas, e.g. producing outflows or jets. The advection dominated flow models of [56, 55] therefore find a radial dependence of mass accretion rate $\dot{M}\propto R^{\alpha}$, typically with $\alpha\sim 0.4$. Within a radius of about 10 RS, where $R_{S}=2GM/c^{2}$ is the Schwarzschild radius, the accretion flow more closely follows the standard Shakura-Sunyaev description of a constant accretion rate with radius down to the last stable orbit ($\sim 3R_{S}$). In terms of the classical Bondi description of quasi-spherical capture, the loss of accreted matter can be associated with the accretion eigenvalue: $\lambda\approx\left(\frac{10R_{S}}{R_{D}}\right)^{\alpha},$ (4.1) where $R_{D}$ is the outer radius of the accretion disk formed. For $\alpha$=0.4, the value of $\lambda$=0.05 chosen for the analysis in this paper therefore corresponds to an outer disk radius of $R_{D}\sim$2$\times$104 $R_{S}$, about 8 orders of magnitude smaller than the Bondi radius. In this picture the accretion flow on large scales follows the Bondi quasi-spherical flow for most of the radial distance, until the advection-dominated accretion disk is formed. The radiative efficiency for the ADAF spectra in figure 4 is the integral over these curves and has been calculated through numerical simulations by [56]. For this work I use a digitized version of the highest efficiency curve in their figure 1, with a viscous heating parameter $\delta$=0.5111Please take note that the definition of $\dot{m}$ between these authors and the analysis presented here differs by a factor of 10.. A maximum radiative efficiency of $\eta$$\sim$0.08 is achieved for log($\dot{m}$)>–1.6. We can now calculate the radiative efficiency for every mass and redshift bin from the normalized accretion rate in figure 3 (left). The result is shown in figure 3 (right). It turns out that above redshifts z$\gtrsim$20 and black hole masses MPBH>100 M⊙, which dominate the contribution to the extragalactic background light, the radiative efficiencies are relatively large (typically >3%). Figure 5: Density-weighted bolometric luminosity of single PBH as a function of mass for different redshifts indicated (left), and as a function of redshift for different mass bins indicated (right). Figure 6: Density-weighted bolometric flux of single PBH as a function of mass for different redshifts indicated (left), and as a function of redshift for different mass bins indicated (right). We now have the ingredients to calculate the bolometric luminosity and flux expected from the baryon accretion onto the assumed PBH mass spectrum over cosmic time. For every black hole of mass MPBH I calculate the expected bolometric luminosity L${}_{bol}=\dot{m}~{}\eta~{}L_{Edd}$, where LEdd=1.26$\times$10${}^{38}~{}M_{PBH}/M_{\odot}$ erg/s is the Eddington luminosity, and the normalized mass accretion rate $\dot{m}$ as well as the radiation efficiency $\eta$ are taken from the data in figure 3. In every mass bin, the relative number density of PBH compared to those of 1 M⊙ is nPBH=fPBH/MPBH, where fPBH is the PBH mass function from figure 1. For every mass and redshift bin I thus multiply the bolometric luminosity with this relative number density in order to obtain the density-weighted luminosity $\langle L_{bol}\rangle^{*}$ for an equivalent PBH of 1 M⊙. This quantity is shown in figure 5 as a function of PBH mass (left) and redshift (right). It shows that the largest contribution to the PBH luminosity over cosmic time comes from PBH in the mass range MPBH=103-7 at redshifts z>100\. The Chandrasekhar PBH mass peak is subdominant in this representation. The total PBH luminosity deposited in the baryonic gas at high redshifts is important for the pre-ionization and X–ray heating of the intergalactic medium discussed in section 6. To calculate the contribution of PBH accretion to the extragalactic background light we need to convert the density-weighted luminosities in Figure 5 to bolometric fluxes using the luminosity distance $D_{L}$ at the corresponding redshift: $\langle F_{bol}\rangle^{*}$=$\langle L_{bol}\rangle^{*}$/(4$\pi~{}D_{L}^{2}$). This quantity is shown in figure 6 as a function of PBH mass (left) and redshift (right). It shows that the largest contribution to the extragalactic surface brightness is produced at a redshift z$\approx$20 from PBH in the mass range MPBH=102-5, and a similar contribution from the Chandrasekhar mass peak. SMBH at M${}_{PBH}\sim 10^{6.5}$ have a notable contribution around z$\sim$10. ## 5 The contribution of PBH to the extragalactic background light To calculate the surface brightness per redshift shell in a particular observed frequency band [$\nu_{1}$;$\nu_{2}$] of the electromagnetic spectrum, I take into account the spectral shape and the fraction of the radiation falling into the rest frame frequency band [$\nu_{1}$/(1+z);$\nu_{2}$/(1+z)]. The exact spectral shape is not so important for this derivation, it is mainly used to calculate the bolometric corrections, i.e. the fraction of the total luminosity falling into the various frequency bands as a function of redshift. The ADAF spectra in figure 4, in particular those at high $\dot{m}$ values, can be approximated by power laws with an exponential cutoff at $\sim$200 keV. Following [37] and [8], I assume a power law slope of –1 (corresponding to a flat line in figure 4). Below a critical frequency $\nu_{c}$ the power law spectrum is cut off by synchrotron self-absorption into a power law with a steep slope of approximately +1.86. As can be seen in figure 4 (right), $\nu_{c}$ depends on MPBH and can be approximated by log($\nu_{c}$)$\approx$14.82–0.4log(MPBH/M⊙). The bolometric corrections are then obtained by integrating the analytic normalized spectra over the observed frequency bands. For the 2–5$\mu$m band we have to consider in addition the Lyman-$\alpha$ break, which produces a sharp cutoff at z$\gtrsim$30 (see e.g. [28, 67]). These bolometric corrections are shown in figure 7 (left) for the 2–5$\mu$m NIR band, the 0.5–2 keV and the 2–10 keV X–ray bands, respectively. To predict the surface brightness of all PBH across cosmic time in these observed frequency bands, the total flux per PBH in figure 6 (right) has to be multiplied with the bolometric correction and the PBH surface density in a particular redshift shell. Using the critical mass density of the universe $\rho_{crit}$=1.26$\times 10^{20}M_{\odot}~{}{\rm Gpc}^{-3}$ and the Dark Matter density $\Omega_{DM}$=0.264, as well as the reference mass 1M⊙, a comoving PBH space density of $n_{PBH}=3.32\times 10^{19}(1+z)^{3}~{}{\rm Gpc}^{-3}$ is obtained. For every redshift shell [z+$\Delta$z] the PBH space density is multiplied with the volume of the shell [V(z+$\Delta$z)–V(z)] and divided by 4$\pi$ deg2 to obtain the number of PBH per deg2. Figure 7 (right) shows the derived surface brightness as a function of redshift (per $\Delta$z=1 interval) for the three spectral bands considered here. The emission in all three bands peaks around z$\approx$20 with a FWHM of $\Delta$z$\approx$[-3;+6]. Figure 7: Left: The bolometric correction, i.e. the fraction of the total luminosity falling into the respective observed frequency band as a function of redshift, for the 2–5$\mu$m NIR band, as well as the 0.5–2 and 2–10 keV X–ray bands. Right: Predicted surface brightness of the PBH in the same observed bands as a function of redshift (per $\Delta$z=1). The curves in figure 7 (right) can now be integrated to predict the total PBH contribution to the extragalactic background light as SB2-5μ$\approx$10-13, SB0.5-2keV$\approx$1.9$\times$10-13, and SB2-10keV$\approx$1.3$\times$10-13 erg cm-2 s-1 deg-2, respectively. The minimum amount of X–ray surface brightness necessary to explain the CXB/CIB cross-correlation signal observed by [23] in the 0.5–2 keV band has been discussed by [29]. This is $9\times 10^{-14}$ erg cm-2 s-1 deg-2, corresponding to roughly 1% of the total CXB signal in this band. The 0.5–2 keV PBH contribution predicted for an accretion eigenvalue of $\lambda$=0.05 in equation (3.11) is thus about a factor of 2 larger than the observed CXB fluctuation signal, which could well be consistent, given the coherence between the CXB and CIB signals. As discussed above, there is a marginally significant diffuse CXB remaining after accounting for all discrete source contributions [31, 34]. Extrapolating into the X–ray bands considered here, this residual flux corresponds to $\approx$(7$\pm$3) and $\approx$(9$\pm$20)$\times 10^{-13}$ erg cm-2 s-1 deg-2 in the 0.5–2 keV and 2–10 keV band, respectively. Assuming the $\lambda$=0.05 value, the predicted PBH contribution is therefore well below the upper limit (15–25%) of any unresolved component left in the CXB. The main result of this paper is therefore, that the assumed PBH population for the dark matter can indeed explain the X–ray fluctuation signal, with a Bondi accretion eigenvalue of $\lambda$=0.05. The flux measured in the 2–5$\mu$m CIB fluctuations at angular scales >100" is about 1 nW m-2 sr-1 [68], or 3$\times 10^{-10}$ erg cm-2 s-1. The cross- correlated CIB/CXB fluctuations contribute about 10% to the total CIB fluctuations [23], i.e. 3$\times 10^{-11}$ erg cm-2 s-1. Therefore the predicted PBH contribution to these CIB fluctuations is only about 0.5% for $\lambda$=0.05. It is argued in [6] that PBH in the early universe could amplify the cosmic power spectrum at small spatial scales (see below). Together with the pre-ionization of the intergalactic medium discussed below, the PBH can therefore significantly increase the associated star formation. The NIR emission in this picture would then be dominated by early star formation associated with PBH instead of direct PBH emission. ## 6 Discussion ### 6.1 Linear versus post-linear growth In this simplified treatment I only consider the linear evolution of the power spectrum above the virialization redshift around z$\approx$20 (see figure 2 right). On sufficiently large scales the initial power spectrum has been very precisely determined as nearly scale invariant with overdensities of 10-4 [35], and the PBH density field is expected to follow the standard adiabatic perturbations. On small scales the power spectrum is only poorly constrained and could be significantly amplified by the discrete nature of the PBH population itself [6, 69, 70]. Poisson variations in the density of PBH will introduce non-linear growth of density fluctuations and the corresponding velocity dispersion already well before the virialization redshift z$\sim$20 discussed above. However, from numerical simulations [70] conclude that the nonlinear velocity perturbations introduced by >20 M⊙ PBH are too small to dominate the relative velocities between baryons and PBH at z$\gtrsim$100 [see also 71]. However, non-linear effects definitely become more important at lower redshifts (see above) and could effectively reduce the Bondi capture rate. ### 6.2 Magnetic fields in the early universe The accretion mechanism assumed in the Bondi capture model only works, if there is a rough equipartition between the kinetic energy and magnetic fields in the accreted gas, as it is the case in the turbulent interstellar medium of our Galaxy. It is therefore justified to ask, whether this mechanism can also work at high redshifts, where the existence and magnitude of magnetic fields is still unclear. Magnetic fields are present at almost every scale of the low redshift universe, from stars and planets to galaxies and clusters of galaxies and possibly even in the intergalactic medium in voids of the cosmic web, as well as in high-redshift galaxies. [72] and [73] review the observations and possible origin of magnetic fields. There is a surprising similarity between the relatively strong magnetic fields measured in our own Galaxy (0.3–0.4 nT) and other nearby galaxies ($\sim$1 nT) with magnetic fields measured in clusters of galaxies (0.1–1 nT), as well as in high redshift galaxies ($\sim$1 nT), when the universe was only about 1/3 of its current age. There are even indications of magnetic fields of order $\gtrsim$10-20 T in cosmic voids derived from the gamma ray emission of blazars [74]. One can come to the conclusion that the origin of cosmic magnetism on the largest scales of galaxies, galaxy clusters and the general intergalactic medium is still an open problem [75]. It is usually assumed that primordial or cosmic seed fields are amplified over time through the galactic dynamo effect to produce the rather strong fields observed in local galaxies. In this picture it is, however, unclear how similar fields can be created in such different settings (e.g. clusters) and different cosmic times (high-redshift galaxies). An interesting possibility is therefore that cosmic magnetic fields could be remnants from the early universe, or created in a process without galactic dynamos. Assuming equipartition, the energy density in the CMB photons would correspond to a magnetic field of about 0.3 nT. Magnetic fields of $10^{-20}$ T, as observed in cosmic voids today, would only require a minute fraction of $10^{-10}$ of this energy density in the early universe to be channeled into magnetic fields. Figure 8: Left: The Bondi radius for a 104 M⊙ (thin blue) and 1 M⊙ (thick blue) PBH compared to the proton (red) and electron (green) Larmor radius, assuming a magnetic field of B=$10^{-20}$ T, as observed in local galaxy voids. Right: Baryon ionization/heating fraction $\chi_{e}$ as a function of redshift. The thin dash-dotted line shows the residual ionization left over from the radiation dominated era [76]. The red curve shows the ionization fraction from UV photons produced by accreting PBH. The blue curve shows the corresponding heating fraction by >1 keV X–ray photons. The thick dashed black line shows one of the models consistent with the Planck satellite data [35] (see text). The green hatched areas shows the range of high–redshift ionization fractions considered in [16]. Here I argue, that PBH could play a role in amplifying early magnetic seed fields and sustaining them until the epoch of galaxy formation. I compare the Bondi radius in eq. (3.12) and figure 8 (left) with the Larmor radius $r_{L}={\frac{m~{}v_{\bot}}{|q|~{}B}},$ (6.1) which determines the gyro motion of particles moving through a magnetic field. Here $m$ is the mass of the particle (either proton or electron), $v_{\bot}$ is the velocity component of the particle perpendicular to the magnetic field, $|q|$ is the absolute electric charge of the particle, and $B$ is the magnetic field. Assuming a seed field of $B$=10-20 T and approximating the velocity with the sound speed $v_{\bot}$$\approx$$c_{s}$ yields the gyro radius for both protons and electrons. The proton gyro radius is about a factor of 2000 larger than the electron gyro radius. Figure 8 (left) shows the Bondi radius as well as the proton and electron Larmor radii as a function of redshift. If the gyro radius is smaller than the Bondi radius, the respective particle is easily accreted by the PBH. If, however, the gyro radius is larger than the Bondi radius, the particle will first not be easily accreted, but rather spiral around the PBH. From 8 we see, that at redshifts z$\gtrsim 70$ and PBH masses in the range MPBH$\approx$0.3–500 for our assumed magnetic field strength the proton Larmor radius is larger than the Bondi radius, while the electron Larmor radius is smaller than the Bondi radius. There is still a substantial fraction of residual electrons and protons/helium ions from the era before recombination (see the dash-dotted curve in 8 right from [76]). These electrons have therefore no problem being accreted, while for certain PBH masses protons resist the accretion. This will create a net electric current, which in turn will increase the average magnetic field strength until the proton gyro radius becomes smaller than the Bondi radius. This way the PBH can amplify the average magnetic field. The supersonic motion between baryon gas and PBH discussed above is expected to be coherent over large scales (of the order of Mpc) and can therefore induce large-scale ordered magnetic fields. A further magnetic field amplification occurs, as discussed above, in the accretion funnel, when magnetic fields are dissipated through reconnection and ejected with the plasma. In a sense, the ubiquitous PBH can possibly create their own magnetic field and distribute it throughout the universe. It is, however, plausible to assume, that magnetic fields in the early universe should be smaller than today, and the fractions of ionized baryons is less. This could also explain a rather small Bondi accretion eigenvalue required to match the observations. ### 6.3 Re-Ionization Next I turn to the contribution of PBH accretion to the re-ionization and re- heating history of the universe. At z$\approx$1089, when the photons decoupled from the baryons and formed the CMB radiation, the universe became predominantly neutral. Afterwards the universe entered a long period of “darkness”, in which the residual ionization left over from the primordial photon-baryon fluid diminished (see figure 8 right), the background photons cooled down, and any higher-frequency emission was quickly absorbed in the atomic gas. In the model described here the first sources to illuminate the “dark ages” would be the PBH accreting from the surrounding gas. Their ultraviolet emission, above the Hydrogen ionization energy of 13.6 eV, would start to re-ionize small regions around each PBH. However, in the beginning the density of the surrounding gas is still so high that the ionized matter quickly recombines. As long as the re-combination time is much shorter than the Hubble time at the corresponding epoch, UV photons from the PBH cannot penetrate the surrounding medium, but instead produce an ionized bubble growing with time. In this type of ionization equilibrium the number of ionizing photons $N_{ion}$ required to overcome recombination is given as the ratio between the Hubble time $t_{H}(z)$ and the recombination time $t_{rec}(z)$ at the particular epoch, and can be derived from equations (2) and (3) in [77] as $N_{ion}=t_{H}/t_{rec}=max[1,0.59~{}\left(\frac{1+z}{7}\right)^{1.5}].$ (6.2) At a redshift z=1000, $N_{ion}$ is about 1000, and reaches a level of unity at z$\lesssim$10 for the assumed set of parameters. For this calculation I ignore clumping of the ionized gas. In reality the effective clumping factor is relatively large for reionization at high redshift because the ionizing sources are more numerous in the filaments of the cosmic web, but must also ionize a correspondingly larger fraction of dense gas in the filaments, and thus ionization is slowed down. At lower redshift, when molecular gas and stars have already formed, not all UV photons will escape the dense regions. The effective escape fraction is one of the largest uncertainties in our current understanding of re-ionization. For simplicity, I assume an escape fraction $f_{esc}$=0.1 for UV photons, and $f_{esc}$=1 for X–ray photons, independent of redshift. To calculate the history of pre-ionization by PBH I integrate the above normalized ADAF model for frequencies log($\nu$)>15.52 Hz, corresponding to the hydrogen ionization energy of 13.6 eV. To calculate the number of ionizing photons per PBH of reference mass 1 M⊙ I take the density-weighted luminosity $\langle L_{bol}\rangle^{*}$ from figure 5 (right). To determine the average space density of ionizing photons I multiply with the average comoving space density of PBH (assuming the reference mass 1 M⊙): $n_{PBH}=1.06\times 10^{-54}~{}\left(\frac{1+z}{1000}\right)^{3}~{}{\rm cm}^{-3},$ (6.3) and with the escape fraction $f_{esc}$ and finally divide by $N_{ion}$ from eq. (6.3) and the average density of baryons given in equation (2.10) to determine the ionization rate of baryons over cosmic time. The red curve in figure 8 (right) shows the cumulative ionization fraction $\chi_{e}$ as a function of redshift for the accretion eigenvalue $\lambda$=0.05. A maximum cumulative ionization fraction of $\sim$2.8%, is reached at a redshift z$\approx$10\. This can be compared to one of the recent models determined from the Planck satellite data [35]. Here the 1$\sigma$ upper percentile of the FlexKnot model in their figure 45, which is consistent with the ionization optical depth determined from the most recent Planck data, is shown as dashed curve. A high-redshift contribution to the ionization history of the universe has also been discussed by [78] and [16]. The range of $\chi_{e}$ values assumed in the latter work is shown as green hatched region in figure 8 (right). For the choice of $\lambda$=0.05, the UV emission from the PBH population assumed in the toy model is therefore fully consistent with the observational constraints from Planck. ### 6.4 X–ray heating The role of X–ray heating in shaping the early universe has been discussed by [79]. Compared to UV photons, X–ray photons have a much larger mean free path and can therefore ionize regions far away from the source. In addition, most of the X–ray energy gets deposited into heating up the gas. In order to estimate the amount of X–ray heating of the gas I applied the same mechanism as for the UV photons above, but integrating the above ADAF model for frequencies log($\nu$)>17.68 Hz, corresponding to 2 keV. I assume an escape fraction of $f_{esc}$=1 and $N_{ion}$=1. The blue curve in figure 8 (right) shows the cumulative 2 keV heating fraction per baryon as a function of redshift for the assumed accretion eigenvalue of $\lambda$=0.05. The maximum cumulative heating fraction is $\sim$1.6%. X–rays from PBH therefore have only a small contribution to the pre-ionization of the universe as a whole, but can be responsible for a part of the pre-heating of gas observed in the "entropy floor" of groups of galaxies. [80] reviewed the energetics of groups and clusters of galaxies, which cannot be reproduced by simple models, where the gas density is proportional to dark matter density. [81] and [82] argued that the gas must have been pre-heated before falling into the cluster potential. X–ray observations of groups of galaxies with ROSAT by [83] confirmed the need for a non-gravitational entropy injection in the group gas. These authors coined the term "entropy floor", which amounts to an energy of about 2 keV per baryon injected into the group gas. The pre-heating of the gas by PBH, albeit only contributing to a small fraction of the total baryon content of the universe, could have played an important role in heating the high-density regions, which first formed groups and clusters. ### 6.5 Cosmological 21-cm signal Figure 9: Density-weighted 1.4 GHz (observed) luminosity of a single PBH as a function of mass for different redshifts indicated. The red-shifted 21-cm line can provide important new constraints on the physical processes in the early universe [see e.g. 84, 8]. The Experiment to Detect the Global EoR Signature (EDGES) has measured a strong, sky-averaged 21-cm absorption line profile after subtracting the Galactic synchrotron emission [85]. The signal is centered at a frequency around 78 MHz and covers a broad range in redshift z=15–20. The signal may be due to ultraviolet light from first objects in the universe altering the emission of the 21-cm line by lowering the spin temperature of neutral hydrogen relative to the CMB. However, the signal is about three times larger than that expected from the standard $\Lambda$CDM cosmology, which led some authors to suggest new dark matter physics [e.g. 86]. Instead of new physics, an increased 21-cm radio background contribution above the CMB at the epoch around z=15–20 could also explain the EDGES signal. Indeed, [87] estimate the additional 21-cm radio background from accretion onto putative radio-loud intermediate-mass black holes (IMBH) forming in first molecular cooling halos at redshifts z=16–30. This could be sufficient to explain the EDGES feature, however, it requires extreme assumptions about the radio loudness of the IMBH population. Instead of assuming an interpretation in terms of mini-QSOs from IMBH grown through accretion, I estimate here, whether PBH accretion could have a significant contribution to the EDGES signal. A full treatment of this effect for the PBH toy model is beyond the scope of this paper, but similar to the treatment of the PBH contribution to the CXB and CIB derived in section 5, I can estimate the PBH contribution to the observed low-frequency cosmic radio background, and thus to the EDGES signal. The balloon-borne double-nulled Absolute Radiometer for Cosmology, Astrophysics and Diffuse Emission (ARCADE2) instrument has measured the absolute temperature of the sky at frequencies 3, 8, 10, 30, and 90 GHz, and detected a significant excess over the CMB blackbody spectrum at a temperature of 2.731K [88]. Combining the ARCADE2 measurements with lower frequency data from the literature, the excess brightness temperature can be characterized as a power law TR=1.19 ($\nu$/1 GHz)-2.62 K, which translates into a radio spectrum with a slope of -0.62 and a normalization of 3$\times$10-22 W m-2 sr-1 at 1.4 GHz. This cosmic radio synchrotron background is substantially larger than that expected from an extrapolation of the observed radio counts [89], and thus presents a major new challenge in astrophysics. [90] found that the global 21cm signal can be significantly amplified by an excess background radiation compared to the standard $\Lambda$CDM models, especially in absorption. Assuming that only 10% of the radio synchrotron background originates at large redshifts, they predict a 21cm feature almost an order of magnitude stronger than that expected purely from the CMB. Interpolating between their calculations for 0.1% and 10% excess background I find, that an excess high-redshift radiation field of about 5% of the observed radio synchrotron background is sufficient to explain the EDGES findings. In order to calculate the expected PBH contribution to the radio background I assume that each black hole has a radio emission following the fundamental plane relation between X-ray luminosity, radio luminosity and black hole mass found by [91]. I use the parameterization for radio-quiet AGN from [92]: log(LR)=0.85 log(LX)+0.12 log(MPBH), where LR is the 1.4 GHz radio luminosity in units of 1040 erg/s, LX is the 0.1–2.4 keV X–ray luminosity in units of 1044 erg/s, and MPBH is the PBH mass in units of 108 M⊙. The X–ray luminosity is calculated from the bolometric luminosity shown in figure 5 (right). Assuming the ADAF radiation spectrum above, the fraction of the bolometric luminosity radiated in the 0.1-2.4 keV band is 0.23. For the radio spectrum I assume a power law with spectral index -0.62. This means that the bolometric correction is $\propto$(1+z)0.38. The radio luminosities derived this way as a function of PBH mass and redshift are shown in figure 9. Multiplying these luminosities with the PBH density over cosmic time, converting into observed fluxes and integrating over mass and redshift I obtain a contribution of radio-quiet PBH to the observed radio background of $\sim$3$\times$10-25 W m-2 sr-1 at 1.4 GHz, i.e. a fraction of 0.1% of the observed synchrotron radio background. Most of this additional radiation field is accumulated at redshifts z$\gtrsim$20\. Following [90], this excess radio flux would increase the depth of the 21cm absorption line only by about 30%. If, however, some fraction of the PBH would be radio-loud (e.g. 5% with 1000 times higher fluxes), like observed in the AGN population, the 5% excess high-redshift radio background flux necessary to explain the EDGES feature could be easily achieved by PBH. ### 6.6 Primordial Black Holes in the Galactic Center Next I discuss some observational effects of the putative PBH population residing in the Galactic Center region. First, assuming a Milky Way dark matter halo of $\sim$10${}^{12}M_{\odot}$, the PBH mass spectrum from section 2 (figure 1) indeed predicts about one supermassive PBH with a mass $\gtrsim 10^{6.5}M_{\odot}$, consistent with the Sgr A∗ SMBH in the center of our Galaxy [93]. To estimate the density of dark matter and baryons in the Galactic bulge region itself, I refer to dynamical models of the Milky Way’s center, using the density of red clump giant stars stars measured in infrared photometric surveys, as well kinematic radial velocity measurements of M-giant stars in the Galactic bulge/bar constructed in [94]. From N–body simulations of stellar populations for barred spiral discs in different dark matter halos these authors were able to determine with high precision the mass in a volume of ($\pm 2.2\times\pm 1.4\times\pm 1.2$ kpc3) centered on the Galactic Bulge/Bar. The total mass is (1.84$\pm$0.07)$\times$1010 M⊙. Depending on the assumed model, about 9–30% consists of dark matter, i.e. 1.7–5.5$\times$109 M⊙. Applying the above PBH mass spectrum, we thus expect 5–10 intermediate- mass PBH with MPBH>104 M⊙ in the Galactic bulge region, but zero with MPBH>105 M⊙. Recent high-resolution observations of high-velocity compact clouds (HVCC) in the central molecular zone of our Milky Way with the Atacama Large Millimeter/submillimeter Array (ALMA) have indeed identified five promising IMBH candidates, wandering through the patchy ISM in the Galactic Center [see 95]. The most compelling case is HCN–0.044–0.009, which shows two dense molecular gas streams in Keplerian orbits around a dark object with a mass MIMBH=(3.2$\pm$0.6)$\times$104 M⊙ [96]. The semimajor axis of these Keplerian streams are around 2 and 5$\times$1017 cm. Another interesting case is the infrared and radio object IRS13E, a star cluster close to the Galactic Center potentially hosting an IMBH [97]. ALMA observations identified a rotating, ionized gas ring around IRS13E [98], with an orbit radius of 6$\times$1015 cm and a rotation velocity of $\sim$230 km/s. This is thus another promising IMBH candidate with a mass of MIMBH=2.4$\times$104 M⊙. Two of the five IMBH candidate sources in [95] are possibly associated with X–ray sources detected in the deep Chandra images of the Galactic Center [99]. IRS13E has the X–ray counterpart CXOGC 174539.7-290029 with an X–ray luminosity L2-10keV$\approx$3$\times$1030 erg/s, and CO–0.31+0.11 has the possible X–ray counterpart CXOGC 174426.3-290816 with an X–ray luminosity L2-10keV$\approx$4$\times$1029 erg/s. The other three sources have X–ray upper limits in the range of several 1030 erg/s. Assuming a bolometric correction factor of 1/30 for the 2–10 keV range, the combination of the mass accretion eigenvalue $\lambda$ and the radiative efficiency $\eta$ therefore has to be extremely small, on the order of 3$\times$10-11. This is about a factor of 100 lower than the 2$\times$10-9 LEdd derived for the Galactic Center black hole Sgr A∗ [55]. Even assuming a very low efficiency ADAF model, a steady-state accretion solution is unlikely for these objects. The solution of this puzzle may come from the fact, that the velocity and density gradients of the gas in the Galactic Center region are so large, that the angular momentum forces any accreted matter into Keplerian motion well outside the classical Bondi radius [see 37]. Indeed, the orbital periods and lifetimes of the Keplerian streams around HVCCs are in the range 104-5 years, and thus accretion is expected to be highly variable on very long time scales. Another possibility to halt accretion for a considerable time is the feedback created by outflows during efficient accretion events. Numerical simulations of the gas dynamics in the center of the Galaxy [100] show that the outflows significantly perturb the gas dynamics near the Bondi radius and thus substantially reduce the capture rate. The net result of both these effects would be a highly variable, low duty cycle bursty accretion onto the IMBH and SMBH in the Galactic Center, consistent with the extremely low accretion efficiencies observed. The accretion limits for black holes in the mass range MPBH=20–100 M⊙ derived from deep Chandra and radio observations of the Galactic Center [45], are already shown in figure 1 to be consistent with the assumed PBH mass spectrum. Recent NuSTAR observations of the Galactic Center, including the effects of gas turbulence and the uncertainties related to the dark matter density profile even further weaken these constraints [101]. At any rate, the assumed PBH mass distribution of section 2 is fully consistent with the observational constraints for all PBH masses >20 M⊙ in the Galactic Center. Finally, I check the PBH predictions for lower masses against the Galactic ridge X–ray emission (GRXE), an unresolved X–ray glow at energies above a few keV discovered almost 40 years ago and found to be coincident with the Galactic disk. The GRXE in the 2–10 keV band has a background-corrected surface brightness of (7.1$\pm$0.5)$\times 10^{-11}$ erg cm-2 s-1 deg-2 which was largely resolved into discrete sources [102], with the brightest source having an X–ray luminosity of about 1032 erg s-1, and the minimum detectable luminosity around 1030 erg s-1. The integrated emission has a strong iron line from hot plasma at 6.7 keV, and the authors interpret the X–ray emission as coming from a large population of cataclysmic variables and coronally active stars. Using the mass determination in the Galactic bulge/bar above I find that the average baryon density in this region is in the range 17–22 cm-3. However, most of these baryons are locked up in stars. In order to estimate the physical conditions of the gas in the Galactic Bulge/Bar region I follow [103]. According to these authors, there are four phases of the interstellar medium in the Galactic center region: (1) a cold molecular phase in Giant Molecular Clouds with temperatures around 50 K and gas densities n=103.5-4 cm-3 covering a volume fraction around 1%; (2) a warm molecular phase with temperatures around 150 K and gas density n=102.5 cm-3, covering a volume fraction of $\sim$10%; (3) an atomic phase with temperatures around 500-1000 K and density $\sim$10 cm-3, covering a volume fraction around 70%, and (4) ionized gas with temperatures 104-8 K and an unknown density. Depending on the temperature of the interstellar medium, the sound speeds are in the range $c_{s}$=1–100 km/s. The stellar velocity dispersion in the central region of our Galaxy is in the range 100–140 km/s [104], while the dark matter velocity dispersion is about 110 km/s [105]. In the spirit of the discussion leading up to equation 3.9 and figure 2 (right) above, I assume an effective velocity for Bondi accretion $v_{eff}\approx$50 km/s and $\lambda$=0.1. As shown in figures 5 and 6, the PBH emissivity for the assumed mass spectrum is typically dominated by objects with MPBH>100, which already are discussed above. Indeed, calculating the Bondi accretion rates and radiative efficiencies for objects with MPBH<100 for the four ISM phases in the Galactic Center I obtain negligible PBH contributions to the total GRXE brightness. Some individual MPBH$\sim$100 M⊙ objects in high density regions could in principle have X–ray luminosities up to $L_{2-10keV}$=1033 erg/s, more luminous than the brightest X–ray source detected in the Galactic Ridge survey [102], but taking into account the strong variability and small duty cycle expected for this class of objects, their absence in the surveys is understandable. Some of the fainter unidentified sources in the current deep X–ray surveys could indeed be accreting PBH and future large X–ray observatories like ATHENA [106] or LYNX [107] should be able to identify more. See also [108] for future searches in the IR and sub-mm region. ## 7 Conclusions and Outlook The interpretation of cold dark matter as the sum of contributions of different mass PBH families [1] could explain a number of so far unsolved mysteries, like e.g. the massive seed black holes required to create the supermassive black holes in the earliest QSOs [13], the ubiquitous massive LIGO/VIRGO massive binary black holes [e.g. 6], or even the putative "Planet X" PBH in our own Solar System [14]. The most abundant family of PBH should be around the Chandrasekhar mass (1.4 M⊙). This prediction may already have been vindicated by the recent OGLE/GAIA discovery of a sizeable population of putative black holes in the mass range 1-10 M⊙ [15]. Here I estimate the contribution of baryon accretion onto the overall PBH population to various cosmic background radiations, concentrating first on the crosscorrelation signal between the CXB and the CIB fluctuations discovered in deep Chandra and Spitzer surveys [23]. Assuming Bondi capture and advection dominsted disk accretion with reasonable parameters like baryon density and the effective relative velocity between baryons and PBH over cosmic time, as well as appropriate accretion and radiation efficiencies, I indeed predict a contribution of PBH consistent with the residual X–ray fluctuation signal. This signal peaks at redshifts z$\approx$17–30. The PBH contribution to the 2–5 $\mu$m CIB fluctuations, however, is only about 1%, so that these would have to come from star formation processes associated with the PBH. I discuss a number of other phenomena, which could be significantly affected by the PBH accretion. Magnetic fields are an essential ingredient in the Bondi accretion process, and I argue that the PBH can play an important role in amplifying magnetic seed fields in the early universe and maintaining them until the galactic dynamo processes set in. Next I study the contribution of the assumed PBH population to the re-ionization history of the universe and find that they do not conflict with the stringent ionization limits set by the most recent Planck measurements [35]. X–ray heating from the PBH population can provide a contribution to the entropy floor observed in groups of galaxies [83]. The tantalizing redshifted 21-cm absorption line feature observed by EDGES [85] could well be connected to the radio emission contributed by PBH to the cosmic background radiation. Finally, the number of IMBH and the diffuse X–ray emission in the Galactic Center region are not violated by the PBH dark matter, on the contrary, some of the discrete sources in the resolved GRXE could be accreting PBH. It is obvious, that our simple PBH toy model for the dark matter requires significantly more work to turn it into quantitative predictions. Real magnetohydrodynamic simulations of the whole PBH mass spectrum including their own hierarchical clustering would be required to obtain the full history of their contribution to the cosmic backgrounds. The exciting EDGES discovery definitely requires a full-blown analysis of the radio contribution of PBH to the cosmic background. Future X–ray observations with eROSITA and ATHENA, infrared wide field surveys with Euclid and WFIRST, and microlensing observations with WFIRST will provide important additional diagnostics in this exciting and dramatically developing PBH field (see [109, 110]). ## Acknowledgments I am thankful to Juan García-Bellido for sharing a digital copy of the new running spectral index PBH mass distribution model in figure 1 in advance of publication, as well as many very useful discussions about PBH. I am indebted to Matthias Bartelmann for computing the small-scale non-linear relative velocity dispersion (figure 2 right) and providing very valuable comments and corrections to the manuscript. I would like to thank Sergey Karpov for very helpful discussions and inputs about their spherical accretion model. I would also like to thank my colleagues Nico Cappelluti, Sasha Kashlinsky and Alexander Knebe for very helpful discussions and contributions. 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2024-09-04T02:54:59.069368

2003.05151

arxiv-papers

11institutetext: Department of Future Technologies, University of Turku, Turku, Finland # Predicting the Amount of GDPR Fines Jukka Ruohonen Kalle Hjerppe {juanruo<EMAIL_ADDRESS> ###### Abstract The General Data Protection Regulation (GDPR) was enforced in 2018. After this enforcement, many fines have already been imposed by national data protection authorities in the European Union (EU). This paper examines the individual GDPR articles referenced in the enforcement decisions, as well as predicts the amount of enforcement fines with available meta-data and text mining features extracted from the enforcement decision documents. According to the results, articles related to the general principles, lawfulness, and information security have been the most frequently referenced ones. Although the amount of fines imposed vary across the articles referenced, these three particular articles do not stand out. Furthermore, good predictions are attainable even with simple machine learning techniques for regression analysis. Basic meta- data (such as the articles referenced and the country of origin) yields slightly better performance compared to the text mining features. ###### Keywords: T ext mining Legal mining Data protection Law enforcement ## 1 Introduction Data protection has a long history in the EU. In particular, the GDPR repealed the earlier Directive 95/46/EC. Although this directive laid down much of the legal groundwork for EU-wide data protection and privacy, its national adaptations, legal interpretations, and enforcement varied both across the member states and different EU institutions [10]. In short: it was a paper tiger. In contrast, Regulation (EU) 2016/679, the GDPR, is a regulation; it is binding throughout the EU with only a minimal space for national adaptations. In practice, only a few Articles (A) in the GDPR provide some but limited room for national maneuvering; these include A6 with respect to relaxation in terms of other legal obligations or public interests, A9 in terms of sensitive data, and A10 regarding criminal matters. Thus, in general, this particular legislation should be interpreted and enforced uniformly through the European Union by national data protection authorities whose formal powers are defined in A58. In practice, however, already the resources and thus the actual power for enforcement vary across the member states [1, 7]. Coupled with a lack of previous research on the enforcement of the GDPR, this variance provides a motivation for the present work to examine the recent enforcement fines imposed according to the conditions specified in A83. In addition, the work is motivated by a tangential question; is it also possible to predict these fines by machine learning methods? To answer to the question, the paper uses meta-data and text miming features extracted from the decision documents released by the national authorities. As such, only black-box predictions are sought; the goal is not to make any legal interpretations whatsoever. Nevertheless, the answer provided still establishes a solid contribution—especially when considering that the paper is presumably the very first to even examine the GDPR fines. As is discussed in Section 2, the black-box approach also places the paper into a specific branch of existing research dealing with legal documents. This section also refines the question into two more specific research questions. Afterwards, the structure is straightforward: the dataset and methods are elaborated in Sections 3 and 4, results are presented in Section 5, and conclusions follow in Section 6. As will be noted in the final section, there are also some lessons that should not be learned from this work. ## 2 Background Legal mining—in lack of a better term—has emerged in recent years as a promising but at times highly contested interdisciplinary field that uses machine learning techniques to analyze various aspects related to law [8]. Although the concrete application domains vary, case law and court cases are the prime examples already because these constitute the traditional kernel of legal scholarship. Within this kernel, existing machine learning applications range from the classification of judges’ ideological positions [12], which may be illegal in some European countries [3], to the prediction of decisions of the European Court of Human Rights [16, 17]. These examples convey the traditional functions of applied machine learning; exploratory data mining and the prediction of the future. There is also another closely related application domain. Again in lack of a better term, data extraction could be a label for this domain: by exploiting the nature of law as an art of persuasion [8], the domain uses distinct information retrieval techniques to extract and quantify textual data from legal documents into structured collections with a predefined logic and semantics [2, 24, 28]. To gain a hint about the extraction, one might consider a legal document to contain some facts, rights, obligations, and prohibitions, statements and modalities about these, and so forth. Although the two application domains are complementary in many respects, the underlying rationales exhibit some notable differences. Oftentimes, the legal mining domain is motivated by a traditional rationale for empirical social science research: to better understand trends and patterns in lawmaking and law enforcement; to contrast these with legal philosophies and theories; and so forth. This rationale extends to public administration: machine learning may ease the systematic archiving of legal documents and the finding of relevant documents, and, therefore, it may also reduce administrative costs [4]. These administrative aspects reflect the goal of building “systems that assist in decision-making”, whereas the predictive legal mining applications seek to build “systems that make decision” [21]. Although the data extraction domain can be motivated by the same administrative rationale, providing data to predictive systems is seldom the intention behind the extraction. Instead, there is a further rationale in this domain: to extract requirements for software and systems in order to comply with the laws from which a given extraction is done [24]. Driven by the genuine interest to facilitate collaboration between lawyers and engineers in order to build law-compliant software and systems [26], this rationale has been particularly prevalent in the contexts of data protection and privacy. For instance, previous work has been done to extract requirements from the Health Insurance Portability and Accountability Act in the United States [2]. Against this backdrop, it is no real surprise that data extraction has been applied also for laws enacted in the EU. While there is previous work for identifying requirements from the GDPR manually [13], there indeed exists also more systematic data extraction approaches [25]. However, neither domain has addressed the enforcement of this EU-wide regulation. In fact, a reasonably comprehensive literature search indicates no previous empirical research on the GDPR’s enforcement. Given this pronounced gap in the existing literature, this paper sets to examine the following two Questions (Q) regarding the enforcement fines: $\textmd{Q}_{1}$: (i) Which GDPR articles have been most often referenced in the recent enforcement cases, (ii) and do the enforcement fines vary across these articles? $\textmd{Q}_{2}$: How well the recent GDPR fines can be predicted in terms of basic available (i) meta-data and (ii) textual traits derived from the enforcement decisions? These two questions place the present work into the legal mining domain. Also the underlying rationales are transferable. For instance, an answer to $\textmd{Q}_{1}$ helps to understand which aspects of the GDPR have been actively enforced during the early roll out of the regulation. Also $\textmd{Q}_{2}$ carries a practical motivation: by knowing whether the penalties are predictable by machine learning techniques, a starting point is available for providing further insights in different practical scenarios. These scenarios range from the automated archival of enforcement decisions and the designation of preventive measures to litigation preparations. However, it is important to remark that the GDPR’s enforcement is done by national data protection authorities. Although the focus on public administration is maintained nevertheless, documents about the enforcement decisions reached by these authorities should not be strictly equated to law-like legal documents. This point provides an impetus to move forward by elaborating the dataset used. ## 3 Data The dataset is based on a GDPR enforcement tracker that archives the fines and penalties imposed by the European data protection authorities [5]. This tracker is maintained by an international law firm for archiving many of the known enforcement cases. Each case is accompanied by meta-data supplied by the firm as well as a link to the corresponding decision from a national authority. In addition to potentially missing cases due to the lack of publicly available information, the archival material is unfortunately incomplete in many respects. The reason originates from the incoherent reporting practices of the European data protection authorities. Therefore, all cases were obtained from the tracker, but the following four steps were followed to construct a sample for the analysis: 1. 1. To maintain coherence between $\textmd{Q}_{1}$ and $\textmd{Q}_{2}$, only those cases were included that had both meta-data and links to the decisions available. In terms of the former, some cases lacked meta-data about the fines imposed, the particular GDPR articles referenced in the decisions, and even links to the decisions. 2. 2. To increase the quality of the sample, only those cases were included that were accompanied with more or less formal documents supplied on the official websites of the data protection authorities. Thus, those cases are excluded whose archival material is based online media articles, excerpts collected from annual reports released by the authorities, and related informal sources. 3. 3. If two or more cases were referenced with the same decision, only one decision document was included but the associated meta-data was unified into a single case by merging the articles references and totaling the fines imposed. 4. 4. All national decisions written in languages other than English were translated to English with Google Translate. In general, such machine translation is necessary due to the EU-wide focus of the forthcoming empirical analysis. Given these restrictions, the sample amounts to about 72% of all cases archived to the tracker at the time of data collection. Even with these precautions, it should be stressed that the quality of the sample is hardly optimal. While the accuracy of the meta-data supplied by the firm is taken for granted, there are also some issues with the quality of the publicly available decisions. The authorities in some countries (e.g., Hungary and Spain) have released highly detailed and rigorous documents about their decisions, while some other authorities (e.g., in Germany) have opted for short press releases. Although most of the documents were supplied in the portable document format (PDF) and informally signed by the authorities, it should be thus stressed that the data quality is not consistent across the European countries observed. In addition, it is worth remarking the detail that scanned PDF documents (as used, e.g., in Portugal) had to be excluded due to the automatic data processing. While these data quality issues underline the paper’s exploratory approach, these carry also political and administrative ramifications that are briefly discussed later on in Section 6. ## 4 Methods Descriptive statistics and regression analysis are used for answering to the two questions asked. In terms of Question $\textmd{Q}_{1}$, dummy variables for the GDPR articles referenced are simply regressed against the logarithm of the fines imposed by using the conventional analysis-of-variance (ANOVA). As many of the cases reference multiple articles, it should be remarked that these dummy variables are not so-called fixed effects. The methods for answering to the second Question $\textmd{Q}_{2}$ require a more thorough elaboration. In addition to (i) the GDPR articles, the meta-data aspects include dummy variables for the following features: (ii) the year of a given enforcement case; (iii) the country in which the given fine was imposed; and (iv) the sector of the violating organization. The last feature was constructed manually by using five categories: individuals, public sector (including associations), telecommunications, private sector (excluding telecommunications), and unknown sector due to the lack of meta-data supplied in the enforcement tracker. In total, these features amount to $49$ dummy variables. The textual aspects for $\textmd{Q}_{2}$ are derived from the translated decisions. Seven steps were used for pre-processing: (a) all translated decision documents were lower-cased and (b) tokenized according to white space and punctuation characters; (c) only alphabetical tokens recognized as English words were included; (d) common and custom stopwords were excluded; (e) tokens with lengths less than three characters or more than twenty characters were excluded; (f) all tokens were lemmatized into their common English dictionary forms; and, finally, (g) those lemmatized tokens were excluded that occurred in the whole decision corpus in less than three times. A common natural language processing library [22] was used for this processing together with a common English dictionary [20]. In addition to the stopwords supplied in the library, the twelve most frequent tokens were used as custom excluded stopwords: data, article, personal, protection, processing, company, authority, regulation, information, case, art, and page. After this pre- processing, the token-based term frequency (TF) and term frequency inverse document frequency (TF-IDF) were calculated from the whole corpus constructed (for the exact formulas used see, e.g., [23]). These common information retrieval statistics are used for evaluating the other part in $\textmd{Q}_{2}$. In general, TF-IDF is often preferred as it penalizes frequently occurring terms. Sparsity is the biggest issue for prediction. There are only $154$ observations but already the meta-data amounts to $49$ independent variables—and the TF and TF-IDF each to $4189$ independent variables. Fortunately, the problem is not uncommon, and well-known solutions exist for addressing it. Genomics is a good example about the application domains riddled with the problem; within this domain, it is not uncommon to operate with datasets containing a few thousand observations and tens of thousands of predictors [6]. Dimension reduction is the generic solution in this domain and other domains with similar problems. Thus, three common dimension reduction methods for regression analysis are used: principal component regression (PCR), partial least squares (PLS), and ridge regression (for a concise overview of these methods see, e.g., [11]). In essence, PCR uses uncorrelated linear combinations as the independent variables; PLS is otherwise similar but also the dependent variable is used for constructing the combinations. Ridge regression is based on a different principle: the dimensionality is reduced by shrinking some of the regression coefficients to zero. In general, all three methods are known to yield relatively similar results in applied work. In terms of practical computation, the number of components for the PCR and PLS models, and the shrinkage parameter for the ridge regression, is optimized during the training while the results are reported with respect to a test set containing 20% of the enforcement cases. Centering (but not scaling) is used prior to the training with a $5$-fold cross-validation. Computation is carried out with the caret package [14] in conjunction with the pls [18] and foba [30] packages. Although root-mean-square errors (RMSEs) are used for optimizing the training, the results are summarized with mean absolute errors (MAEs) due to their straightforward interpretability. These are defined as the arithmetic means of the absolute differences between the observed and predicted fines in the test set. ## 5 Results The GDPR fines imposed vary greatly. As can be seen from Fig. 1, a range from about $e^{6}$ euros to $e^{12}$ euros capture the majority of the enforcement fines observed. This range amounts roughly from about four hundred to $163$ thousand euros. That said, the distribution has a fairly long tail; also a few large, multi-million euro fines are present in the sample. Therefore, the sample cannot be considered biased even though the restrictions discussed in Section 3 exclude some of the largest enforcement cases, including the announcements about the intention to fine the British Airways and Marriott International by the Information Commissioner’s Office in the United Kingdom. Although these two excluded cases are—at least at the time of writing—preliminary announcements, they are still illuminating in the sense that both were about large-scale data breaches. Figure 1: Enforcement Fines in the Sample However, the GDPR’s corresponding A32 for information security has not been the most frequently referenced article in the recent enforcement cases. Instead, A5 and A6, which address the general principles and lawfulness of personal data processing, have clearly been the most referenced individual articles, as can be seen from Fig. 2. These two articles account for as much as 87% of all $252$ references made in the $154$ enforcement cases. More than six references have been made to A13 (informing obligations to data subjects), A15 (right to access), A21 (right to object), and A17 (right to erasure). These references indicate that enforcement has been active also with respect to the rights granted by the GDPR for individual data subjects. Furthermore, less frequent references have been made in the decisions to numerous other articles. These include the obligations to designate data protection officers (A37), conduct impact assessments (A35), and consult supervisory authorities (A36), to name three examples. While the principles, lawfulness, and information security account for the majority, the less frequent but still visible references to more specific articles hint that the regulation’s whole scope is slowly being enforced by the European authorities. Figure 2: Referenced GDPR Articles in the Enforcement Cases Turning to the second part of $\textmd{Q}_{1}$, the regression coefficients from the log-linear ANOVA model are visualized in Fig. 3 (the intercept is present in the model but not shown in the figure, and A36 is omitted as the single reference made to the article corresponds with the single reference made to A35 in the same decision; the dummy variable for A35 thus captures the effect of both articles). As can be seen, the confidence intervals (CIs) are quite wide for the articles referenced only infrequently, and only six coefficients are statistically significant at the conventional threshold. Thus, some care is required for interpretation. Figure 3: Enforcement Fines Across Articles (logarithm, ANOVA, 95% CIs) When looking at the coefficients with relatively tight CIs, it is evident that variation is present but the magnitude of this variation is not substantial. Most of the coefficients remain in the range $[-5,5]$. However, together all the references do yield a decent model; an $F$-test is statistically significant and the coefficient of determination is large ($R^{2}\simeq 0.44$). To put aside the statistical insignificance, it is also interesting to observe that some of the coefficients have negative signs, meaning that some references indicate smaller fines compared to the average. Among these are the conditions for consent (A7), sensitive data (A9), transparency (A12), and informing (A13), as well as the already noted right to access (A15), proper notifications about data breaches (A33), and the powers granted for the supervisory authorities (A58). Finally, the magnitude of the coefficient ($1.52$) for the information security article (A32) is significant but does not stand out in terms of magnitude. When compared to cases without a reference to this article, only about $1.5\%$ higher fines have been imposed in cases referencing A32. Figure 4: Prediction Performance (logarithm, MAEs) Figure 5: Observed and Predicted Values in the Test Set The results regarding $\textmd{Q}_{2}$ are summarized in Fig. 4 (the MAEs for the training refer to the best cross-validated models). Three noteworthy observations can be drawn from this summary. First and foremost, the prediction performance is generally decent: the best-performing cases all yield MAEs roughly between $1.3$ and $1.5$ for the log-transformed fines. These average prediction errors seem also reasonable when taking a closer look at the actual predictions—except for the outlying large fines. Take Fig. 5 as a brief example; the figure displays the observed fines and the predicted fines based on the PLS and ridge regression estimators for the first meta-data model. Even though most of the predicted observations are fairly close to the observed fines, the test set also contains one five million euro fine that is quite severely underestimated by both regression estimators. The underestimations amount to over $246$ thousand euros. Though, when a magnitude is measured in millions, it is a matter of interpretation whether an error measured in hundreds of thousands is large, small, or something else. Second, there are some interesting differences between the regression estimators. In particular, PLS and ridge regression exhibit relatively large differences between training and testing. The explanation relates to the RMSE- based optimization during training. For instance, PCR was estimated with only one component for the first meta-data model and three components for the remaining three models, whereas two components were picked for all four PLS models. Last but not least, the smallest MAE for the test set is outputted by ridge regression using only the $49$ meta-data variables. The second and third models containing the TF and TF-IDF variables both perform worse. Furthermore, the fourth model, which contains the meta-data and TF-IDF variables, indicates that the text mining features tend to slightly weaken the predictions. It is also worth remarking that some redundancy is present among the meta-data variables; comparable performance is obtained with only $17$ meta-data variables that are left after prior pre-processing with the caret’s nearZeroVar function. All this said, the overall interpretation should be less explicit when considering the practical motivation for $\textmd{Q}_{2}$ noted in Section 2. If only the decision documents are available without any prior work to manually construct the meta-data from these, even the simple text mining features could be used for black-box predictions. ## 6 Conclusion This paper explored two questions. The answers to these can be summarized as follows. First: regarding $\textmd{Q}_{1}$, the articles related to the general principles (A5), lawfulness (A6), and information security (A32) have been most frequently referenced by the national data protection authorities during the early enforcement period observed in this paper. Although also the enforcement fines vary across the various GDPR articles referenced in the authorities’ decisions, the effects of these three articles do not stand out in particular. A good corollary question for further work would be to examine the future evolution of these references; a hypothesis is that the regulation’s enforcement is slowly moving from the principles and lawfulness conditions to more specific elements. Then: regarding $\textmd{Q}_{2}$, it is possible to obtain decent predictions even with standard machine learning techniques for regression analysis. Basic meta-data (i.e., articles referenced, year of enforcement, country or origin, and industry sector) seems to provide slightly better predictive performance compared to basic text mining features (i.e., TF and TF-IDF) extracted from the decision documents. Yet, even the text mining features seem sufficient for blind black-box predictions. There are also many potential ways to improve the predictions reported, including those related regression analysis (such as using specific sparse-PLS estimators) and text mining (such as using word embeddings). Data mining techniques (such as topic modeling) could be used also for better understanding the nuances behind the decisions. An alternative path forward would be to extend the specific data extraction approaches discussed in Section 2 to the enforcement decisions. However, the motivation to move forward is undermined by practical problems. As was remarked in Section 3, already the quality of data is a problem of its own. Recently, the enforcement of the GDPR has been fiercely criticized by some public authorities and pundits alike. The reasons are many: a lack of transparency and cooperation between national data protection authorities, diverging legal interpretations, cultural conflicts, the so-called “one-stop- shop” system, old-fashioned information systems and poor data exchange practices, and so on and so forth [27]. The data collection used for the present work testifies on behalf of the criticism: the decision documents released by the national authorities have varied wildly in terms of quality and rigor. Some national authorities have even hidden their decisions from public scrutiny. A paradox is present: although A15 grants a right for data subjects to access their personal data, the same subjects may need to exercise their separate freedom of information rights to obtain cues about decisions reached by national authorities. Four legs good, two legs bad. Finally, it is necessary to briefly point out the bigger issues affecting the legal mining and data extraction domains—and, therefore, also the present work. For one thing, the practical usefulness of legal expert systems has been questioned for a long time. The artificial intelligence hype has not silenced the criticism [15]. Like with the “code is law” notion, which has never existed in reality [19], there are also many philosophical counterarguments against the legal mining and data extraction domains [8, 9, 21]. It is problematic at best to codify the methodology of a scholarly discipline into rigid schemas in order to nurse the methodological requirements of another discipline; legal reasoning is distinct from other types of reasoning exercised in empirical sciences; and so forth. Law is not code. But code is increasingly used to predict law enforcement decisions. The legal mining domain, in particular, is frequently involved with a motivation to build “a system that could predict judicial decisions automatically” but with a provision that there is “no intention of creating a system that could replace judges” [17]. Such system-building leads to another delicate paradox. 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2024-09-04T02:54:59.080131

2003.05174

arxiv-papers

# Delay-Adaptive Learning in Generalized Linear Contextual Bandits Jose Blanchet Department of Management Science and Engineering, Stanford University, USA. Email<EMAIL_ADDRESS>Renyuan Xu Mathematical Institute, University of Oxford, UK. Email<EMAIL_ADDRESS>Zhengyuan Zhou Stern School of Business, New York University, USA. Email<EMAIL_ADDRESS> ###### Abstract In this paper, we consider online learning in generalized linear contextual bandits where rewards are not immediately observed. Instead, rewards are available to the decision maker only after some delay, which is unknown and stochastic. We study the performance of two well-known algorithms adapted to this delayed setting: one based on upper confidence bounds, and the other based on Thompson sampling. We describe modifications on how these two algorithms should be adapted to handle delays and give regret characterizations for both algorithms. Our results contribute to the broad landscape of contextual bandits literature by establishing that both algorithms can be made to be robust to delays, thereby helping clarify and reaffirm the empirical success of these two algorithms, which are widely deployed in modern recommendation engines. ## 1 Introduction The growing availability of user-specific data has welcomed the exciting era of personalized recommendation, a paradigm that uncovers the heterogeneity across individuals and provides tailored service decisions that lead to improved outcomes. Such heterogeneity is ubiquitous across a variety of application domains (including online advertising, medical treatment assignment, product/news recommendation ([LCLS2010], [BCN2012],[chapelle2014],[bastani2015online],[SBF2017])) and manifests itself as different individuals responding differently to the recommended items. Rising to this opportunity, contextual bandits ([besbes2009dynamic, rigollet2010nonparametric, goldenshluger2011note, hsu2014taming, agrawal2016efficient]) have emerged to be the predominant mathematical formalism that provides an elegant and powerful formulation: its three core components, the features (representing individual characteristics), the actions (representing the recommendation), and the rewards (representing the observed feedback), capture the salient aspects of the problem and provide fertile ground for developing algorithms that balance exploring and exploiting users’ heterogeneity. As such, the last decade has witnessed extensive research efforts in developing effective and efficient contextual bandits algorithms. In particular, two types of algorithms–upper confidence bounds (UCB) based algorithms ([LCLS2010, FCGS2010, chu2011contextual, JBNW2017, LLZ2017]) and Thompson sampling (TS) based algorithms ([AG2013a, AG2013b, RV2014, russo2016information, agrawal2017thompson])–stand out from this flourishing and fruitful line of work: their theoretical guarantees have been analyzed in many settings, often yielding (near-)optimal regret bounds; their empirical performance have been thoroughly validated, often providing insights into their practical efficacy (including the consensus that TS based algorithms, although sometimes suffering from intensive computation for posterior updates, are generally more effective than their UCB counterparts, whose performance can be sensitive to hyper-parameter tuning). To a large extent, these two family of algorithms have been widely deployed in many modern recommendation engines. However, a key assumption therein–both the algorithm design and their analyses–is that the reward is immediately available after an action is taken. Although useful as a first-step abstraction, this is a stringent requirement that is rarely satisfied in practice, particularly in large-scale systems where the time-scale of a single recommendation is significantly smaller than the time-scale of a user’s feedback. For instance, in E-commerce, a recommendation is typically made by the engine in milliseconds, whereas a user’s response time (i.e. to buy a product or conversion) is typically much larger, ranging from hours to days, sometimes even to weeks. For instance, a thorough empirical study in [chapelle2014] found that more than 10% of the conversions in Criteo (a real-time bidding company) were at least 2 weeks old. Furthermore, [chapelle2014] found that the delay distribution from the company’s data follows the exponential distribution closely and hence does have heavy tails. Similarly, in clinical trials, it is infeasible to immediately observe and hence take into account the medical outcome after applying a treatment to a patient–collecting medical feedback can be a time- consuming and often random process; and in general, it is common to have applied trial treatments to a large number of patients, with individual medical outcomes only available much later at different, random points in time. In both the E-commerce ([KCW2001, chapelle2014])and the clinical trials cases ([CC2011]), a random and often significantly delayed reward is present. Further, such delays empirically often follow a heavy tail distribution, and hence a priori can have substantially negative impact on the learning performance. Consequently, to understand such impact of delays, adjustments in classical formulations must be made, both at the algorithmic level and at the analysis level. ### 1.1 Related Work In the past five years or so, the problem of learning on bandits with delays has received increasing attention and has been studied in several different settings in the existing literature, where most of the efforts have concentrated on the multi-armed bandits setting, including both the stochastic multi-armed bandits and the adversarial multi-armed bandits. For stochastic multi-armed bandits with delays, [JGS2013] show a regret bound $O(\log T+\mathbb{E}[\tau]+\sqrt{\log T\mathbb{E}[\tau]})$ where $\mathbb{E}[\tau]$ is the mean of the iid delays. [DKVB2014] consider Gaussian Process bandits with a bounded stochastic delay. [MLBP2015] follow the work of [JGS2013] and propose a queue-based multi-armed bandit algorithm to handle delays. [PASG2017] match the same regret bound as in [JGS2013] when feedback is not only delayed but also anonymous. For adversarial multi-armed bandits with delays, [NAGS2010] establish the regret bound of $\mathbb{E}[R_{T}]\leq O(\tau_{\text{const}})\times\mathbb{E}[R^{\prime}_{T}(\frac{T}{\tau_{\text{const}}})]$ for Markov decision process, where $\tau_{\text{const}}$ is the constant delay and $R^{\prime}_{T}$ is the regret without delays. [CGM2019] consider adversarial bandits with fixed constant delays on the network graph, with a minimax regret of the order $\tilde{O}\sqrt{(K+\tau_{\text{const}})T}$, where $K$ is the number of arms. Another related line of work to adversarial multi- armed bandits is adversarial learning with full information, where the rewards for all arms are observed. Different variants of this problems in the delayed setting have been studied by [WO2002], [mesterharm2005], [QK2015] and [GST2016]. On the other hand, learning in contextual bandits with delays are much less explored. [JGS2013] consider learning on adversarial contextual bandits with delays and establish an expected regret bound $\mathbb{E}\left[R_{T}\right]\leq(1+\mathbb{E}[M_{T}^{*}])\times\mathbb{E}\left[R^{\prime}_{T}\left(\frac{T}{1+\mathbb{E}[M_{T}^{*}]}\right)\right]$ by using a black-box algorithm, where $M_{T}^{*}$ is the running maximum number of delays up to round $T$. [DHKKLRZ2011] consider stochastic contextual bandits with a fixed constant delay. The reward model they consider is general (i.e. not necessarily parametric); however, they require the policy class to be finite. In particular, they obtain the regret bound $O(\sqrt{K\log N}(\tau_{\text{const}}+\sqrt{T}))$, where $N$ is the number of policies and $\tau_{\text{const}}$ is again the fixed constant delay. Finally, we also note that there is a growing literature on offline contextual bandits (for a highly incomplete list, see [dudik2011doubly, swaminathan2015batch, athey2017efficient, zhou2018offline, kitagawa2018should, off-policy-evaluation-slate-recommendation, deep-learning-logged-bandit- feedback]). This is a setting where all the data has been collected upfront and a policy needs to be learned from this batch data at once. Although sharing the same primitives (contexts, actions and rewards), this problem has important differences from the online setting. In particular, the exploration part is missing in this problem and a separate set of challenges exist in the offline case. In this setting, delays would have no impact since all the rewards will have been collected at the end (except perhaps at the tail of the batch). ### 1.2 Our Contributions In this paper, we consider learning on generalized linear (stochastic) contextual bandits with stochastic unbounded delays. Our contributions are two-fold. First, we design two delay-adaptive algorithms for generalized linear contextual bandits, one based on UCB, the other based on TS. We refer to the two variants as Delayed UCB (DUCB, as given in Algorithm 1) and Delayed TS (DTS, as given in Algorithm 2) respectively. DUCB requires a carefully designed delay-adaptive confidence parameter, which depends on how many rewards are missing up to the current time step. In contrast, DTS is a straightforward adaptation that incorporates the delayed rewards as they become available. Second, we give regret characterizations of both DUCB and DTS under (1) independent stochastic, unbounded delays that can have heavy tails, (2) unbounded Markov delays that can have near-heavy tails (tails that are arbitrarily close to exponential tails), and (3) unbounded delays with any dependency structure that have light (sub-Gaussian) tails. In particular, as a special case of our results, when the delays are iid with mean $\mu_{I}$, we have a high-probability regret bound of $\tilde{O}\left(\left(\sigma_{G}\sqrt{d}+\mu_{I}d+d\right)\sqrt{T}\right)$ on DUCB, where $\sigma_{G}$ is a parameter characterizing the tail bound of the delays and $d$ is the feature dimension. For comparison, the state-of-the-art regret bound of UCB on generalized linear contextual bandits without delays is $\tilde{O}\left(d\sqrt{T}\right)$ ([FCGS2010, LLZ2017]). For DTS, we have the Bayesian regret bound of $\tilde{O}\left(\left(\sigma_{G}\sqrt{d}+\mu_{I}\sqrt{d}+d\right)\sqrt{T}\right)$. For comparison, the state-of-the-art Bayesian regret bound of TS on generalized linear contextual bandits without delays is $\tilde{O}\left(d\sqrt{T}\right)$ ([RV2014, russo2016information]). The regret bounds we have obtained highlight the dependence on the delays in two ways: one is how much delay is present on average, the other is how heavy the tail of the distribution is. Both factors contribute to the degradation of the regret bounds: that the average delay enlarges regret is intuitive; that the tail influences regret is because a more likely large delay (at the far right end of a tail) can delay the learning for that context significantly, particularly in the early stages when the decision maker is unsure about the underlying parameter is. To the best of our knowledge, these regret bounds provide the first theoretical characterizations in generalized linear contextual bandits with large delays. Our results contribute to the broad landscape of contextual bandits literature by establishing that both algorithms are robust to delays, thereby helping clarify and reaffirm the empirical success of these two algorithms, which are widely deployed in modern recommendation engines. Some of the initial results have appeared in the conference version [zhou2019]. Our work here provides a comprehensive treatment of learning in generalized linear contextual bandits with large delays that incorporates substantially more in-depth inquiries on several fronts. First, we consider the heavier-tailed delays that include exponential distributions whereas [zhou2019] only dealt with light-tailed delays that are either sub-Gaussian or have ($1+q$)-th moment (for some $q>0$). This relaxation is important both from an empirical standpoint and from a theoretical standpoint. Empirically, as mentioned earlier, the field study in [chapelle2014] found that the delay distribution from the company’s data follows the exponential distribution closely, rather than a sub-Gaussian distribution that is commonly assumed in the bandits literature. Theoretically, establishing guarantees in this larger- delay regime requires us to develop a new (and arguably more elegant) argument from that in [zhou2019], which is not applicable here. We explain the technical difficulty in more detail in Section 3.3. Second, the sole focus of [zhou2019] is on adapting and analyzing UCB-based algorithms. However, as mentioned earlier, it is known that Thompson sampling often achieves superior empirical performance, despite the fact that their theoretical bounds (when no delays are present) may not match exactly those of the UCB algorithms. Furthermore, TS-based algorithms do not suffer from hyper-parameter tuning and can effectively incorporate prior and can therefore significantly outperform (when priors are available and correct). Consequently, in this paper, in addition to adapting and analyzing the UCB-based algorithms, we also discuss (in Section 4) the adaptation of TS-based algorithms in the delayed feedback setting and obtain regret bounds that characterize the corresponding performance. Finally, we move beyond the regime of the independent delay setting studied in [zhou2019], and instead consider (in Section 5) the much more general and realistic history-dependent delays setting. We give regret bounds of both UCB-based algorithms and TS-based algorithms, under both the Markov delays assumption and the general stationary delays assumption. We also highlight, in this unified presentation, the comparison of the various regret bounds as the assumption on delays get progressively weakened. ## 2 Problem Setup In this section, we describe the formulation for learning in generalized linear contextual bandits (GLCB) in the presence of delays. We start by reviewing the basics of generalized linear contextual bandits, followed by a description of the delay model. Before proceeding, we first fix some notation. For a vector $x\in\mathbb{R}^{d}$, we use $\|x\|$ to denote its $l_{2}$-norm and $x^{\prime}$ its transpose. $\mathbb{B}^{d}:=\\{x\in\mathbb{R}^{d}:\|x\|\leq 1\\}$ is the unit ball centered at the origin. The weighted $l_{2}$-norm associated with a positive- definite matrix $A$ is defined by $\|x\|_{A}:=\sqrt{x^{\prime}Ax}$. The minimum and maximum singular values of a matrix $A$ are written as $\lambda_{\min}(A)$ and $\|A\|$ respectively. For two symmetric matrices $A$ and $B$ the same dimensions, $A\succeq B$ means that A-B is positive semi- definite. For a real-valued function f, we use $\dot{f}$ and $\ddot{f}$ to denote its first and second derivatives. Finally, $[n]:=\\{1,2,\cdots,n\\}$. ### 2.1 Generalized Linear Contextual Bandits #### Decision procedure. We consider the generalized linear contextual bandits problem with $K$ actions. At each round $t$, the agent observes a context consisting of a set of $K$ feature vectors $x_{t}:=\\{x_{t,a}\in\mathbb{R}^{d}|a\in[K]\\}$, which is drawn iid from an unknown distribution $\gamma$ with $\|x_{t,a}\|\leq 1$. Each feature vector $x_{t,a}$ is associated with an unknown stochastic reward $y_{t,a}\in[0,1]$. If the agent selects one action $a_{t}$, there is a resulting reward $y_{t,a_{t}}\in[0,1]$ associated. In the standard contextual bandits setting, the reward is immediately observed after the decision is made and the observed reward can be utilized to make decision in the next round. Although it is generally understood in the contextual bandits literature, for completeness, here we briefly discuss the meaning of the above quantities, as well as where they come from. In general, at each round $t$, an individual characterized by $v_{t}$ (a list of characteristics associated with that individual) is drawn from a population and becomes available. When the decision maker decides to apply action $a_{t}$ (one of the available $K$ actions) to this individual, then a reward $y_{t}(v_{t},a_{t})$ is obtained: this reward can depend stochastically on both the individual characteristics $v_{t}$ and the selected action $a_{t}$. However, in practice, for both modelling and computational reasons, one often first featurizes the individual characteristics and the actions. In particular, with sufficient generality, one assumes $\mathbf{E}[y_{t}(v_{t},a_{t})\mid v_{t},a_{t}]=g_{\theta}(\phi(v_{t},a_{t}))$, where $g_{\theta}(\cdot)$ is the parametrized mean reward function and $\phi(v_{t},a_{t})$ extracts the features from the given raw individual characteristics $v_{t}$ and action $a_{t}$. In the above formulation, as is standard in the contextual bandits literature, we assume the feature map $\phi(\cdot)$ is known and given and $x_{t,a}=\phi(v_{t},a)$. If $V_{t}$ is already a vector in Euclidean space, then a common choice for the feature extractor is $\phi(v_{t},a)=[\mathbf{0},\dots,\mathbf{0},v_{t},\mathbf{0},\dots,\mathbf{0}]$: that is, a $Kd$-dimensional vector with all zeros except at the $a$-th block. #### Relationship between reward $Y$ and context $X$. In terms of the relationship between $Y_{t,a}$ and $X_{t,a}$, we follow the standard generalized linear contextual bandits literature ([FCGS2010, LLZ2017]). Define $\mathcal{H}^{0}_{t}=\\{(s,x_{s},a_{s},y_{s,a_{s}}),s\leq t-1\\}\cup\\{x_{t}\\}$ as the information available at the beginning of round $t$. The agent maximizes the cumulative expected rewards over $T$ rounds with information $\mathcal{H}^{0}_{t}$ at each round $t$ ($t\geq 1$). Suppose the agent takes action $a_{t}$ at round $t$. Denote by $X_{t}=x_{t,a_{t}}$, $Y_{t}=y_{t,a_{t}}$ and we assume the conditional distribution of $Y_{t}$ given $X_{t}$ is from the exponential family. Therefore its density is given by $\displaystyle\mathbb{P}_{\theta^{*}}(Y_{t}|X_{t})=\exp\left(\frac{Y_{t}X_{t}^{\prime}\theta^{*}-m(X_{t}^{\prime}\theta^{*})}{h(\eta)}+A(Y_{t},\eta)\right).$ (1) Here, $\theta^{*}$ is an unknown number under the frequentist setting; $\eta\in\mathbb{R}^{+}$ is a given parameter; $A$, $m$ and $h$ are three normalization functions mapping from $\mathbb{R}$ to $\mathbb{R}$. For exponential families, $m$ is infinitely differentiable, $\dot{m}(X^{\prime}\theta^{*})=\mathbb{E}[Y|X]$, and $\ddot{m}(X^{\prime}\theta^{*})=\mathbb{V}(Y|X)$. Denote $g(X^{\prime}\theta^{*})=\mathbb{E}[Y|X]$ , one can easily verify that $g(x^{\prime}\theta)=x^{\prime}\theta$ for linear model, $g(x^{\prime}\theta)=\frac{1}{1+\exp(-x^{\prime}\theta)}$ for logistic model and $g(x^{\prime}\theta)=\exp(x^{\prime}\theta)$ for Poisson model. In the generalized linear model (GLM) literature ([NW1972, McCullagh2018]), $g$ is often referred to as the inverse link function. Note that (1) can be rewritten as the GLCB form, $\displaystyle Y_{t}=g(X_{t}^{\prime}\theta^{*})+\epsilon_{t},$ (2) where $\\{\epsilon_{t},t\in[T]\\}$ are independent zero-mean noise, $\mathcal{H}^{0}_{t}$-measurable with $\mathbb{E}[\epsilon_{t}|{\mathcal{H}^{0}_{t}}]=0$. Data generated from (1) automatically satisfies the sub-Gaussian condition: $\displaystyle\mathbb{E}\left[\exp({\lambda\epsilon_{t}})|{\mathcal{H}^{0}_{t}}\right]\leq\exp\left({\frac{\lambda^{2}\hat{\sigma}^{2}}{2}}\right).$ (3) Throughout the paper, we denote $\hat{\sigma}>0$ as the sub-Gaussian parameter of the noise $\epsilon_{t}$. ###### Remark 1 In this paper, we focus on the GLM with exponential family (1). In general, one can work with model (2) under the sub-Gaussian assumption (3). Our analysis will still hold by considering maximum quasi-likelihood estimator for (2). See more explanations in Section 3.1. ### 2.2 The Delay Model Unlike the traditional setting where each reward is immediately observed, here we consider the case where stochastic and unbounded delays are present in revealing the rewards. Let $T$ be the number of total rounds. At round $t$, after the agent takes action $a_{t}$, the reward $y_{t,a_{t}}$ may not be available immediately. Instead, it will be observed at the end of round $t+D_{t}$ where $D_{t}$ is the delay at time $t$. We assume $D_{t}$ is a non- negative random number which is independent of $\\{D_{s}\\}_{s\leq t-1}$ and $\\{x_{s},y_{s,a_{s}},a_{s}\\}_{s\leq t}$. First, we define the available information for the agent at each round. #### Information structure under delays. At any round $t$, if $D_{s}+s\leq t-1$ (reward occurred in round $s$ is available at the beginning of round $t$), then we call $(s,x_{s},y_{s,a_{s}},a_{s})$ the complete information tuple at round $t$. If $D_{s}+s\geq t$, we call $(s,x_{s},a_{s})$ the incomplete information tuple at the beginning of round $t$. Define $\mathcal{H}_{t}=\left\\{(s,x_{s},y_{s,a_{s}},a_{s})\,\,|\,\,s+D_{s}\leq t-1\right\\}\cup\left\\{(s,x_{s},a_{s})\,\,|\,\,s\leq t-1,s+D_{s}\geq t\right\\}\cup\left\\{x_{t}\right\\},$ then $\mathcal{H}_{t}$ is the information (filtration) available at the beginning of round $t$ for the agent to choose action $a_{t}$. In other words, $\mathcal{H}_{t}$ contains all the incomplete and complete information tuples up to round $t-1$ and the content vector $x_{t}$ at round $t$. Moreover define $\displaystyle\mathcal{F}_{t}=\\{(s,x_{s},a_{s},y_{s,a_{s}})\,\,|\,\,s+D_{s}\leq t\\}.$ (4) Then $\mathcal{F}_{t}$ contains all the complete information tuples $(s,x_{s},a_{s},y_{s,a_{s}})$ up to the end of round $t$. Denote $\mathcal{I}_{t}=\mathcal{F}_{t}-\mathcal{F}_{t-1}$, $\mathcal{I}_{t}$ is the new complete information tuples revealed at the end of round $t$. #### Performance criterion. Under the frequentist setting, assume there exists an unknown true parameter $\theta^{*}\in\mathbb{R}^{d}$. The agent’s strategy can be evaluated by comparing her rewards to the best reward. To do so, define the optimal action at round $t$ by $a_{t}^{*}=\arg\max_{a\in[K]}g(x_{t,a}^{\prime}\theta^{*})$. Then, the agent’s total regret of following strategy $\pi$ can be expressed as follows $R_{T}(\pi):=\sum_{t=1}^{T}\left(g\left(x_{t,a^{*}_{t}}^{\prime}\theta^{*}\right)-g\left(x_{t,a_{t}}^{\prime}\theta^{*}\right)\right),$ where $a_{t}\sim\pi_{t}$ and policy $\pi_{t}$ maps $\mathcal{H}_{t}$ to the probability simplex $\Delta^{K}:=\\{(p_{1},\cdots,p_{K})\,\,|\,\,\sum_{i=1}^{K}p_{i}=1,p_{i}\geq 0\\}$. Note that $R_{T}(\pi)$ is in general a random variable due to the possible randomness in $\pi$. #### Assumptions. Throughout the paper, we assume the following assumption on distribution $\gamma$ and function $g$, which is standard in the generalized linear bandit literature ([FCGS2010, LLZ2017, JBNW2017]). ###### Assumption 1 (GLCB) * • $\lambda_{\min}(\mathbb{E}[\frac{1}{K}\sum_{a\in[K]}x_{t,a}x_{t,a}^{\prime}])\geq\sigma_{0}^{2}$ for all $t\in[T]$. * • $\kappa:=\inf_{\\{\|x\|\leq 1,\|\theta-\theta^{*}\|\leq 1\\}}\dot{g}(x^{\prime}\theta)>0$. * • $g$ is twice differentiable. $\dot{g}$ and $\ddot{g}$ are upper bounded by $L_{g}$ and $M_{g}$, respectively. In addition, we assume the delay sequence $\\{D_{t}\\}_{t=1}^{T}$ satisfies the following assumption. ###### Assumption 2 (Delay) Assume $\\{D_{t}\\}_{t=1}^{T}$ are independent non-negative random variables with tail-envelope distribution $(\xi,\mu,M)$. That is, there exists a constant $M>0$ and a distribution $\xi$ with mean $\mu<\infty$ such that for any $m\geq M$ and $t\in[T]$, $\mathbb{P}(D_{t}\geq m)\leq\mathbb{P}(D\geq m),$ where $D\sim\xi$. Furthermore, assume there exists $q\geq 0$ such that $\mathbb{P}(D-\mu\geq x)\leq\exp\left(\frac{-x^{1+q}}{2\sigma^{2}}\right),$ where $\mathbb{E}[D]=\mu$. Assumption 2 includes the most common delay patterns in real-world applications. $D$ is sub-Gaussian when $q=1$ and $D$ has exponential delays when $q=0$. When $D_{t}$’s are iid, the following condition guarantees Assumption 2: $\mathbb{P}(D_{t}-\mathbb{E}[D_{t}]\geq x)\leq\exp\left(\frac{-x^{1+q}}{2\tilde{\sigma}^{2}}\right),$ with some $\tilde{\sigma}>0$ and $q\geq 0$. We summarize the parameter definition in Table LABEL:tab:parameters. (See Section LABEL:app:table.) Note that with Assumption 2, we do not need to assume all delays have identical distributions, as long as they are independent over time. Since there exists an envelop distribution $\xi$ uniformly dominating the tail probability of all delays, we can get a handle on the tail of all the delay distributions. This can be viewed as the regularity condition on the delays. ## 3 Delayed Upper Confidence Bound (DUCB) for GLCB In this section, we propose a UCB type of algorithm for GLCB adapting the delay information in an online version. Let us first introduce the maximum likelihood estimator we adopt and then state the main algorithm. ### 3.1 Maximum Likelihood Estimators (MLEs). Denote $T_{t}=\\{s:s\leq t-1,D_{s}+s\leq t-1\\}$ as the set containing timestamps with complete information tuples at the beginning of round $t$. We use data with timestamps in $T_{t}$ to construct the MLE. Suppose we have independent samples of $\\{Y_{s}:s\in T_{t}\\}$ condition on $\\{X_{s}:s\in T_{t}\\}$. The log-likelihood function of $\theta$ under (1) is $\displaystyle\log l\left(\theta\,\,|\,\,T_{t}\right)$ $\displaystyle=$ $\displaystyle\sum_{s\in T_{t}}\left[\frac{Y_{s}X_{s}^{\prime}\theta-m(X_{s}^{\prime}\theta)}{v(\eta)}+B(Y_{s},\eta)\right]$ $\displaystyle=$ $\displaystyle\frac{1}{v(\eta)}\sum_{s\in T_{t}}\left[Y_{s}X_{s}^{\prime}\theta-m(X_{s}^{\prime}\theta)\right]+\text{constant}.$ Therefore, the MLE can be defined as $\hat{\theta}_{t}\in\arg\max_{\theta\in\Theta}\sum_{s\in T_{t}}\left[Y_{s}X_{s}^{\prime}\theta-m(X_{s}^{\prime}\theta)\right].$ Since $m$ is differentiable with $\ddot{m}\geq 0$, the MLE can be written as the solution of the following equation $\displaystyle\sum_{s\in T_{t}}(Y_{s}-g(X_{s}^{\prime}\theta))X_{s}=0,$ (5) which is the estimator we use in Step 4 of Algorithm 1. Note that, the general GLCB, a semi-parametric version of the GLM, is obtained by assuming only that $\mathbb{E}[Y|X]=g(X^{\prime}\theta^{*})$ (see (2)) without further assumptions on the conditional distribution of $Y$ given $X$. In this case, the estimator obtained by solving (5) is referred to as the maximum quasi-likelihood estimator. It is well-documented that this estimator is consistent under very general assumptions as long as matrix $\sum_{s\in T_{t}}X_{s}X_{s}^{\prime}$ tends to infinity as $t\rightarrow\infty$ ([CHY1999, FCGS2010]). ### 3.2 Algorithm: DUCB-GLCB Denote $G_{t}=\sum_{s=1}^{t-1}\mathbb{I}\\{s+D_{s}\geq t\\}$ as the number of missing reward when the agent is making a prediction at round $t$. Further denote $W_{t}=\sum_{s\in T_{t}}X_{s}X_{s}^{\prime}$ as the matrix consisting feature information with timestamps in $T_{t}$ and $V_{t}=\sum_{s=1}^{t-1}X_{s}X_{s}^{\prime}$ as the matrix consisting all available features at the end of round $t-1$. Then the main algorithm is defined as follows. Algorithm 1 DUCB-GLCB 1: Input: the total rounds $T$ , model parameters $d$ and $\kappa$, and tuning parameters $\tau$ and $\delta$. 2: Initialization: randomly choose $\alpha_{t}\in[K]$ for $t\in[\tau]$, set $V_{\tau+1}=\sum_{i=1}^{\tau}X_{s}X_{s}^{\prime}$, $T_{\tau+1}:=\\{s\,:\,s\leq\tau,s+D_{s}\leq\tau\\}$, $G_{\tau+1}=\tau-|T_{\tau+1}|$ and $W_{\tau+1}=\sum_{s\in T_{\tau+1}}X_{s}X_{s}^{\prime}$ 3: for $t=\tau+1,\tau+2,\cdots,T$ do 4: Update Statistics: calculate the MLE $\hat{\theta}_{t}$ by solving $\sum_{s\in T_{t}}(Y_{s}-g(X_{s}^{\prime}\theta))X_{s}=0$ 5: Update Parameter: $\beta_{t}=\frac{\hat{\sigma}}{\kappa}\sqrt{\frac{d}{2}\log\left(1+\frac{2(t-G_{t})}{d}\right)+\log(\frac{1}{\delta})}+{\sqrt{G_{t}}}$ 6: Select Action: choose $a_{t}=\arg\max_{a\in[K]}\left(x_{t,a}^{\prime}\hat{\theta}_{t}+\beta_{t}\|x_{t,a}\|_{V_{t}^{-1}}\right)$ 7: Update Observations: $X_{t}\leftarrow x_{t,a_{t}}$, $V_{t+1}\leftarrow V_{t}+X_{t}X_{t}^{\prime}$ and $T_{t+1}\leftarrow T_{t}\cup\\{s\,:\,s+D_{s}=t\\}$, $G_{t+1}=t-|T_{t+1}|$, and $W_{\tau+1}=W_{\tau}+\sum_{s:s+D_{s}=t}X_{s}X_{s}^{\prime}$ 8: end for ###### Remark 2 (Comparison to UCB-GLM Algorithm in [LLZ2017]) We make several adjustments to the UCB-GLM Algorithm in [LLZ2017]. First, in step 4 (statistics update), we only use data with timestamps in $T_{t}$ to calculate the estimator using MLE. In this step, using data without reward will cause bias in the estimation. Second, when selecting the action in step 5, parameter $\beta_{t}$ is updated adaptively at each round whereas in [LLZ2017], the corresponding parameter is constant over time. Moreover, in step 4, we choose to use $V_{t}$ to normalize the context vector $X_{t,a}$ instead of $W_{t}$. ### 3.3 Preliminary Analysis Denote $G_{t}^{*}=\max_{1\leq s\leq t}G_{s}$ as the running maximum number of missing reward up to round $t$. The property of $G_{t}$ and $G^{*}_{t}$ is the key to analyze the regret bound for both UCB and Thompson sampling algorithms. We next characterize the tail behavior of $G_{t}$ and $G^{*}_{t}$. ###### Proposition 1 (Properties of $G_{t}$ and $G_{t}^{\star}$) Assume Assumption 2. Denote $\sigma_{G}=\sigma\sqrt{2+q}$. Then, 1. 1. $G_{t}$ is sub-Gaussian. Moreover, for all $t\geq 1$, with probability $1-\delta$ $\displaystyle{G}_{t}\leq 2(\mu+M)+\sigma_{G}\sqrt{2\log\left(\frac{1}{\delta}\right)}+2\sigma_{G}^{2}\log C_{3}+1,$ (6) where $C_{3}=2\sigma^{2}+1$. 2. 2. With probability $1-\delta$, $\displaystyle{G}_{T}^{*}$ $\displaystyle\leq$ $\displaystyle 2(\mu+M)+\sigma_{G}\sqrt{2\log T}+2\sigma_{G}^{2}\log C_{3}$ (7) $\displaystyle+\sigma_{G}\sqrt{2\log\left(\frac{1}{\delta}\right)+2\log C_{3}\sigma_{G}\sqrt{2\log T}+2\log C_{3}}+1,$ where $G_{T}^{*}=\max_{1\leq s\leq T}G_{s}$. 3. 3. Define $W_{t}=\sum_{s\in T_{t}}X_{s}X_{s}^{\prime}$ where $X_{t}$ is drawn iid. from some distribution $\gamma$ with support in the unit ball $\mathbb{B}_{d}$. Furthermore, let $\Sigma:=\mathbb{E}[X_{t}X_{t}^{\prime}]$ be the second moment matrix, and $B$ and $\delta>0$ be two positive constants. Then there exist positive, universal constants $C_{1}$ and $C_{2}$ such that $\lambda_{\min}(W_{t})\geq B$ with probability at least $1-2\delta$, as long as $\displaystyle t\geq\left(\frac{C_{1}\sqrt{d}+C_{2}\sqrt{\log(\frac{1}{\delta})}}{\lambda_{\min}(\Sigma)}\right)^{2}+\frac{2B}{\lambda_{\min}(\Sigma)}+2(\mu+M)+\sigma_{G}\sqrt{2\log\left(\frac{1}{\delta}\right)}+2\sigma_{G}^{2}\log C_{3}+1.$ (8) A special case of Proposition 1-1 is when $D_{i}$’s are iid and $q=0$. Now assume $D_{i}\sim D$ are iid with exponential-decays: $\displaystyle\mathbb{P}(D-\mu_{I}\geq t)\leq\exp(-\frac{t}{2\sigma_{I}^{2}}),$ (9) and $\mu_{I}=\mathbb{E}D$. Then with probability $1-\delta$, we have $\displaystyle G_{t}-\mu_{I}\leq 2\sigma_{I}\sqrt{\log\left(\frac{1}{\delta}\right)}+1+4\sigma_{I}^{2}\log(2\sigma_{I}^{2}).$ (10) At a high level, the proof utilizes the fact that, with high probability, there will be a lot of zero terms in the summation $G_{t}=\sum_{s=1}^{t-1}\mathbb{I}(s+D_{s}\geq s)$ when $t$ is large. This is done by designing a sequence of stopping times for the successes. We highlight the idea by showing result (10) for the special case when $D_{t}$’s are iid and $q=0$. The full version of the proof is deferred to Appendix LABEL:proof. ###### Sketch of the proof.. Define $V=\sum_{i=1}^{\infty}\mathbb{I}(D_{i}-\mu_{I}\geq i)$ where $D_{i}\sim D$ are iid that satisfies (9). Now let us define the following sequence of stopping times, $(k\geq 1)$, $T(k)=\inf\\{t>T(k-1):D_{t}\geq t\\},$ where $T(k)$ is the time of the $k^{\text{t}h}$ success. Therefore, $\displaystyle\mathbb{P}(V\geq j)$ $\displaystyle=$ $\displaystyle\mathbb{P}(T(1)<\infty,T(2)<\infty,\cdots,T(j-1)<\infty,T(j)<\infty)$ (11) $\displaystyle=$ $\displaystyle\Pi_{k=1}^{j}\mathbb{P}\left(T(k)<\infty|T(i)<\infty\,\,\text{ for }\,\,i\leq k-1\right)$ $\displaystyle=$ $\displaystyle\Pi_{k=2}^{j}\mathbb{P}\left(T(k)<\infty|T(k-1)<\infty\right)\mathbb{P}\left(T(1)<\infty\right)$ (12) $\displaystyle\leq$ $\displaystyle\Pi_{k=1}^{j}\left(\sum_{i=k}^{\infty}\exp\left(-\frac{i}{2\sigma_{I}^{2}}\right)\right)$ (13) $\displaystyle\leq$ $\displaystyle\Pi_{k=1}^{j}\left(2\sigma_{I}^{2}\exp\left(-\frac{k-1}{2\sigma_{I}^{2}}\right)\right)$ (14) $\displaystyle=$ $\displaystyle(2\sigma_{I}^{2})^{j}\exp\left(-\frac{(j-1)j}{4\sigma_{I}^{2}}\right)$ (15) (11) holds by tower property. (12) holds since event $\\{T(k)<\infty|T(k-1)<\infty\\}$ is equivalent to event $\\{T(k)<\infty|T(j)<\infty\,\,{\text{f}or}\,\,j\leq k-1\\}$. Condition on $T(k-1)<\infty$, we have $\mathbb{P}\left(T(k)<\infty|T(k-1)<\infty\right)\leq\mathbb{P}(\cup_{j\geq k}\mathbb{I}(D_{j}\geq j))\leq\sum_{i=k}^{\infty}\exp\left(-\frac{i}{2\sigma_{I}^{2}}\right)$. The last inequality holds by the union bound. Therefore (13) holds. Finally, 14 holds by integration. Given (14), $V$ is sub-Gaussian and with probability $1-\delta$, $V\leq 2\sigma_{I}\sqrt{\log\left(\frac{1}{\delta}\right)}+1+4\sigma_{I}^{2}\log(2\sigma_{I}^{2}).$ Similarly, we can show that, for any $t\geq 1$, $G_{t}$ is sub-Gaussian. With probability $1-\delta$, we have $G_{t}-\mu_{I}\leq 2\sigma_{I}\sqrt{\log\left(\frac{1}{\delta}\right)}+1+4\sigma_{I}^{2}\log(2\sigma_{I}^{2}).$ ∎ Note that $G_{t}$ is sub-Gaussian even when $D$ has near-heavy-tail distribution ($p\in[0,1)$). ###### Remark 3 The proof of Proposition 1 is simple but essential. It fully utilizes the property that the sequence in $V$ has a lot of zero terms (with high probability). In particular, one will not be able to fully obtain the result if one uses the standard approach and directly works at the level of “the-sum- of-sub-Guassians-is-sub-Gaussian" and thereafter analyzing sum of sub-Gaussian constants, which is the method used in [zhou2019]. In order to drive this point home, we provide an approach in this direction using Hoeffding bound (Theorem LABEL:thm9). See Appendix LABEL:further_G. With such a approach, one can only handle the case when $q>0$, which excludes the most difficult scenario with exponential delays. With Hoeffding bound, the sub-Gaussian parameter for $V$ is of the form $\sigma=\sqrt{\sum_{i=1}^{\infty}\sigma_{i}^{2}}$ where $\sigma$ is the sub- Gaussian parameter for indicator function $\mathbb{I}(G_{i}\geq i)$. Intuitively speaking, this Hoeffding bound does not take into consideration of the sparsity in the sequence. Therefore, the argument cannot reach the limit for $q=0$. ### 3.4 Regret Bounds ###### Theorem 1 Assume Assumptions 1-2. Fix any $\delta$. There exists a universal constant $C:=C(C_{1},C_{2},M,\mu,\sigma_{0},\hat{\sigma},\,\sigma,\kappa)>0,$ such that if we run DUCB-GLCB with $\tau:=C\left(d+\log(\frac{1}{\delta})\right)$ and $\beta_{t}=\frac{\hat{\sigma}}{\kappa}\sqrt{\frac{d}{2}\log\left(1+\frac{2(t-G_{t})}{d}\right)+\log(\frac{1}{\delta})}+G_{t}$, then, with probability at least $1-5\delta$, the regret of the algorithm is upper bounded by $\displaystyle R_{T}$ $\displaystyle\leq$ $\displaystyle\tau+L_{g}\left[4\sqrt{\mu+M}\sqrt{Td\log\left(\frac{T}{d}\right)}+2^{7/4}\sqrt{\sigma_{G}}(\log T)^{1/4}\sqrt{d\log\left(\frac{T}{d}\right)T}+\frac{2d\hat{\sigma}}{\kappa}\log\left(\frac{T}{d\delta}\right)\sqrt{T}\right.$ (16) $\displaystyle\,\,+\left.2\sqrt{2Td\log\left(\frac{T}{d}\right)}\left(\sqrt{\sigma_{G}}\left({2\log\left(\frac{1}{\delta}\right)+2\log C_{3}\sigma_{G}\sqrt{2\log T}+2\log C_{3}}\right)^{1/4}\right.\right.$ $\displaystyle\,\,\left.\left.+\sqrt{1+2\sigma_{G}^{2}\log C_{3}}\right)\right]$ For parameter definition, we refer to Table LABEL:tab:parameters in Section LABEL:app:table. The proof of Theorem 1 consists of three steps. The first step is to construct a confidence ball associated with the adaptive parameter $\beta_{t}$ and show that the true parameter falls into the confidence ball with high probability. The second step is to upper bound the normalized context sequence $\sum_{t=\tau+1}^{\tau+n}\|X_{t}\|_{V_{t}^{-1}}$. And the last step is to utilize the property of $G_{t}$ and $G^{*}_{t}$ proved in Proposition 1. The details is deferred to Appendix LABEL:proof. Given the high probability bound in Theorem 1, one can show the expected regret bound without much of work. ###### Corollary 1 (Expected regret) Assume Assumptions 1-2. The expected regret is bounded by $\displaystyle\mathbb{E}[R_{T}]={O\left(d\sqrt{T}\log(T)+\sqrt{\sigma_{G}}\sqrt{Td}(\log(T))^{3/4}+(\sqrt{\mu+M}+\sigma_{G})\sqrt{Td\log\left({T}\right)}\right).}$ (17) Given the result in (16), (17) holds by choosing $\delta=\frac{1}{T}$ and using the fact that $R_{T}\leq T$. The highest order term $O(d\sqrt{T}\log(T))$ does not depend on delays. This result is in line with the non-contextual stochastic bandit literature ([JGS2013]). Delay impacts the expected regret bound in two folds. First, the sub-Gaussian parameter $\sigma_{G}$ and the mean-related parameter ${\mu+M}$ appears in the second-highest order term. Second, the sub-Gaussian parameter ${\sigma_{G}}$ appears in the third-order term. Note that here we include the log factors in deciding the highest order term, the second highest order term and so on. If we exclude the log terms, then both delay parameters impact the regret bound multiplicatively. ### 3.5 Tighter Regret Bounds for Special Cases When the sequence $\\{D_{s}\\}_{s=1}^{T}$ satisfies some specific assumptions, we are able to provide tighter high probability bounds on the regret. ###### Proposition 2 Given Assumptions 1-2, we have the following results. 1. 1. If there exists a constant $D_{\max}>0$ such that $\mathbb{P}(D_{s}\leq D_{\max})=1$ for all $s\in[T]$. Fix $\delta$. There exists a universal constant $C>0$ such that by taking $\tau=D_{\max}+C(d+\log(\frac{1}{\delta}))$, with probability $1-3\delta$, the regret of the algorithm is upper bounded by $\displaystyle R_{T}\leq\^{A}~\tau+L_{g}\left(2{\sqrt{D_{\max}}}\sqrt{2Td\log\left(\frac{T}{d}\right)}+\frac{2d\hat{\sigma}}{\kappa}\log\left(\frac{T}{d\delta}\right)\sqrt{T}\right).$ (18) 2. 2. Assume $D_{1},\cdots,D_{T}$ are iid non-negative random variables with mean $\mu_{I}$. There exists $C>0$ such that by taking $\tau:=C\left(d+\log(\frac{1}{\delta})\right)$, with probability $1-5\delta$, the regret of the algorithm is upper bounded by $\displaystyle R_{T}\leq$ $\displaystyle\tau+L_{g}\left[{4\sqrt{\mu_{I}}}\sqrt{Td\log\left(\frac{T}{d}\right)}+{2^{7/4}\sqrt{\sigma_{G}}(\log T)^{1/4}}\sqrt{d\log\left(\frac{T}{d}\right)T}+\frac{2d\hat{\sigma}}{\kappa}\log\left(\frac{T}{d\delta}\right)\sqrt{T}\right.$ $\displaystyle\,\,+\left.2\sqrt{2Td\log\left(\frac{T}{d}\right)}\left(\sqrt{\sigma_{G}}\left({2\log\left(\frac{1}{\delta}\right)+2\log C_{3}\sigma_{G}\sqrt{2\log T}+2\log C_{3}}\right)^{1/4}\right.\right.$ $\displaystyle\,\,\left.\left.+\sqrt{1+2\sigma_{G}^{2}\log C_{3}}\right)\right]$ When delays $\\{D_{s}\\}_{s=1}^{T}$ are bounded by $D_{\max}$, the delay paramter $D_{\max}$ only appears in the term $\sqrt{Td\log{{T}}}$ and does not affect the highest order term $d\log(\frac{T}{d\delta})\sqrt{T}$. Compared to (17), there is no regret term on the order of $O(\sqrt{Td}\left(\log(T)\right)^{3/4})$ in (18). This is because we can provide a smaller number on the right hand side of (8) when delays are bounded. When delays are iid, $\mu+M$ is replaced by $\mu_{I}$, which is the common expectation of all the random delays. We refer to Appendix LABEL:proof for the proof of Proposition 2. ## 4 Delayed Thompson Sampling (DTS) for GLCB In section 3, under the frequentist set-up, we assume there exists a true parameter $\theta^{*}$ and use UCB to encourage exploration and construct the confidence interval for $\theta^{*}$. On the contrary, posterior sampling does not make use of upper confidence bounds to encourage exploration and instead relies on randomization. In this section, we operate in the Bayesian decision making setting and assume the decision maker is equipped with a prior distribution on $\theta^{*}$. In this setting, the standard performance metric is Bayesian regret, defined as follows: $R^{B}_{T}(\pi)=\mathbb{E}_{\theta^{*},x}[R_{T}(\pi,\theta^{*})]=\sum_{t=1}^{T}\mathbb{E}_{\theta^{*},x}\left[g\left(x_{t,a^{*}_{t}(\theta^{*})}^{\prime}\theta^{*}\right)-g\left(\ x_{t,a_{t}}^{\prime}\theta^{*}\right)\right],$ where $a_{t}\sim\pi_{t}$. Next, we present the Thompson sampling algorithm when adapted to the delayed setting. Algorithm 2 provides a formal description. Algorithm 2 DTS-GLCB 1: Input: the total rounds $T$ , tuning parameter $\tau$, prior $Q_{0}$ 2: Initialization: randomly choose $\alpha_{t}\in[K]$ for $t\in[\tau]$ 3: Update information: $\mathcal{F}_{\tau}$ according to (4). 4: if $\mathcal{I}_{\tau}=\emptyset$ then 5: $Q_{1}(\theta)=Q_{0}(\theta)$ 6: else 7: $Q_{1}(\theta)\propto Q_{0}(\theta|\mathcal{F}_{\tau})\Pi_{(s,x_{s},a_{s},y_{s,a_{s}})\in\mathcal{I}_{\tau}}\mathbb{P}(y_{s,a_{s}}|\theta,x_{s,a_{s}})$ 8: end if 9: for $t=1,2,\cdots,T-\tau$ do 10: Sample Model: $\hat{\theta}_{t+\tau}\sim Q_{t}$ 11: Select Action: $\bar{a}_{t+\tau}\in\arg\max_{a\in[K]}\left\langle x_{t+\tau,a},\hat{\theta}_{t+\tau}\right\rangle$ 12: Update information: $\mathcal{F}_{t+\tau}$ according to (4). Define $\mathcal{I}_{t+\tau}:=\mathcal{F}_{t+\tau}-\mathcal{F}_{t+\tau-1}$ as the new information at round $t+\tau$ 13: if $\mathcal{I}_{t+\tau+1}=\emptyset$ then 14: $Q_{t+1}(\theta)=Q_{t}(\theta)$ 15: else 16: $Q_{t+1}(\theta)\propto Q_{t}(\theta|\mathcal{F}_{\tau+t})\Pi_{(s,x_{s},a_{s},y_{s,a_{s}})\in\mathcal{I}_{t+\tau+1}}\mathbb{P}(y_{s,a_{s}}|\theta,x_{s,a_{s}})$ 17: end if 18: end for ###### Remark 4 Note that in Algorithm 2, there is an exploration period of length $\tau$. The posterior distribution employed at round $\tau+1$ is conditioned on observations made over the first $\tau$ time rounds. Another point to note is that Algorithm 2 is kept at an abstract level. The exact computation depends on the prior chosen and the exponential family. Note that every exponential family has a conjugate prior ([DY1979]), which admits efficient posterior update. Section 4.1 provides a concrete example on linear contextual bandits, which is a simple special case. We use this special case to illustrate how one can perform efficient incremental update in the presence of delays. ### 4.1 Delayed Thompson Sampling For Linear Contextual Bandits When $g(x)=x$ and $m(x)=\frac{x^{2}}{2}$, (1) reduces to $\displaystyle\mathbb{P}(Y|X)=\exp\left(\frac{YX^{\prime}\theta^{*}-(X^{\prime}\theta^{*})^{2}/2}{h(\eta)}+A(Y,\eta)\right).$ (19) Recall, from Bayes’ theorem, the posterior distribution is equal to the product of the likelihood function $\theta\rightarrow\mathbb{P}(y|\theta)$ and prior $\mathbb{P}(\theta)$, normalized by the probability of the data $\mathbb{P}(y)$: $\mathbb{P}(\theta|y)=\frac{\mathbb{P}(y|\theta)\mathbb{P}(\theta)}{\int\mathbb{P}(y|\theta)\mathbb{P}(\theta^{\prime})d\theta^{\prime}}$ Different choices of the prior distribution $\mathbb{P}(\theta)$ may make the integral more or less difficult to calculate. Moreover, the product $\mathbb{P}(y|\theta)\mathbb{P}(\theta)$ may take one form or another. But for certain choices of the prior, the posterior will have the same form as the prior, with possibly different parameter values. Such a choice is a conjugate prior. The conjugate prior, giving a closed-form expression for the posterior, makes Thompson sampling efficient to update. Further notice that, every exponential family has a conjugate prior ([DY1979]). Now we consider the normal conjugate prior for the linear model (19). Let $B_{t}=aI_{d}+\sum_{s\in T_{t}}x_{s,a_{s}}x_{s,a_{s}}^{\prime}$ and $\theta_{t}=B_{t}^{-1}\left(\sum_{s\in T_{t}}x_{s,a_{s}}y_{s,a_{s}}\right)$. Given the linear model (19), suppose we have $Y|X$ is Gaussian with $\mathcal{N}(X^{\prime}{\theta},v^{2})$. If the prior for $\theta$ at round $t$ is given by $\mathcal{N}(\theta_{t},v^{2}B_{t}^{-1})$, then it is easy to verify that the posterior distribution at around $t+1$ is $\mathcal{N}(\theta_{t+1},v^{2}B_{t+1}^{-1})$. Then Algorithm 2 becomes Algorithm 3 DTS-LCB 1: Input: the total rounds $T$ , constant $v>0$ and $a\geq 0$, tuning parameter $\tau$, conjugate prior $\mathcal{N}(\theta_{0},v^{2}B_{0}^{-1})$ with $\theta_{0}=0$ and $B_{0}=aI_{d}$, $f_{0}=0$ 2: Initialization: randomly choose $\alpha_{t}\in[K]$ for $t\in[\tau]$ 3: Update information: $\mathcal{F}_{\tau}$ according to (4). 4: if $\mathcal{I}_{t}=\emptyset$ then 5: $\theta_{1}=\theta_{0}$, $B_{1}=B_{0}$ 6: else 7: $B_{1}=\sum_{(s,x_{s},a_{s},y_{s,a_{s}})\in\mathcal{I}_{t}}x_{s,a_{s}}x_{s,a_{s}}^{T}$, $f_{1}=\sum_{(s,x_{s},a_{s},y_{s,a_{s}})\in\mathcal{I}_{t}}x_{s,a_{s}}y_{s,a_{s}}$ and $\theta_{1}=B_{1}^{-1}f_{1}$ 8: end if 9: for $t=1,2,\cdots,T-\tau$ do 10: Sample Model: $\hat{\theta}_{t+\tau}\sim\mathcal{N}(\theta_{t},v^{2}B_{t}^{-1})$ 11: Select Action: $\bar{a}_{t+\tau}\in\arg\max_{a\in[K]}\left\langle x_{t+\tau,a},\hat{\theta}_{t+\tau}\right\rangle$ 12: Update information: $\mathcal{F}_{t+\tau}$ according to (4). Define $\mathcal{I}_{t+\tau}:=\mathcal{F}_{t+\tau}-\mathcal{F}_{t+\tau-1}$ as the new information at round $t+\tau$ 13: if $\mathcal{I}_{t+\tau+1}=\emptyset$ then 14: $B_{t+1}=B_{t}$ 15: $\theta_{t+1}=\theta_{t}$ 16: else 17: $B_{t+1}=B_{t}+\sum_{(s,x_{s},a_{s},y_{s,a_{s}})\in\mathcal{I}_{t+\tau}}x_{s,a_{s}}x_{s,a_{s}}^{T}$,$f_{t+1}=f_{t}+\sum_{(s,x_{s},a_{s},y_{s,a_{s}})\in\mathcal{I}_{t+\tau}}x_{s,a_{s}}y_{s,a_{s}}$, and $\theta_{t+1}=B_{t+1}^{-1}f_{t+1}$ 18: end if 19: end for Note that the update (line 17) is on the incremental form which is practically efficient. ### 4.2 Regret Bounds Denote $\pi_{\tau}^{\text{PS}}$ as the posterior sampling policy described in Algorithm 2 with an exploration period $\tau$. We have the following result. ###### Theorem 2 Assume Assumptions 1-2. There exists a universal constant $C:=C(C_{1},C_{2},M,\mu,\sigma_{0},\sigma_{G},\,\sigma,\kappa)>0$, such that if we run exploration with $\tau:=C\left(d+\log(\frac{1}{\delta})\right)$, $\displaystyle R^{B}_{T}(\pi_{\tau}^{\text{PS}})={O\left(d\log T\sqrt{T}+\sqrt{\sigma_{G}}\sqrt{Td}(\log(T))^{3/4}+(\sqrt{\mu+M}+\sigma_{G})\sqrt{dT\log\left(T\right)}\right).}$ (20) For parameter definition, we refer to Table LABEL:tab:parameters in Section LABEL:app:table. We follow the steps in [RV2014] to prove the Bayesian regret bound in Theorem 2. The idea is the follows. We first decompose the Bayesian regret and the UCB the regret and build a connection between them. We then provide the Bayesian regret bound by utilizing a sequence of upper confidence bounds. We defer the details to Appendix LABEL:proof. When $\\{D_{s}\\}_{s=1}^{T}$ satisfies some specific assumptions, we are able to provide tighter Bayesian regret bounds. ###### Corollary 2 Assume Assumptions 1-2, we have the following result: 1. 1. If there exists a constant $D_{\max}>0$ such that $\mathbb{P}(D_{s}\leq D_{\max})=1$ for all $s\in[T]$. Then, $R^{B}_{T}(\pi_{\tau}^{\text{PS}})=O\left(d\log T\sqrt{T}+{\sqrt{D_{\max}}}\sqrt{dT\log T}\right).$ 2. 2. Assume $D_{1},\cdots,D_{T}$ are iid non-negative random variables with mean $\mu_{I}$. Then, $R^{B}_{T}(\pi_{\tau}^{\text{PS}})={O\left(d\log T\sqrt{T}+\sqrt{\sigma_{G}}\sqrt{Td}(\log(T))^{3/4}+(\sqrt{\mu_{I}}+\sigma_{G})\sqrt{dT\log\left(T\right)}\right).}$ We defer the proof of Corollary 2 to Appendix LABEL:proof. The results in Theorem 2 and Corollary 2 are comparable to the results in Section 3. ## 5 Extensions: History-dependent Delays In previous sections, we have analyzed the regret bounds for both DUCB-GLCB and DTS-GLCB when delays are independent. In practice, such independence assumption may not hold and current delays may depend on historical delays. In this section, we explore two types of dependency structures for the delays. In section 5.1, we discuss Markov delays where the stationary distribution is near-heavy-tail. In section LABEL:sec:random_delay, we discuss delays with random dependency structures but under a stronger assumption on the stationary distribution, which is lighter-than-sub-Gaussian. ### 5.1 Markov Delays ###### Assumption 3 (Markov Delay) Let $\\{D_{t}\\}_{t=1}^{T}$ be a stationary Markov chain on the general state space $\mathcal{X}=\mathbb{N}^{+}$ with invariant distribution $\pi$. Given $D\sim\pi$ with $\mu_{M}=\mathbb{E}[D]$, we further assume that $\mathbb{P}(D-\mu_{M}\geq x)\leq\exp\left(\frac{-x^{1+q}}{2\sigma_{M}^{2}}\right),$ for some $q>0$ and $\sigma_{M}>0$. Under Assumption 3, the stationary distribution $\pi$ can have near-heavy-tail property when $q$ is small. Recall that $G_{t}=\sum_{s=1}^{t-1}\mathbb{I}\\{s+D_{s}\geq t\\}$ is the number of missing reward and $G_{t}^{*}=\max_{1\leq s\leq t}G_{t}$ is the running maximum number of missing reward. Under Assumption 3, $G_{t}$ and $G_{t}^{*}$ has the following properties and again this is the key to analyze regret bounds for both DUCB and DTS. ###### Proposition 3 (Properties of $G_{t}$ and $G_{t}^{\star}$ under Markov delays) Assume Assumption 3 and $l_{2}$-spectral gap $1-\lambda\in(0,1]$. Then, 1. 1. For any $0<\delta<1$ and any $t$ we have, with probability at least $1-\delta$, $\displaystyle G_{t}-\mu_{M}\leq A_{2}(\lambda)\log\left(\frac{1}{\delta}\right)+\sqrt{2A_{1}(\lambda)\mu_{M}\log\left(\frac{1}{\delta}\right)},$ (21) where $A_{1}(\lambda)=\frac{1+\lambda}{1-\lambda}$ and $A_{2}(\lambda)=\frac{1}{3}\mathbb{I}(\lambda=0)+\frac{5}{1-\lambda}\mathbb{I}(\lambda>0)$. 2. 2. With probability at least $1-\delta$, $\displaystyle G_{T}^{*}\leq\mu_{M}+A_{2}(\lambda)\log\left(\frac{T}{\delta}\right)+\sqrt{2A_{1}(\lambda)\mu_{M}\log\left(\frac{T}{\delta}\right)},$ (22) where $G_{T}^{*}=\max_{1\leq t\leq T}G_{t}$. 3. 3. Define $W_{t}=\sum_{s\in T_{t}}X_{s}X_{s}^{\prime}$ where $X_{t}$ is drawn iid. from some distribution $\gamma$ with support in the unit ball $\mathbb{B}_{d}$. Furthermore, let $\Sigma:=\mathbb{E}[X_{t}X_{t}^{\prime}]$ be the second moment matrix, and $B$ and $\delta>0$ be two positive constants. Then there exist positive, universal constants $C_{1}$ and $C_{2}$ such that $\lambda_{\min}(W_{t})\geq B$ with probability at least $1-2\delta$, as long as $\displaystyle t\geq\left(\frac{C_{1}\sqrt{d}+C_{2}\sqrt{\log(\frac{1}{\delta})}}{\lambda_{\min}(\Sigma)}\right)^{2}+\frac{2B}{\lambda_{\min}(\Sigma)}+\mu_{M}+A_{2}(\lambda)\log\left(\frac{1}{\delta}\right)+\sqrt{2A_{1}(\lambda)\mu_{M}\log\left(\frac{1}{\delta}\right)}.$ (23) $\lambda$ is the $l_{2}$-spectral gap of the transition probability. We refer the formal concepts and the definition of $l_{2}$-spectral gap to [JSF2018, Section 2.2]. Proposition 3-1 is proved by utilizing the Berstein’s inequality for general Markov chains ([JSF2018, Theorem 1.1]) and Proposition 3-2 is proved by applying union bound. ###### Proof. Proof of Proposition 3. Recall $G_{t}=\sum_{s=1}^{t-1}\mathbb{I}\\{D_{s}\geq t-s\\}$. Define $f_{i}(D_{i})=\mathbb{I}\\{D_{s}\geq t-s\\}-p_{i}$ with $p_{i}=\mathbb{P}(D_{s}\geq t-s)$. Then $\mathbb{E}[f_{i}(D_{i})]=0$, $\mathbb{V}[f_{i}(D_{i})]=p_{i}(1-p_{i})\leq p_{i}$, and $\sum_{s=1}^{t-1}\mathbb{V}[f_{i}(D_{i})]\leq\sum_{s=1}^{t-1}p_{s}<\mu_{M}$. From [JSF2018, Theorem 1.1], we have $\displaystyle\mathbb{P}\left(\sum_{s=1}^{t-1}f_{i}(D_{i})>x\right)\leq\exp\left(-\frac{x^{2}}{2(A_{1}(\lambda)\mu_{M}+A_{2}(\lambda)x)}\right).$ (24) Note that the right hand side in (24) is independent of $t$. Technically speaking, this is because the summation of the variance $\sum_{s=1}^{t-1}\mathbb{V}[f_{i}(D_{i})]$ is upper bounded by $\mu_{D}$ which is independent of $t$. Therefore, Property 1 in Proposition 3 holds for any $t\geq 1$. Property 2 holds by the union bound and Property 1, $\displaystyle\mathbb{P}\left(\max_{1\leq t\leq T}G_{t}>\mu_{M}+A_{2}(\lambda)\log\left(\frac{T}{\delta}\right)+\sqrt{2A_{1}(\lambda)\mu_{M}\log\left(\frac{T}{\delta}\right)}\right)$ $\displaystyle\leq\sum_{t=1}^{T}\mathbb{P}\left(G_{t}>\mu_{M}+A_{2}(\lambda)\log\left(\frac{T}{\delta}\right)+\sqrt{2A_{1}(\lambda)\mu_{M}\log\left(\frac{T}{\delta}\right)}\right)\leq T\times\frac{\delta}{T}.$ Therefore, the following holds with probability no smaller than $1-\delta$, $\mathbb{P}\left(\max_{1\leq t\leq T}G_{t}<\mu_{M}+A_{2}(\lambda)\log\left(\frac{T}{\delta}\right)+\sqrt{2A_{1}(\lambda)\mu_{M}\log\left(\frac{T}{\delta}\right)}\right).$ ∎

2024-09-04T02:54:59.095205

2003.05196

arxiv-papers

# Uncovering the Data-Related Limits of Human Reasoning Research: An Analysis based on Recommender Systems Nicolas Riesterer Cognitive Computation Lab, University of Freiburg, email: <EMAIL_ADDRESS>Daniel Brand Cognitive Computation Lab, University of Freiburg, email<EMAIL_ADDRESS>Marco Ragni Cognitive Computation Lab, University of Freiburg, email<EMAIL_ADDRESS> ###### Abstract Understanding the fundamentals of human reasoning is central to the development of any system built to closely interact with humans. Cognitive science pursues the goal of modeling human-like intelligence from a theory- driven perspective with a strong focus on explainability. Syllogistic reasoning as one of the core domains of human reasoning research has seen a surge of computational models being developed over the last years. However, recent analyses of models’ predictive performances revealed a stagnation in improvement. We believe that most of the problems encountered in cognitive science are not due to the specific models that have been developed but can be traced back to the peculiarities of behavioral data instead. Therefore, we investigate potential data-related reasons for the problems in human reasoning research by comparing model performances on human and artificially generated datasets. In particular, we apply collaborative filtering recommenders to investigate the adversarial effects of inconsistencies and noise in data and illustrate the potential for data-driven methods in a field of research predominantly concerned with gaining high-level theoretical insight into a domain. Our work (i) provides insight into the levels of noise to be expected from human responses in reasoning data, (ii) uncovers evidence for an upper-bound of performance that is close to being reached urging for an extension of the modeling task, and (iii) introduces the tools and presents initial results to pioneer a new paradigm for investigating and modeling reasoning focusing on predicting responses for individual human reasoners. ## 1 Introduction The goal of human-level AI is currently approached from two directions: solving tasks with a performance similar to or even exceeding the one of humans [3], and understanding human cognition to a level that allows for an application to real-world problems [18]. While the first direction has seen major progress mainly fueled by the development of high-performant data-driven methods over the course of the last years, the second lags behind. Gaining insight into the processes underlying human cognition is the core focus of cognitive science, the inter-disciplinary research area at the junction of artificial intelligence, cognitive psychology, and neuroscience. Currently, this field of research is focused mainly on the psychological questions related to cognition. As a consequence, there is a distinct lack of readily available computational models developed for use in real-world applications such as human-like assistant systems. In this article we propose the use of methods from information retrieval to perform data analyses for investigating the remaining potential in modeling human cognition. In particular, we apply models from the family of collaborative filtering recommendation systems (for introduction see [15, 12]) to re-evaluate the theory-focused state of the art and illustrate the potential of a more data- driven approach to modeling human reasoning in one of its core domains: syllogistic reasoning. Syllogisms are one of the core domains of human reasoning research. They are concerned with categorical assertions of the form “All A are B; All B are C” consisting of two premises featuring a quantifier out of “All”, “Some”, “Some not”, and “No”, and three terms, A, B, and C, two of which are uniquely tied to their respective premise. Depending on the arrangement of terms, the syllogism is said to be in one of four figures (_A-B;B-C_ , _B-A;C-B_ , _B-A;B-C_ , _A-B;C-B_). When presented with syllogistic problems, the goal is to determine the logically valid conclusion out of the nine possibilities constructed by relating the two end terms of the premises via one of the four quantifiers (eight options), or to respond with “No Valid Conclusion” (NVC) if nothing else can be concluded. Featuring 64 distinct problems with nine conclusion options, the domain is well-defined, small enough to gain interpretable insight, but more detailed than most of its alternatives such as conditional reasoning (“If it rains, then the street is wet; It rains”). The domain of human syllogistic reasoning has seen an increase of interest in modeling over the last years. A meta-analysis compiled a list consisting of twelve accounts trying to provide explanations for the behavior of humans which differs drastically from formal logics [5]. However, since cognitive science follows a strongly theory-driven perspective on modeling, the focus of interest often rests on analyzing and comparing specific properties of models instead of their general predictive performance. Recent work identified a lack of predictive accuracy of cognitive models which raises concerns about their general expressiveness [13]. In this article, we briefly analyze the predictive accuracy of the state of the art in modeling human syllogistic reasoning and compare the results with data-driven models. In particular, we apply collaborative filtering-based recommender systems which exhibit properties making them promising tools for cognitive research. We leverage these properties to test structural assumptions about the syllogistic domain to analyze the data’s information content and the impact of noise on model performance. Finally, the implications for modeling reasoning and working with human data in general are discussed and ideas for improving the cognitive modeling problem are proposed. ## 2 Related Work Computational modeling has become one of the prime choices for formalizing knowledge and understanding about a domain of interest. By implementing intuition and assumptions into computationally tractable models, competing theories can be evaluated, progress in the understanding of a domain can be monitored, and finally, real-world applications can be solved [9]. The field of syllogistic reasoning has seen a rise of computational models. From initially only verbally described abstract theories [16], a recent meta- analysis compiled a list of twelve theoretical accounts for syllogistic reasoning, seven of which could be specified via tables relating syllogistic problems with sets of possible conclusions [5]. While these prediction tables still are far off fully specified implementations of the theoretical foundations, they can serve as a starting point for conducting model evaluation and comparison. The authors of the meta-analysis used the prediction data in order to determine strengths and weaknesses of the competing approaches when compared to dichotomized human response data via classification metrics (hits, misses, and false alarms). They found that while the approaches all exhibit distinct properties with respect to predictive precision, no single model could be determined as an overall winner. A recent analysis focusing on combining individual models’ strengths while avoiding their weaknesses took the evaluation of models one step further by avoiding the data aggregation step and focusing on the performance obtained from querying models for individual response predictions instead [13]. Their work revealed substantial lack of predictive performance of state-of-the-art models for syllogistic reasoning. Simultaneously, the authors demonstrated that data-driven modeling in form of a predictor portfolio, could be applied successfully to increase the predictive accuracy on the task. Information systems and machine learning as the fields concerned with data- driven model construction and optimization have seen an astonishing increase in popularity over the last years. Parts of this success have been due to an integration of features related to personalities of individuals [10]. Still, even though they share methods such as clustering, principle component analyses or mixed models, they have yet to enter the domain of cognitive research. Collaborative filtering as one of the default methods in the field of recommender systems has been successfully applied to model human reasoning before [6]. What makes this kind of memory-based collaborative filtering approaches promising for cognitive research in general is their high predictive capabilities paired with the similarity to the core assumption of cognitive science, that groups of people share similar reasoning patterns. Since recommendations are extracted from similarities between different features of the data or the users themselves, they allow both for an analysis of the data underlying the recommendation process, and an analysis of high- level theoretical assumptions which can be integrated directly into the model’s algorithmic structure (e.g., the integration of user personality [10, 7, 4]). The following sections contrast the models from cognitive science with collaborative filtering-based approaches in a general benchmarking setting for syllogistic reasoning based on predictive accuracy. ## 3 Benchmarking Syllogistic Models Figure 1: Overview over the model evaluation procedure. The benchmark selects a task which is fed to the model in order to obtain a prediction (black arrows). Simultaneously, by being based on experimental data, it simulates querying a human for a response (red arrows). After obtaining the model prediction, the true response is revealed to the model in an adaption step. The true (human) and model conclusions are collected and ultimately evaluated in terms of predictive accuracy. To gain an overview over the state-of-the-art’s performance in the prediction task, we performed a benchmark analysis using data obtained from an online experiment conducted on Amazon Mechanical Turk consisting of $139$ reasoners which responded to all $64$ syllogisms. Evaluations were computed relying on leave-one-out crossvalidation, i.e., by testing one reasoner and supplying the remaining $138$ as training data. The model evaluation procedure is inspired by a live prediction scenario where model predictions are retrieved simultaneously to the human reasoner selecting a conclusion. This is illustrated by Figure 1. In particular, our benchmark simulates this experiment by passing the tasks to a model generating predictions (black arrows). After a prediction is obtained, the model is supplied with the true response obtained from the human reasoner (red arrows). This allows models to perform an adaptation to an individual’s reasoning processes. Predictions and true responses are collected and finally compared to compute the predictive accuracy as the average number of hits. We included the cognitive models (matching, atmosphere, probability heuristics model, PHM; mental models theory, MMT; PSYCOP, conversion, verbal models) supplied with the meta-analysis on syllogistic reasoning by extracting the prediction tables [5]. Additionally, we included two baseline models, _Random_ and _MFA_. Random represents a lower bound of predictive performance defined by the strategy that always picks a random response out of the nine options. MFA denotes the most-frequent answer strategy which generates predictions by responding with the conclusion most frequently occurring in the training data. Finally, we included two variants of memory-based collaborative filtering. The user-based variant (UBCF) generates its prediction based on the responses of other users weighted by the similarity computed as the number of matching responses. The item-based variant (IBCF) compiles an item x item matrix $\mathbf{M}$ of corresponding responses (i.e., who responded with $x$ to syllogism $A$ also responded with $y$ to $B$) and a user vector $\mathbf{u}$ consisting of the user’s previous responses. The prediction is generated by selecting the highest-rated response for a syllogism from the result of the matrix-vector multiplication $\mathbf{M}\times\mathbf{u}$. Figure 2: Accuracies of models for human syllogistic reasoning. The plot includes cognitive models based on prediction tables reported by a recent meta-analysis by Khemlani & Johnson-Laird (2012; Probability Heuristics Model, PHM; Mental Models Theory, MMT; Matching, Atmosphere, PSYCOP, Conversion, VerbalModels), baseline models (Most Frequent Answer, MFA; Random), as well as user-based collaborative filtering (UBCF) and item-based collaborative filtering (IBCF). Figure 2 depicts the result of the benchmark analysis. The image highlights the difference between cognitive models and the recommenders. This is not too surprising since most cognitive models were not introduced with predictive performance in mind. They were originally based on some statistical effect (e.g., illicit conversion, a bias towards misinterpreting the direction of the input premises [1]) or a high-level cognitive theory (e.g., PSYCOP which assumes that reasoning is the result of interactions between different mental rules [14]) and are analyzed with respect to their qualities in reproducing aggregate effects of data. Still, the gap between cognitive models and data- driven approaches calls for a re-thinking of the goals of cognitive science. If the high-level insight cannot be integrated into successful models, their analysis is of limited use for advancing the understanding of human cognition. When observing the plot, special emphasis should be placed on MFA, the baseline model responding with the most-frequent answer of the training dataset. In terms of data-driven approaches, the MFA represents an upper bound of performance for models which do not take inter-individual differences into consideration. Since the cognitive models we considered for our analysis lack computational mechansisms for handling differences between reasoners, they are not expected to score higher than the $45\%$ achieved by MFA. In general, models can only hope to score higher if they rely on an active adaption to information about an individual’s reasoning processes such as previous responses or other personality traits known to influence cognition such as working memory capacity [17]. Being defined on an explicit database of information, collaborative filtering is an ideal tool for data analysis and modeling. They allow researchers to directly incorporate knowledge about the domain into the recommendation process and thereby to directly evaluate the value of findings in rigorous modeling scenarios. However, since this transformation of abstract findings is out of scope for this article and remains a challenge for future research, we do not focus on proposing an optimal recommender. We rather intend to highlight the method’s potential for future research in the domain by illustrating the levels of performance standard domain-agnostic implementations can achieve. Our benchmark shows that even domain-agnostic recommenders outperform cognitive models. Still, they do not manage to significantly surpass MFA. This could mean (i) that these models fail to recognize the reasoning strategies underlying the data, or (ii) that human reasoning is too irregular, i.e., too prone to uncontrollable noise for the approaches to succeed. In the following section we analyze artificially generated data in order to gain further information about the reasons behind the limited predictive performance of syllogistic models. ## 4 Simulation Analysis A core assumption of cognitive science is that reasoning is the result of different processes [2]. Depending on the individual state of the reasoner (e.g., previous experience or concentration), thorough inferences based on the rules underlying formal logics can be conducted or simple heuristic rules can be applied to reach a conclusion. Consequently, when assessing reasoning data, it is usually assumed that the data at hand is the result of multiple interleaved strategies which need to be disentangled in order to allow for an interpretable analysis. ### 4.1 Entropy Analysis High information content in data is essential for the success of data-driven methods. If the data consists mostly of random effects with little structure, patterns cannot be recognized to base future predictions on. A common measure of information is the Shannon entropy $S$: $S=-\sum_{i}p_{i}\log_{2}p_{i}$ Entropy can be understood as a measure of unpredictability of a state defined via the probabilities $p_{i}$. In the case of syllogisms, entropy has previously been applied to quantify the difficulty of the 64 problems [5]. Higher entropy results from a more uniform spread of probability mass over the nine conclusion options and thus serves as an indicant for a more difficult task. Figure 3: Relationship between syllogistic problems of varying entropy and model performances. Dotted lines represent interpolations between the data points. Figure 3 depicts the entropies of syllogistic problems with corresponding model performances. It shows a distinct gap in performance between the recommenders (IBCF, UBCF) and the remaining models. For low entropies, the recommenders are able to leverage the information encoded in the data resulting in high predictive accuracies. For higher entropies they are unable to maintain their initial distance to the cognitive models which are much more stable overall. Entropy in reasoning data can originate from (i) inconsistencies in the response behavior of individual human reasoners or (ii) interactions between independent reasoning strategies. The former point is a general issue of psychological and cognitive research since human participants are prone to lose attention due to boredom or fatigue. As a result, inconsistent and even conflicting data of single individuals can emerge [11]. Especially for collaborative filtering-based models this introduces substantial problems since users might not even be useful predictors for themselves. The latter point is a core challenge of cognitive science. Since reasoners differ with respect to their levels of education and experience with the task [8], recorded datasets are likely to be the result of a large number of individual strategies. For modeling purposes, the implications of both points differ greatly. Since inconsistencies due to lack of attention lead to behavior similar to guessing, it is unlikely for models to capture these effects by relying on behavioral data alone. Interactions between different strategies, on the other hand, are much more likely to be disentangled given additional insight into the domains and inter-individual differences between reasoners. Unfortunately, though, with the limited features currently contained in reasoning datasets, i.e., the responses, it is impossible to safely attribute the entropy of the data to either point. In the following sections, we therefore focus on collaborative filtering to shed light on the general capabilities of data-driven models in trying to uncover additional information about the problems of the domain. ### 4.2 Strategy Simulation Even though data-driven recommenders are able to achieve higher accuracies when compared to cognitive models, they are still far from perfectly predicting an individual reasoner. To investigate the remaining potential in the syllogistic domain, we need to gain an understanding of potential issues with the data. This second analysis considers artificial data with controlled levels of noise. Four of the cognitive models from the literature (Atmosphere, Matching, First-Order Logic, Conversion) were implemented and assigned to one of the four figures, respectively. By permuting the model-figure assignment and generating the corresponding response data we obtain $256$ artificial reasoners featuring interleaving strategies. The informativeness of this data is reduced by additionally introducing varying levels of random noise obtained from replacing conclusions with a random choice out of the nine conclusion options. With increasing levels of noise, the data should be less accessible for data-driven models. Figure 4: Strategy reconstruction performance of models based on artificial reasoning data with different levels of noise added by replacing a certain proportion of responses with a random choice from the nine conclusion options. The left and right images contrast performance with the raw noise proportions and entropies, respectively. Figure 4 depicts the performance of the baseline and data-driven models on the artificial data. The left image plots the different noise levels against predictive accuracies. It shows that a decrease in response consistency has drastic effects on the models’ capabilities to correctly predict responses due to the lack of information contained in the training data. The nearly linear relationship between the levels of noise and performance suggests that the models are stable in performance given the amount of reconstructable information. Consequently, they allow for a data-analytic assessment of “noise” in the data they are supplied with. In the case of syllogistic reasoning this means that close to $50\%$ of the data would effectively be indistinguishable from random noise. Explanations for this could be numerous ranging from too little data with respect to the number of possible reasoning strategies, over a lack of descriptive features, to guessing-like behavior, i.e. strategy-less decision-making on the side of study participants. The right image of Figure 4 presents a different perspective on the impact of noisy data by computing corresponding entropies. Again, it shows that entropy is tightly linked to predictive accuracies. By comparison with the Figure 3, some interesting conclusions can be drawn. In general, IBCF scores lower on the artificial data than on human data. Since IBCF is based on item-item dependencies, it is unable to directly exploit structural patterns of the data. It bases its predictions solely on information about “reasoners responding x to problem A also respond with y to problem B”. Higher performance on the human data therefore suggests the existence of preferential clusters of reasoners which exhibit similar response behavior. Since the artificial data does not feature such groups but puts more focus on the structural information by evenly distributing the permutations of model-figure combinations, IBCF is at a disadvantage. While we cannot formally attribute the entropy observed in the human data to inconsistencies due to random noise, or varying overlap between distinct reasoning strategies, the properties of IBCF suggest the existence of key responses or groups exhibiting similar research patterns in the data which allow the method to perform some form of clustering to boost its accuracy. This can be interpreted as soft evidence for the second hypothesis, that the current problem with modeling syllogistic reasoning stems from the fact that features allowing for a disentanglement of strategies are scarce. A possibility to overcome these problems for the short-term progress of the field is by explicitly integrating assumptions about the structural properties of the data into models. If accurate enough, they should be able to boost models’ capabilities to disentangle the overlapping strategies and allow for a general improvement of performance. Additionaly, the converse is true: if high-level theoretical assumptions lead to a significant improvement of the predictor, the theory is on the right track. ### 4.3 Potential for Better Predictions It appears as if a lack of information preventing the identification of strategies limits the potential of modeling in the domain of syllogistic reasoning. In general, there are two options to tackle this problem: improving models and extending the problem domain. There exist many possibilities to increase the predictive capabilities of models. On the one hand, additional features known for influencing reasoning patterns such as education [8] or working memory [17] can be integrated into the data to boost a model’s ability to identify patterns. On the other hand, the model can be supplied with background information about the problem domain. Since cognitive science has a history of in-depth data analysis there is a lot of potential for integrating theoretical findings into models. We propose the use of collaborative filtering as an accessible tool for cognitive scientists to transform abstract insight into testable models. Figure 5 illustrates the potential of recommenders for insight-driven research by contrasting item-based collaborative filtering (IBCF) and user-based collaborative filtering (UBCF) with variants of them tuned to the structure of the artificially generated data, i.e., the observation that syllogisms of the same figure rely on the same inference mechanism. The plot highlights that this additional information about the data is able to push both IBCF and UBCF far beyond their initial performance. Especially for IBCF, the explicit integration of the structural foundation of the data lifts its performance to the same levels of UBCF. The gap between the domain-agnostic and tuned variants remains clearly visible even for high levels of noise. Even though explicit information about the structure of human data can only be approximated from theoretical insight into the domain, this shows that recommenders would be a useful tool for assessing the quality of assumptions. The second option to improve modeling of human reasoning is to extend the domain in question. If information about individuals is accumulated even across the borders of reasoning domains, models have more data to recognize descriptive patterns in. Additionally, it is possible to include distinctive background information about individuals such as personality traits. This approach has proven to boost performance in recommendation scenarios before and is likely to generalize to the reasoning domain [4, 7]. However, since the extension of the domain is out of scope for this work, we leave this idea open for future research. For research on human reasoning this final analysis shows that there exist data-driven methods which benefit from the integration of the kind of information that is usually uncovered in cognitive science and psychology. By integrating correlative insight into these kinds of models, the value of the findings can be directly assessed in benchmarking evaluations. Paired with more informative problem domains obtained from a unification of multiple domains of reasoning, or the addition of personality features about individual reasoners, data-driven and theory-driven research can collaborate to overcome the distance between the current state of the art and the goal of human-level AI. ## 5 Conclusion Cognitive models for human syllogistic reasoning achieve unsatisfying accuracies when applied in a prediction setting. While the reasons for this could be numerous, it is interesting to see that data-driven recommenders based on collaborative filtering do not perform substantially better on an absolute scale. This raises concerns about the data foundation of reasoning research which is usually composed solely of reasoning problems along with the corresponding human responses. Our results obtained from comparison with artificially generated data suggest that data-driven models are unable to identify and successfully exploit patterns in the structure of human reasoning datasets when, in theory, they should be able to. The two most likely explanations for this are noise in form of inconsistencies in the response behavior of humans, or a lack of distinctive features preventing data-driven approaches to identify the patterns required for successful predictions. Figure 5: Comparison of item-based collaborative filtering (IBCF) and user- based collaborative filtering (UBCF) variants on artificially generated reasoning data. Fit-versions denote implementations where structural properties of the artificial data were actively integrated. In order to advance the predictive performance to levels which are relevant for applications in the era of human-level AI, reasoning research needs to address its current shortcomings. Potential solutions include the improvement of models by a better integration of domain-specific insight as well as an active consideration of inter-individual differences, and the extension of the task for example by including other domains of reasoning, recording more comprehensive datasets, and leaving behind the current focus on data aggregation. For integrating insight into models, we propose collaborative filtering recommenders as a general-purpose research method. On a technical level, they are easy to implement and understand, and outperform the current state of the art even in their domain-agnostic form. By integrating additional information about the domain (even if just on the level of correlations by weighting the dependencies between different features of the data), they allow for a transformation of abstract hypotheses into testable assumptions for modeling. Consequently, recommenders exhibit useful properties with respect to comprehensibility, especially in contrast to other methods from machine learning such as neural networks. As an example, they can naturally be applied to clustering contexts where stereotypical users are sought after. Generally, we see a need for an increased focus on predictive accuracies for individual reasoners to allow more comprehensive benchmarking, to allow for a more accessible interpretation of the results, and ultimately to enable the models for real-world application. To facilitate this shift in perspective for other researchers, we released the tools driving our predictive analysis as a general benchmarking framework111https://github.com/CognitiveComputationLab/ccobra. Only if the different disciplines of cognitive science find together to compete in modeling on unified informative domains using expressive and standardized metrics such as predictive performance, will human reasoning enter a level of progress relevant for human-level AI applications. This paper was supported by DFG grants RA 1934/3-1, RA 1934/2-1 and RA 1934/4-1 to MR. ## References * [1] Loren J Chapman and Jean P Chapman, ‘Atmosphere effect re-examined.’, Journal of Experimental Psychology, 58(3), 220, (1959). * [2] Jonathan St.B.T. Evans, ‘Dual-process theories of reasoning: Contemporary issues and developmental applications’, Developmental Review, 31(2-3), 86–102, (2011). * [3] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun, ‘Delving deep into rectifiers: Surpassing human-level performance on imagenet classification’, in Proceedings of the 2015 IEEE International Conference on Computer Vision (ICCV), ICCV ’15, pp. 1026–1034, Washington, DC, USA, (2015). IEEE Computer Society. * [4] Rong Hu and Pearl Pu, ‘Enhancing collaborative filtering systems with personality information’, in Proceedings of the Fifth ACM Conference on Recommender Systems, RecSys ’11, pp. 197–204, New York, NY, USA, (2011). ACM. * [5] Sangeet Khemlani and P. N. Johnson-Laird, ‘Theories of the syllogism: A meta-analysis.’, Psychological Bulletin, 138(3), 427–457, (2012). * [6] Ilir Kola and Marco Ragni, ‘Predict the individual reasoner: A new approach’, in Lecture Notes in Computer Science, 401–414, Springer International Publishing, (2018). * [7] Orestis Nalmpantis and Christos Tjortjis, ‘The 50/50 recommender: A method incorporating personality into movie recommender systems’, in Engineering Applications of Neural Networks, 498–507, Springer International Publishing, (2017). * [8] M. F. Nehrke, ‘Age, sex, and educational differences in syllogistic reasoning’, Journal of Gerontology, 27(4), 466–470, (1972). * [9] Allen Newell, ‘You can’t play 20 questions with nature and win: Projective comments on the papers of this symposium’, in Visual Information Processing, 283–308, Elsevier, (1973). * [10] David M. Pennock, Eric Horvitz, Steve Lawrence, and C. Lee Giles, ‘Collaborative filtering by personality diagnosis: A hybrid memory- and model-based approach’, in Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence, UAI’00, pp. 473–480, San Francisco, CA, USA, (2000). Morgan Kaufmann Publishers Inc. * [11] Marco Ragni, Nicolas Riesterer, Sangeet Khemlani, and Phil Johnson-Laird, ‘Individuals become more logical without feedback’, in Proceedings of the 40th Annual Conference of the Cognitive Science Society, eds., Tim Rogers, Marina Rau, Jerry Zhu, and Chuck Kalish, pp. 1584–1589, Austin, TX, (2018). Cognitive Science Society. * [12] Francesco Ricci, Lior Rokach, and Bracha Shapira, ‘Introduction to recommender systems handbook’, in Recommender Systems Handbook, 1–35, Springer US, (2010). * [13] Nicolas Riesterer, Daniel Brand, and Marco Ragni, ‘The predictive power of heuristic portfolios in human syllogistic reasoning’, in Proceedings of the 41st German Conference on AI, eds., Frank Trollmann and Anni-Yasmin Turhan, pp. 415–421, Berlin, Germany, (2018). Springer. * [14] Lance J Rips, The psychology of proof: Deductive reasoning in human thinking, Mit Press, 1994. * [15] Badrul Munir Sarwar, George Karypis, Joseph A Konstan, John Riedl, et al., ‘Item-based collaborative filtering recommendation algorithms.’, Www, 1, 285–295, (2001). * [16] Ron Sun, ‘Theoretical status of computational cognitive modeling’, Cognitive Systems Research, 10(2), 124–140, (2009). * [17] Heinz-Martin Süß, Klaus Oberauer, Werner W Wittmann, Oliver Wilhelm, and Ralf Schulze, ‘Working-memory capacity explains reasoning ability—and a little bit more’, Intelligence, 30(3), 261–288, (2002). * [18] Ingo J. Timm, Steffen Staab, Michael Siebers, Claudia Schon, Ute Schmid, Kai Sauerwald, Lukas Reuter, Marco Ragni, Claudia Niederée, Heiko Maus, Gabriele Kern-Isberner, Christian Jilek, Paulina Friemann, Thomas Eiter, Andreas Dengel, Hannah Dames, Tanja Bock, Jan Ole Berndt, and Christoph Beierle, ‘Intentional forgetting in artificial intelligence systems: Perspectives and challenges’, in Lecture Notes in Computer Science, 357–365, Springer International Publishing, (2018).

2024-09-04T02:54:59.108757

2003.05208

arxiv-papers

# In Situ Network and Application Performance Measurement on Android Devices and the Imperfections Mohammad A. Hoque University of Helsinki, Finland <EMAIL_ADDRESS>, Ashwin Rao University of Helsinki, Finland <EMAIL_ADDRESS>and Sasu Tarkoma University of Helsinki, Finland <EMAIL_ADDRESS> ###### Abstract. Understanding network and application performance are essential for debugging, improving user experience, and performance comparison. Meanwhile, modern mobile systems are optimized for energy-efficient computation and communications that may limit the performance of network and applications. In recent years, several tools have emerged that analyze network performance of mobile applications in situ with the help of the VPN service. There is a limited understanding of how these measurement tools and system optimizations affect the network and application performance. In this study, we first demonstrate that mobile systems employ energy-aware system hardware tuning, which affects application performance and network throughput. We next show that the VPN-based application performance measurement tools, such as Lumen, PrivacyGuard, and Video Optimizer, aid in ambiguous network performance measurements and degrade the application performance. Our findings suggest that sound application and network performance measurement on Android devices requires a good understanding of the device, networks, measurement tools, and applications. ## 1\. Introduction In situ Internet traffic measurement tools, such as Video Optimizer (VoP) (Qian:2011:PRU, ), Lumen (Razaghpanah:2017, ), PrivacyGuard (PvG) (Song:2015:PVP, ), and MopEye (Wu:2017:MOM:3154690, ), are essential for debugging, improving user experience, and performance comparison of mobile applications. The alternative is rooting the device and using _tcpdump_ for offline analysis. The above traffic measurement tools shed light on the network and application performance. However, they may also contribute to imperfect and ambiguous results, as we might measure something which we do not intend to measure. Studying the sources of these imperfections is vital to calibrate the measurement procedures and to improve the tools. At present, there is a limited understanding of the impact of in situ mobile Internet traffic measurement tools and how device hardware optimization affects the network and application performance. In this work, we quantify the performance impact of system hardware optimization and also evaluate the impact of VoP, Lumen, and PvG on network performance metrics, and application traffic. We focus on these three applications, as they exemplify state-of-the-art traffic measurement and analysis tools. These tools have similar designs and use the Android VPN interface. However, they do not route the traffic to a remote VPN server. VoP (vop, ), formerly known as ARO (Qian:2011:PRU, ), is a popular open-source tool for collecting traffic from mobile devices without rooting the device, and it also enables various diagnosis and optimization of applications, network, CPU and GPU (vop, ) through offline analysis. In contrast, Lumen and PvG are two online traffic analysis tool helping users to find privacy leaking incidents. Lumen also provides insights on the TLS usage of mobile applications (Razaghpanah:2017, ), the CDN usage by mobile applications (8485872, ), and the DNS (Almeida2017DissectingDS, ). MopEye is another similar application. It is currently unavailable in the Google Play Store and also in popular source code hosting websites, such as GitHub. This article investigates the imperfections in traffic measurements on Android devices due to system optimization and in situ traffic measurement tools. We demonstrate that sound Internet traffic measurement requires a thorough understanding of the device, tools, and applications. Note that we do not aim to establish whether a particular tool is the best or worst. Our key observations are as follows. _(1)_ Mobile systems employ CPU and WiFi transmit power optimization triggered by the battery level. We observe that the CPU optimization techniques, such as CPU hot-plugging and dynamic frequency scaling, mostly affect network I/O, while WiFi optimization, i.e., dynamic modulation scheme, affects the uplink throughput. These optimizations deteriorate application performance and network throughput. Charging the device, when the battery level is below 20%, does not improve the network performance. Therefore, one must be aware of the adaptive performance characteristics of mobile devices while conducting experiments (Section 2). _(2)_ Although it is expected that VPN-based tools would provide degraded network performance as the packets spend more time on the device (Qian:2011:PRU, ; Razaghpanah:2017, ; Song:2015:PVP, ), we may estimate ambiguous latency and throughput in the presence of the VPN-based tools. For example, in the presence of PvG, SpeedCheck (speedcheck, ) estimates on-device latency instead of the network latency. Similarly, VoP doubles the uplink throughput estimates. The sources of these ambiguities are the implementation of the measurement tools, as we present in Section 3. VoP also delays the outgoing traffic, and PvG delays the incoming traffic. Therefore, to avoid such pitfalls in network and application performance measurements, one must have a good understanding of these applications and tools. _(3)_ Furthermore, all these VPN-based applications fail to apply the application intended optimization through socket options and thus affect the application performance, as we demonstrate for the outgoing TCP traffic in Section 3. Finally, we summarize the sources of the above ambiguous or imperfect measurement results (Section 4). Figure 1. Impact of battery level. We consider two battery level (L) ranges, L$\leq$20% & L$>$20%, on Nexus 6 over WiFi (W) and LTE (4G). ## 2\. Impact of System Optimization Android devices may come with advanced CPU governors that save energy by hot plugging and unplugging of CPU cores, as supported by modern Linux kernels (cpuhotplug, ). Apart from workload characteristics, the devices may also consider the status of the battery to employ the CPU cores. We look into the impact of such off-the-shelf system optimization on network latency and throughput on Nexus 6. During our measurements with Nexus 6, we have found that two of the four cores remain offline when the battery discharges to below 20%, and the active cores operate at the maximum frequency of 1.73 GHz. When the battery level is above 20%, all the four cores become active, and their maximum operating frequency increases to 2.65 GHz. Therefore, the battery level also prompts dynamic frequency scaling. We performed the following measurements to quantify the impact of this optimization on the network traffic characteristics. Specifically, we used SpeedCheck (speedcheck, ) (paid) and measured the latency and throughput on Nexus 6 (Android 7.0) when the battery levels were above 20% and below 20%. We performed the measurements using both WiFi and LTE. Each of the above four scenarios was repeated ten times, and the results are presented in Figure 1. Figure 1 shows that while hot unplugging of CPU cores on Android has a negligible impact on the latency, its impacts on throughput is significant. The availability of additional CPU cores, when the battery level is above 20%, improves the I/O performance across the two access technologies, WiFi and LTE. Furthermore, WiFi uplink throughput improves almost four times when the battery level is above 20% compared to when it is below 20%. The closer inspections of the MAC layer frames revealed that WiFi radio of the Nexus 6 switches from _802.11ac_ to _802.11g_ mode when the battery level drops below 20%. These performance limiting optimizations also affected the device responsiveness for various applications, such as browsing and streaming. This also implies that modern Android devices adapt the physical layer mechanisms similar to the iOS devices111https://www.forbes.com/sites/ewanspence/2017/12/20/apple-iphone- kill-switch-ios-degrade-cripple-performance-battery/ to avoid unexpected shutdown of the devices (8720247, ) and to improve battery life. Figure 2. Impact of battery level on LTE modulation scheme. These snapshots are from a single uplink and downlink throughput measurement. Figure 1 also depicts that the downloading speed of SpeedCheck over LTE doubles when the battery level is higher than 20%. Similar to WiFi, we further looked into the physical layer modulation scheme used by the mobile device in the LTE network. We rooted Nexus 6 and installed Network Signal Guru (netsiguru, ) that samples LTE physical layer parameters after every 500 ms. Figure 2 shows that the modulation schemes were always 16QAM (Quadrature Amplitude Modulation) and 64QAM for uplink and downlink, respectively, during the throughput measurements. The other attributes in the figure are discussed in section 6. Nexus 6 employs three optimization techniques, triggered by the battery level, which affect the network and application performance. Charging the device, when the battery level is below 20%, does not improve the throughput either on WiFi or LTE and application performance. The optimization may vary from device to device. ## 3\. Impact of Measurement Tools Figure 3. The system components of VoP, Lumen, and PvG for Android.The newly created sockets are protected so that the Forwarder generated packets are not in a loop. Figure 4. Impact on LTE network latency and throughput. We used SpeedCheck and SpeedTest on Nexus 6 in the presence of Lumen (Lum.), VoP, PvG, and Baseline, i.e., without any localhost VPN. ### 3.1. In-situ Traffic Measurement Tools The forwarder and the packet inspector are two components of the VPN-based in situ traffic measurement tools exemplified by VoP, Lumen, and PvG, as shown in Figure 3. The primary role of the forwarder is to forward (i) the packets received from Android applications to the Internet, and (ii) the packets received from the Internet to the Android applications. The forwarder also copies those packets to the inspection queue to isolate traffic analysis from the path of the packet. The forwarder essentially creates a new TCP socket on seeing a TCP SYN packet from the VPN interface. The forwarder in Lumen and VoP establish a socket connection with the remote server using connect() API before sending SYN-ACK to the application. PvG, on the other hand, establishes socket connection after replying with SYN-ACK. Later, we demonstrate how these implementations affect network performance measurements. The forwarder creates a new UDP socket when it detects a new UDP flow. These newly created sockets are protected so that packets from the newly created flows do not loop the _tun_ interface (vpnprot, ). A packet inspector is responsible for inspecting the packets in its queue. In the case of Lumen and PvG, the packet inspector performs the privacy analysis on the packets, whereas the VoP’s inspector sends packets to the desktop application. In the later sections, we quantify the impact of VoP, Lumen, and PvG on (a) the network performance, and (b) the network characteristics of applications. ### 3.2. Addressing Biases We took the following steps to ensure that the measurement results presented in the upcoming sections are not the artifacts of misconfigured tools and the measurement setup. (i) Battery level. For the upcoming measurements, we ensured that the devices had more than 80% charge. This is because mobile devices might restrict resources based on the battery level, as we have shown in section 2. (ii) Throughput throttling. VoP also offers to throttle downlink and uplink traffic. All the measurements in this paper were conducted without any throughput throttling. (iii) Software Auto Update. During the experiments, application and the auto system updates were disabled on mobile devices. (iv) Advertisements. We have purchased without ad subscriptions of SpeedCheck and SpeedTest to avoid any biases caused by the free versions. (a) Baseline (b) VoP (c) Lumen Figure 5. Inter-packets gaps of the VoIP applications. Baseline refers to the measurements without any localhost VPN. ### 3.3. Impact on Network Performance This section explores the network performance using SpeedCheck (speedcheck, ) and SpeedTest (speedtest, ). These two applications work as the traffic load generator without any VPN-based tools and in the presence of the listed VPN applications. Without any VPN scenario gives the baseline performance. SpeedCheck connects to its servers in Germany, and SpeedTest connects to the severs in the LTE operator network within a few kilometers from the mobile device. The measurements were repeated ten times. _(1) Latency._ Figure 4 (left) compares the network latency reported by two applications in the presence of the VPN-based tools. From the _tcpdump_ traces, we have identified that SpeedTest uses 10-12 requests/responses of few bytes (less than 100 Bytes) over a TCP connection to estimate the latency. SpeedTest estimates the baseline latency of 16-18 ms. This is expected, as the server was located at the operator’s network. It experiences 3-5 ms additional latency in the presence of Lumen and PvG, whereas VoP increases the latency by three-fold. This is due to the energy optimization strategy adopted by VoP, which we discuss in the upcoming sections. In contrast, SpeedCheck reports the median baseline network latency of about 45 ms. From the corresponding _tcpdump_ traces, we have identified 10 empty and consecutive TCP flows (without any data exchange) for each latency measurements. These flows suggest that SpeedCheck uses TCP connect() API to measure the latency. Both VoP and Lumen increase the median latency significantly. We speculate that these two take more time to set up new TCP flows. However, SpeedCheck underestimates the latency in the presence of PvG, which is the consequence of the sending SYN-ACK by the PvG forwarder before the connection is established with the remote server, as discussed in Section 3.1. _(2) Uplink Throughput._ Figure 4 (center) depicts that SpeedTest estimates higher uplink baseline throughput, as the server is in the LTE operator network. It uses multiple parallel TCP connections to estimate the throughput. Both Lumen and PvG reduce the throughput of SpeedTest/SpeedCheck by half compared to the baseline measurements. However, Lumen severely affects the uplink throughput measurements of the SpeedCheck. It uses a single TCP connection and sends a large amount of data. From an exception in the debug log, we characterized that Lumen’s forwarder cannot handle such volume. Interestingly, VoP doubles the uplink throughput of both applications. _(3) Downlink Throughput._ Figure 4 (right) demonstrates that SpeedTest measures similar downlink throughput in the presence of the VPN tools to the baseline. Lumen aids the highest throughput measurements with SpeedCheck. However, VoP and PvG degrade the throughput of SpeedCheck significantly. The typical network measurement tools, such as SpeedCheck and SpeedTest, can have different methods to estimate the latency and throughput. While their baseline estimates are reasonable, their estimates vary according to the implementation of the VPN tools. ### 3.4. Impact on Realtime Application (UDP) In this section, we investigate the traffic from three realtime applications; IMO, WhatsApp, and Skype. The versions of the apps used are presented in Table 1. While these applications fall into the broad category of messaging applications, their varying traffic characteristics help us to study the impact of the design of VoP and Lumen. We could not use these applications in the presence of PvG in several trials. We used a rooted Nexus 6 (Android 7.0) and a non-rooted LG G5 (Android 8.0) for these measurements. These apps exchange bi-directional encrypted UDP traffic. The conversations were two minutes long over LTE, and we ran 3 iterations in each of the following scenarios. We investigate their inter-packet gaps and bitrates. As the baseline, we initiated conversations between Nexus 6 and LG G5 using these apps without VoP or Lumen and captured traffic using _tcpdump_ on Nexus 6. We then repeated the experiments with VoP running on Nexus 6 and collected traffic from VoP. Finally, we used Lumen. Since Lumen does not store traffic, we captured traffic with _tcpdump_ on Nexus 6. | Baseline | VoP | Lumen ---|---|---|--- Application | (in/out) | (in/out) | (in/out) WhatsApp (v2.18) | 21/24 kbps | 23/16 kbps | 20/22 kbps IMO (v9.8) | 14/15 kbps | 14/13 kbps | 13/14 kbps Skype (v8.41) | 60/50 kbps | 55/44 kbps | 48/44 kbps Table 1. Average bitrates of UDP traffic flows from VoIP applications. _Baseline Results._ Figure 5(a) shows that IMO has the highest inter-packet gaps, and Skype packets have the smallest gaps. These apps also have distinct data rates with Skype having the highest data rate, as shown in Table 1. _Impact of VoP._ Compared to the baseline packet-gaps in Figure 5(a), VoP significantly alters the inter-packet gaps of outgoing UDP packets, as shown in Figure 5(b). Most of the outgoing packets across all applications have an inter-packet gap of about 100 ms. In contrast, the incoming packets have had similar distributions to the baseline. This delay is similar to the latency measurements with VoP discussed earlier. Table 1 shows that the outgoing data rates of Skype and Whatsapp reduce significantly, which we speculate to be a consequence of the delays introduced by VoP. _Impact of Lumen._ Figure 5(c) shows that with Lumen the inter-packet gaps of the outgoing packets are similar to the baseline measurements. Besides, the applications experience similar bitrates to the baseline and when using Lumen as shown in Table 1. ### 3.5. Impact on Realtime Application (TCP) We used Periscope (v1.24) to study the impact of VoP and Lumen on realtime TCP flows. Periscope’s live broadcast did not work in the presence of PvG. Periscope broadcasts over LTE across three different scenarios. We capture traffic on Nexus 6 using _tcpdump_ for baseline and Lumen scenarios. Similar to our observations for UDP traffic, we observed 100 ms inter-packet gap, as shown in Figure 6 (left). From the distribution of packet size in Figure 6 (right) (collected by VoP), we notice that more than 70% packets captured by VoP are larger than 1500 bytes. From Traffic traces, we have identified that VoP creates packets of a maximum of 65549 bytes for Periscope, and the uplink throughput measurements flow from SpeedCheck. Figure 6. Properties of uplink Periscope TCP flows. From the source code in Github, we have identified that VoP forwarder implements the maximum segment of 65535 bytes for the TCP flows. It accumulates traffic from the client application, and the segments reach the maximum size very quickly with very high bitrate traffic. This also explains how VoP aids in higher uplink throughput measurements presented in Section 3.3. Nevertheless, these massive TCP segments are eventually fragmented once written to the socket. Lumen has a very negligible impact on packets. ### 3.6. Analysis with Socket Options In this section, we investigate the performance of the VPN-based tools in processing the flows with TCP_NODELAY (Nagel’s algorithm) socket option on Nexus 6. We specifically look into this option, as it has a direct impact on the local delay and thus affects the performance of web browsing and other realtime applications, such as live broadcasting, crypto/stock exchange applications, on mobile devices. We developed a separate traffic generating application that creates two blocking TCP sockets enabled and disabled Nagle’s algorithm. The application sends 1300 bytes data over LTE after every 20 ms to a remote server at the university campus. The application also receives data from the remote server after every 20 ms in separate TCP sessions. Figure 7. Distributions of the outgoing packet gaps observed at the network interface. Figure 8. Distributions of incoming packet gaps observed at the network interface and application. _Performance of VPN-based Tools._ Figure 7(a) compares the outgoing inter- packet gap of the application flows; having Nagel’s algorithm enabled and disabled. When Nagel’s algorithm is enabled, more than 70% of the packets sent from the application have more than 20 ms delays at the network layer. In the presence of VPN applications, disabling Nagel’s algorithm by the application does not improve the delay compared to the baseline (Figure 7(b)). Interestingly, VoP’s packet gap reduces, as it receives packets from the local TCP/IP stack without delay. From traffic traces, we have identified that these VPN-based tools do not disable Nagel’s algorithm while establishing socket connections. Figure 8 shows the performance of the VPN applications for incoming traffic. The application receives data at almost similar gaps observed at the network interface. However, in the presence of PvG, the application receives 40% packets at late. The packet-gaps patterns suggest that it uses a fixed interval to read the VPN interface similar to VoP. The investigations in this section reveal that the VPN-based tools do not set the TCP/IP socket options as intended by the other user applications. Consequently, they can misguide the developers and degrade application performance. For example, SpeedTest disables Nagel’s algorithm or sets the TCP_NODELAY socket option to send tiny packets to measure the network latency. Findings in this section explain the higher latency experienced by SpeedTest in Section 3.3. ## 4\. Sources of Imperfection Mobile system optimizations affect downlink and uplink throughput, whereas the VPN-based tools mostly affect the uplink throughput and latency, i.e., they mostly affect the outgoing traffic. In this section, we summarize the sources of such measurement results. _Energy-Aware Optimization._ Energy-aware system optimization can affect the network performance by limiting the network I/O and by applying adaptive modulation schemes. Therefore, it is wise to perform such measurements when the battery is fully charged. VoP, Lumen, and PvG rely on different sleeping techniques to optimize their energy usage. The additional latency introduced by VoP on outgoing packets is the artifact of using a fixed sleep interval of 100 ms in the main VPN thread. This delay further contributes to large outgoing packets for higher bitrate uplink traffic and energy consumption for fragmentation. PvG also introduces a fixed delay for the incoming traffic. Regardless, these delays affect not only the quality of the measurements but also the quality of experience when using other user applications. _Forwarder._ In situ VPN-based measurement tools are middleboxes that tap the packets using the VPN interface. These applications, therefore, implement a forwarder which primarily consists of three threads: the main VPN thread, and two-socket reader/writer threads. The reader/writer threads continuously iterate through a list of live sockets, which contributes to the delays. The forwarder also implements a flow state machine for each flow and constructs/de-constructs the packets. The implementation of the forwarder affects the latency and throughput measurements. We have also shown that the characteristics of the newly created flows and their packet headers might not be the same as those generated by the applications. The reason is that the socket options must be set before the connection establishment. ## 5\. Conclusions In this preliminary work, we investigated the challenges in measuring network performance in the presence of system optimizations and state-of-the-art application performance measurement tools on Android devices. System optimizations limit the performance of the hardware components and thus the applications, which in turn result in confusing measurement results. It can be argued that VoP is mostly for the developers, and therefore, incurring higher delays should not a problem. Similarly, frequent massive content uploading is rare, and 3-4 ms additional latency is acceptable. Nevertheless, these imperfections can significantly affect the outcome of traffic measurement studies. An acceptable latency also depends on the application type. A user can benefit significantly from 1-millisecond latency improvement for the financial and other realtime applications. Therefore, there is still room for improvement in such tools. For instance, VoP and PvG can follow Lumen’s adaptive sleeping algorithm for reducing the gaps in the outgoing and incoming packets, respectively. All of them can adopt some default socket options to mitigate the performance issues with the outgoing TCP traffic. Along with the measurement tools, it is necessary to understand the presence of various system optimization techniques which may affect network performance. ## References * [1] SPEEDCHECK - Speed Test. https://play.google.com/store/apps/details?id=org.speedspot.speedanalytics. [Online; accessed 7-August-2019]. * [2] Speedtest by Ookla. https://play.google.com/store/apps/details?id=org.zwanoo.android.speedtest.gworld. [Online; accessed 11-August-2019]. * [3] VPN - Android Developers. https://developer.android.com/guide/topics/connectivity/vpn. [Online; accessed 23-January-2019]. * [4] AT&T Video Optimizer. https://developer.att.com/video-optimizer, 2019. [Online; accessed 7-August-2019]. * [5] Network Signal Guru. https://play.google.com/store/apps/details?id=com.qtrun.QuickTest, 2019\. * [6] Mario Almeida, Alessandro Finamore, Diego Perino, Narseo Vallina-Rodriguez, and Matteo Varvello. Dissecting DNS Stakeholders in Mobile Networks. In Proceedings of CoNEXT ’17, pages 28–34. ACM, 2017. * [7] Mohammad Kawser, Nafiz Imtiaz Bin Hamid, Md Nayeemul Hasan, M Shah Alam, and M Musfiqur Rahman. Downlink snr to cqi mapping for different multiple antenna techniques in lte. International Journal of Information and Electronics Engineering, 2:756–760, 09 2012. * [8] The kernel development community. CPU hotplug in the Kernel. https://www.kernel.org/doc/html/latest/core-api/cpu_hotplug.html, 2019. [Online; accessed 11-September-2019]. * [9] F. Michclinakis, H. Doroud, A. Razaghpanah, A. Lutu, N. Vallina-Rodriguez, P. Gill, and J. Widmer. The Cloud that Runs the Mobile Internet: A Measurement Study of Mobile Cloud Services. In IEEE INFOCOM 2018 - IEEE Conference on Computer Communications, pages 1619–1627, April 2018. * [10] Feng Qian, Zhaoguang Wang, Alexandre Gerber, Zhuoqing Mao, Subhabrata Sen, and Oliver Spatscheck. Profiling Resource Usage for Mobile Applications: A Cross-layer Approach. In Proceedings of MobiSys’11, pages 321–334. ACM, 2011. * [11] Abbas Razaghpanah, Arian Akhavan Niaki, Narseo Vallina-Rodriguez, Srikanth Sundaresan, Johanna Amann, and Phillipa Gill. Studying TLS Usage in Android Apps. In Proceedings of CoNEXT ’17, pages 350–362. ACM, 2017. * [12] Yihang Song and Urs Hengartner. Privacyguard: A vpn-based platform to detect information leakage on android devices. In Proceedings of the 5th Annual ACM CCS Workshop on Security and Privacy in Smartphones and Mobile Devices, SPSM ’15, pages 15–26, New York, NY, USA, 2015. ACM. * [13] Y. Sun, L. Kong, H. Abbas Khan, and M. G. Pecht. Li-ion battery reliability – a case study of the apple iphone®. IEEE Access, 7:71131–71141, 2019. * [14] Daoyuan Wu, Rocky K. C. Chang, Weichao Li, Eric K. T. Cheng, and Debin Gao. MopEye: Opportunistic Monitoring of Per-app Mobile Network Performance. In Proceedings of USENIX ATC ’17, pages 445–457. USENIX Association, 2017. * [15] Jim Zyren. Overview of the 3GPP long term evolution physical layer. 01 2007. ## 6\. LTE Radio Resource Allocation In LTE networks, Physical Resource Block (RB) is considered as the unit of the radio resource. With 5 MHz bandwidth, there are 25 RBs. In an RB, there are 12 sub-carriers in the frequency domain. Each of the RBs can have either $7\times 12$ or $14\times 12$ resource elements (REs), where 7 and 14 are the symbols, in the time domain, over 0.5 and 1 ms respectively using normal cyclic prefix (CP) [15]. Now the amount of bits an RB can carry depends on the channel quality indicator (CQI) notification from the UE. Essentially, each CQI maps to a modulation and coding scheme according to Table 2. CQI indicates not only the channel quality but also a device’s capability whether the device can receive data of a particular modulation and coding scheme or not. The equations to compute the number bits an RB can hold for a certain CQI, and the number of RBs is required by an eNB to transmit a packet can be expressed as the followings. (1) $RB_{bits}=RE_{bits}\times n\times t_{s}\\\ =C_{CQI}\times M_{bits}\times n\times t_{s}$ In equation1, $M_{bits}$ is the bits for a modulation scheme, $n$ is the number of usable REs, and $t_{s}$ is the duration of time slot (0.5 or 1 ms). (2) $RB_{n}=(PacketSize_{bits}+RLC_{bits}+MAC_{bits})/RE_{bits}$ CQI | Modulation | Real Bits ($N_{m}$) | $C_{CQI}=N/1024$ ---|---|---|--- 1 | QPSK | 78 | 0.0762 2 | QPSK | 120 | 0.1171 3 | QPSK | 193 | 0.1884 4 | QPSK | 308 | 0.3 5 | QPSK | 449 | 0.4384 6 | QPSK | 602 | 0.5879 7 | 16QAM | 378 | 0.3691 8 | 16QAM | 490 | 0.4785 9 | 16QAM | 616 | 0.6015 10 | 64QAM | 466 | 0.4550 11 | 64QAM | 567 | 0.5537 12 | 64QAM | 666 | 0.6503 13 | 64QAM | 772 | 0.7539 14 | 64QAM | 873 | 0.8525 15 | 64QAM | 948 | 0.9258 Table 2. Channel Quality Index (CQI), Modulation Scheme, and Coding Rate mapping [7]. Figure 9 shows the usage of the Modulation Scheme and the number of resource blocks for a large file download on Nexus 6 with CQI11. LTE supports QPSK, 16QAM, and 64QAM, i.e., each RE can carry a maximum of 2, 4, and 6 bits accordingly. Let us consider the duration of 1 RB is 1 ms ($t_{s}$), and there are 168 REs. Nevertheless, mostly 120 REs ($n$) are available for carrying data traffic. For CQI11, the modulation scheme is 64QAM and the effective code rate $C_{CQI}=N_{m}/1024=0.55$. Therefore, an RE can hold only, $RE_{bits}=C_{CQI}\times M_{bits}=0.55\times 6$, 3.32 bits and an RB can hold $n\times RE_{bits}=398$ bits. Figure 9. LTE throughput and other network parameter observed on a mobile device using Network Signaling Guru [5]. The number of RBs required for a packet in a downlink can be computed using equation 2 by considering the additional bits for RLC and MAC headers. However, the network may not allocate the RBs according to the CQI. It may have other complex resource scheduling algorithms, as it has to deal with various types of traffic and users. The number of uplink RBs also may vary. ## 7\. Application ⬇ 1int val = 1; 2// Disabling Nagel’s Algorithm 3setsockopt(sockfd,SOL_TCP,TCP_NODELAY,&one,sizeof(one)); 4if (connect(sockfd, &servaddr, sizeof(servaddr)) < 0) 5 LOGE("[***Server Connect Error***"); 6for (int i = 0; i < 5000; i++) { 7 usleep(20000); 8 char *daat = rand_string(1300); 9 gettimeofday(&tv, NULL); 10 times[i] = (tv.tv_sec*1000000LL+tv.tv_usec)/1000; 11 n = write(sockfd,daat, 1300); 12 if (n < 0){ 13 LOGE("Error sendto %s", strerror(errno)); 14 break; 15 } 16} Listing 1: TCP sending code with/without Nagel’s algorithm. ⬇ 1int BUFSIZE = 4096; 2if (connect(sockfd, &servaddr, sizeof(servaddr)) < 0) 3 LOGE("[***Server Connect Error***"); 4while (true) { 5 bzero(buf, BUFSIZE); 6 n = read(sockfd, buf, BUFSIZE); 7 if (n > 0) { 8 gettimeofday(&tvo, NULL); 9 times[i]=(tvo.tv_sec*1000000LL+tvo.tv_usec)/1000; 10 i = i+1;} 11 else 12 break; 13 if (i==5000) 14 break; 15} Listing 2: TCP receiving code.

2024-09-04T02:54:59.119667

2003.05227

arxiv-papers

A spatio-temporal model to understand forest fires causality in Europe Oscar Rodriguez de Rivera1,*, Antonio López-Quílez2, Marta Blangiardo3, Martyna Wasilewska1 1 Statistical Ecology @ Kent, National Centre for Statistical Ecology. School of Mathematics, Statistics and Actuarial Science, University of Kent. Canterbury, UK. 2 Dept. Estadística i Investigació Operativa; Universitat de València. Valencia, Spain. 2 Faculty of Medicine, School of Public Health; Imperial College of London. London, UK. *<EMAIL_ADDRESS> ###### Abstract Forest fires are the outcome of a complex interaction between environmental factors, topography and socioeconomic factors (Bedia _et al_ , 2014). Therefore, understand causality and early prediction are crucial elements for controlling such phenomenon and saving lives. The aim of this study is to build spatio-temporal model to understand causality of forest fires in Europe, at NUTS2 level between 2012 and 2016, using environmental and socioeconomic variables. We have considered a disease mapping approach, commonly used in small area studies to assess the spatial pattern and to identify areas characterised by unusually high or low relative risk. _K_ eywords Hierarchical Bayesian models $\cdot$ disease mapping $\cdot$ integrated nested laplace approximation $\cdot$ forest fires $\cdot$ causality $\cdot$ spatio-temporal model ## 1 Introduction Nowadays, wildfires have become one of the most significant disturbances worldwide (Flannigan _et al_ , 2009; Pechony and Shindell, 2010; Pausas _et al_ , 2012; Cardil _et al_ , 2014; Boer _et al_ , 2017; Molina _et al_ , 2018). The combination of a longer drought period and a higher woody biomass and flammability of dominant species creates an environment conducive to fire spread (Piñol _et al_ , 1998; Millán _et al_ , 2005). Furthermore, vegetation pattern changes with the abandonment of traditional rural activities plays a direct role in the increase of fire severity and ecological and economic fire impacts (Flannigan _et al_ , 2009; Chuvieco _et al_ , 2014). Fire behavior exceeds most frequently firefighting capabilities and fire agencies have trouble in suppressing flames while providing safety for both firefighters and citizens (Werth _et al_ , 2016). Many areas across the world have seen a rise in extreme fires in recent years. Those include South America and southern and western Europe. They also include unexpected places above the Arctic Circle, like the fires in Sweden during the summer of 2018 (de Groot _et al_ , 2013; European Commission, 2019). Extreme fire events, which are also referred to as “megafires”, are becoming frequent on a global scale; recent fires in Portugal, Greece, Amazone and other areas confirm this fact. There is not complete agreement on the term “megafires”, which often refers to catastrophic fire events in terms of human casualties, economic losses or both (San-Miguel-Ayanz _et al_ , 2013b). Climate change will reduce fuel moisture levels from present values around the Mediterranean region and the region will become drier, increasing the weather- driven danger of forest fires. The countries in highest danger are Spain, Portugal, Turkey, Greece, parts of central and southern Italy, Mediterranean France and the coastal region of the Balkans, according to recent research of the Joint Research Centre (JRC) (De Rigo _et al_ , 2017). Most international reports on biomass burning recognize the importance of the human factors in fire occurrence (Food, 2007). Although fire is a natural factor in many ecosystems, human activities play a critical role in altering natural fire conditions, either by increasing ignitions (Leone _et al_ , 2003), or by suppressing natural fires (Johnson _et al_ , 2001; Keeley _et al_ , 1999). Both factors are contradictory, and act mainly through the mixture of fire policy practices, on one hand, and land uses and demographic changes on the other. Most developed countries have maintained for several decades a fire suppression policy, which has lead to almost total fire exclusion. The long term impact of that policy has implied an alteration of traditional fire regimes, commonly by increasing average burn severity and size, as a result of higher fuel accumulation (Pyne, 2001), although other authors are more critical about the real implication of fire suppression policy (Johnson _et al_ , 2001), or they tend to put more emphasis on the impact of climate changes (Westerling _et al_ , 2006). For developing countries, fire is still the most common tool for land clearing, and therefore it is strongly associated to deforestation, especially in Tropical areas (Cochrane _et al_ , 1999; DeFries _et al_ , 2002). The traditional use of fire in shifting cultivation has turned in the last decades to permanent land use change, in favour of cropland and grasslands. In addition, fire is a traditional tool to manage permanent grasslands, which are burned annually to favour new shoots and improve palatability (Hobbs _et al_ , 1991; Chuvieco _et al_ , 2010). Global and local implications of changing natural fire circumstances have been widely recognized, with major effects on air quality, greenhouse gas emissions, soil degradation and vegetation succession (Goetz _et al_ , 2006; Parisien _et al_ , 2006; Randerson _et al_ , 2005). The role of human activities in changing those conditions has not been assessed at global scale. Several local studies have identified factors that are commonly associated to human fire ignition, such as distance to roads, forest-agricultural or forest- urban interfaces, land use management, and social conflicts (unemployment, rural poverty, hunting disputes,) (Leone _et al_ , 2003; Martínez _et al_ , 2009; Vega-García _et al_ , 1995). On the other hand, humans not only cause fires, but they suffer their consequences as well. Fire is recognized as a major natural hazard (Food, 2007), which imply severe losses of human lives, properties and other socio-economic values (Radeloff _et al_ , 2005; Reisen and Brown, 2006). Fire is no longer a significant part of the traditional systems of life; however, it remains strongly tied to human activity (Leone _et al_ , 2009). Knowledge of the causes of forest fires and the main driving factors of ignition is an indispensable step toward effective fire prevention (Ganteaume _et al_ , 2013). It is widely recognized that current fire regimes are changing as a result of environmental and climatic changes (Pausas and Keeley 2009) with increased fire frequency in several areas in the Mediterranean Region of Europe (Rodrigues _et al_ , 2013). In Mediterranean-type ecosystems, several studies have indicated that these changes are mainly driven by fire suppression policies (Minnich, 1983), climate (Pausas _et al_ , 2012), and human activities (Bal _et al_ , 2011). Human drivers mostly have a temporal dimension, which is why an historical/temporal perspective is often required (Zumbrunnen _et al_ , 2011; Carmona _et al_ , 2012). In Mediterranean Europe, increases in the number of fires have been detected in some countries, including Portugal and Spain (San-Miguel-Ayanz _et al_ , 2013a; Rodrigues _et al_ , 2013). In addition, a recent work by Turco _et al_ (2016) suggests huge spatial and temporal variability in fire frequency trends especially in the case of Spain, where increasing and decreasing trends were detected depending on the analysis period and scale. This increase in wildfire frequency and variability, with its associated risks to the environment and society (Moreno _et al_ , 2011, 2014), calls for a better understanding of the processes that control wildfire activity (Massada _et al_ , 2013). In recent decades, major efforts have been made to determine the influence of climate change on natural hazards, and to develop models and tools to properly characterize and quantify changes in climatic patterns. While physical processes involved in ignition and combustion are theoretically simple, understanding the relative influence of human factors in determining wildfire is an ongoing task (Mann _et al_ , 2016). It is clear that human-caused fires that occur repeatedly in a given geographical area are not simply reducible to individual personal factors, and thus subject to pure chance. They are usually the result of a spatial pattern, whose origin is in the interaction of environmental and socioeconomic conditions (Koutsias _et al_ , 2015). This is particularly true in human- dominated landscapes such as Spain, where anthropogenic ignitions surpass natural ignitions, and humans interact to a large degree with the landscape, changing its flammability, and act as fire initiators or suppressors. In such cases, human influence may cause sudden changes in fire frequency, intensity, and burned area size (Pezzatti _et al_ , 2013). Fire is an integral component of Mediterranean ecosystems since at least the Miocene (Dubar _et al_ , 1995). Although humans have used fires in the region for tens of thousands of years (Goren-Inbar _et al_ , 2004), it is only in the last 10,000 or so that man has significantly influenced fire regime (Daniau _et al_ , 2010). The use of fire as a management tool has persisted until these days, although the second half of the past century saw a major change and a regime shift due to abandonment of many unproductive lands (Moreno _et al_ , 1988; Pausas _et al_ , 2012). Although fire still is a traditional management tool in some rural areas for control of vegetation and enhancement of pastures for cattle feed, most fires these days are no longer related to the management of the land (San-Miguel-Ayanz _et al_ , 2012, 2013a, 2013b). The European Mediterranean region is a highly populated area where nearly 200 Million people live in just 5 European Union countries, Portugal, Spain, France, Italy and Greece. Population density varies but remains very high with about 2500 inhabitants/km2 in the French Riviera (with peaks of up to 750,000 tourists per day during the summer) (Corteau, 2007) versus an average of 111 inhabitants/km2 in the region. The region is characterized by an extensive wildland urban interface (WUI). Large urban areas have expanded into the neighboring wildland areas, where expensive households are built. The WUI has been further increased by the construction of second holiday homes in the natural environment. Fire prone areas along the Mediterranean coast have been extensively built up, reducing in some cases the availability of fuels, but increasing largely the probability of fire ignition by human causes (Ganteaume _et al_ , 2013). In other areas of the same region, abandonment of the rural environment has lead to low utilization of forests, which are generally of limited productivity, and the subsequent accumulation of fuel loads (San- Miguel-Ayanz _et al_ , 2012, 2013a, 2013b; Moreira _et al_ , 2011). The combination of the above factors converts the European Mediterranean region in a high fire risk area (Sebastián-López _et al_ , 2008), especially during the summer months when low precipitations and very high temperatures favor fire ignition and spread. About 65,000 fires take place every year in the European region, burning, on average, around half a million ha of forest areas (European Commission, 2011). Approximately 85% of the total burnt area occurs in the EU Mediterranean region (San-Miguel-Ayanz _et al_ , 2010). Although fires ignite and spread under favorable conditions of fuel availability and low moisture conditions, ignition is generally caused by human activities. Over 95% of the fires in Europe are due to human causes. An analysis of fire causes show that the most common cause of fires is “agricultural practices”, followed by “negligence” and “arson” (Vilar Del Hoyo _et al_ , 2009; Reus Dolz _et al_ , 2003). Most fires in the region are small, as a fire exclusion (extinction) policy prevails in Europe. Fires are thus extinguished as soon as possible, and only a small percentage escapes the initial fire attack and the subsequent firefighting operations. An enhanced international collaboration for firefighting exists among countries in the European Mediterranean region. This facilitates the provision of additional firefighting means to those in a given country from the neighbouring countries in case of large fire events. The trend of large fires, those larger than 500 ha, is shown quite stable in the last decades (San-Miguel-Ayanz _et al_ , 2010). However, among these large fires, several fire episodes caused catastrophic damages and the loss of human lives (San-Miguel-Ayanz _et al_ , 2013a). A first step is to identify all the factors linked to human activity, establishing their relative importance in space and time (Martínez _et al_ , 2009, 2013). According to Moreno _et al_ (2014), the number of fires over the past 50 years in Spain has increased, driven by climate and land-use changes. However, this tendency has been recently reversed due to fire prevention and suppression policies. This highlights the influence of changes in the role of human activities as some of the major driving forces. For instance, changes in population density patterns—both rural and urban—and traditional activities have been linked to an increase in intentional fires. In this sense, several works have previously investigated the influence of human driving factors of wildfires in Spain. These works have explored in detail a wide range of human variables (Martínez _et al_ , 2009; Chuvieco _et al_ , 2010) and methods. Specifically, Generalized Linear Models (Vilar Del Hoyo _et al_ , 2009; Martínez _et al_ , 2009; Moreno _et al_ , 2014), machine learning methods (Lee _et al_ , 1996; Rodrigues and de la Riva, 2014), and more spatial-explicit models like Geographically Weighted Regression (Martínez _et al_ , 2013; Rodrigues _et al_ , 2014) have previously been employed. However, all these approaches could be considered as stationary from a temporal point of view, since they are based on ‘static’ fire data information summarized or aggregated for a given time span. However, the influence of human drivers cannot be expected to be stationary (Rodrigues _et al_ , 2016). Zumbrunnen _et al_ (2011) stress the importance of dealing with the temporal dimension of human drivers of wildfires. Therefore, exploring temporal changes in socioeconomic or anthropogenic drivers of wildfire will enhance our understanding of both current and future patterns of fire ignition, and thus help improve suppression and prevention policies (Rodrigues _et al_ , 2016). Disease risk mapping analyses can help to better understand the spatial variation of the disease, and allow the identification of important public health determinants (Moraga, 2018). Spatio-temporal disease mapping models are a popular tool to describe the pattern of disease counts and to identify regions with an unusual incidence levels, time trend or both (Schrödle and Held, 2011). This class of models is usually formulated within a hierarchical Bayesian framework with latent Gaussian model (Besag _et al_ , 1991). Several proposals have been made including a parametric (Bernardinelli _et al_ , 2014) and nonparametric (Knorr-Held, 2000; Lagazio _et al_ , 2003; Schmid and Held, 2004) formulation of the time trend and the respective space-time interactions. Areal disease data often arise when disease outcomes observed at point level locations are aggregated over subareas of study region due to constraints such as population confidentiality. Producing disease risk estimates at area level is complicated by the fact that raw rates can be very unstable in areas with small population and for rare circumstances, an also by the presence of spatial autocorrelation that may exist due to spatially correlated risk factors (Leroux _et al_ , 2000). Thus, generalised linear mixed models are often used to obtain disease risk estimates since they enable to improve local estimates by accommodating spatial correlation and the effects of explanatory variables. Bayesian inference in these models can be performed using Integrated Nested Laplace Approximation (INLA) approach (Rue _et al_ , 2009) which is a computational alternative to the commonly used Markov chain Monte Carlo methods (MCMC); INLA allows to run fast approximate Bayesian inference in latent Gaussian models. INLA is implemented in the INLA package for the R programming language, that provides an easy way to fit models via inla() function, which works in a similar way as other functions to fit models, such as glm() or gam() (Palmi- Perales _et al_ , 2019). Statistical reporting in the European Union is done according to the Nomenclature of Units for Territorial Statistics (NUTS) system. The NUTS is a five-level hierarchical classification based on three regional levels and two local levels. Each member state is divided into a number of NUTS-1 regions, which in turn are divided into a number of NUTS-2 regions and so on. There are 78 NUTS-1 regions, 210 NUTS-2 and 1093 NUTS-3 units within the current 15 EU countries (Eurostat, 2002). In this paper, we explore the application of these models to understand forest fires causality using environmental and socio-economic variables. We will work with areal data using the number of forest fires at NUTS-2 regional level in Europe and consider forest fires between 2012 and 2016. ## 2 Material and Methods We extend the analysis of globalization to the NUTS-2 regions of the 27 countries of the European Union (EU-27), as not all the regions have been included due to absence of information (forest fires or socio-economic data) (Figure 1). Figure 1: Study area, in grey administrative areas included in the analysis. Our main data set comprises the number of fires in Europe at NUTS-2 level, requested to the European Forest Fire Information System (EFFIS) (San-Miguel- Ayanz _et al_ , 2012). We have chosen this level due to the variables that we are interested to analyse (socioeconomic and environmental). In order to summarise the forest fires in Europe we can see that the number of forest fires and the area affected have decreased between 2012 and 2014. However, the minimum was achieved in 2014, with a subsequent increase during 2015 and 2016 (Figure 2). Figure 2: Summary of number of forest fires (left) and area affected between 2012 and 2016 (right) The following environmental variables were obtained from the AGRI4CAST Resources Portal: Maximum air temperature (∘C); Minimum air temperature (∘C); Mean air temperature (∘C); Mean daily wind speed at 10 m. (m/s); Vapour pressure (hPa); Daily precipitation (mm/day); Potential evaporation from the water surface (mm/day); Potential evaporation from moist bare soil surface (mm/day); Potential evapotranspiration from crop canopy (mm/day); Total global radiation (kJ/m2/day). For each region we have the average by year. In Figure 3 we can see the average of the different variables by year for all the NUTS 2 regions. Figure 3: Trend of average of NUTS 2 regions by year of environmental variables between 2012 and 2015. (a) Maximum air temperature by year; (b) Miminum air temperature;(c) Mean air temperature; (d) Mean daily wind speed; (e) Vapour pressure; (f) Daily precipitation; (g) Potential evaporation from the water surface; (g) Potential evaporation from moist bare soil surface; (h) Potential evapotranspiration from crop canopy; (j)Total global radiation. The following socio-economic variables were obtained from Eurostat: Active population (*1000 employed persons), Woodland (*1000 hectares of Woodland in the area), Manufactured (*1000 employed persons working in manufactured products from woodland); Forestry (*1000 employed persons working in Forest sector); Economic aggregates of forestry (million euro) and Unemployment (%). In this case we have included in our model totals values by year and region. In order to summarise the different variables we have included in Figure 4 the total values by year for all the variables except for Unemployment where we have done the average for all regions by year. Figure 4: Trend the socioeconomic variables between 2012 and 2016. Totalisers by year in the following graphs: (a) Active population; (c) Economic aggregates of forestry; (d) Employed persons working in Forest sector; (e) Employed persons working in manufactured products from woodland); and (b) Average of Unemployment. ### 2.1 Spatio-Temporal model Here we consider a disease mapping approach, commonly used in small area studies to assess the spatial pattern of a particular outcome and to identify areas characterised by unusually high or low relative risk (Lawson, 2013; Pascutto _et al_ , 2000). For the _i-th_ area, the number of forest fires $y_{i}$ is modelled as $y_{it}\sim Poisson(\lambda_{it});\lambda_{it}=E_{it}\rho_{it}$ (1) where the $E_{it}$ are the expected number of forest fires and $\rho_{it}$ is the rate. We specify a log-linear model on $\rho_{i}$ and include spatial, temporal and a space-time interaction, which would explain differences in the time trend for different areas. We use the following specification to explain these differences: $\rho_{it}=\alpha+\upsilon_{i}+\nu_{i}+\gamma_{t}+\phi_{t}+\delta_{it},$ (2) There are several ways to define the interaction term: here, we assume that the two unstructured effects $\nu_{i}$ and $\phi_{t}$ interact. We re-write the precision matrix as the product of the scalar $\tau_{\nu}$ (or $\tau_{\phi}$) and the so called structure matrix $\textbf{\emph{F}}_{\nu}$ (or $\textbf{\emph{F}}_{\phi}$), which identifies the neighboring structure; here the structure matrix $\textbf{\emph{F}}_{\delta}$ can be factorised as the Kronecker product of the structure matrix for $\nu$ and $\phi$ (Clayton, 1996): $\textbf{\emph{F}}_{\phi}=\textbf{\emph{F}}_{\nu}\otimes\textbf{\emph{F}}_{\phi}=\textbf{\emph{I}}\otimes\textbf{\emph{I}}=\textbf{\emph{I}}$ (because both $\nu$ and $\phi$ are unstructured). Consequently, we assume no spatial and/or temporal structure on the interaction and therefore $\delta_{it}\sim Normal(0,\tau_{\phi})$ — see Knorr-Held (2000) for a detailed description of other specifications. In the model presented we assume the default specification of R-INLA for the distribution of the hyper-parameters; therefore, log$\tau_{\upsilon}$ $\sim$ logGamma(1,0.0005) and log$\tau_{\nu}$ $\sim$ logGamma(1,0.0005). In addition we specify a logGamma(1,0.0005) prior on the log-precision of the random walk and of the two unstructured effects (Blangiardo and Cameletti, 2015). To evaluate the fit of this model, we have applied the Watanabe-Akaike information criterion (WAIC) (Watanabe, 2010). WAIC was suggested as an appropriate alternative for estimating the out-of-sample expectation in a fully Bayesian approach. This method starts with the computed log pointwise posterior predictive density and then adds a correction for the effective number of parameters to adjust for overfitting (Gelman and Shalizi, 2013). Watanabe-Akaike information criterion works on predictive probability density of detected variables rather than on model parameter; hence, it can be applied in singular statistical models (i.e. models with non-identifiable parameterization) (Li _et al_ , 2016). We have used Integrated Nested Laplace Approximation (INLA) implemented in R-INLA within the R statistical software (version 3.6.0). ## 3 Results In this section, we show how the forest fires have evolved between 2012 and 2016. Analysing the temporal trend, we can see graphically (Figure 5), the posterior temporal trend for forest fires in Europe. In this graph we show how the number of forest fires tend to be reduced over time. Figure 5: Global linear temporal trend for number of forest fires in Europe at NUTS2 region level. The solid line identifies the posterior mean for $\beta_{t}$ , while the dashed lines are the 95% credibility intervals. Analysing the posterior distribution of forest fires (Figure 6) in Europe we can see that there is a “hot point” in western of the continent (North of Portugal and North West of Spanish peninsula). Also, as we can see, in general, the predicted number of forest fires is low in central Europe. Figure 6: Map of the number of forest fires posterior distribution by region. Comparing the different years, as we pointed previously, during 2014 the number of forest fires decreased in all areas except in some regions of Spain and Sicily. In addition, analysing the number of forest fires by region we can see that the region with stronger variations is the North region from Portugal. In Figure 7 we can see a more detailed map focused in Mediterranean countries (France, Greece, Italy and Spain). In this case it is clear that variability in France is almost inexistent only with some increase in the number of forest fires in Southern regions in 2016. The results from the data available for Greece, show that there are not big changes during the time analysed. However, Italy and Spain show more fluctuations during this period. The Southern part of Italy shows great changes along the time, starting with almost 150 forest fires in Sicily in 2012 to reduce until about 30 forest fires in 2015 and increase again in 2016 (67 forest fires). Similarly, in Spain the Northwest region shows several fluctuations. However, in Spain higher number of forest fires affects more regions. Figure 7: Detail of posterior distribution of forest fires in the Mediterranean region. As we can see in Table1, several variables are affecting the quantity of forest fires. But two of them have more impact (have higher values) than the others. Evaporation in water surface (EvaporationW) is affecting positively the volume of forest fires at region level. On the other hand, and with similar magnitude but negative sign, Evapotranspiration from crop canopy (Evapotrans.) is affecting negatively the presences of forest fires. Table 1: Posterior estimates summary (Mean, Standard deviation and 95% Credible Interval). Fixed effects and hyperparameters for spatio-temporal model. Fixed effects | | | | ---|---|---|---|--- | mean | sd | 0.025quant | 0.975quant Active | 0.3045 | 0.1739 | -0.0375 | 0.6468 Aggregates | -0.2919 | 0.1568 | -0.6008 | 0.0152 Forestry | 0.6073 | 0.3321 | -0.058 | 1.2472 Manufactured | -0.9303 | 0.3944 | -1.717 | -0.1669 MaxTemperature | 0.1761 | 0.14 | -0.0999 | 0.4503 MinTemperature | 0.584 | 0.2153 | 0.1631 | 1.0083 AvgTemperature | -0.1396 | 0.4931 | -1.1094 | 0.8277 Wind | 0.5609 | 0.2512 | 0.0655 | 1.0522 Presion | -0.28 | 0.2978 | -0.8653 | 0.3046 Precipitation | -0.0224 | 0.0984 | -0.2158 | 0.1707 Evapotrans | -23.6247 | 9.8466 | -43.0572 | -4.3758 EvaporationW | 24.5911 | 10.907 | 3.2597 | 46.0829 EvaporationS | 0.9008 | 0.6832 | -0.4398 | 2.2435 Radiation | -0.2677 | 0.9264 | -2.0907 | 1.5486 Woodland | 0.7649 | 0.2249 | 0.3331 | 1.2172 Model hyperparameters | | | | | mean | sd | 0.025quant | 0.975quant Precision for AREA_ID | 2.02E-01 | 3.85E-02 | 0.1347 | 2.85E-01 Precision for Year | 1.14E-01 | 6.61E-02 | 0.0299 | 2.80E-01 Precision for AREA_ID.YEAR | 1.59E+00 | 2.66E-01 | 1.1262 | 2.17E+00 However, evapotranspiration from crop canopy is having a negative effect in forest fires quantity. In this group we need to highlight variables having more impact (higher values) than the others. Evaporation in water surface (EvaporationW) is affecting positively the volume of forest fires at region level. On the other hand, and with similar magnitude but negative sign, Evapotranspiration from crop canopy (Evapotrans) is affecting negatively the presences of forest fires. The rest of the variables that are affecting positively the amount of forest fires are Minimum temperature at 10 m. (MinTemperature) and Mean daily wind speed at 10 m (WIND). Finally, Manufactured is affecting negatively the quantity of forest fires. Graphical representation of estimation for the fixed effects is presented in Figure 8. This chart presents the variables and their relationships with forest fires. Variables distributed in a positive side contribute to higher number of forest fires; the opposite, with variables with negative distribution. Variables present in both areas (positive and negative) do not have a clear relationship with answer. Figure 8: Graphical representation of fixed effecs. Evaporation in water surface (EvaporationW) and Evapotranspiration from crop canopy (Evapotrans) were excluded in order to obtain more detail of the rest of the variables. ## 4 Conclusions We have built spatio-temporal models to predict the quantity of forest fires in Europe at NUTS-2 regional level. We have shown the relationship between the different variables and the number of forest fires by region. We have shown that this relationship not only is between some of the variables (fixed effects), but also the evolution of forest fires along the time is affected not only by time and spatial effects but also by the combination of both (Precision for AREA_ID.YEAR). Initially our main objective in this project was to apply these models to Europe with more granularity, assuming that more local information will help to understand better the causality of forest fires. However due to data availability it was not possible to develop the project in that way. Currently not all the socioeconomic data is available for all the NUTS-3 regions in a continuous timestamp, being this characteristic necessary to carry a spatio- temporal analysis. Also, several factors can affect in different ways depending of the area. In our case, variables have been assumed in a scale that in some of the cases local information can help to understand cause-effect of forest fires. Analysing the models, we believe that the use of spatio-temporal models is an advantage for the understanding of the different dynamics, given that the temporal and spatio-temporal perspective is not very frequent analysing forest hazards. Summarising, we can generalise that not only environmental factors but also socioeconomic variables are affecting the causality of forest fires. However, more data and more granularity in the analysis in needed in order to understand this causality. Landscapes became more hazardous with the time, since land abandonment led to an increase in forest area. Treeless areas burned proportionally more than treed ones (Urbieta _et al_ , 2019). Fires in southern Europe have more preference shrublands than for forest types (Moreira _et al_ , 2011; Oliveira _et al_ , 2014), but may vary along locations (Moreno _et al_ , 2011). This could be due to a change in the ignition patterns owing to shifts in the wildland-agricultural and wildland-urban interfaces (Rodrigues _et al_ , 2014; Modugno _et al_ , 2016). The most vulnerable landscapes were those with diversity of land uses, with forest-agriculture mixtures (Ortega _et al_ , 2012). For these reasons, inclusion of vegetation to analyse causality needs to be studied. Fire trends can be affected by changes in ignition cause. In European Mediterranean countries, a minor percentage of fires are caused by lightning, and most are caused by people. Fires of these two sources tend to occur at different locations (Vázquez and Moreno, 1998), which could affect the vegetation they burn and the difficulty of extinction. However, no changes between these two sources have been observed (Ganteaume _et al_ , 2013). Regarding people-caused fires, the majority of them are voluntary, followed by negligence (Urbieta _et al_ , 2019). In recent times, negligence fires are increasing and voluntary ones decreasing (Ganteaume _et al_ , 2013). Whether this is differentially affecting the number of fires trends, is something that needs research (Urbieta _et al_ , 2019). Spatio-temporal models and the R-INLA package appear to offer additional benefits beyond the traditional analysis used to understand the causes of this hazards. The combination of using a complex spatial latent field to capture spatial processes and an underlying simple additive regression model for the response variables relationship to the different factors, means that the fixed effects are potentially more straightforward to interpret (Golding and Purse, 2016). R-INLA models are extremely flexible in their specifications, with spatial autocorrelation and observer bias being straightforwardly incorporated as random effects, while standard error distributions, such as Gaussian, Poisson, Binomial, and a variety of zero-inflated models, can be used interchangeably (Rue _et al_ , 2009). ## References * Bal _et al_ (2011) Bal, M.C., Pelachs, A., Perez-Obiol, R., Julia, R. and Cunill, R., 2011. 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2024-09-04T02:54:59.132455

2003.05230

arxiv-papers

# Inequalities for generalized matrix function and inner product Yongtao Li Yongtao Li, College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, 410082, P.R. China<EMAIL_ADDRESS>, Yang Huang Yang Huang, School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, P.R. China<EMAIL_ADDRESS>, Lihua Feng Lihua Feng, School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, P.R. China<EMAIL_ADDRESS>and Weijun Liu† Weijun Liu, School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, P.R. China<EMAIL_ADDRESS> ###### Abstract. We present inequalities related to generalized matrix function for positive semidefinite block matrices. We introduce partial generalized matrix functions corresponding to partial traces and then provide an unified extension of the recent inequalities due to Choi [6], Lin [14] and Zhang et al. [19, 5]. We demonstrate the applications of a positive semidefinite $3\times 3$ block matrix, which motivates us to give a simple alternative proof of Dragomir’s inequality and Krein’s inequality. ###### Key words and phrases: Block matrices; Positive semidefinite; Generalized matrix function; Partial traces; Partial determinants; Dragomir’s inequality; Krein’s inequality. ###### 2010 Mathematics Subject Classification: 47B65, 15B42, 15A45 ## 1\. Introduction Let $G$ be a subgraph of the symmetric group $S_{n}$ on $n$ letters and let $\chi$ be an irreducible character of $G$. For any $n\times n$ complex matrix $A=[a_{ij}]_{i,j=1}^{n}$, the generalized matrix function of $A$ (also known as immanant) afforded by $G$ and $\chi$ is defined as $\mathrm{d}_{\chi}^{G}(A):=\sum\limits_{\sigma\in G}\chi(\sigma)\prod\limits_{i=1}^{n}a_{i\sigma(i)}.$ Some specific subgroups $G$ and characters $\chi$ lead to some acquainted functionals on the matrix space. For instance, If $G=S_{n}$ and $\chi$ is the signum function with value $\pm 1$, then the generalized matrix function becomes the usual matrix determinant; By setting $\chi(\sigma)\equiv 1$ for each $\sigma\in G=S_{n}$, we get the permanent of the matrix; Setting $G=\\{e\\}\subset S_{n}$ defines the product of the main diagonal entries of the matrix (also known as the Hadamard matrix function). Let $A$ and $B$ be $n\times n$ positive semidefinite matrices. It is easy to prove by simultaneous diagonalization argument that $\det(A+B)\geq\det(A)+\det(B).$ (1) There are many extensions and generalizations of (1) in the literature. For example, a remarkable extension (e.g., [17, p. 228]) says that $\mathrm{d}_{\chi}^{G}(A+B)\geq\mathrm{d}_{\chi}^{G}(A)+\mathrm{d}_{\chi}^{G}(B).$ (2) Recently, Paksoy, Turkmen and Zhang [19] provided a natural extension of (2) for triple matrices by embedding the vectors of Gram matrices into a “sufficiently large” inner product space and using tensor products. More precisely, if $A,B$ and $C$ are positive semidefinite, they showed $\mathrm{d}_{\chi}^{G}(A+B+C)+\mathrm{d}_{\chi}^{G}(C)\geq\mathrm{d}_{\chi}^{G}(A+C)+\mathrm{d}_{\chi}^{G}(B+C).$ (3) Their approach to establish (3) is algebraic as well as combinatorial. Soon after, Chang, Paksoy and Zhang [5, Theorem 3] presented a further improvement of (3) by considering the tensor products of operators as words on certain alphabets, which states that $\displaystyle\mathrm{d}_{\chi}^{G}(A+B+C)+\mathrm{d}_{\chi}^{G}(A)+\mathrm{d}_{\chi}^{G}(B)+\mathrm{d}_{\chi}^{G}(C)$ (4) $\displaystyle\quad\geq\mathrm{d}_{\chi}^{G}(A+B)+\mathrm{d}_{\chi}^{G}(A+C)+\mathrm{d}_{\chi}^{G}(B+C).$ We remark here that (4) is indeed an improvement of (3) since $\displaystyle\mathrm{d}_{\chi}^{G}(A+B+C)+\mathrm{d}_{\chi}^{G}(C)-\mathrm{d}_{\chi}^{G}(A+C)-\mathrm{d}_{\chi}^{G}(B+C)$ $\displaystyle\quad\geq\mathrm{d}_{\chi}^{G}(A+B)-\mathrm{d}_{\chi}^{G}(A)-\mathrm{d}_{\chi}^{G}(B)\geq 0.$ We use the following standard notation. The set of $m\times n$ complex matrices is denoted by $\mathbb{M}_{m\times n}$. If $m=n$, we use $\mathbb{M}_{n}$ instead of $\mathbb{M}_{n\times n}$ and if $n=1$, we use $\mathbb{C}^{m}$ instead of $\mathbb{M}_{m\times 1}$. The identity matrix of $\mathbb{M}_{n}$ is denoted by $I_{n}$, or simply by $I$ if no confusion is possible. We use $\mathbb{M}_{m}(\mathbb{M}_{n})$ for the set of $m\times m$ block matrices with each block being $n$-square. By convention, if $X\in\mathbb{M}_{n}$ is positive semidefinite, we write $X\geq 0$. For two Hermitian matrices $A$ and $B$ of the same size, $A\geq B$ means $A-B\geq 0$. It is easy to verify that $\geq$ is a partial ordering on the set of Hermitian matrices, referred to Löwner ordering. On the other hand, Lin and Sra [16] gave the following extension of (1), i.e., if $A=[A_{ij}],B=[B_{ij}]\in\mathbb{M}_{m}(\mathbb{M}_{n})$ are block positive semidefinite matrices, then ${\det}_{2}(A+B)\geq{\det}_{2}(A)+{\det}_{2}(B),$ (5) where $\det_{2}(A)=[\det A_{ij}]_{i,j=1}^{m}\in\mathbb{M}_{m}$ and $\geq$ stands for the Löwner ordering. The paper is organized as follows. In Section 2, we briefly review some basic definitions and properties of tensor product in Multilinear Algebra Theory. In Section 3, we extend the above-cited results (2), (3), (4) and (5) to block positive semidefinte matrices (Theorem 3.5 and Corollary 3.6). As byproducts, some new inequalities related to trace, determinant and permanent are also included. In Section 4, we investigate the applications of a positive semidefinite $3\times 3$ block matrix and provide a short proof of Dragomir’s inequality (Theorem 4.4). In Section 5, we present a simple proof of Krein’s inequality (Theorem 5.1), and then we also provide some new triangle inequalities. ## 2\. Preliminaries Before starting our results, we first review some basic definitions and notations of Multilinear Algebra Theory [17]. Let $X\otimes Y$ denote the Kronecker product (tensor product) of $X$ with $Y$, that is, if $X=[x_{ij}]_{i,j=1}^{m}\in\mathbb{M}_{m}$ and $Y\in\mathbb{M}_{n}$, then $X\otimes Y\in\mathbb{M}_{m}(\mathbb{M}_{n})$ whose $(i,j)$-block is $x_{ij}Y$. Let $\otimes^{r}A:=A\otimes\cdots\otimes A$ denote the $r$-fold tensor power of $A$. We denote by $\wedge^{r}A$ the $r$th antisymmetric tensor power (or $r$th Grassmann power) of $A$, which is the same as the $r$th multiplicative compound matrix of $A$, and denote by $\vee^{r}A$ the $r$th symmetric tensor power of $A$; see [1, p. 18] for more details. We denote by $e_{r}(A),s_{r}(A)$ the $r$th elementary symmetric and $r$th complete symmetric function of the eigenvalues of $A$ (see [11, p. 54]). Trivially, $e_{1}(A)=s_{1}(A)=\mathrm{tr}(A)$ and $e_{n}(A)=\det(A)$ for $A\in\mathbb{M}_{n}$. Let $V$ be an $n$-dimensional Hilbert space and $\otimes^{n}V$ be the tensor product space of $n$ copies of $V$. Let $G$ be a subgroup of the symmetric group $S_{n}$ and $\chi$ be an irreducible character of $G$. The symmetrizer induced by $\chi$ on the tensor product space $\otimes^{n}V$ is defined by its action $S(v_{1}\otimes\cdots\otimes v_{n}):=\frac{1}{|G|}\sum\limits_{\sigma\in G}\chi(\sigma)v_{\sigma^{-1}(1)}\otimes\cdots\otimes v_{\sigma^{-1}(n)}.$ (6) All elements of the form (6) span a vector space, denoted by $V_{\chi}^{n}(G)\subset\otimes^{n}V$, which is called the space of the symmetry class of tensors associated with $G$ and $\chi$ (see [17, p. 154, 235]). It is easy to verified that $V^{n}_{\chi}(G)$ is an invariant subspace of $\otimes^{n}V$ under the tensor operator $\otimes^{n}A$. For a linear operator $A$ on $V$, the induced operator $K(A)$ of $A$ with respect to $G$ and $\chi$ is defined to be $K(A)=(\otimes^{n}A)\big{|}_{V^{n}_{\chi}(G)}$, the restriction of $\otimes^{n}A$ on $V_{\chi}^{n}(G)$. The induced operator $K(A)$ is closely related to generalized matrix function. Let $e_{1},e_{2},\ldots,e_{n}$ be an orthonormal basis of $V$ and $P$ be a matrix representation of the linear operator $A$ on $V$ with respect to the basis $e_{1},\ldots,e_{n}$. Then $\mathrm{d}_{\chi}^{G}\left(P^{T}\right)=\frac{|G|}{\mathrm{deg}(\chi)}\langle K(A)e^{*},e^{*}\rangle,$ (7) where $\mathrm{deg}(\chi)$ is the degree of $\chi$ and $e^{*}:=e_{1}*e_{2}*\cdots*e_{n}$ is the decomposable symmetrized tensor of $e_{1},\ldots,e_{n}$ (see [17, p. 227, 155]). Now, we list some basic properties of tensor product for our latter use. ###### Proposition 2.1. (see [1, pp. 16–20]) Let $A,B$ and $C$ be $n\times n$ matrices. Then * (1) $\otimes^{r}(AB)=(\otimes^{r}A)(\otimes^{r}B),\wedge^{r}(AB)=(\wedge^{r}A)(\wedge^{r}B)$ and $\vee^{r}(AB)=(\vee^{r}A)(\vee^{r}B)$. * (2) $\mathrm{tr}(\otimes^{r}A)=(\mathrm{tr}A)^{r}:=p_{r}(A),\mathrm{tr}(\wedge^{r}A)=e_{r}(A)$ and $\mathrm{tr}(\vee^{r}A)=s_{r}(A)$. * (3) $\det(\otimes^{r}A)=(\det A)^{rn^{r-1}},\det(\wedge^{r}A)=(\det A)^{{n-r\choose r-1}}$ and $\det(\vee^{r}A)=(\det A)^{\frac{r}{n}{n+r-1\choose r}}$. Furthermore, if $A,B$ and $C$ are positive semidefinite matrices, then * (4) $A\otimes B,A\wedge B$ and $A\vee B$ are positive semidefinite. * (5) $\otimes^{r}(A+B)\geq\otimes^{r}A+\otimes^{r}B,\wedge^{r}(A+B)\geq\wedge^{r}A+\wedge^{r}B$ and $\vee^{r}(A+B)\geq\vee^{r}A+\vee^{r}B$. Finally, we introduce the definition of partial traces, which comes from Quantum Information Theory [20, p. 12]. Given $A\in\mathbb{M}_{m}(\mathbb{M}_{n})$, the first partial trace (map) $A\mapsto\mathrm{tr}_{1}(A)\in\mathbb{M}_{n}$ is defined as the adjoint map of the imbedding map $X\mapsto I_{m}\otimes X\in\mathbb{M}_{m}\otimes\mathbb{M}_{n}$. Correspondingly, the second partial trace (map) $A\mapsto\mathrm{tr}_{2}(A)\in\mathbb{M}_{m}$ is defined as the adjoint map of the imbedding map $Y\mapsto Y\otimes I_{n}\in\mathbb{M}_{m}\otimes\mathbb{M}_{n}$. Therefore, we have $\langle I_{m}\otimes X,A\rangle=\langle X,\mathrm{tr}_{1}(A)\rangle,\quad\forall X\in\mathbb{M}_{n},$ and $\langle Y\otimes I_{n},A\rangle=\langle Y,\mathrm{tr}_{2}(A)\rangle,\quad\forall Y\in\mathbb{M}_{m}.$ Assume that $A=[A_{ij}]_{i,j=1}^{m}$ with $A_{ij}\in\mathbb{M}_{n}$, then the visualized forms of the partial traces are actually given in [2, Proposition 4.3.10] as $\mathrm{tr}_{1}{(A)}=\sum\limits_{i=1}^{m}A_{ii},\quad\mathrm{tr}_{2}{(A)}=\bigl{[}\mathrm{tr}A_{ij}\bigr{]}_{i,j=1}^{m}.$ Under the above definition, it follows that both $\mathrm{tr}_{1}(A)$ and $\mathrm{tr}_{2}(A)$ are positive semidefinite whenever $A$ is positive semidefinite; see, e.g., [24, p. 237]. ## 3\. Partial Matrix Functions For $A=[A_{ij}]_{i,j=1}^{m}\in\mathbb{M}_{m}(\mathbb{M}_{n})$, suppose that $A_{ij}=\bigl{[}a_{rs}^{ij}\bigr{]}_{r,s=1}^{n}$. Setting $G_{rs}:=\bigl{[}a_{rs}^{ij}\bigr{]}_{i,j=1}^{m}\in\mathbb{M}_{m}.$ Then we can verify that $\mathrm{tr}_{1}(A)=\sum_{i=1}^{m}A_{ii}=\sum_{i=1}^{m}\bigl{[}a_{rs}^{ii}\bigr{]}_{r,s=1}^{n}=\left[\begin{matrix}\sum\limits_{i=1}^{m}a_{rs}^{ii}\end{matrix}\right]_{r,s=1}^{n}=\bigl{[}\mathrm{tr}\,G_{rs}\bigr{]}_{r,s=1}^{n},$ Motivated by this relation, we next introduce the following definition. ###### Definition 3.1. Let $\Gamma:\mathbb{M}_{p}\to\mathbb{M}_{q}$ be a matrix function. The first and second partial matrix functions of $\Gamma$ are defined by $\Gamma_{1}(A):=\bigl{[}\Gamma(G_{rs})\bigr{]}_{r,s=1}^{n}~{}~{}~{}\text{and}~{}~{}~{}\Gamma_{2}(A):=\bigl{[}\Gamma(A_{ij})\bigr{]}_{i,j=1}^{m}.$ Clearly, when $\Gamma=\mathrm{tr}$, this definition coincides with that of partial traces; when $\Gamma=\det$, it identifies with the partial determinants, which were introduced by Choi in [6] recently. Let $A=[A_{ij}]_{i,j=1}^{m}\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite block matrix. It is well known that both $\det_{2}(A)=[\det A_{ij}]_{i,j=1}^{m}$ and $\mathrm{tr}_{2}(A)=[\mathrm{tr}A_{ij}]_{i,j=1}^{m}$ are positive semidefinite matrices; see, e.g., [24, p. 221, 237]. Whereafter, Zhang [25, Theorem 3.1] extends the positivity to generalized matrix function via generalized Cauchy-Binet formula, more precisely, $\mathrm{d}_{\chi}^{G}{}_{2}(A)=[\mathrm{d}_{\chi}^{G}(A_{ij})]_{i,j=1}^{m}$ is also positive semidefinite. We extend the positivity to more matrix functionals. ###### Proposition 3.2. Let ${A}\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. If $\Gamma$ is one of the functionals $\mathrm{tr},\det,\mathrm{per},\mathrm{d}_{\chi}^{G},p_{r},e_{r}$ and $s_{r}$, then $\Gamma_{1}(A)$ and $\Gamma_{2}(A)$ are positive semidefinite. ###### Proof. We denote by $\widetilde{A}=[G_{rs}]_{r,s=1}^{n}\in\mathbb{M}_{n}(\mathbb{M}_{m})$, and then it is easy to see that $\widetilde{\widetilde{A}}=A$ and $\Gamma_{1}(A)=\Gamma_{2}(\widetilde{A})$. Moreover, $\widetilde{A}$ and $A$ are unitarily similar; see [6, Theorem 7] for more details. Thus, we only need to show $\Gamma_{2}(A)$ is positive semidefinite. It is similar with the approach in [25], we omit the details of proof. ∎ The following Lemma 3.3 plays a key step in our extension (Theorem 3.5), it could be found in [3] or [5], we here provide a proof for convenience of readers. ###### Lemma 3.3. Let $A,B,C$ be positive semidefinite matrices of same size. Then for every positive integer $r$, we have $\displaystyle\otimes^{r}(A+B+C)+\otimes^{r}A+\otimes^{r}B+\otimes^{r}C$ $\displaystyle\quad\geq\otimes^{r}(A+B)+\otimes^{r}(A+C)+\otimes^{r}(B+C).$ The same result is true for $\wedge^{r}$ and $\vee^{r}$. ###### Proof. The proof is by induction on $r$. The base case $r=1$ holds with equality, and the case $r=2$ is easy to verify. Assume the required result holds for $r=m\geq 2$, that is $\displaystyle\otimes^{m}(A+B+C)+\otimes^{m}A+\otimes^{m}B+\otimes^{m}C$ $\displaystyle\quad\geq\otimes^{m}(A+B)+\otimes^{m}(A+C)+\otimes^{m}(B+C).$ For $r=m+1$, we get from Proposition 2.1 that $\displaystyle\otimes^{m+1}(A+B+C)$ $\displaystyle\quad=\bigl{(}\otimes^{m}(A+B+C)\bigr{)}\otimes(A+B+C)$ $\displaystyle\quad\geq\bigl{(}\otimes^{m}(A+B)+\otimes^{m}(A+C)+\otimes^{m}(B+C)-\otimes^{m}A-\otimes^{m}B-\otimes^{m}C\bigr{)}$ $\displaystyle\quad\quad\,\otimes(A+B+C)$ $\displaystyle\quad=\otimes^{m+1}(A+B)+\otimes^{m+1}(A+C)+\otimes^{m+1}(B+C)$ $\displaystyle\quad\quad-\otimes^{m+1}A-\otimes^{m+1}B-\otimes^{m+1}C$ $\displaystyle\quad\quad+\bigl{(}\otimes^{m}(A+B)\bigr{)}\otimes C+\bigl{(}\otimes^{m}(A+C)\bigr{)}\otimes B+\bigl{(}\otimes^{m}(B+C)\bigr{)}\otimes A$ $\displaystyle\quad\quad-\bigl{(}\otimes^{m}A\bigr{)}\otimes(B+C)-\bigl{(}\otimes^{m}B\bigr{)}\otimes(A+C)-\bigl{(}\otimes^{m}C\bigr{)}\otimes(A+B).$ It remains to show that $\displaystyle\bigl{(}\otimes^{m}(A+B)\bigr{)}\otimes C+\bigl{(}\otimes^{m}(A+C)\bigr{)}\otimes B+\bigl{(}\otimes^{m}(B+C)\bigr{)}\otimes A$ $\displaystyle\quad\geq\bigl{(}\otimes^{m}A\bigr{)}\otimes(B+C)+\bigl{(}\otimes^{m}B\bigr{)}\otimes(A+C)+\bigl{(}\otimes^{m}C\bigr{)}\otimes(A+B).$ This follows immediately by the superadditivity (5) in Proposition 2.1. ∎ We require one more lemma for our purpose. ###### Lemma 3.4. ([2, p. 93]) Let $A=[A_{ij}]_{i,j=1}^{m}\in\mathbb{M}_{m}(\mathbb{M}_{n})$. Then $[\otimes^{r}A_{ij}]_{i,j=1}^{m}$ is a principal submatrix of $\otimes^{r}A$ for every positive integer $r$. Now, we present our main result, which is an unified extension of (4) and (5). ###### Theorem 3.5. Let ${A},B,C\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. If $\Gamma$ is one of the functionals $\mathrm{tr},\det,\mathrm{per},\mathrm{d}_{\chi}^{G},p_{r},e_{r}$ and $s_{r}$, then $\displaystyle\Gamma_{1}(A+B+C)+\Gamma_{1}(A)+\Gamma_{1}(B)+\Gamma_{1}(C)$ $\displaystyle\quad\geq\Gamma_{1}(A+B)+\Gamma_{1}(A+C)+\Gamma_{1}(B+C),$ and $\displaystyle\Gamma_{2}(A+B+C)+\Gamma_{2}(A)+\Gamma_{2}(B)+\Gamma_{2}(C)$ $\displaystyle\quad\geq\Gamma_{2}(A+B)+\Gamma_{2}(A+C)+\Gamma_{2}(B+C).$ ###### Proof. We only show that the desired result holds for $\Gamma=\mathrm{d}_{\chi}^{G}$ and $\Gamma=e_{r}$ since other case of functionals can be proved similarly. It suffices to show the second desired result by exchanging the role of $\widetilde{A}$ and $A$. By Lemma 3.3, we have $\displaystyle\otimes^{r}(A+B+C)+\otimes^{r}A+\otimes^{r}B+\otimes^{r}C$ $\displaystyle\quad\geq\otimes^{r}(A+B)+\otimes^{r}(A+C)+\otimes^{r}(B+C),$ which together with Lemma 3.4 leads to the following $\displaystyle[\otimes^{r}(A_{ij}+B_{ij}+C_{ij})]_{i,j=1}^{m}+[\otimes^{r}A_{ij}]_{i,j=1}^{m}+[\otimes^{r}B_{ij}]_{i,j=1}^{m}+[\otimes^{r}C_{ij}]_{i,j=1}^{m}$ $\displaystyle\quad\geq[\otimes^{r}(A_{ij}+B_{ij})]_{i,j=1}^{m}+[\otimes^{r}(A_{ij}+C_{ij})]_{i,j=1}^{m}+[\otimes^{r}(B_{ij}+C_{ij})]_{i,j=1}^{m}.$ By restricting above inequality to the symmetry class $V_{\chi}^{G}(V)$, we get $\displaystyle[K(A_{ij}+B_{ij}+C_{ij})]_{i,j=1}^{m}+[K(A_{ij})]_{i,j=1}^{m}+[K(B_{ij})]_{i,j=1}^{m}+[K(C_{ij})]_{i,j=1}^{m}$ $\displaystyle\quad\geq[K(A_{ij}+B_{ij})]_{i,j=1}^{m}+[K(A_{ij}+C_{ij})]_{i,j=1}^{m}+[K(B_{ij}+C_{ij})]_{i,j=1}^{m}.$ By combining (7), the second desired result in the case of $\Gamma=\mathrm{d}_{\chi}^{G}$ follows. By the same way, it follows that $\displaystyle[\wedge^{r}(A_{ij}+B_{ij}+C_{ij})]_{i,j=1}^{m}+[\wedge^{r}A_{ij}]_{i,j=1}^{m}+[\wedge^{r}B_{ij}]_{i,j=1}^{m}+[\wedge^{r}C_{ij}]_{i,j=1}^{m}$ $\displaystyle\quad\geq[\wedge^{r}(A_{ij}+B_{ij})]_{i,j=1}^{m}+[\wedge^{r}(A_{ij}+C_{ij})]_{i,j=1}^{m}+[\wedge^{r}(B_{ij}+C_{ij})]_{i,j=1}^{m}.$ By taking trace blockwise and using Proposition 2.1, it yields the second desired result in the case of $\Gamma=e_{r}$. ∎ From Theorem 3.5, one could get the following Corollary 3.6. ###### Corollary 3.6. Let ${A},B,C\in\mathbb{M}_{m}(\mathbb{M}_{n})$ be positive semidefinite. If $\Gamma$ is one of the functionals $\mathrm{tr},\det,\mathrm{per},\mathrm{d}_{\chi}^{G},p_{r},e_{r}$ and $s_{r}$, then $\displaystyle\Gamma_{1}(A+B+C)+\Gamma_{1}(C)\geq\Gamma_{1}(A+C)+\Gamma_{1}(B+C),$ and $\displaystyle\Gamma_{2}(A+B+C)+\Gamma_{2}(C)\geq\Gamma_{2}(A+C)+\Gamma_{2}(B+C).$ In particular, by setting $m=1$ and $\Gamma=\det$ in Theorem 3.5 and Corollary 3.6, which yields the following renowned determinantal inequalities, $\displaystyle\det(A+B+C)+\det A+\det B+\det C$ $\displaystyle\quad\geq\det(A+B)+\det(A+C)+\det(B+C),$ and $\det(A+B+C)+\det C\geq\det(A+C)+\det(B+C).$ We remark that these two inequalities could be proved by using a majorization approach of eigenvalues. It is more elementary and totally different from our method. We refer to [14] and [24, p. 215] for more details. ## 4\. Positivity and Dragomir’s inequality Recently, positive semidefinite $3\times 3$ block matrices are extensively studied, such a partition leads to versatile and elegant theoretical inequalities; see, e.g., [15, 9]. Assume that $X,Y,Z$ are matrices with appropriate size, then it follows from Section 3 that the $3\times 3$ matrix $\begin{bmatrix}\Gamma(X^{*}X)&\Gamma(X^{*}Y)&\Gamma(X^{*}Z)\\\ \Gamma(Y^{*}X)&\Gamma(Y^{*}Y)&\Gamma(Y^{*}Z)\\\ \Gamma(Z^{*}X)&\Gamma(Z^{*}Y)&\Gamma(Z^{*}Z)\end{bmatrix}$ (8) is positive semidefinite whenever $\Gamma$ is selected for trace and determinant. Different size of matrices in (8) will yield a large number of interesting triangle inequalities. In particular, if $X,Y,Z$ are column vectors, say $u,v,w\in\mathbb{C}^{n}$, it is easy to see that $\begin{bmatrix}\mathrm{Re}(u^{*}u)&\mathrm{Re}(u^{*}v)&\mathrm{Re}(u^{*}w)\\\\[2.84544pt] \mathrm{Re}(v^{*}u)&\mathrm{Re}(v^{*}v)&\mathrm{Re}(v^{*}w)\\\\[2.84544pt] \mathrm{Re}(w^{*}u)&\mathrm{Re}(w^{*}v)&\mathrm{Re}(w^{*}w)\end{bmatrix}$ (9) is positive semidefinite; see [13, 4] for more applications. In this section, we provide two analogous results (Corollary 4.2 and Proposition 4.3) of the above (9). Based on this result, we then give a short proof of Dragomir’s inequality (Theorem 4.4). The following Lemma is an Exercise in [2, p. 26], we will present a detailed proof. ###### Lemma 4.1. Let $A=[a_{ij}]$ be a $3\times 3$ complex matrix and let $|A|=\bigl{[}|a_{ij}|\bigr{]}$ be the matrix obtained from $A$ by taking the absolute values of the entries of $A$. If $A$ is positive semidefinite, then $|A|$ is positive semidefinite. ###### Proof. We first note that the positivity of $A$ implies all diagonal entries of $A$ are nonnegative. If a diagonal entry of $A$ is zero, as $A$ is positive semidefinite, then the entire row entries and column entries of $A$ are zero and it is obvious that the positivity of $\begin{bmatrix}\begin{smallmatrix}a&c\\\ \overline{c}&b\end{smallmatrix}\end{bmatrix}$ implies the positivity of $\begin{bmatrix}\\!\begin{smallmatrix}|a|&|c|\\\ |\overline{c}|&|b|\end{smallmatrix}\\!\end{bmatrix}$. Without loss of generality, we may assume that $a_{ii}>0$ for every $i=1,2,3$. Let $D=\mathrm{diag}\bigl{\\{}a_{11}^{-1/2},a_{22}^{-1/2},a_{33}^{-1/2}\bigr{\\}}$ and observe that $D^{*}|A|D=|D^{*}AD|$. By scaling, we further assume that $A=\begin{bmatrix}1&a&b\\\ \overline{a}&1&c\\\ \overline{b}&\overline{c}&1\end{bmatrix}.$ Recall that $X\geq 0$ means $X$ is positive semidefinite. Our goal is to prove $\begin{bmatrix}1&a&b\\\ \overline{a}&1&c\\\ \overline{b}&\overline{c}&1\end{bmatrix}\geq 0\Rightarrow\begin{bmatrix}1&|a|&|b|\\\ |\overline{a}|&1&|c|\\\ |\overline{b}|&|\overline{c}|&1\end{bmatrix}\geq 0.$ (10) Assume that $a=|a|e^{i\alpha}$ and $b=|b|e^{i\beta}$, and denote $Q=\mathrm{diag}\left\\{1,e^{-i\alpha},e^{-i\beta}\right\\}$. By a direct computation, we obtain $Q^{*}AQ=\begin{bmatrix}1&|a|&|b|\\\ |a|&1&ce^{i(\alpha-\beta)}\\\ |b|&\overline{c}e^{i(\beta-\alpha)}&1\end{bmatrix}.$ Since $Q^{*}AQ\geq 0$, taking the determinant leads to the following $1+|a||b|\left(ce^{i(\alpha-\beta)}+\overline{c}e^{i(\beta-\alpha)}\right)\geq|a|^{2}+|b|^{2}+|c|^{2}.$ Note that $2|c|\geq 2\,\mathrm{Re}\left(ce^{i(\alpha-\beta)}\right)\geq\left(ce^{i(\alpha-\beta)}+\overline{c}e^{i(\beta-\alpha)}\right)$, then $1+2|a||b||c|\geq|a|^{2}+|b|^{2}+|c|^{2},$ which is actually $\det|A|\geq 0$. Combining $1-|a|^{2}\geq 0$, that is, every principal minor of $|A|$ is nonnegative, then $|A|\geq 0$. Thus, the desired statement (10) now follows. ∎ Remark. We remark that the converse of Lemma 4.1 is not true, additionally, the statement not holds for $4\times 4$ case. For example, setting $B=\begin{bmatrix}1&-1&-1\\\ -1&1&-1\\\ -1&-1&1\end{bmatrix},\quad C=\begin{bmatrix}10&3&-2&1\\\ 3&10&0&9\\\ -2&0&10&4\\\ 1&9&4&10\end{bmatrix}.$ We can see that both $|B|$ and $C$ are positive semidefinite, however, $B$ and $|C|$ are not positive semidefinite, because $\det B=-4$ and $\det|C|=-364$. By the positivity of Gram matrix and Lemma 4.1, we get the following corollary. ###### Corollary 4.2. If $u,v$ and $w$ are vectors in $\mathbb{C}^{n}$, then $\begin{bmatrix}\bigl{|}u^{*}u\bigr{|}&\bigl{|}u^{*}v\bigr{|}&\bigl{|}u^{*}w\bigr{|}\\\\[4.26773pt] \bigl{|}v^{*}u\bigr{|}&\bigl{|}v^{*}v\bigr{|}&\bigl{|}v^{*}w\bigr{|}\\\\[4.26773pt] \bigl{|}{w^{*}u}\bigr{|}&\bigl{|}{w^{*}v}\bigr{|}&\bigl{|}{w^{*}w}\bigr{|}\end{bmatrix}$ is a positive semidefinite matrix. ###### Proposition 4.3. If $u,v$ and $w$ are vectors in $\mathbb{R}^{n}$ such that $u+w=v$, then $\begin{bmatrix}{u^{*}u}&{u^{*}v}&-{u^{*}w}\\\\[2.84544pt] {v^{*}u}&{v^{*}v}&{v^{*}w}\\\\[2.84544pt] -{w^{*}u}&{w^{*}v}&{w^{*}w}\end{bmatrix}$ is a positive semidefinite matrix. ###### Proof. We choose an orthonormal basis of $\mathrm{Span}\\{u,v,w\\}$, then we may assume that $u,v$ and $w$ are vectors in $\mathbb{R}^{3}$ and form a triangle on a plane. We denote the angle of $u,v$ by $\alpha$, angle of $-u,w$ by $\beta$ and angle of $-w,-v$ by $\gamma$, respectively. Note that $\alpha+\beta+\gamma=\pi$, then we have $\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma+2\cos\alpha\cos\beta\cos\gamma=1.$ By computing the principal minor, it follows that $R:=\begin{bmatrix}1&\cos\alpha&\cos\beta\\\ \cos\alpha&1&\cos\gamma\\\ \cos\beta&\cos\gamma&1\end{bmatrix}$ is positive semidefinite. Setting $S=\mathrm{diag}\\{\left\lVert u\right\rVert,\left\lVert v\right\rVert,\left\lVert w\right\rVert\\}$. Thus $S^{T}RS$ is positive semidefinite. This completes the proof. ∎ Dragomir [7] established the following inequality (Theorem 4.4) related to inner product of three vectors, which yields some improvements of Schwarz’s inequality; see, e.g.,[8]. We here give a short proof using Corollary 4.2. ###### Theorem 4.4. Let $u,v$ and $w$ be vectors in an inner product space. Then $\displaystyle\left(\left\lVert u\right\rVert^{2}\left\lVert w\right\rVert^{2}-\bigl{|}\left\langle u,w\right\rangle\bigr{|}^{2}\right)\left(\left\lVert w\right\rVert^{2}\left\lVert v\right\rVert^{2}-\bigl{|}\left\langle w,v\right\rangle\bigr{|}^{2}\right)$ $\displaystyle\quad\geq\bigl{|}\left\langle u,w\right\rangle\left\langle w,v\right\rangle-\left\langle u,v\right\rangle\left\langle w,w\right\rangle\bigr{|}^{2}.$ ###### Proof. Without loss of generality, by scaling, we may assume that $u,v$ and $w$ are unit vectors. We now need to prove $\left(1-\bigl{|}\left\langle u,w\right\rangle\bigr{|}^{2}\right)\left(1-\bigl{|}\left\langle w,v\right\rangle\bigr{|}^{2}\right)\geq\left(\bigl{|}\left\langle u,w\right\rangle\bigr{|}\bigl{|}\left\langle w,v\right\rangle\bigr{|}-\bigl{|}\left\langle u,v\right\rangle\bigr{|}\right)^{2},$ which is equivalent to showing $1+2\bigl{|}\left\langle u,v\right\rangle\bigr{|}\bigl{|}\left\langle v,w\right\rangle\bigr{|}\bigl{|}\left\langle w,u\right\rangle\bigr{|}\geq\bigl{|}\left\langle u,v\right\rangle\bigr{|}^{2}+\bigl{|}\left\langle v,w\right\rangle\bigr{|}^{2}+\bigl{|}\left\langle w,u\right\rangle\bigr{|}^{2}.$ (11) By Corollary 4.2, it follows that $\begin{bmatrix}1&\bigl{|}\left\langle u,v\right\rangle\bigr{|}&\bigl{|}\left\langle u,w\right\rangle\bigr{|}\\\\[4.26773pt] \bigl{|}\left\langle v,u\right\rangle\bigr{|}&1&\bigl{|}\left\langle v,w\right\rangle\bigr{|}\\\\[4.26773pt] \bigl{|}\left\langle w,u\right\rangle\bigr{|}&\bigl{|}\left\langle w,v\right\rangle\bigr{|}&1\end{bmatrix}$ is positive semidefinite. Taking determinant on this matrix yields (11). ∎ Recently, Zhang gave the following inequality (see [25, Theorem 5.1] ), if $u,v$ and $w$ are all unit vectors in an inner product space, then $1+2\,\mathrm{Re}\left(\left\langle u,v\right\rangle\left\langle v,w\right\rangle\left\langle w,u\right\rangle\right)\geq\bigl{|}\left\langle u,v\right\rangle\bigr{|}^{2}+\bigl{|}\left\langle v,w\right\rangle\bigr{|}^{2}+\bigl{|}\left\langle w,u\right\rangle\bigr{|}^{2}.$ (12) Inequality (11) seems weaker than (12). Actually, it is not difficult to prove that (11) and (12) are mutually equivalent, we leave the details for the interested reader. ## 5\. Some Triangle inequalities Let $V$ be an inner product space with the inner product $\left\langle\cdot,\cdot\right\rangle$ over the real number field $\mathbb{R}$ or the complex number field $\mathbb{C}$. For any two nonzero vectors $u,v$ in $V$, there are two defferent ways to define the angle between the vectors $u$ and $v$ in terms of the inner product, such as, $\displaystyle\Phi(u,v):=\arccos\frac{\mathrm{Re}\left\langle u,v\right\rangle}{\left\lVert u\right\rVert\left\lVert v\right\rVert},$ and $\Psi(u,v):=\arccos\frac{\bigl{|}\left\langle u,v\right\rangle\bigr{|}}{\left\lVert u\right\rVert\left\lVert v\right\rVert}.$ Both these definitions are frequently used in the literature, and there are various reasons and advantages that the angles are defined in these ways; see, e.g., [13, 4, 18] for recent studies. The angles $\Phi$ and $\Psi$ are closely related, but not equal unless $\left\langle u,v\right\rangle$ is a nonnegative number. We can easily see that $0\leq\Phi\leq\pi$ and $0\leq\Psi\leq\pi/2$, and $\Phi(u,v)\geq\Psi(u,v)$ for all $u,v\in V$, since $\mathrm{Re}\left\langle u,v\right\rangle\leq\left|\left\langle u,v\right\rangle\right|$ and $f(x)=\arccos x$ is a decreasing function in $x\in[-1,1]$. It is easy to verify that $\Psi(u,v)=\min\limits_{|p|=1}\Phi(pu,v)=\min\limits_{|q|=1}\Phi(u,qv)=\min\limits_{|p|=|q|=1}\Phi(pu,qv).$ (13) There exist two well known triangle inequalities for $\Phi$ and $\Psi$ in the literature, we will state it as the following Theorem 5.1. ###### Theorem 5.1. Let $u,v$ and $w$ be vectors in an inner product space. Then $\displaystyle\Phi(u,v)$ $\displaystyle\leq\Phi(u,w)+\Phi(w,v),$ (14) and $\displaystyle\Psi(u,v)$ $\displaystyle\leq\Psi(u,w)+\Psi(w,v).$ (15) The first inequality (14) is attributed to Krein who stated it without proof in [12], and proved first by Rao [21] and [10, p. 56], whose proof boils down to the positivity of the matrix (9). We remark that (14) on the real field could be seen in [24, p. 31]. For the second one, Lin [13] observed that (15) can be deduced from (14) because of the relation (13). It is noteworthy that either Corollary 4.2 or Theorem 4.4 also guarantees (15). Indeed, by Theorem 4.4, we can obtain $\displaystyle\left(\left\lVert u\right\rVert^{2}\left\lVert w\right\rVert^{2}-\bigl{|}\left\langle u,w\right\rangle\bigr{|}^{2}\right)^{1/2}\left(\left\lVert w\right\rVert^{2}\left\lVert v\right\rVert^{2}-\bigl{|}\left\langle w,v\right\rangle\bigr{|}^{2}\right)^{1/2}$ $\displaystyle\quad\geq\bigl{|}\left\langle u,w\right\rangle\left\langle w,v\right\rangle\bigr{|}-\bigl{|}\left\langle u,v\right\rangle\left\langle w,w\right\rangle\bigr{|}.$ By dividing with $\left\lVert u\right\rVert\left\lVert v\right\rVert\left\lVert w\right\rVert^{2}$, we have $\frac{\left|\left\langle u,v\right\rangle\right|}{\left\lVert u\right\rVert\left\lVert v\right\rVert}\geq\frac{\left|\left\langle u,w\right\rangle\right|}{\left\lVert u\right\rVert\left\lVert w\right\rVert}\frac{\left|\left\langle w,v\right\rangle\right|}{\left\lVert w\right\rVert\left\lVert v\right\rVert}-\sqrt{1-\frac{\left|\left\langle u,w\right\rangle\right|}{\left\lVert u\right\rVert\left\lVert w\right\rVert}}\cdot\sqrt{1-\frac{\left|\left\langle w,v\right\rangle\right|}{\left\lVert w\right\rVert\left\lVert v\right\rVert}},$ which is equivalent to $\displaystyle\cos\Psi(u,v)$ $\displaystyle\geq\cos\Psi(u,w)\cos\Psi(w,v)-\sin\Psi(u,w)\sin\Psi(w,v)$ $\displaystyle=\cos(\Psi(u,w)+\Psi(w,v)).$ Thus, (15) follows by the decreasing property of cosine on $[0,\pi]$. To end this paper, we present a new proof of inequality (14) and (15), which can be viewed as a generalization of the method in [24, p. 31], and then we also provide some new angle inequalities. ###### Proof of Theorem 5.1. We here only prove (15), since (14) can be proved in a slight similar way. Because the desireed inequality involves only three vectors $u,v$ and $w$, we may focus on the subspace spaned by $u,v$ and $w$, which has dimension at most $3$. We may further choose an orthonormal basis (a unit vector in the case of dimension one) of this subspace $\mathrm{Span}\\{u,v,w\\}$. Assume that $u,v$ and $w$ have coordinate vectors $x,y$ and $z$ under this basis, respectively. Then the desired inequality holds if and only if it holds for complex vectors $x,y$ and $z$ with the standard product $\left\langle x,y\right\rangle=\overline{y_{1}}x_{1}+\overline{y_{2}}x_{2}+\cdots+\overline{y}_{n}x_{n}.$ That is to say, our mian goal is to show the following $\Psi(x,y)\leq\Psi(x,z)+\Psi(z,y),\quad\forall\,x,y,z\in\mathbb{C}^{3}.$ (16) We next prove the inequality (16) in two steps. If the inner product space is a Euclidean space (i.e., an inner product space over field $\mathbb{R}$). Then the problem is reduced to $\mathbb{R},\mathbb{R}^{2}$ or $\mathbb{R}^{3}$ depending on whether the dimension of $\mathrm{Span}\\{u,v,w\\}$ is $1,2$ or $3$, respectively. In this real case, one can draw a simple graph to get the result. If the inner product space is an unitary space (i.e., an inner product space over field $\mathbb{C}$). We now do some technical tricks. We note that the desired inequality (16) is not changed if we replace $x,y$ with $\omega x,\delta y$ for any complex numbers $\omega,\delta$ satisfying $|\omega|=|\delta|=1$. Therefore, we may assume further that both $\left\langle x,z\right\rangle$ and $\left\langle z,y\right\rangle$ are real numbers. Let $x=X_{1}+iX_{2},y=Y_{1}+iY_{2}$ and $z=Z_{1}+iZ_{2}$ for some vectors $X_{i},Y_{i},Z_{i}\in\mathbb{R}^{3}(i=1,2)$ and denote by $X=\begin{bmatrix}X_{1}\\\ X_{2}\end{bmatrix},\quad Y=\begin{bmatrix}Y_{1}\\\ Y_{2}\end{bmatrix},\quad Z=\begin{bmatrix}Z_{1}\\\ Z_{2}\end{bmatrix}.$ Note that $X,Y,Z\in\mathbb{R}^{6}$, then by the previous statement for Euclidean space, we get $\Psi(X,Y)\leq\Psi(X,Z)+\Psi(Z,Y).$ (17) Since $\left\langle x,z\right\rangle$ and $\left\langle z,y\right\rangle$ are real numbers, we have $\displaystyle\left\langle x,z\right\rangle$ $\displaystyle=\mathrm{Re}\left\langle x,z\right\rangle=Z_{1}^{T}X_{1}+Z_{2}^{T}X_{2}=\left\langle X,Z\right\rangle,$ $\displaystyle\left\langle z,y\right\rangle$ $\displaystyle=\mathrm{Re}\left\langle z,y\right\rangle=Y_{1}^{T}Z_{1}+Y_{2}^{T}Z_{2}=\left\langle Z,Y\right\rangle,$ $\displaystyle\left\langle x,y\right\rangle$ $\displaystyle=Y_{1}^{T}X_{1}+Y_{2}^{T}X_{2}+i(Y_{1}^{T}X_{2}-Y_{2}^{T}X_{1}).$ It is easy to see that $\left\lVert x\right\rVert=\left\lVert X\right\rVert,\left\lVert y\right\rVert=\left\lVert Y\right\rVert$ and $\left\lVert z\right\rVert=\left\lVert Z\right\rVert$. Thus, $\Psi(x,z)=\Psi(X,Z),\quad\Psi(z,y)=\Psi(Z,Y).$ (18) Since $f(t)=\mathrm{arccos}\,(t)$ is a decreasing function in $t\in[-1,1]$, we get $\Psi(x,y)=\arccos\frac{\bigl{|}\left\langle x,y\right\rangle\bigr{|}}{\left\lVert x\right\rVert\left\lVert y\right\rVert}\leq\frac{\bigl{|}Y_{1}^{T}X_{1}+Y_{2}^{T}X_{2}\bigr{|}}{\left\lVert X\right\rVert\left\lVert Y\right\rVert}=\Psi(X,Y).$ (19) Combining (17), (18) and (19), we can get the desired inequality (16). ∎ Using the same idea of the proof of Theorem 5.1, one could also get the following Proposition 5.2. ###### Proposition 5.2. Let $u,v$ and $w$ be vectors in an inner product space. Then $\displaystyle\left|\Theta(u,v)-\Theta(v,w)\right|\leq\Theta(u,w)\leq\Theta(u,v)+\Theta(v,w),$ $\displaystyle 0\leq\Theta(u,v)+\Theta(v,w)+\Theta(w,u)\leq 2\pi.$ Moreover, the above inequalities hold for $\Psi$. The following inner product inequality is the main result in [22] and also can be found in [24, p. 195], it is derived as a tool in showing a trace inequality for unitary matrices. Of course, the line of proof provided here is quite different and simple. ###### Corollary 5.3. Let $u,v$ and $w$ be vectors in an inner product space over $\mathbb{C}$. Then $\sqrt{1-\frac{\left|\left\langle u,v\right\rangle\right|^{2}}{\left\lVert u\right\rVert^{2}\left\lVert v\right\rVert^{2}}}\leq\sqrt{1-\frac{\left|\left\langle u,w\right\rangle\right|^{2}}{\left\lVert u\right\rVert^{2}\left\lVert w\right\rVert^{2}}}+\sqrt{1-\frac{\left|\left\langle w,v\right\rangle\right|^{2}}{\left\lVert w\right\rVert^{2}\left\lVert v\right\rVert^{2}}}.$ Moreover, inequality holds if we replace $|\cdot|$ with $\mathrm{Re}\,(\cdot)$. ###### Proof. For brevity, we denote $\alpha,\beta,\gamma$ by the angles $\Psi(u,v),\Psi(u,w)$, $\Psi(w,v)$ or $\Phi(u,v),\Phi(u,w)$, $\Phi(w,v)$, respectively. By Proposition 5.2, we have $\displaystyle\frac{\alpha}{2}\leq\frac{\beta+\gamma}{2}\leq\pi-\frac{\alpha}{2},\quad 0\leq\frac{|\beta-\gamma|}{2}\leq\frac{\alpha}{2}\leq\frac{\pi}{2}.$ Then $\displaystyle 0\leq\sin\frac{\alpha}{2}\leq\sin\frac{\beta+\gamma}{2},\quad 0\leq\cos\frac{\alpha}{2}\leq\cos\frac{\beta-\gamma}{2}.$ The required inequality can be written as $\sin\alpha\leq 2\sin\frac{\beta+\gamma}{2}\cos\frac{\beta-\gamma}{2}=\sin\alpha+\sin\beta.$ This completes the proof. ∎ ## Acknowledgments The first author would like to expresses sincere thanks to Professor Fuzhen Zhang for his kind help and valuable discussion [23] before its publication, which considerably improves the presentation of our manuscript. Finally, all authors are grateful for valuable comments and suggestions from anonymous reviewer. This work was supported by NSFC (Grant No. 11671402, 11871479), Hunan Provincial Natural Science Foundation (Grant No. 2016JJ 2138, 2018JJ2479) and Mathematics and Interdisciplinary Sciences Project of Central South University. ## References * [1] R. Bhatia, Matrix Analysis, GTM 169, Springer-Verlag, New York, 1997. * [2] R. Bhatia, Positive Definite Matrices, Princeton University Press, Princeton, 2007. * [3] W. Berndt, S. Sra, Hlawka-Popoviciu inequalities on positive definite tensors, Linear Algebra Appl. 486 (2015) 317–327. * [4] D. Castano, V. E. Paksoy, F. Zhang, Angles, trangle inequalities, correlation matrices and metric-preserving and subadditive functions, Linear Algebra Appl. 491 (2016) 15–29. * [5] H. Chang, V. E. Paksoy, F. Zhang, An inequality for tensor product of positive operators and its applications, Linear Algebra Appl. 498 (2016) 99–105. * [6] D. Choi, Inequalities related to trace and determinant of positive semidefinite block matrices, Linear Algebra Appl. 532 (2017) 1–7. * [7] S. S. Dragomir, Some refinements of Schwarz inequality 13–16, Simpozionul de Matematici si Aplicatii. Timisoara Romania, 1985. * [8] S. S. Dragomir, Improving Schwarz inequality in inner product spaces, Linear and Multilinear Algebra 67 (2) (2019) 337–347. * [9] S. W. Drury, Positive semidefiniteness of a $3\times 3$ matrix related to partitioning, Linear Algebra Appl. 446 (2014) 369–376. * [10] K.E. Gustafson, D.K.M. Rao, Numerical Range, Springer, New York, 1997. * [11] R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2013. * [12] M. G. Krein, Angular localization of the spectrum of a multiplicative integral in a Hilbert space, Funct. Anal. Appl. 3 (1969) 89–90. * [13] M. Lin, Remarks on Krein’s inequality, Math. Intelligencer 34 (1) (2012) 3–4. * [14] M. Lin, A determinantal inequality for positive definite matrices, Electron. J. Linear Algebra 27 (2014) 821–826. * [15] M. Lin, P. Driessche, Positive semidefinite $3\times 3$ block matrices, Electron. J. Linear Algebra 27 (2014) 827–836. * [16] M. Lin, S. Sra, A proof of Thompson’s determinantal inequality, Math. Notes 99 (2016) 164–165. * [17] R. Merris, Multilinear Algebra, Gordon & Breach, Amsterdam, 1997. * [18] Z. Otachel, Inequalities for angles between subspaces with applications to Cauchy-Schwarz inequality in inner product spaces, Math. Inequal. Appl. 23 (2020) 487–495. * [19] V. Paksoy, R. Turkmen, F. Zhang, Inequalities of generalized matrix functions via tensor products, Electron. J. Linear Algebra 27 (2014) 332–341. * [20] D. Petz, Quantum Information Theory and Quantum Statistics. Theoretical and Mathematical Physics, Springer, Berlin, 2008. * [21] D.K. Rao, A triangle inequality for angles in a Hilbert space, Rev. Colombiana Mat. X (1976) 95–97. * [22] B.-Y. Wang, F. Zhang, A trace inequality for unitary matrices, Amer. Math. Monthly 101 (1994) 453–455. * [23] F. Zhang, Matrix Gems, private communication. * [24] F. Zhang, Matrix Theory: Basic Results and Techniques, 2nd edition, Springer, New York, 2011. * [25] F. Zhang, Positivity of matrices with generalized matrix functions, Acta Math. Sinica 28(9) (2012) 1779–1786.

2024-09-04T02:54:59.142476

2003.05232

arxiv-papers

# Spin Seebeck Effect near the Antiferromagnetic Spin-Flop Transition Derek Reitz Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA Junxue Li Wei Yuan Jing Shi Department of Physics and Astronomy, University of California, Riverside, California 92521, USA Yaroslav Tserkovnyak Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA ###### Abstract We develop a low-temperature, long-wavelength theory for the interfacial spin Seebeck effect (SSE) in easy-axis antiferromagnets. The field-induced spin- flop (SF) transition of Néel order is associated with a qualitative change in SSE behavior: Below SF, there are two spin carriers with opposite magnetic moments, with the carriers polarized along the field forming a majority magnon band. Above SF, the low-energy, ferromagnetic-like mode has magnetic moment opposite the field. This results in a sign change of the SSE across SF, which agrees with recent measurements on Cr2O3/Pt and Cr2O3/Ta devices [Li et al., Nature 578, 70 (2020)]. In our theory, SSE is due to a Néel spin current below SF and a magnetic spin current above SF. Using the ratio of the associated Néel to magnetic spin-mixing conductances as a single constant fitting parameter, we reproduce the field dependence of the experimental data and partially the temperature dependence of the relative SSE jump across SF. Introduction.—SSE involves transfer of spin angular momentum between a magnet and a metal via thermal spin fluctuations at their interface. In a typical experiment, a heat flux injected across the interface pumps a spin current into the metal, which is then converted into a transverse electric voltage $V_{\mathrm{SSE}}$ by spin-orbit interactions. This spin-current generation can be broadly attributed to two sources: One is due to a thermal gradient inside the magnet, which produces bulk magnon transport Adachi _et al._ (2010); Rezende _et al._ (2014, 2016a); Flebus _et al._ (2017); Prakash _et al._ (2018); Luo _et al._ (2019) and results in interfacial spin accumulation. The other is due to the interfacial temperature discontinuity, which produces spin pumping directly Xiao _et al._ (2010). SSE has been studied in ferromagnets Slachter _et al._ (2010); Uchida _et al._ (2010a), ferrimagnets Miao _et al._ (2016); Geprägs _et al._ (2016); Ohnuma _et al._ (2013), paramagnets Wu _et al._ (2015); Li _et al._ (2019); Yamamoto _et al._ (2019), and recently in antiferromagnets Seki _et al._ (2015); Wu _et al._ (2016); Li _et al._ (2020); Rezende _et al._ (2016b); Troncoso _et al._ (2020) as well as noncollinear magnets Flebus _et al._ (2019); Ma _et al._ (2020). The sign of $V_{\mathrm{SSE}}$ is determined by the polarization of the spin current along the applied magnetic field and the effective spin Hall angle of the metal detector. Fixing the spin Hall angle and the gyromagnetic ratio, the observed sign of the underlying spin current turns out to contain valuable information about the nature of spin order in the magnet and its nonequilibrium transport properties. Collinear ferromagnets (FMs) or noncollinear systems with weak ferromagnetic order have their net spin ordering along the magnetic field, whereas the elementary low-energy magnon excitations yield average spin polarization in the opposite direction. We can also imagine another class of systems, whose intrinsic excitations form spin-degenerate bands, with the degeneracy lifted by Zeeman splitting. The majority species, polarized along the field, may then determine the sign of the spin current, thus ending up opposite to the FM case. In our formalism, uniaxial AFs fall in this latter, majority-species scenario below SF, switching to the ferromagnetic-like SSE behavior above SF. Unlike argued in Ref. Hirobe _et al._ (2017), therefore, the SSE with the sign opposite to the FM case is a not a unique signature of correlated spin liquids, but can be expected to be a rather generic low-temperature signature of materials lacking FM order. Theoretically, there is at present no consensus on the “correct” sign of the SSE in antiferromagnets. Rezende et al. Rezende _et al._ (2016b) developed a magnon transport theory for uniaxial AFs below SF and concluded it falls into the majority-species scenario (i.e., SSE opposite to the FM case), but did not consider the sign when comparing their theory to experiment. Yamamoto et al. Yamamoto _et al._ (2019) used the fluctuation-dissipation theorem in a Landau-Ginzburg theory for easy-axis AFs below SF to study SSE around the Néel temperature $T_{N}$, concluding paramagnets and AFs below SF both have the same sign, but that it is the same as FMs. Here, we determine the sign within a low-temperature, long-wavelength theory for the interfacial SSE and show it changes across SF, in agreement with recent experiments. The quantitative aspects of the SSE over a broad range of temperatures and magnetic fields also appear in general agreement with the data. Spin pumping near SF transition.—In easy-axis AFs, when the Zeeman energy due to an applied field along the easy axis exceeds the anisotropy energy, there is a metamagnetic phase transition called spin flop (SF). Below SF (state I), the Néel order aligns with the easy axis, and there is a small net magnetization due to remnant longitudinal magnetic susceptibility Nordblad _et al._ (1979); Foner (1963). Dynamically, there are two circularly-polarized spin-wave modes with opposite handedness. When quantized, they correspond to magnons with magnetic moment parallel or antiparallel to the order parameter, each forming a gas (with equal and opposite chemical potentials, if driven slightly out of equilibrium Flebus (2019)). Above SF (state II), the Néel order reorients into the hard plane, and the spins cant giving net magnetization along the easy axis, due to a sizeable transverse magnetic susceptibility. There are now two distinct spin-wave modes at long wavelengths: a ferromagnetic-like mode ($\omega\rightarrow\gamma B$ when applied field $B\rightarrow\infty$) and a low-energy Goldstone mode associated with the U(1)-symmetry breaking Néel orientation in the hard plane. See Fig. 1. Figure 1: $k=0$ resonance frequencies are plotted for an easy-axis AF: $\omega_{1}$ and $\omega_{2}$ below spin flop and $\omega_{3}$ and $\omega_{4}$ above spin flop. $B$ is the applied magnetic field, $B_{c}=(\gamma s)^{-1}\sqrt{K_{1}/\chi}$ is the spin-flop field (which is about 6 Tesla for Cr2O3) according to the energy (3), and $\omega_{0}=\gamma B_{c}$ is the gap in I. The $\omega_{1}$ mode is right-hand circularly polarized and $\omega_{2}$ is left-hand circularly polarized in $\delta\boldsymbol{l}$ and $\delta\boldsymbol{m}$ (however the magnitude of $\delta\boldsymbol{m}$ is a factor $\chi K_{1}$ smaller than $\delta\boldsymbol{l}$ below SF, so it is omitted from the Figure). $\omega_{3}$ is linearly polarized in $\delta\boldsymbol{l}$ and $\delta\boldsymbol{m}$ so it does not produce spin currents bey . $\omega_{4}$ is linearly polarized in $\delta\boldsymbol{l}$ and elliptically polarized in $\delta\boldsymbol{m}$. The spin-current density pumped across the interface consist of the Néel, $\boldsymbol{J}_{l}$, and magnetic, $\boldsymbol{J}_{m}$, contributions: $\boldsymbol{J}_{l}=(\hbar g^{\uparrow\downarrow}_{l}/4\pi)\,\boldsymbol{l}\times\partial_{t}\boldsymbol{l},~{}~{}~{}\boldsymbol{J}_{m}=(\hbar g^{\uparrow\downarrow}_{m}/4\pi)\,\boldsymbol{m}\times\partial_{t}\boldsymbol{m},$ (1) where $g^{\uparrow\downarrow}$ is the respective (real part of the dimensionless) interfacial spin-mixing conductance per unit area. Thermal agitations in the metal held at temperature $T_{e}$ and in the AF at $T_{a}$ produce contributions $\boldsymbol{J}_{e}$ and $\boldsymbol{J}_{a}$ to the spin current, respectively. The spin Seebeck coefficient $S$ can be defined as the net spin current $J_{s}$ (projected onto the direction of the applied field) across the interface, divided by the temperature drop $\delta T=T_{a}-T_{e}$: $S\equiv J_{s}/\delta T=[J_{a}(T_{a})-J_{e}(T_{e})]/\delta T\to\partial_{T}J_{a}(T),$ (2) in linear response, where $J_{a}=J_{l}+J_{m}$ and $J_{e}(T)=J_{a}(T)$, in thermal equilibrium. In this paper, we investigate the signatures of SF in the SSE. In state I, there are two components of the Néel spin current that contribute oppositely to the SSE. With respect to increasing field, the (anti)parallel mode (decreases) increases in frequency. The antiparallel mode thus has greater thermal occupation at finite field, producing a net Néel spin current antiparallel to the field Ohnuma _et al._ (2013); Rezende _et al._ (2016b). In state II, there is only a magnetic spin current parallel to the field from the FM-like mode. Therefore, the SSE changes sign across SF. Spin-wave modes.—Following standard procedure Andreev and Marchenko (1980), we construct the low-energy long-wavelength theory for AF dynamics in terms of the Lagrangian density $\mathcal{L}(\boldsymbol{l},\boldsymbol{m})=s\boldsymbol{m}\cdot(\boldsymbol{l}\times\partial\boldsymbol{l}/\partial t)-E$. The energy density is given here by $E(\boldsymbol{l},\boldsymbol{m})=A(\gradient{\boldsymbol{l}})^{2}/2+\boldsymbol{m}^{2}/2\chi- K_{1}l_{z}^{2}/2-b\,\boldsymbol{m}\cdot\hat{\textbf{z}},$ (3) for a bipartite easy-axis AF subjected to a collinear magnetic field. The AF state is parametrized by directional Néel order $\boldsymbol{l}$ and normalized spin density $\boldsymbol{m}=\mathbf{s}/s$ ($\mathbf{s}$ being the spin density and $s\equiv\hbar S/V$, for spin $S$ and volume $V$ per site), in a nonlinear $\sigma$ model with constraint $\boldsymbol{l}^{2}=1$ and $\boldsymbol{l}\cdot\boldsymbol{m}=0$. We work well below the ordering temperature $T_{N}$, retaining the lowest-order gradient term of the Néel order with spin stiffness $A$. $\chi$ is the transverse magnetic susceptibility, $K_{1}$ the easy-axis anisotropy, and $b\equiv\gamma sB$, in terms of the magnetic field $B$ applied along the easy axis in the $\hat{\textbf{z}}$ direction (where $\gamma$ is the gyromagnetic ratio, whose sign is lumped into the value of $B$; i.e. when $\gamma<0$, our $B$ has opposite sign to the applied field). The Euler-Lagrange equations of motion may be extended to include dissipative forces $\partial\mathcal{F}/\partial\dot{\boldsymbol{m}}$ and $\partial\mathcal{F}/\partial\dot{\boldsymbol{l}}$ from the Rayleigh dissipation functional $\mathcal{F}=\alpha\dot{\boldsymbol{l}}^{2}/2+\widetilde{\alpha}\dot{\boldsymbol{m}}^{2}/2$, parametrized by Gilbert damping constants $\alpha$ and $\widetilde{\alpha}$. The ground states I and II are $(\boldsymbol{l}_{0},\boldsymbol{m}_{0})_{\mathrm{I}}=(\hat{\textbf{z}},0)$ and $(\boldsymbol{l}_{0},\boldsymbol{m}_{0})_{\mathrm{II}}=(\hat{\textbf{y}},\chi b\hat{\textbf{z}})$, with the critical field $B_{c}$ marking the jump from I to II. Spin waves are linear excitations, $\boldsymbol{l}=\boldsymbol{l}_{0}+\delta\boldsymbol{l}$ and $\boldsymbol{m}=\boldsymbol{m}_{0}+\delta\boldsymbol{m}$, satisfying the equations of motion. The dispersions are $\displaystyle\omega_{1k},\omega_{2k}=\mp\gamma B+\sqrt{(\gamma B_{c})^{2}+(ck)^{2}},$ (4a) $\displaystyle\omega_{3k}=ck,~{}~{}~{}\omega_{4k}=\sqrt{\gamma^{2}B^{2}-\gamma^{2}B_{c}^{2}+(ck)^{2}},$ (4b) where $c=s^{-1}\sqrt{A/\chi}$ is the speed of the large-$k$ AF spin waves. The six Cartesian components of $\delta\boldsymbol{l}$ and $\delta\boldsymbol{m}$ reduce to four independent and two slave variables, after applying the nonlinear constraints. Correspondingly, there are four spin-wave modes with momentum $k$, as shown in Fig. 1 (for consistency of the gradient expansion, we require $k\ll a^{-1}$, the inverse lattice spacing). $\omega_{1k}$ and $\omega_{2k}$ are waves with circularly precessing $\delta\boldsymbol{l}$ and $\delta\boldsymbol{m}$ in the plane perpendicular to $\boldsymbol{l}_{0,\mathrm{I}}$. $\omega_{3k}$ has linearly polarized $\delta\boldsymbol{l}(t)\propto e^{i\omega_{3k}t}\hat{\textbf{x}}$ and $\delta\boldsymbol{m}(t)\propto(\omega_{3k}/\omega_{x})e^{i(\omega_{3k}t-\pi/2)}\hat{\textbf{z}}$ bey . $\omega_{4k}$ has linearly polarized $\delta\boldsymbol{l}(t)\propto e^{i\omega_{4k}t}\hat{\textbf{z}}$ and elliptically polarized $\delta\boldsymbol{m}(t)\propto(\omega_{4k}/\omega_{x})e^{i\omega_{4k}t}\hat{\textbf{x}}-\chi be^{i(\omega_{3k}t-\pi/2)}\hat{\textbf{y}}$, where $\omega_{x}\equiv 1/\chi s$. Additional anisotropy energy $-K_{2}l_{y}^{2}/2$ within the easy plane will slightly shift the ground states, gap $\omega_{3}$, and introduce ellipticities in precession. When $k_{B}T\gg(\hbar/s)\sqrt{K_{2}/\chi}$, however, these modifications are negligible k2_ . Main results.—A thermal heat flux driven across the AF interface with a metal is given in the bulk by $-\sigma\gradient{T}$ and at the interface by $-\kappa\delta T$, where $\sigma$ and $\kappa$ are, respectively, the bulk and interfacial (Kapitza) thermal conductivities. $\delta T$ here is the temperature difference between phonons in the AF and electrons in the metal, $\delta T=T_{p}-T_{e}$ Xiao _et al._ (2010); Adachi _et al._ (2011). The Kapitza resistance ($\kappa^{-1}$) is large when there is poor phonon-phonon and phonon-electron interfacial coupling. For a fixed heat flux, this results in a larger $\delta T$, which drives the local SSE. The temperature gradient $\gradient{T}$ inside the magnet, furthermore, generates a bulk spin current, which flows towards the interface and contributes to the measured SSE Hoffman _et al._ (2013). We will specialize to the limit, in which the local spin pumping $\propto\delta T$ dominates, which corresponds to the case of an opaque interface and/or short spin-diffusion length in the AF. Equipped with the theory for AF dynamics, based on the Hamiltonian (3), we can use thermodynamic fluctuation-dissipation relations in order to convert magnetic response into thermal noise. The spin Seebeck coefficient (2) can then be evaluated by averaging Eqs. (1) over thermal fluctuations, whose spectral features follow the spin-wave dispersions discussed above. Carrying out this program, we arrive at the following final results (with the details of the derivations discussed later): Below spin flop (state I), $S_{\mathrm{I}}=\frac{g^{\uparrow\downarrow}_{l}\hbar^{2}}{2\pi\chi s^{2}}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\omega_{2k}\partial_{T}n_{\rm BE}(\omega_{2k})-\omega_{1k}\partial_{T}n_{\rm BE}(\omega_{1k})}{\omega_{1k}+\omega_{2k}},$ (5) and above spin flop (state II), $S_{\mathrm{II}}=\frac{g^{\uparrow\downarrow}_{m}\hbar^{2}\chi\gamma B}{2\pi}\int\frac{d^{3}k}{(2\pi)^{3}}\omega_{4k}\partial_{T}n_{\rm BE}(\omega_{4k}),$ (6) where $n_{\rm BE}(\omega)=(e^{\hbar\omega/k_{B}T}-1)^{-1}$ is the Bose- Einstein distribution function. We may evaluate the Seebeck coefficients analytically when $k_{B}T\gg\hbar\gamma B_{c}$. Since they are both linear in $B$, we compare the field slopes which go as $\partial_{B}S_{\mathrm{I}}\propto g^{\uparrow\downarrow}_{l}T$ and $\partial_{B}S_{\mathrm{II}}\propto g^{\uparrow\downarrow}_{m}T^{3}$: $v(T)\equiv-\frac{\partial_{B}S_{\mathrm{I}}}{\partial_{B}S_{\mathrm{II}}}\approx\frac{g^{\uparrow\downarrow}_{l}}{g^{\uparrow\downarrow}_{m}}\left(\frac{\hbar/\chi s}{k_{B}T}\right)^{2}\sim\frac{g^{\uparrow\downarrow}_{l}}{g^{\uparrow\downarrow}_{m}}\left(\frac{T_{N}}{T}\right)^{2}.$ (7) The ratio $v(T)$ contains the square of exchange ($\propto T_{N}$) to thermal energy in $v(T)$ (for the complete expressions, see s_I ). Note that for the applicability of our long-wavelength description, we require that $T\ll T_{N}$, throughout. Comparison to experiment.—In a conventional measurement scheme, the (longitudinal) SSE is revealed in a Nernst geometry as a lateral voltage induced perpendicular to the magnetic field applied in the plane of the magnetic interface Uchida _et al._ (2010a). This voltage is understood to arise from the inverse spin Hall effect associated with the thermally injected spin current. Normalizing the SSE voltage by the input thermal power $P_{\mathrm{in}}$, this gives $\frac{V_{\mathrm{SSE}}}{P_{\mathrm{in}}}=S(B,T)\frac{2e}{\hbar}\frac{\lambda^{*}}{wt}\frac{\rho(T)}{\kappa^{*}(T)},$ (8) where the materials-dependent interfacial spin-to-charge conversion lengthscale $\lambda^{*}$ can be loosely broken down into a product of an effective spin-diffusion length (a.k.a. spin-memory loss) $\lambda_{\mathrm{sd}}$ in the (heavy) normal metal and the effective spin Hall angle $\theta_{\mathrm{sH}}$, which converts the spin-current density $J_{\mathrm{s}}$ injected into the normal metal into the lateral charge- current density $J_{c}=(2e/\hbar)\theta_{\mathrm{sH}}J_{s}$. The total charge current is $I_{c}=w\lambda_{\mathrm{sd}}J_{c}$ when $\lambda_{\mathrm{sd}}\ll t$, the thickness of the metal film, where $w$ is the heterostructure width transverse to the injected charge current. In the open circuit, the underlying spin Hall motive force Uchida _et al._ (2010b) is balanced by the detectable voltage $V_{\mathrm{SSE}}=\rho lI_{c}/wt$, along the length $l$, where $\rho$ is the normal-metal resistivity. Putting everything together and expressing the spin current in terms of the Seebeck coefficient (2), we get the SSE voltage (8) normalized by the input power $P_{\mathrm{in}}=\kappa(T_{p}-T_{\mathrm{e}})lw$. $\kappa^{*}=\kappa(T_{p}-T_{e})/(T_{a}-T_{e})$ is an effective Kapitza conductance, which can be reduced relative to $\kappa$, if the lengthscale for the magnon-phonon equilibration that controls the temperature mismatch $T_{a}-T_{p}$ in the AF is long compared to $\sigma/\kappa$. Kapitza conductances for metal-insulator interfaces have been investigated in Refs. Stoner _et al._ (1992); Stevens _et al._ (2005); Hohensee _et al._ (2015); Lu _et al._ (2016), yielding nontrivial temperature dependences. The parameters for Cr2O3 are: $\sqrt{A}/a=(\chi\gamma s)^{-1}\approx 500$ T, $B_{\mathrm{c}}\approx 6$ T, $\gamma\approx\gamma_{e}$ Li _et al._ (2020) (where $\gamma_{e}$ is the free-electron value), $K_{2}\approx 0$ Foner (1963); for the Cr2O3/Pt and Cr2O3/Ta devices: $w=0.2$ mm, $t=5$ nm, the resistivity of the strips are $\rho_{\mathrm{Pt}}\approx 7\times 10^{-6}~{}\Omega\cdot$m and $\rho_{\mathrm{Ta}}\approx 9\times 10^{-5}~{}\Omega\cdot$m Li _et al._ (2020) at $T=75$ K, we take $\lambda^{*}$ from spin-pumping experiments: $\lambda_{\mathrm{Pt}}^{*}\sim 0.1$ nm Sinova _et al._ (2015) and $\lambda_{\mathrm{Ta}}^{*}\sim-0.04$ nm Hahn _et al._ (2013); Gómez _et al._ (2014); Yu _et al._ (2018), we approximate $g^{\uparrow\downarrow}_{m}$ for Pt and Ta with YIG/Pt’s: $g^{\uparrow\downarrow}_{m}\sim 10$ nm-2 Zhang _et al._ (2015). Figure 2: Theoretical spin Seebeck coefficients below, Eq. (5), and above, Eq. (6), spin flop for Cr2O3 are compared to experimental data from Li et al. Li _et al._ (2020). (a) and (b): The ratio $g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$ is fit to the relative slopes across SF. c) $S(T)$ is plotted until $T=80$ K; at higher temperatures, the long-wavelength theory loses quantitative accuracy. (d) Dispersions below SF are plotted. The majority spin carrier has magnetic moment along the field, which determines the polarization of the spin current. The comparison of the Seebeck coefficients (5), (6) (which may be evaluated analytically s_I ) to the data Li _et al._ (2020) is shown in Figs. 2(a)-(b). We use the slope of experimental $V_{\mathrm{SSE}}/P_{\mathrm{in}}$ in I to determine $\kappa_{\mathrm{Pt}}^{*}\sim 10^{9}$ W/m${}^{2}\cdot$K and $\kappa_{\mathrm{Ta}}^{*}\sim 10^{10}$ W/m${}^{2}\cdot$K at $T=75$ K, which are within 1-2 orders of magnitude of Stoner et al. measurements Stoner _et al._ (1992) of $\kappa$ in diamond$|$heavy-metal films. We also use an independent measurement of crystalline Cr2O3’s bulk thermal conductivity $\sigma$ Yuan _et al._ (2018), giving us an associated length scale $\sigma/\kappa_{\mathrm{Pt}}^{*}\approx 400$ nm and $\sigma/\kappa_{\mathrm{Ta}}^{*}\approx 60$ nm. Since the thin-film resistivities in our samples are about ten times larger than those in Refs. Vlaminck _et al._ (2013); Dutta _et al._ (2017) for Pt, from which we use the values for $\lambda_{\mathrm{Pt}}^{*}$ and $g^{\uparrow\downarrow}_{m}$ which go into determining $\kappa_{\mathrm{Pt}}^{*}$, the latter can only be taken as giving us a rough order-of-magnitude guidance. It should be safe to suppose that $\rho$, $\kappa^{*}$, and $g^{\uparrow\downarrow}$ are largely field independent, so that the field dependence in $V_{\mathrm{SSE}}/P_{\mathrm{in}}$ comes from $S$. The relative value of $S(B)$ across SF is determined theoretically up to the ratio $g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$ gl_ , which is a property of the interfaces. Several values are chosen in plotting Fig. 2. The best fit is determined by comparing theoretical $v(T)$ s_I , defined in Eq. 7, to the data at $T=75$ K. Note that $S|_{B=0}=0$, as expected on symmetry grounds. However, it is nontrivial that the $S_{\rm II}(B)$ dependence extrapolates to zero at zero field, both experimentally and in our theory. The temperature dependence in the calculated spin Seebeck coefficient $S$ enter through the magnon occupation number in the fluctuation-dissipation relation (9). The overall temperature dependence of the measured SSE is, furthermore, convoluted with thermal and charge conductivities. There are also slower temperature dependences in various parameters, such as $\chi(T)$ Foner (1963), which can complicate a detailed analysis. By looking at the slope ratio $v(T)$, however, we can eliminate the common prefactor associated with the heat-to-spin-to-charge conversions [see Eq. (8)], if the signal is dominated by the interfacial thermal bias. The experimental $v(T)$ for a bulk Cr2O3/Pt sample is plotted in Fig. 3 along with theoretical curves. The experimental data points for $v(T)$ are obtained by fitting a linear-in-field line to $V_{\mathrm{SSE}}$ in states I and II and taking the ratio of the slopes; for the theoretical curves see s_I . At low temperatures $T<7$ K, the theoretical slopes start becoming nonlinear [so that $S_{\mathrm{I}},S_{\mathrm{II}}$ must be evaluated numerically using Eqs. (5), (6)], with $S_{\mathrm{II}}(B)$ at large fields being the first portion of $S(B)$ to become nonlinear. Nonlinearities in $V_{\mathrm{SSE}}(B)$ are also observed experimentally above SF at $T=5$ K Li _et al._ (2020). While we see qualitative agreement, it appears there are additional spin Seebeck contribution(s) not captured by our formalism. The latter can stem from a bulk SSE in state I Lebrun _et al._ (2018), since thermal magnons polarized along the Néel order can diffuse over long distances Prakash _et al._ (2018). In particular, an additional linear in $T$ contribution to $S_{\mathrm{I}}$ would affect the estimate of $g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$ from the low-$T$ data, while a cubic contribution would explain the constant offset in $v(T)$ at larger temperatures. There may also be additional contributions in I and II due to other types of dynamics associated with interfacial inhomogeneities and locally uncompensated moments. In order to fit the totality of experimental data with our interfacial SSE-based model, we would require different values of $g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$ as a function of temperature. In particular, the data shown in Fig. 2a for $T=75$ K (corresponding to the largest temperature data point in Fig. 3) is well reproduced by taking $g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}\approx 15$, while the low temperature dependence of the data follows $v(T)\approx 160/T^{2}$ corresponding to $g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}\approx 300$. Although the order-of-magnitude estimate for the mixing conductance ratio and the trend in $v(T)$ as a function of temperature are reasonably captured by our simple model, a more complete theory (accounting for the bulk spin transport as well as for disorder-induced mesoscopic effects at the interface) is needed for developing a detailed quantitative understanding. Figure 3: The ratio of the spin Seebeck coefficient field slopes $v(T)$. Experimental data is from the same device as in Fig. 2(a) and is obtained from the slopes of linear-in-field fit lines, as discussed in the text. Theoretical curves are based on Eq. (7), evaluated here s_I ; plotted for various $g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$. The dashed line shows an approximate fit to the data. Theoretical formalism.—We calculate the spin currents in Eqs. (1) by averaging over thermal fluctuations of the magnetic variables. The latter can be obtained from the symmetrized fluctuation-dissipation theorem: $\left\langle\delta\phi_{i}\delta\phi_{j}\right\rangle=\frac{i\hbar}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\left[\chi_{ji}^{*}(\mathbf{k},\omega)-\chi_{ij}(\mathbf{k},\omega)\right]N(\omega),$ (9) where $\delta\phi_{i}$ stands for a Cartesian component of $\boldsymbol{l}$ or $\boldsymbol{m}$ and $\chi_{ij}$ is the corresponding linear-response function. $N(\omega)\equiv n_{\rm BE}(\omega)+1/2$ accounts for thermal fluctuations associated with occupied modes, according to the Bose-Einstein distribution function $n_{\rm BE}$, with $1/2$ reflecting the zero-point motion Landau and Lifshitz (1980). The dynamic susceptibility tensor is defined by $\delta\phi_{i}=\chi_{ij}\xi_{j}$, for the field $\xi_{j}$ thermodynamically conjugate to $\phi_{j}$. Our system is driven according to the energy density $E(B,t)=E(B)-\boldsymbol{m}\cdot\boldsymbol{h}(t)-\boldsymbol{l}\cdot\boldsymbol{g}(t)$, where $\boldsymbol{g}$ and $\boldsymbol{h}$ are conjugate to $\boldsymbol{l}$ and $\boldsymbol{m}$, respectively. The off-diagonal components of the Néel response $\chi^{(l)}_{ij}$ thus determine the Néel pumping as $\left\langle\boldsymbol{l}\times\partial\boldsymbol{l}/\partial t\right\rangle_{k}\to i\omega\epsilon^{ijk}\left\langle l_{i}l_{j}\right\rangle$ (in terms of the Levi-Civita tensor $\epsilon^{ijk}$, and upon the Fourier transform), and similarly for the magnetic response, $\chi^{(m)}_{ij}$. The components contributing to spin currents in I are $\displaystyle\chi_{xy}^{(l)}=-\frac{i}{2s^{2}\chi\omega_{0k}}\left(\frac{1}{\omega-\omega_{1k}+i\epsilon}-\frac{1}{\omega-\omega_{2k}+i\epsilon}\right),$ (10a) $\displaystyle\chi_{xy}^{(m)}=\chi^{2}K_{1}^{2}\chi_{xy}^{(l)},$ (10b) where $\omega_{0k}=\sqrt{(\gamma B_{c})^{2}+(ck)^{2}}$ and the dispersions are given in Eq. (4). According to Eq. (10a), the fluctuations perpendicular to $\boldsymbol{l}_{0,\mathrm{I}}=\hat{\textbf{z}}$ at $\omega_{1k}$ and $\omega_{2k}$ produce opposite contributions to the spin currents. The magnetic fluctuations in I in, e.g. Cr2O3, are a factor $(\chi K_{1})^{2}\sim 10^{-7}$ smaller than the Néel fluctuations and will be neglected. In II, $\delta\boldsymbol{l}$ is linearly polarized in the $\omega_{3k}$ and $\omega_{4k}$ modes, so Néel fluctuations do not produce spin currents bey . $\delta\boldsymbol{m}$ is elliptically polarized in the $\omega_{4k}$ mode, with magnetic fluctuations producing a spin current according to $\chi_{xy}^{(m)}=i\gamma\chi B\left(\frac{1}{\omega-\omega_{4k}+i\epsilon}\right).$ (11) Without dissipation, the poles $\chi_{ij}\propto 1/(\omega-\omega_{k}+i\epsilon)$ at the resonance frequencies are shifted by positive infinitesimal $\epsilon$. With dissipation, we end up with Lorentzians centered at these poles, whose widths are determined by bulk Gilbert damping and the effective damping due to interfacial spin pumping Tserkovnyak _et al._ (2002); Hoffman _et al._ (2013). When these resonance modes’ quality factors are large, however, their spectral weight is sharp and may be simply integrated over. We will assume this is the case, allowing us to neglect dissipation and simply use the infinitesimal $\epsilon$. Conclusion and outlook.—Our theory specializes to SSE from spin currents produced by an interfacial thermal bias. The formalism may be extended to account for bulk thermal gradients, which produce nonequilibrium interfacial spin accumulation $\boldsymbol{\mu}$. However, determining $\boldsymbol{\mu}$ requires complimenting the interfacial transport with coupled spin and heat transport in the bulk Prakash _et al._ (2018), which is beyond our present scope. The purely local SSE studied here should quantitatively model SSE for interfaces with large interfacial thermal resistances and weak interfacial spin coupling. In this regime, SSE would provide a noninvasive probe of the magnet’s transverse components of $\chi_{ij}$, much like scanning tunneling microscopy is an interfacial probe of an electron density of states Tersoff and Hamann (1983). We have discussed two classes of systems which produce different signs for SSE. The FM-like class involves spin excitations with magnetic moment opposite the order parameter, such as in FMs, uniaxial AFs above SF, and DMI AFs. Another class involves degenerate spin excitations, whose degeneracy is lifted by magnetic field. The majority carrier, which has magnetic moment along the magnetic field, can then dominates spin transport. In our low-temperature, long-wavelength theory we have shown that uniaxial AFs below SF belong to this class. However, when the bulk SSE contribution is significant, this reasoning alone may not determine the sign. Since the majority band reaches the edge of the BZ faster than the minority, it may suffer greater umklapp scattering at elevated temperatures, which would lower its conductivity. A full transport theory is then required to determine the SSE sign, as a function of temperature. By comparing $v(T)\equiv-\partial_{B}S_{\mathrm{I}}/\partial_{B}S_{\mathrm{II}}$ in experiment to our theory as a function of $T$, we see some discrepancy. Our theory predicts $v\propto 1/T^{2}$, while the Cr2O3/Pt sample indicates $v(T)\approx 0.7+160/T^{2}$. The constant offset could stem from a bulk Seebeck contribution in I at higher $T$ whose coefficient goes as $T^{3}$. Above SF, bulk contributions to SSE can be expected to be reduced, since spin transport is then normal to the Néel order. $v(T)$ may also have contributions from paramagnetic impurities or other extrinsic surface modes, or be convoluted with temperature dependence in $g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$. The magnitude of $g^{\uparrow\downarrow}_{l}$ and $g^{\uparrow\downarrow}_{m}$ can, furthermore, vary from one sample to another due to the amount of disorder in the interfacial exchange coupling Takei _et al._ (2014); Troncoso _et al._ (2020). While our theory well reproduces the temperature dependence at low $T$, a different value of $g^{\uparrow\downarrow}_{m}/g^{\uparrow\downarrow}_{l}$ is needed to consistently explain higher temperature data. Looking forward, a more complete theory is called for which includes SSE contributions from both the interface and the bulk, in addition to the dynamical effects of disorder at the interface. The sensitivity of the SSE to the preparation and quality of the interface may complicate the analysis based on the measured $v(T)$ across the SF. We recall that Seki et al. Seki _et al._ (2015) did not observe a significant SSE in I at low temperatures in Cr2O3/Pt. Wu et al. Wu _et al._ (2016) observed SSE with nonlinear field dependence and ferromagnetic sign signature on both sides of SF in MnF2/Pt. Ferromagnetic sign in I was also observed in an etched- interface Cr2O3/Pt sample by Li et al. Li _et al._ (2020). 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2024-09-04T02:54:59.151337

2003.05234

arxiv-papers

# Compact gradient $\rho$-Einstein soliton is isometric to the Euclidean sphere Absos Ali Shaikh1, Chandan Kumar Mondal2, Prosenjit Mandal3 Department of Mathematics, University of Burdwan, Golapbag, Burdwan-713104, West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract. In this paper we have investigated some aspects of gradient $\rho$-Einstein Ricci soliton in a complete Riemannian manifold. First, we have proved that the compact gradient $\rho$-Einstein soliton is isometric to the Euclidean sphere by showing that the scalar curvature becomes constant. Second, we have showed that in a non-compact gradient $\rho$-Einstein soliton satisfying some integral condition, the scalar curvature vanishes. ††footnotetext: $\mathbf{2020}$ Mathematics Subject Classification: 53C20; 53C21; 53C44. Key words and phrases: Gradient $\rho$-Einstein Ricci soliton; scalar curvature; Riemannian manifold. ## 1\. Introduction and preliminaries A $1$-parameter family of metrics $\\{g(t)\\}$ on a Riemannian manifold $M$, defined on some time interval $I\subset\mathbb{R}$ is said to satisfy Ricci flow if it satisfies $\frac{\partial}{\partial t}g_{ij}=-2R_{ij},$ where $R_{ij}$ is the Ricci curvature with respect to the metric $g_{ij}$. Hamilton [9] proved that for any smooth initial metric $g(0)=g_{0}$ on a closed manifold, there exists a unique solution $g(t)$, $t\in[0,\epsilon)$, to the Ricci flow equation for some $\epsilon>0$. A solution $g(t)$ of the Ricci flow of the form $g(t)=\sigma(t)\varphi(t)^{*}g(0),$ where $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is a positive function and $\varphi(t):M\rightarrow M$ is a 1-parameter family of diffeomorphisms, is called a Ricci soliton. It is known that if the initial metric $g_{0}$ satisfies the equation (1) $Ric(g_{0})+\frac{1}{2}\pounds_{X}g_{0}=\lambda g_{0},$ where $\lambda$ is a constant and $X$ is a smooth vector field on $M$, then the manifold $M$ admits Ricci soliton. Therefore, the equation (1), in general, is known as Ricci soliton. If $X$ is the gradient of some smooth function, then it is called gradient Ricci soliton. For more results of Ricci soliton see [2, 7, 8]. In 1979, Bourguignon [1] introduced the notion of Ricci-Bourguignon flow, where the metrics $g(t)$ is evolving according to the flow equation $\frac{\partial}{\partial t}g_{ij}=-2R_{ij}+2\rho Rg_{ij},$ where $\rho$ is a non-zero scalar constant and $R$ is the scalar curvature of the metric $g(t)$. Following the Ricci soliton, Catino and Mazzier [4] gave the definition of gradient $\rho$-Einstein soliton, which is the self-similar solution of Ricci-Bourguignon flow. This soliton is also called gradient Ricci-Bourguignon soliton by some authors. ###### Definition 1.1. [4] Let $(M,g)$ be a Riemannian manifold of dimension $n$, $(n\geq 3)$, and let $\rho\in\mathbb{R}$, $\rho\neq 0$. Then $M$ is called gradient $\rho$-Einstein soliton, denoted by $(M,g,f,\rho)$, if there is a smooth function $f:M\rightarrow\mathbb{R}$ such that (2) $Ric+\nabla^{2}f=\lambda g+\rho Rg,$ for some constant $\lambda$. The soliton is trivial if $\nabla f$ is a parallel vector field. The function $f$ is known as $\rho$-Einstein potential function. If $\lambda>0$ $(\text{resp.}=0,<0)$, then the gradient $\rho$-Einstein soliton $(M,g,f,\rho)$ is said to be shrinking (resp. steady or expanding) . On the other hand, the $\rho$-Einstein soliton is called gradient Einstein soliton, gradient traceless Ricci soliton or gradient Schouten soliton if $\rho=1/2,1/n$ or $1/2(n-1)$. Later, this notion has been generalized in various directions such as $m$-quasi Einstein manifold [11], $(m,\rho)$-quasi Einstein manifold [12], Ricci-Bourguignon almost soliton [13]. Catino and Mazzier [4] showed that compact gradient Einstein, Schouten or traceless Ricci soliton is trivial. They classified three-dimensional gradient shrinking Schouten soliton and proved that it is isometric to a finite quotient of either $\mathbb{S}^{3}$ or $\mathbb{R}^{3}$ or $\mathbb{R}\times\mathbb{S}^{2}$. Huang [10] deduced a sufficient condition for the compact gradient shrinking $\rho$-Einstein soliton to be isometric to a quotient of the round sphere $\mathbb{S}^{n}$. ###### Theorem 1.1. [10] Let $(M,g,f,\rho)$ be an $n$-dimensional $(4\leq n\leq 5)$ compact gradient shrinking $\rho$-Einstein soliton with $\rho<0$. If the following condition holds $\displaystyle\Big{(}\int_{M}|W+\frac{\sqrt{2}}{\sqrt{n}(n-2)}Z\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g|^{2}\Big{)}^{\frac{2}{n}}$ $\displaystyle+\sqrt{\frac{(n-4)^{2}(n-1)}{8(n-2)}}\lambda vol(M)^{\frac{2}{n}}$ $\displaystyle\leq\sqrt{\frac{n-2}{32(n-1)}}Y(M,[g]),$ where $Z=Ric-\frac{R}{n}g$ is the trace-less Ricci tensor, $W$ is the Weyl tensor and $Y(M,[g])$ is the Yamabe invariant associated to $(M,g)$, then $M$ is isometric to a quotient of the round sphere $\mathbb{S}^{n}$. In 2019, Mondal and Shaikh [14] proved the isometry theorem for gradient $\rho$-Einstein soliton in case of conformal vector field. In particular, they proved the following result: ###### Theorem 1.2. [14] Let $(M,g,f,\rho)$ be a compact gradient $\rho$-Einstein soliton. If $\nabla f$ is a non-trivial conformal vector field, then $M$ is isometric to the Euclidean sphere $\mathbb{S}^{n}$. Dwivedi [13] proved an isometry theorem for gradient Ricci-Bourguignon soliton. ###### Theorem 1.3. [13] A non-trivial compact gradient Ricci-Bourguignon soliton is isometric to an Euclidean sphere if any one of the following holds (1) $M$ has constant scalar curvature. (2) $\int_{M}g(\nabla R,\nabla f)\leq 0$. (3) $M$ is a homogeneous manifold. We note that Catino et. al. [5] proved many results for gradient $\rho$-Einstein soliton in non-compact manifold. ###### Theorem 1.4. Let $(M,g,f,\rho)$ be a complete non-compact gradient shrinking $\rho$-Einstein soliton with $0<\rho<1/2(n-1)$ bounded curvature, non-negative radial sectional curvature, and non-negative Ricci curvature. Then the scalar curvature is constant. In this paper, we have showed that a non-trivial compact gradient $\rho$-Einstein soliton is isometric to an Euclidean sphere. The main results of this paper are as follows: ###### Theorem 1.5. A nontrivial compact gradient $\rho$-Einstein soliton has constant scalar curvature and therefore $M$ is isometric to an Euclidean sphere. We have also showed that in a non-compact gradient $\rho$-Einstein soliton satisfying some conditions the scalar curvature vanishes. ###### Theorem 1.6. Suppose $(M,g,f,\rho)$ is a non-compact gradient non-expanding $\rho$-Einstein soliton with non-negative scalar curvature. If $\rho>1/n$ and the $\rho$-Einstein potential function satisfies (3) $\int_{M-B(p,r)}d(x,p)^{-2}f<\infty,$ then the scalar curvature vanishes in $M$. ## 2\. Proof of the results ###### Proof of the Theorem 1.5. Since the gradient $\rho$-Einstein soliton is non-trivial, it follows that $\rho\neq 1/n$, see [4]. Taking the trace of (2) we get (4) $R+\Delta f=\lambda n+\rho Rn.$ From the commutative equation, we obtain (5) $\Delta\nabla_{i}f=\nabla_{i}\Delta f+R_{ij}\nabla_{j}f.$ By using contracted second Bianchi identity, we have $\displaystyle\Delta\nabla_{i}f=\nabla_{j}\nabla_{j}\nabla_{i}f$ $\displaystyle=$ $\displaystyle\nabla_{j}(\lambda g_{ij}+\rho Rg_{ij}-R_{ij})$ $\displaystyle=$ $\displaystyle\nabla_{i}(\rho R-\frac{1}{2}R).$ and $\nabla_{i}\Delta f=\nabla_{i}(\lambda n+\rho Rn-R)=\nabla_{i}(\rho Rn-R).$ Therefore, (5) yields (6) $(n-1)\rho\nabla_{i}R-\frac{1}{2}\nabla_{i}R+R_{ij}\nabla_{j}f=0,$ Taking covariant derivative $\nabla_{l}$, we get $(n-1)\rho\nabla_{l}\nabla_{i}R-\frac{1}{2}\nabla_{l}\nabla_{i}R+\nabla_{l}R_{ij}\nabla_{j}f+R_{ij}\nabla_{l}\nabla_{j}f=0.$ Taking trace in both sides, we obtain (7) $((n-1)\rho-\frac{1}{2})\Delta R+\frac{1}{2}g(\nabla R,\nabla f)+R(\lambda n+\rho Rn-R)=0.$ Now integrating using divergence theorem we get $\displaystyle\int_{M}R(\lambda n+\rho Rn-R)$ $\displaystyle=$ $\displaystyle-\int_{M}((n-1)\rho-\frac{1}{2})\Delta R-\frac{1}{2}\int_{M}g(\nabla R,\nabla f)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int_{M}R\Delta f=\frac{1}{2}\int_{M}R(\lambda n+\rho Rn-R).$ The above equation is true only if (8) $\int_{M}R(\lambda n+\rho Rn-R)=0,$ which implies (9) $\int_{M}R\Big{(}R+\frac{\lambda n}{n\rho-1}\Big{)}=0,$ Again integrating (4), we obtain (10) $\int_{M}\Big{(}R+\frac{\lambda n}{n\rho-1}\Big{)}=0.$ Therefore, (9) and (10) together imply that $\int_{M}\Big{(}R+\frac{\lambda n}{n\rho-1}\Big{)}^{2}=0.$ Hence, $R=\lambda n/(1-\rho n)$. Then from Theorem 1.3 we can conclude our result. ∎ ###### Proof of the Theorem 1.6. From (4) we get $(n\rho-1)R=\Delta f-\lambda n.$ Since $\lambda\geq 0$, the above equation implies that (11) $(n\rho-1)R\leq\Delta f.$ Now, we consider the cut-off function, introduced in [6], $\varphi_{r}\in C^{2}_{0}(B(p,2r))$ for $r>0$ such that $\begin{cases}0\leq\varphi_{r}\leq 1&\text{ in }B(p,2r)\\\ \varphi_{r}=1&\text{ in }B(p,r)\\\ |\nabla\varphi_{r}|^{2}\leq\frac{C}{r^{2}}&\text{ in }B(p,2r)\\\ \Delta\varphi_{r}\leq\frac{C}{r^{2}}&\text{ in }B(p,2r),\end{cases}$ where $C>0$ is a constant. Then for $r\rightarrow\infty$, we have $\Delta\varphi^{2}_{r}\rightarrow 0$ as $\Delta\varphi^{2}_{r}\leq\frac{C}{r^{2}}$. Then we calculate (12) $\displaystyle(n\rho-1)\int_{M}R\varphi^{2}_{r}\leq\int_{M}\varphi^{2}_{r}\Delta f$ $\displaystyle=$ $\displaystyle\int_{B(p,2r)-B(p,r)}f\Delta\varphi_{r}^{2}$ (13) $\displaystyle\leq$ $\displaystyle\int_{B(p,2r)-B(p,r)}f\frac{C}{r^{2}}\rightarrow 0,$ as $r\rightarrow\infty$. Hence, we obtain (14) $(n\rho-1)\lim_{r\rightarrow\infty}\int_{B(p,r)}R\leq 0.$ Since $\rho>1/n$, it follows that $\lim_{r\rightarrow\infty}\int_{B(p,r)}R\leq 0.$ But $R$ is non-negative everywhere in $M$. Therefore, $R\equiv 0$ in $M$. ∎ ## 3\. acknowledgment The third author gratefully acknowledges to the CSIR(File No.:09/025(0282)/2019-EMR-I), Govt. of India for financial assistance. ## References * [1] Bourguignon, J. P., Ricci curvature and Einstein metrics. Global differential geometry and global analysis, Berlin, 1979, 42–63, Lecture Notes in Math. 838, Springer, Berlin, 1981. * [2] Cao, H. D., Recent progress on Ricci solitons, in: Recent Advances in Geometric Analysis, Adv. Lectures Math., 11 (2010), 1–38. * [3] Cao, H. D. and Zhou, D., On complee gradient shrnking Ricci solitons. J. Diff. Geom., 85 (2010), 175–183. * [4] Catino, G. and Mazzieri, L., Gradient Einstein solitons, Nonlinear Anal., 132 (2016), 66–94. * [5] Catino, G., Mazzieri, L. and Mongodi, S., Rigidity of gradient Einstein shrinkers, Commun. Contemp. Math., 17(6) (2015), 1–18. * [6] Cheeger, J. and Colding, T. H., Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Math., 144(1) (1996), 189–237. * [7] Chow, B. and Knopf, D., The Ricci flow: An introduction, mathematical surveys and monographs. Amer. Math. Soc., 110, 2004. * [8] Fang, F. Q., Man, J. W. and Zhang, Z. L., Complete gradient shrinking Ricci solitons have finite topological type. C. R. Acad. Sci. Paris, Ser. I 346(1971), 653–656. * [9] Hamilton, R. S., Three-manifolds with positive Ricci curvature, J. Differ. Geom., 17 (1982), 255–306. * [10] Huang, G., Integral pinched gradient shrinking $\rho$-Einstein solitons, J. Math. Ann. Appl. 451(2) (2017), 1045–1055. * [11] Hu, Z., Li, D. and Xu, J., On generalized $m$-quasi-Einstein manifolds with constant scalar curvature, J. Math. Ann. Appl. 432(2) (2015), 733–743. * [12] Huang, G. and Wei, Y., The classification of $(m,ρ)$-quasi-Einstein manifolds, Ann. Glob. Anal. geom. 44 (2013), 269–282. * [13] Dwivedi, S., Some results on Ricci-Bourguignon and Ricci-Bourguignon almost solitons, arXiv:1809.11103. * [14] Mondal, C. K. and Shaikh, A. A., Some results on $\eta$-Ricci Soliton and gradient $rho$-Einstein soliton in a complete Riemannian manifold, Comm. Korean Math. Soc. 34(4) (2019), 1279–1287.

2024-09-04T02:54:59.159884

2003.05263

arxiv-papers

# Spectrum and Analytic Functional Calculus in Real and Quaternionic Frameworks Florian-Horia Vasilescu Department of Mathematics, University of Lille, 59655 Villeneuve d’Ascq, France <EMAIL_ADDRESS> ###### Abstract We present an approach to the spectrum and analytic functional calculus for quaternionic linear operators, following the corresponding results concerning the real linear operators. In fact, the construction of the analytic functional calculus for real linear operators can be refined to get a similar construction for quaternionic linear ones, in a classical manner, using a Riesz-Dunford-Gelfand type kernel, and considering spectra in the complex plane. A quaternionic joint spectrum for pairs of operators is also discussed, and an analytic functional calculus is constructed, via a Martinelli type kernel in two variables. Keywords: spectrum in real and quaternionic contexts; holomorphic stem functions; analytic functional calculus for real and quaternionic operators AMS Subject Classification: 47A10; 30G35; 47A60 Keywords: real and quaternionic operators; spectra; analytic functionalcalculus. Mathematics Subject Classification 2010: 47A10; 47A60; 30G35 ## 1 Introduction In this text we consider ${\mathbb{R}}$-, ${\mathbb{C}}$-, and ${\mathbb{H}}$-linear operators, that is, real, complex and quaternionic linear operators, respectively. While the spectrum of a linear operator is traditionally defined for complex linear operators, it is sometimes useful to have it also for real linear operators, as well as for quaternionic linear ones. The definition of the spectrum for a real linear operator goes seemingly back to Kaplansky (see [10]), and it can be stated as follows. If $T$ is a real linear operator on the real vector space ${\mathcal{V}}$, a point $u+iv$ ($u,v\in{\mathbb{R}}$) is in the spectrum of $T$ if the operator $(u-T)^{2}+v^{2}$ is not invertible on ${\mathcal{V}}$, where the scalars are identified with multiples of the identity on ${\mathcal{V}}$. Although this definition involves only operators acting in ${\mathcal{V}}$, the spectrum is, nevertheless, a subset of the complex plane. As a matter of fact, a motivation of this choice can be illustrated via the complexification of the space ${\mathcal{V}}$ (see Section 2). The spectral theory for quaternionic linear operators is largely discussed in numerous work, in particular in the monographs [5] and [4], and in many of their references as well. In these works, the construction of an analytic functional calculus (called $S$-analytic functional calculus) means to associate to each function from the class of the so-called slice hyperholomorphic or slice regular functions a quaternionic linear operator, using a specific noncommutative kernel. The idea of the present work is to replace the class of slice regular functions by a class holomorphic functions, using a commutative kernel of type Riesz- Dunford-Gelfand. These two classes are isomorphic via a Cauchy type transform (see [21]), and the image of the analytic functional calculus is the same, as one might expect (see Remark 8). As in the case of real operators, the verbatim extension of the classical definition of the spectrum for quaternionic operators is not appropriate, and so a different definition using the squares of operators and real numbers was given, which can be found in [5] (see also [4]). We discuss this definition in our framework (see Definition 1), showing later that its ”complex border“ contains the most significant information, leading to the construction of an analytic functional calculus, equivalent to that obtained via the slice hyperholomorphic functions. In fact, we first consider the spectrum for real operators on real Banach spaces, and sketch the construction of an analytic functional calculus for them, using some classical ideas (see Theorem 2). Then we extend this framework to a quaternionic one, showing that the approach from the real case can be easily adapted to the new situation. As already mentioned, and unlike in [5] or [4], our functional calculus is obtained via a Riesz-Dunford-Gelfand formula, defined in a partially commutatative context, rather than the non-commutative Cauchy type formula used by previous authors. Our analytic functional calculus holds for a class of analytic operator valued functions, whose definition extends that of stem functions, and it applies, in particular, to a large family of quaternionic linear operators. Moreover, we can show that the analytic functional calculus obtained in this way is equivalent to the analytic functional calculus obtained in [5] or [4], in the sense that the images of these functional calculi coincide (see Remark 8). We finally discuss the case of pairs of commuting real operators, in the spirit of [20], showing some connections with the quaternionic case. Specifically, we define a quaternionic spectrum for them and construct an analytic functional calculus using a Martinelli type formula, showing that for such a construction only a sort of ”complex border“ of the quaternionic spectrum should be used. This work is just an introductory one. Hopefully, more contributions on this line will be presented in the future. ## 2 Spectrum and Conjugation Let ${\mathcal{A}}$ be a unital real Banach algebra, not necessarily commutative. As mentioned in the Introduction, the (complex) spectrum of an element $a\in{\mathcal{A}}$ may be defined by the equality $\sigma_{\mathbb{C}}(a)=\\{u+iv;(u-a)^{2}+v^{2}\,\,{\rm is\,\,not\,\,invertible},u,v\in{\mathbb{R}}\\},$ (1) This set is conjugate symmetric, that is $u+iv\in\sigma_{\mathbb{C}}(a)$ if and only if $u-iv\in\sigma_{\mathbb{C}}(a)$. A known motivation of this definition comes from the following remark. Fixing a unital real Banach algebra ${\mathcal{A}}$, we denote by ${\mathcal{A}}_{\mathbb{C}}$ the complexification of ${\mathcal{A}}$, which is given by $A_{\mathbb{C}}={\mathbb{C}}\otimes_{\mathbb{R}}{\mathcal{A}}$, written simply as ${\mathcal{A}}+i{\mathcal{A}}$, where the sum is direct, identifying the element $1\otimes a+i\otimes b$ with the element $a+ib$, for all $a,b\in{\mathcal{A}}$. Then ${\mathcal{A}}_{\mathbb{C}}$ is a unital complex algebra, which can be organized as a Banach algebra, with a (not necessarily unique) convenient norm. To fix the ideas, we recall that the product of two elements is given by $(a+ib)(c+id)=ac-bd+i(ad+bc)$ for all $a,b,c,d\in{\mathcal{A}}$, and the norm may be defind by $\|a+ib\|=\|a\|+\|b\|$, where $\|*\|$ is the norm of ${\mathcal{A}}$. In the algebra ${\mathcal{A}}_{\mathbb{C}}$, the complex numbers commute with all elements of ${\mathcal{A}}$. Moreover, we have a conjugation given by ${\mathcal{A}}_{\mathbb{C}}\ni a+ib\mapsto a-ib\in{\mathcal{A}}_{\mathbb{C}},\,a,b\in{\mathcal{A}},$ which is a unital conjugate-linear automorphism, whose square is the identity. In particular, an arbitrary element $a+ib$ is invertible if and only if $a-ib$ is invertible. The usual spectrum, defined for each element $a\in{\mathcal{A}}_{\mathbb{C}}$, will be denoted by $\sigma(a)$. Regarding the algebra ${\mathcal{A}}$ as a real subalgebra of ${\mathcal{A}}_{\mathbb{C}}$, one has the following. ###### Lemma 1 For every $a\in{\mathcal{A}}$ we have the equality $\sigma_{\mathbb{C}}(a)=\sigma(a)$. Proof. The result is well known but we give a short proof, because a similar idea will be later used. Let $\lambda=u+iv$ with $u,v\in{\mathbb{R}}$ arbitrary. Assuming $\lambda-a$ invertible, we also have $\bar{\lambda}-a$ invertible. From the obvious identity $(u-a)^{2}+v^{2}=(u+iv-a)(u-iv-a),$ we deduce that the element $(u-a)^{2}+v^{2}$ is invertible, implying the inclusion $\sigma_{\mathbb{C}}(a)\subset\sigma(a)$. Conversely, if $(u-a)^{2}+v^{2}$ is invertible, then both $u+iv-a,u-iv-a$ are invertible via the decomposition from above, showing that we also have $\sigma_{\mathbb{C}}(a)\supset\sigma(a)$. ###### Remark 1 The spectrum $\sigma(a)$ with $a\in{\mathcal{A}}$ is always a conjugate symmetric set. We are particularly interested to apply the discussion from above to the context of linear operators. The spectral theory for real linear operators is well known, and it is developed actually in the framework of linear relations (see [1]). Nevertheless, we present here a different approach, which can be applied, with minor changes, to the case of some quaternionic operators. For a real or complex Banach space $\mathcal{V}$, we denote by $\mathcal{B(V)}$ the algebra of all bounded ${\mathbb{R}}$-( respectively ${\mathbb{C}}$-)linear operators on $\mathcal{V}$. As before, the multiples of the identity will be identified with the corresponding scalars. Let ${\mathcal{V}}$ be a real Banach space, and let ${\mathcal{V}}_{\mathbb{C}}$ be its complexification, which, as above, is identified with the direct sum ${\mathcal{V}}+i{\mathcal{V}}$. Each operator $T\in\mathcal{B(V)}$ has a natural extension to an operator $T_{\mathbb{C}}\in\mathcal{B}({\mathcal{V}}_{\mathbb{C}})$, given by $T_{\mathbb{C}}(x+iy)=Tx+iTy,\,x,y\in{\mathcal{V}}$. Moreover, the map $\mathcal{B(V)}\ni T\mapsto T_{\mathbb{C}}\in\mathcal{B}({\mathcal{V}}_{\mathbb{C}})$ is unital, ${\mathbb{R}}$-linear and multiplicative. In particular, $T\in\mathcal{B(V)}$ is invertible if and only if $T_{\mathbb{C}}\in\mathcal{B}({\mathcal{V}}_{\mathbb{C}})$ is invertible. Fixing an operator $S\in\mathcal{B}(\mathcal{V}_{\mathbb{C}})$, we define the operator $S^{\flat}\in\mathcal{B}(\mathcal{V}_{\mathbb{C}})$ to be equal to $CSC$, where $C:{\mathcal{V}}_{\mathbb{C}}\mapsto{\mathcal{V}}_{\mathbb{C}}$ is the conjugation $x+iy\mapsto x-iy,\,x,y\in{\mathcal{V}}$. It is easily seen that the map $\mathcal{B}(\mathcal{V}_{\mathbb{C}})\ni S\mapsto S^{\flat}\in\mathcal{B}(\mathcal{V}_{\mathbb{C}})$ is a unital conjugate- linear automorphism, whose square is the identity on $\mathcal{B}(\mathcal{V}_{\mathbb{C}})$. Because $\mathcal{V}=\\{u\in\mathcal{V}_{\mathbb{C}};Cu=u\\}$, we have $S^{\flat}=S$ if and only if $S(\mathcal{V})\subset\mathcal{V}$. In particular, we have $T_{\mathbb{C}}^{\flat}=T_{\mathbb{C}}$. In fact, because of the representation $S=\frac{1}{2}(S+S^{\flat})+i\frac{1}{2i}(S-S^{\flat}),\,\,S\in\mathcal{B}(\mathcal{V}_{\mathbb{C}}),$ where $(S+S^{\flat})({\mathcal{V}})\subset{\mathcal{V}},i(S-S^{\flat})({\mathcal{V}})\subset{\mathcal{V}}$, the algebras $\mathcal{B}(\mathcal{V}_{\mathbb{C}})$ and $\mathcal{B(V)}_{\mathbb{C}}$ are isomorphic and they will be often identified, and $\mathcal{B(V)}$ will be regarded as a (real) subalgebra of $\mathcal{B}(\mathcal{V})_{\mathbb{C}}$. In particular, if $S=U+iV$, with $U,V\in\mathcal{B(V)}$, we have $S^{\flat}=U-iV$, so the map $S\mapsto S^{\flat}$ is the conjugation of the complex algebra $\mathcal{B}(\mathcal{V})_{\mathbb{C}}$ induced by the conjugation $C$ of ${\mathcal{V}}_{\mathbb{C}}$. For every operator $S\in\mathcal{B}(\mathcal{V}_{\mathbb{C}})$, we denote, as before, by $\sigma(S)$ its usual spectrum. As $\mathcal{B(V)}$ is a real algebra, the (complex) spectrum of an operator $T\in\mathcal{B(V)}$ is given by the equality (1): $\sigma_{\mathbb{C}}(T)=\\{u+iv;(u-T)^{2}+v^{2}\,\,{\rm is\,\,not\,\,invertible},u,v\in{\mathbb{R}}\\}.$ ###### Corollary 1 For every $T\in\mathcal{B}({\mathcal{V}})$ we have the equality $\sigma_{\mathbb{C}}(T)=\sigma(T_{\mathbb{C}})$. ## 3 Analytic Functional Calculus for Real Operators Having a concept of spectrum for real operators, an important step for further development is the construction of an analytic functional calculus. Such a construction has been done actually in the context of real linear relations in [1]. In what follows we shall present a similar construction for real linear operators. Although the case of linear relations looks more general, unlike in [1], we perform our construction using a class of operator valued analytic functions insted of scalar valued analytic functions. Moreover, our arguments look simpler, and the construction is a model for a more general one, to get an analytic functional calculus for quaternionic linear operators. If ${\mathcal{V}}$ is a real Banach space, and so each operator $T\in\mathcal{B}({\mathcal{V}})$ has a complex spectrum $\sigma_{\mathbb{C}}(T)$, which is compact and nonempty, one can use the classical Riesz-Dunford functional calculus, in a slightly generalized form (that is, replacing the scalar-valued analytic functions by operator-valued analytic ones, which is a well known idea). The use of vector versions of the Cauchy formula is simplified by adopting the following definition. Let $U\subset{\mathbb{C}}$ be open. An open subset $\Delta\subset U$ will be called a Cauchy domain (in $U$) if $\Delta\subset\bar{\Delta}\subset U$ and the boundary of $\Delta$ consists of a finite family of closed curves, piecewise smooth, positively oriented. Note that a Cauchy domain is bounded but not necessarily connected. ###### Remark 2 If $\mathcal{V}$ is a real Banach space, and $T\in\mathcal{B(V})$, we have the usual analytic functional calculus for the operator $T_{\mathbb{C}}\in\mathcal{B}(\mathcal{V}_{\mathbb{C}})$ (see [6]). That is, in a slightly generalized form, and for later use, if $U\supset\sigma(T_{\mathbb{C}})$ is an open set in ${\mathbb{C}}$ and $F:U\mapsto\mathcal{B}(\mathcal{V}_{\mathbb{C}})$ is analytic, we put $F(T_{\mathbb{C}})=\frac{1}{2\pi i}\int_{\Gamma}F(\zeta)(\zeta- T_{\mathbb{C}})^{-1}d\zeta,$ where $\Gamma$ is the boundary of a Cauchy domain $\Delta$ containing $\sigma(T_{\mathbb{C}})$ in $U$. In fact, because $\sigma(T_{\mathbb{C}})$ is conjugate symmetric, we may and shall assume that both $U$ and $\Gamma$ are conjugate symmetric. Because the function $\zeta\mapsto F(\zeta)(\zeta- T_{\mathbb{C}})^{-1}$ is analytic in $U\setminus\sigma(T_{\mathbb{C}})$, the integral does not depend on the particular choice of the Cauchy domain $\Delta$ containing $\sigma(T_{C})$. A natural question is to find an appropriate condition to we have $F(T_{\mathbb{C}})^{\flat}=F(T_{\mathbb{C}})$, which would imply the invariance of $\mathcal{V}$ under $F(T_{\mathbb{C}})$. With the notation of Remark 2, we have the following. ###### Theorem 1 Let $U\subset{\mathbb{C}}$ be open and conjugate symmetric. If $F:U\mapsto\mathcal{B}(\mathcal{V}_{\mathbb{C}})$ is analytic and $F(\zeta)^{\flat}=F(\bar{\zeta})$ for all $\zeta\in U$, then $F(T_{\mathbb{C}})^{\flat}=F(T_{\mathbb{C}})$ for all $T\in\mathcal{B}(\mathcal{V})$ with $\sigma_{\mathbb{C}}(T)\subset U$. Proof. We use the notation from Remark 2, assuming, in addition, that $\Gamma$ is conjugate symmetric as well. We put $\Gamma_{\pm}:=\Gamma\cap{\mathbb{C}}_{\pm}$, where ${\mathbb{C}}_{+}$ (resp. ${\mathbb{C}}_{-}$) equals to $\\{\lambda\in{\mathbb{C}};\Im\lambda\geq 0\\}$ (resp. $\\{\lambda\in{\mathbb{C}};\Im\lambda\leq 0\\}$). We write $\Gamma_{+}=\cup_{j=1}^{m}\Gamma_{j+}$, where $\Gamma_{j+}$ are the connected components of $\Gamma_{+}$. Similarly, we write $\Gamma_{-}=\cup_{j=1}^{m}\Gamma_{j-}$, where $\Gamma_{j-}$ are the connected components of $\Gamma_{-}$, and $\Gamma_{j-}$ is the reflexion of $\Gamma_{j+}$ with respect of the real axis. As $\Gamma$ is a finite union of Jordan piecewise smooth closed curves, for each index $j$ we have a parametrization $\phi_{j}:[0,1]\mapsto{\mathbb{C}}$, positively oriented, such that $\phi_{j}([0,1])=\Gamma_{j+}$. Taking into account that the function $t\mapsto\overline{\phi_{j}(t)}$ is a parametrization of $\Gamma_{j-}$ negatively oriented, and setting $\Gamma_{j}=\Gamma_{j+}\cup\Gamma_{j-}$, we can write $F_{j}(T_{\mathbb{C}}):=\frac{1}{2\pi i}\int_{\Gamma_{j}}F(\zeta)(\zeta- T_{\mathbb{C}})^{-1}d\zeta=$ $\frac{1}{2\pi i}\int_{0}^{1}F(\phi_{j}(t))(\phi_{j}(t)-T_{\mathbb{C}})^{-1}\phi_{j}^{\prime}(t)dt$ $-\frac{1}{2\pi i}\int_{0}^{1}F(\overline{\phi_{j}(t)})(\overline{\phi_{j}(t)}-T_{\mathbb{C}})^{-1}\overline{\phi_{j}^{\prime}(t)}dt.$ Therefore, $F_{j}(T_{\mathbb{C}})^{\flat}=-\frac{1}{2\pi i}\int_{0}^{1}F(\phi_{j}(t))^{\flat}(\overline{\phi_{j}(t)}-T_{\mathbb{C}})^{-1}\overline{\phi_{j}^{\prime}(t)}dt$ $+\frac{1}{2\pi i}\int_{0}^{1}F(\overline{\phi_{j}(t)})^{\flat}(\phi_{j}(t)-T_{\mathbb{C}})^{-1}\phi_{j}^{\prime}(t)dt.$ According to our assumption on the function $F$, we obtain $F_{j}(T_{\mathbb{C}})=F_{j}(T_{\mathbb{C}})^{\flat}$ for all $j$, and therefore $F(T_{\mathbb{C}})^{\flat}=\sum_{j=1}^{m}F_{j}(T_{\mathbb{C}})^{\flat}=\sum_{j=1}^{m}F_{j}(T_{\mathbb{C}})=F(T_{\mathbb{C}}),$ which concludes the proof. ###### Remark 3 If ${\mathcal{A}}$ is a unital real Banach algebra, ${\mathcal{A}}_{\mathbb{C}}$ its complexification, and $U\subset{\mathbb{C}}$ is open, we denote by $\mathcal{O}(U,{\mathcal{A}}_{\mathbb{C}})$ the algebra of all analytic ${\mathcal{A}}_{\mathbb{C}}$-valued functions. If $U$ is conjugate symmetric, and ${\mathcal{A}}_{\mathbb{C}}\ni a\mapsto\bar{a}\in{\mathcal{A}}_{\mathbb{C}}$ is its natural conjugation, we denote by $\mathcal{O}_{s}(U,{\mathcal{A}}_{\mathbb{C}})$ the real subalgebra of $\mathcal{O}(U,{\mathcal{A}}_{\mathbb{C}})$ consisting of those functions $F$ with the property $F(\bar{\zeta})=\overline{F(\zeta)}$ for all $\zeta\in U$. Adapting a well known terminology, such functions will be called (${\mathcal{A}}_{\mathbb{C}}$-valued $)$ stem functions. When ${\mathcal{A}}={\mathbb{R}}$, so ${\mathcal{A}}_{\mathbb{C}}={\mathbb{C}}$, the space $\mathcal{O}_{s}(U,{\mathbb{C}})$ will be denoted by $\mathcal{O}_{s}(U)$, which is a real algebra. Note that $\mathcal{O}_{s}(U,{\mathcal{A}}_{\mathbb{C}})$ is also a bilateral $\mathcal{O}_{s}(U)$-module. In the next result, we identify the algebra $\mathcal{B}(\mathcal{V})$ with a subalgebra of $\mathcal{B}(\mathcal{V})_{\mathbb{C}}$. In ths case, when $F\in\mathcal{O}_{s}(U,\mathcal{B}(\mathcal{V})_{\mathbb{C}})$, we shall write $F(T)=\frac{1}{2\pi i}\int_{\Gamma}F(\zeta)(\zeta-T)^{-1}d\zeta,$ noting that the right hand side of this formula belongs to $\mathcal{B}(\mathcal{V})$, by Theorem 1. The properties of the map $F\mapsto F(T)$, which can be called the (left) analytic functional calculus of $T$, are summarized by the following. ###### Theorem 2 Let ${\mathcal{V}}$ be a real Banach space, let $U\subset{\mathbb{C}}$ be a conjugate symmetric open set, and let $T\in\mathcal{B}(\mathcal{V})$, with $\sigma_{\mathbb{C}}(T)\subset U$. Then the assignment ${\mathcal{O}}_{s}(U,\mathcal{B}(\mathcal{V})_{\mathbb{C}})\ni F\mapsto F(T)\in\mathcal{B}(\mathcal{V})$ is an ${\mathbb{R}}$-linear map, and the map ${\mathcal{O}}_{s}(U)\ni f\mapsto f(T)\in\mathcal{B}(\mathcal{V})$ is a unital real algebra morphism. Moreover, the following properties are true: (1) For all $F\in\mathcal{O}_{s}(U,\mathcal{B}(\mathcal{V})_{\mathbb{C}}),\,f\in{\mathcal{O}}_{s}(U)$, we have $(Ff)(T)=F(T)f(T)$. (2) For every polynomial $P(\zeta)=\sum_{n=0}^{m}A_{n}\zeta^{n},\,\zeta\in{\mathbb{C}}$, with $A_{n}\in\mathcal{B}(\mathcal{V})$ for all $n=0,1,\ldots,m$, we have $P(T)=\sum_{n=0}^{m}A_{n}T^{n}\in\mathcal{B}(\mathcal{V})$. Proof. The arguments are more or less standard (see [6]). The ${\mathbb{R}}$-linearity of the maps ${\mathcal{O}}_{s}(U,\mathcal{B}(\mathcal{V})_{\mathbb{C}})\ni F\mapsto F(T)\in\mathcal{B}(\mathcal{V}),\,{\mathcal{O}}_{s}(U)\ni f\mapsto f(T)\in\mathcal{B}(\mathcal{V}),$ is clear. The second one is actually multiplicative, which follows from the multiplicativiry of the usual analytic functional calculus of $T$. In fact, we have a more general property, specifically $(Ff)(T)=F(T)f(T)$ for all $F\in\mathcal{O}_{s}(U,\mathcal{B}(\mathcal{V})_{\mathbb{C}}),\,f\in{\mathcal{O}}_{s}(U)$. This follows from the equalities, $(Ff)(T)=\frac{1}{2\pi i}\int_{\Gamma_{0}}F(\zeta)f(\zeta)(\zeta-T)^{-1}d\zeta=$ $\left(\frac{1}{2\pi i}\int_{\Gamma_{0}}F(\zeta)(\zeta-T)^{-1}d\zeta\right)\left(\frac{1}{2\pi i}\int_{\Gamma}f(\eta)(\eta-T)^{-1}d\eta\right)=F(T)f(T),$ obtained as in the classical case (see [6], Section VII.3), which holds because $f$ is ${\mathbb{C}}$-valued and commutes with the operators in $\mathcal{B}(\mathcal{V})$. Here $\Gamma,\,\Gamma_{0}$ are the boundaries of two Cauchy domains $\Delta,\,\Delta_{0}$ respectively, such that $\Delta\supset\bar{\Delta}_{0}$, and $\Delta_{0}$ contains $\sigma(T)$. Note that, in particular, for every polynomial $P(\zeta)=\sum_{n=0}^{m}A_{n}\zeta^{n}$ with $A_{n}\in\mathcal{B}(\mathcal{V})$ for all $n=0,1,\ldots,m$, we have $P(T)=\sum_{n=0}^{m}A_{n}q^{n}\in\mathcal{B}(\mathcal{V})$ for all $T\in\mathcal{B}(\mathcal{V})$. ###### Example 1 Let $\mathcal{V}={\mathbb{R}}^{2}$, so $\mathcal{V}_{\mathbb{C}}={\mathbb{C}}^{2}$, endowed with its natural Hilbert space structure. Let us first observe that we have $S=\left(\begin{array}[]{cc}a_{1}&a_{2}\\\ a_{3}&a_{4}\end{array}\right)\,\,\Longleftrightarrow S^{\flat}=\left(\begin{array}[]{cc}\bar{a}_{1}&\bar{a}_{2}\\\ \bar{a}_{3}&\bar{a}_{4}\end{array}\right),$ for all $a_{1},a_{2},a_{3},a_{4}\in{\mathbb{C}}$. Next we consider the operator $T\in\mathcal{B}({\mathbb{R}}^{2})$ given by the matrix $T=\left(\begin{array}[]{cc}u&v\\\ -v&u\end{array}\right),$ where $u,v\in{\mathbb{R}},v\neq 0$. The extension $T_{\mathbb{C}}$ of the operator $T$ to ${\mathbb{C}}^{2}$, which is a normal operator, is given by the same formula. Note that $\sigma_{\mathbb{C}}(T)=\\{\lambda\in{\mathbb{C}};(\lambda-u)^{2}+v^{2}=0\\}=\\{u\pm iv\\}=\sigma(T_{\mathbb{C}}).$ Note also that the vectors $\nu_{\pm}=(\sqrt{2})^{-1}(1,\pm i)$ are normalized eigenvectors for $T_{\mathbb{C}}$ corresponding to the eigenvalues $u\pm iv$, respectively. The spectral projections of $T_{\mathbb{C}}$ corresponding to these eigenvalues are given by $E_{\pm}(T_{\mathbb{C}}){\bf w}=\langle{\bf w},\nu_{\pm}\rangle\nu_{\pm}=\frac{1}{2}\left(\begin{array}[]{cc}1&\mp i\\\ \pm i&1\end{array}\right)\left(\begin{array}[]{c}w_{1}\\\ w_{2}\end{array}\right),$ for all ${\bf w}=(w_{1},w_{2})\in{\mathbb{C}}^{2}$. Let $U\subset{\mathbb{C}}$ be an open set with $U\supset\\{u\pm iv\\}$, and let $F:U\mapsto\mathcal{B}({\mathbb{C}}^{2})$ be analytic. We shall compute explicitly $F(T_{\mathbb{C}})$. Let $\Delta$ be a Cauchy domain contained in $U$ with its boundary $\Gamma$, and containing the points $u\pm iv$. Assuming $v>0$, we have $F(T_{\mathbb{C}})=\frac{1}{2\pi i}\int_{\Gamma}F(\zeta)(\zeta- T_{\mathbb{C}})^{-1}d\zeta=$ $F(u+iv)E_{+}(T_{\mathbb{C}})+F(u-iv)E_{-}(T_{\mathbb{C}})=$ $\frac{1}{2}F(u+iv)\left(\begin{array}[]{cc}1&-i\\\ i&1\end{array}\right)+\frac{1}{2}F(u-iv)\left(\begin{array}[]{cc}1&i\\\ -i&1\end{array}\right).$ Assume now that $F(T_{\mathbb{C}})^{\flat}=F(T_{\mathbb{C}})$. Then we must have $(F(u+iv)-F(u-iv)^{\flat})\left(\begin{array}[]{cc}1&-i\\\ i&1\end{array}\right)=(F(u+iv)^{\flat}-F(u-iv))\left(\begin{array}[]{cc}1&i\\\ -i&1\end{array}\right).$ We also have the equalities $\left(\begin{array}[]{cc}1&-i\\\ i&1\end{array}\right)\left(\begin{array}[]{c}1\\\ i\end{array}\right)=2\left(\begin{array}[]{c}1\\\ i\end{array}\right),\,\,\left(\begin{array}[]{cc}1&-i\\\ i&1\end{array}\right)\left(\begin{array}[]{c}1\\\ -i\end{array}\right)=0,$ $\left(\begin{array}[]{cc}1&i\\\ -i&1\end{array}\right)\left(\begin{array}[]{c}1\\\ -i\end{array}\right)=2\left(\begin{array}[]{c}1\\\ -i\end{array}\right),\,\,\left(\begin{array}[]{cc}1&i\\\ -i&1\end{array}\right)\left(\begin{array}[]{c}1\\\ i\end{array}\right)=0,$ Using these equalities, we finally deduce that $(F(u+iv)-F(u-iv)^{\flat})\left(\begin{array}[]{c}1\\\ i\end{array}\right)=0,$ and $(F(u-iv)-F(u+iv)^{\flat})\left(\begin{array}[]{c}1\\\ -i\end{array}\right)=0,$ which are necessary conditions for the equality $F(T_{\mathbb{C}})^{\flat}=F(T_{\mathbb{C}})$. As a matter of fact, this example shows, in particular, that the condition $F(\zeta)^{\flat}=F(\bar{\zeta})$ for all $\zeta\in U$, used in Theorem 1, is sufficient but it might not be always necessary. ## 4 Analytic Functional Calculus for Quaternionic Operators ### 4.1 Quaternionic Spectrum We now recall some known definitions and elementary facts (see, for instance, [5], Section 4.6, and/or [21]). Let ${\mathbb{H}}$ be the abstract algebra of quaternions, which is the four- dimensional ${\mathbb{R}}$-algebra with unit $1$, generated by the ”imaginary units“ $\\{\bf{j,k,l}\\}$, which satisfy ${\bf jk=-kj=l,\,kl=-lk=j,\,lj=-jl=k,\,jj=kk=ll}=-1.$ We may assume that ${\mathbb{H}}\supset{\mathbb{R}}$ identifying every number $x\in{\mathbb{R}}$ with the element $x1\in{\mathbb{H}}$. The algebra ${\mathbb{H}}$ has a natural multiplicative norm given by $\|{\bf x}\|=\sqrt{x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{0}^{2}},\,\,{\bf x}=x_{0}+x_{1}{\bf j}+x_{2}{\bf k}+x_{3}{\bf l},\,\,x_{0},x_{1},x_{2},x_{3}\in{\mathbb{R}},$ and a natural involution ${\mathbb{H}}\ni{\bf x}=x_{0}+x_{1}{\bf j}+x_{2}{\bf k}+x_{3}{\bf l}\mapsto{\bf x}^{*}=x_{0}-x_{1}{\bf j}-x_{2}{\bf k}-x_{3}{\bf l}\in{\mathbb{H}}.$ Note that ${\bf x}{\bf x}^{*}={\bf x}^{*}{\bf x}=\|{\bf x}\|^{2}$, implying, in particular, that every element ${\bf x}\in{\mathbb{H}}\setminus\\{0\\}$ is invertible, and ${\bf x}^{-1}=\|{\bf x}\|^{-2}{\bf x}^{*}$. For an arbitrary quaternion ${\bf x}=x_{0}+x_{1}{\bf j}+x_{2}{\bf k}+x_{3}{\bf l},\,\,x_{0},x_{1},x_{2},x_{3}\in{\mathbb{R}}$, we set $\Re{\bf x}=x_{0}=({\bf x}+{\bf x}^{*})/2$, and $\Im{\bf x}=x_{1}{\bf j}+x_{2}{\bf k}+x_{3}{\bf l}=({\bf x}-{\bf x}^{*})/2$, that is, the real and imaginary part of ${\bf x}$, respectively. We consider the complexification ${\mathbb{C}}\otimes_{\mathbb{R}}{\mathbb{H}}$ of the ${\mathbb{R}}$-algebra ${\mathbb{H}}$ (see also [8]), which will be identified with the direct sum ${\mathbb{M}}={\mathbb{H}}+i{\mathbb{H}}$. Of course, the algebra ${\mathbb{M}}$ contains the complex field ${\mathbb{C}}$. Moreover, in the algebra ${\mathbb{M}}$, the elements of ${\mathbb{H}}$ commute with all complex numbers. In particular, the ”imaginary units“ $\bf j,k,l$ of the algebra ${\mathbb{H}}$ are independent of and commute with the imaginary unit $i$ of the complex plane ${\mathbb{C}}$. In the algebra ${\mathbb{M}}$, there also exists a natural conjugation given by $\bar{\bf a}={\bf b}-i{\bf c}$, where ${\bf a}={\bf b}+i{\bf c}$ is arbitrary in ${\mathbb{M}}$, with ${\bf b},{\bf c}\in{\mathbb{H}}$ (see also [8]). Note that $\overline{\bf a+b}=\bar{\bf a}+\bar{\bf b}$, and $\overline{\bf ab}=\bar{\bf a}\bar{\bf b}$, in particular $\overline{r\bf a}=r\bar{\bf a}$ for all ${\bf a},{\bf b}\in{\mathbb{M}}$, and $r\in{\mathbb{R}}$. Moreover, $\bar{{\bf a}}={\bf a}$ if and only if ${\bf a}\in{\mathbb{H}}$, which is a useful characterization of the elements of ${\mathbb{H}}$ among those of ${\mathbb{M}}$. ###### Remark 4 In the algebra ${\mathbb{M}}$ we have the identities $(\lambda-{\bf x}^{*})(\lambda-{\bf x})=(\lambda-{\bf x})(\lambda-{\bf x}^{*})=\lambda^{2}-\lambda({\bf x}+{\bf x}^{*})+\|{\bf x}\|^{2}\in{\mathbb{C}},$ for all $\lambda\in{\mathbb{C}}$ and ${\bf x}\in{\mathbb{H}}$. If the complex number $\lambda^{2}-2\lambda\Re{\bf x}+\|{\bf x}\|^{2}$ is nonnull, then both element $\lambda-{\bf x}^{*},\,\lambda-{\bf x}$ are invertible. Conversely, if $\lambda-{\bf x}$ is invertible, we must have $\lambda^{2}-2\lambda\Re{\bf x}+\|{\bf x}\|^{2}$ nonnull; otherwise we would have $\lambda={\bf x}^{*}\in{\mathbb{R}}$, so $\lambda={\bf x}\in{\mathbb{R}}$, which is not possible. Therefore, the element $\lambda-{\bf x}\in{\mathbb{M}}$ is invertible if and only if the complex number $\lambda^{2}-2\lambda\Re{\bf x}+\|{\bf x}\|^{2}$ is nonnull. Hence, the element $\lambda-{\bf x}\in{\mathbb{M}}$ is not invertible if and only if $\lambda=\Re{\bf x}\pm i\|\Im{\bf x}\|$. In this way, the spectrum of a quaternion ${\bf x}\in{\mathbb{H}}$ is given by the equality $\sigma({\bf x})=\\{s_{\pm}(\bf x)\\}$, where $s_{\pm}(\bf x)=\Re{\bf x}\pm i\|\Im{\bf x}\|$ are the eigenvalues of $\bf x$ (see also [20, 21]). The polynomial $P_{\bf x}(\lambda)=\lambda^{2}-2\lambda\Re{\bf x}+\|{\bf x}\|^{2}$ is the minimal polynomial of $\bf x$. In fact, the equality $\sigma({\bf y})=\sigma({\bf x})$ for some ${\bf x,y}\in{\mathbb{H}}$ is an equivalence relation in the algebra ${\mathbb{H}}$, which holds if and only if $P_{\bf x}=P_{\bf y}$. In fact, setting $\mathbb{S}=\\{\mathfrak{\kappa}\in{\mathbb{H}};\Re\mathfrak{\kappa}=0,\|\mathfrak{\kappa}\|=1\\}$ (that is the unit sphere of purely imaginary quaternions), representig an arbitrary quaternion $\bf x$ under the form $x_{0}+y_{0}\mathfrak{\kappa}_{0}$, with $x_{0},y_{0}\in{\mathbb{R}}$ and $\mathfrak{\kappa}_{0}\in\mathbb{S}$, a quaternion $\bf y$ is equivalent to $\bf x$ if anf only if it is of the form $x_{0}+y_{0}\mathfrak{\kappa}$ for some $\mathfrak{\kappa}\in\mathbb{S}$ (see [3] or [21] for some details). ###### Remark 5 Following [5], a right ${\mathbb{H}}$-vector space $\mathcal{V}$ is a real vector space having a right multiplication with the elements of ${\mathbb{H}}$, such that $(x+y){\bf q}=x{\bf q}+y{\bf q},\,x({\bf q}+{\bf s})=x{\bf q}+x{\bf s},\,x({\bf q}{\bf s})=(x{\bf q}){\bf s}$ for all $x,y\in\mathcal{V}$ and ${\bf q},{\bf s}\in{\mathbb{H}}$. If $\mathcal{V}$ is also a Banach space the operator $T\in\mathcal{B(V)}$ is right ${\mathbb{H}}$-linear if $T(x{\bf q})=T(x){\bf q}$ for all $x\in\mathcal{V}$ and ${\bf q}\in{\mathbb{H}}$. The set of right ${\mathbb{H}}$ linear operators will be denoted by $\mathcal{B^{\rm r}(V)}$, which is, in particular, a unital real algebra. In a similar way, one defines the concept of a left ${\mathbb{H}}$-vector space. A real vector space $\mathcal{V}$ will be said to be an ${\mathbb{H}}$-vector space if it is simultaneously a right ${\mathbb{H}}$\- and a left ${\mathbb{H}}$-vector space. As noticed in [5], it is the framework of ${\mathbb{H}}$-vector spaces an appropriate one for the study of right ${\mathbb{H}}$-linear operators. If ${\mathcal{V}}$ is ${\mathbb{H}}$-vector space which is also a Banach space, then ${\mathcal{V}}$ is said to be a Banach ${\mathbb{H}}$-space. In this case, we also assume that $R_{\bf q}\in\mathcal{B}({\mathcal{V}})$, and the map ${\mathbb{H}}\ni{\bf q}\mapsto R_{\bf q}\in\mathcal{B}({\mathcal{V}})$ is norm continuous, where $R_{\bf q}$ is the right multiplication of the elements of $\mathcal{V}$ by a given quaternion ${\bf q}\in{\mathbb{H}}$. Similarly, if $L_{\bf q}$ is the left multiplication of the elements of $\mathcal{V}$ by the quaternion ${\bf q}\in{\mathbb{H}}$, we assume that $L_{\bf q}\in\mathcal{B}({\mathcal{V}})$ for all ${\bf q}\in{\mathbb{H}}$, and that the map ${\mathbb{H}}\ni{\bf q}\mapsto L_{\bf q}\in\mathcal{B}({\mathcal{V}})$ is norm continuous. Note also that $\mathcal{B^{\rm r}(V)}=\\{T\in\mathcal{B(V)};TR_{\bf q}=R_{\bf q}T,\,{\bf q}\in{\mathbb{H}}\\}.$ To adapt the discussion regarding the real algebras to this case, we first consider the complexification ${\mathcal{V}}_{\mathbb{C}}$ of ${\mathcal{V}}$. Because ${\mathcal{V}}$ is an ${\mathbb{H}}$-bimodule, the space ${\mathcal{V}}_{\mathbb{C}}$ is actually an ${\mathbb{M}}$-bimodule, via the multiplications $({\bf q}+i{\bf s})(x+iy)={\bf q}x-{\bf s}y+i({\bf q}y+{\bf s}x),(x+iy)({\bf q}+i{\bf s})=x{\bf q}-y{\bf s}+i(y{\bf q}+x{\bf s}),$ for all ${\bf q}+i{\bf s}\in{\mathbb{M}},\,{\bf q},{\bf s}\in{\mathbb{H}},\,x+iy\in{\mathcal{V}}_{\mathbb{C}},\,x,y\in{\mathcal{V}}$. Moreover, the operator $T_{\mathbb{C}}$ is right ${\mathbb{M}}$-linear, that is $T_{\mathbb{C}}((x+iy)({\bf q}+i{\bf s}))=T_{\mathbb{C}}(x+iy)({\bf q}+i{\bf s})$ for all ${\bf q}+i{\bf s}\in{\mathbb{M}},\,x+iy\in{\mathcal{V}}_{\mathbb{C}}$, via a direct computation. Let $C$ be the conjugation of ${\mathcal{V}}_{\mathbb{C}}$. As in the real case, for every $S\in\mathcal{B}({\mathcal{V}}_{\mathbb{C}})$, we put $S^{\flat}=CSC$. The left and right multiplication with the quaternion ${\bf q}$ on ${\mathcal{V}}_{\mathbb{C}}$ will be also denoted by $L_{\bf q},R_{\bf q}$, respectively, as elements of $\mathcal{B}({\mathcal{V}}_{\mathbb{C}})$. We set $\mathcal{B}^{\rm r}({\mathcal{V}}_{\mathbb{C}})=\\{S\in\mathcal{B}({\mathcal{V}}_{\mathbb{C}});SR_{\bf q}=R_{\bf q}S,\,{\bf q}\in{\mathbb{H}}\\},$ which is a unital complex algebra containing all operators $L_{\bf q},{\bf q}\in{\mathbb{H}}$. Note that if $S\in\mathcal{B}^{\rm r}({\mathcal{V}}_{\mathbb{C}})$, then $S^{\flat}\in\mathcal{B}^{\rm r}({\mathcal{V}}_{\mathbb{C}})$. Indeed, because $CR_{\bf q}=R_{\bf q}C$, we also have $S^{\flat}R_{\bf q}=R_{\bf q}S^{\flat}$. In fact, as we have $(S+S^{\flat})({\mathcal{V}})\subset{\mathcal{V}}$ and $i(S-S^{\flat})({\mathcal{V}})\subset{\mathcal{V}}$, it folows that the algebras $\mathcal{B}^{\rm r}({\mathcal{V}}_{\mathbb{C}}),\,\mathcal{B}^{\rm r}({\mathcal{V}})_{\mathbb{C}}$ are isomorphic, and they will be often identified, where $\mathcal{B^{\rm r}(V)}_{\mathbb{C}}=\mathcal{B^{\rm r}(V)}+i\mathcal{B^{\rm r}(V)}$ is the complexification of $\mathcal{B^{\rm r}(V)}$, which is also a unital complex Banach algebra. Looking at the Definition 4.8.1 from [5] (see also [4]), we give the folowing. ###### Definition 1 For a given operator $T\in\mathcal{B^{\rm r}(V)}$, the set $\sigma_{\mathbb{H}}(T):=\\{{\bf q}\in{\mathbb{H}};T^{2}-2(\Re{\bf q})T+\|{\bf q}\|^{2})\,\,{\rm not}\,\,{\rm invertible}\\}$ is called the quaternionic spectrum (or simply the $Q$-spectrum) of $T$. The complement $\rho_{\mathbb{H}}(T)={\mathbb{H}}\setminus\sigma_{\mathbb{H}}(T)$ is called the quaternionic resolvent (or simply the $Q$-resolvent) of $T$. Note that, if ${\bf q}\in\sigma_{\mathbb{H}}(T)$), then $\\{{\bf s}\in{\mathbb{H}};\sigma({\bf s})=\sigma({\bf q})\\}\subset\sigma_{\mathbb{H}}(T)$. Assuming that ${\mathcal{V}}$ is a Banach ${\mathbb{H}}$-space, then $\mathcal{B^{\rm r}(V)}$ is a unital real Banach ${\mathbb{H}}$-algebra (that is, a Banach algebra which also a Banach ${\mathbb{H}}$-space), via the algebraic operations $({\bf q}T)(x)={\bf q}T(x)$, and $(T{\bf q})(x)=T({\bf q}x)$ for all ${\bf q}\in{\mathbb{H}}$ and $x\in{\mathcal{V}}$. Hence the complexification $\mathcal{B^{\rm r}(V)}_{\mathbb{C}}$ is, in particular, a unital complex Banach algebra. Also note that the complex numbers, regarded as elements of $\mathcal{B^{\rm r}(V)}_{\mathbb{C}}$, commute with the elements of $\mathcal{B^{\rm r}(V)}$. For this reason, for each $T\in\mathcal{B^{\rm r}(V)}$ we have the resolvent set $\rho_{\mathbb{C}}(T)=\\{\lambda\in{\mathbb{C}};(T^{2}-2(\Re\lambda)T+|\lambda|^{2})^{-1}\in\mathcal{B^{\rm r}(V)}\\}=$ $\\{\lambda\in{\mathbb{C}};(\lambda- T_{\mathbb{C}})^{-1}\in\mathcal{B^{\rm r}(V}_{\mathbb{C}})\\}=\rho(T_{\mathbb{C}}),$ and the associated spectrum $\sigma_{\mathbb{C}}(T)=\sigma(T_{\mathbb{C}})$. Clearly, there exists a strong connexion between $\sigma_{\mathbb{H}}(T)$ and $\sigma_{\mathbb{C}}(T)$. In fact, the set $\sigma_{\mathbb{C}}(T)$ looks like a ”complex border“ of the set $\sigma_{\mathbb{H}}(T)$. Specifically, we can prove the following. ###### Lemma 2 For every $T\in\mathcal{B^{\rm r}(V)}$ we have the equalities $\sigma_{\mathbb{H}}(T)=\\{{\bf q}\in{\mathbb{H}};\sigma_{\mathbb{C}}(T)\cap\sigma({\bf q})\neq\emptyset\\}.$ (2) and $\sigma_{\mathbb{C}}(T)=\\{\lambda\in\sigma({\bf q});{\bf q}\in\sigma_{\mathbb{H}}(T)\\}.$ (3) Proof. Let us prove (2). If ${\bf q}\in\sigma_{\mathbb{H}}(T)$, and so the $T^{2}-2(\Re{\bf q})T+\|{\bf q}\|^{2}$ is not invertible, choosing $\lambda\in\\{\Re{\bf q}\pm i\|\Im{\bf q}\|\\}=\sigma({\bf q})$, we clearly have $T^{2}-2(\Re\lambda)T+|\lambda|^{2}$ not invertible, implying $\lambda\in\sigma_{\mathbb{C}}(T)\cap\sigma({\bf q})\neq\emptyset$. Conversely, if for some ${\bf q}\in{\mathbb{H}}$ there exists $\lambda\in\sigma_{\mathbb{C}}(T)\cap\sigma({\bf q})$, and so $T^{2}-2(\Re\lambda)T+|\lambda|^{2}=T^{2}-2(\Re{\bf q})T+\|{\bf q}\|^{2}$ is not invertible, implying ${\bf q}\in\sigma_{\mathbb{H}}(T)$. We now prove (3). Let $\lambda\in\sigma_{\mathbb{C}}(T)$, so the operator $T^{2}-2(\Re\lambda)T+|\lambda|^{2}$ is not invertible. Setting ${\bf q}=\Re(\lambda)+\|\Im\lambda\|\kappa$, with $\kappa\in\mathbb{S}$, we have $\lambda\in\sigma({\bf q})$. Moreover, $T^{2}+2(\Re{\bf q})T+\|{\bf q}\|^{2}$ is not invertible, and so ${\bf q}\in\sigma_{\mathbb{H}}(T)$. Conversely, if $\lambda\in\sigma({\bf q})$ for some ${\bf q}\in\sigma_{\mathbb{H}}(T)$, then $\lambda\in\\{\Re{\bf q}\pm i\|\Im({\bf q})\|\\}$, showing that $T^{2}-2\Re(\lambda)T+|\lambda|^{2}=T^{2}+2(\Re{\bf q})T+\|{\bf q}\|^{2}$ is not invertible. Remark As expected, the set $\sigma_{\mathbb{H}}(T)$ is nonempty and bounded, which follows easily from Lemma 2. It is also compact, as a consequence of Definition 1, because the set of invertible elements in $\mathcal{B^{\rm r}(V)}$ is open. We recall that a subset $\Omega\subset{\mathbb{H}}$ is said to be spectrally saturated (see [20],[21]) if whenever $\sigma({\bf h})=\sigma({\bf q})$ for some ${\bf h}\in{\mathbb{H}}$ and ${\bf q}\in\Omega$, we also have ${\bf h}\in\Omega$. As noticed in [20] and [21], this concept coincides with that of axially symmetric set, introduced in [5]. Note that the subset $\sigma_{\mathbb{H}}(T)$ spectrally saturated. ### 4.2 Analytic Functional Calculus If ${\mathcal{V}}$ is a Banach ${\mathbb{H}}$-space, because $\mathcal{B^{\rm r}({\mathcal{V}})}$ is real Banach space, each operator $T\in\mathcal{B^{\rm r}({\mathcal{V}})}$ has a complex spectrum $\sigma_{\mathbb{C}}(T)$. Therefore, applying the corresponding result for real operators, we may construct an analytic functional calculus using the classical Riesz-Dunford functional calculus, in a slightly generalized form. In this case, our basic complex algebra is $\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}}$, endowed with the conjugation $\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}}\ni S\mapsto S^{\flat}\in\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}}$. ###### Theorem 3 Let $U\subset{\mathbb{C}}$ be open and conjugate symmetric. If $F:U\mapsto\mathcal{B}^{\rm r}(\mathcal{V}_{\mathbb{C}})$ is analytic and $F(\zeta)^{\flat}=F(\bar{\zeta})$ for all $\zeta\in U$, then $F(T_{\mathbb{C}})^{\flat}=F(T_{\mathbb{C}})$ for all $T\in\mathcal{B}^{\rm r}(\mathcal{V})$ with $\sigma_{\mathbb{C}}(T)\subset U$. Both the statement and the proof of Theorem 3 are similar to those of Theorem 1, and will be omitted. As in the real case, we may identify the algebra $\mathcal{B}^{\rm r}(\mathcal{V})$ with a subalgebra of $\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}}$. In ths case, when $F\in\mathcal{O}_{s}(U,\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}})=\\{F\in{\mathcal{O}}(U,\mathcal{B}^{\rm r}({\mathcal{V}})_{\mathbb{C}});F(\bar{\zeta})=F(\zeta)^{\flat}\,\,\forall\zeta\in U\\}$ (see also Remark 3), we can write, via the previous Theorem, $F(T)=\frac{1}{2\pi i}\int_{\Gamma}F(\zeta)(\zeta-T)^{-1}d\zeta\in\mathcal{B}^{\rm r}(\mathcal{V}),$ for a suitable choice of $\Gamma$. The next result provides an analytic functional calculus for operators from the real algebra $\mathcal{B}^{\rm r}(\mathcal{V})$. ###### Theorem 4 Let ${\mathcal{V}}$ be a Banach ${\mathbb{H}}$-space, let $U\subset{\mathbb{C}}$ be a conjugate symmetric open set, and let $T\in\mathcal{B}^{\rm r}(\mathcal{V})$, with $\sigma_{\mathbb{C}}(T)\subset U$. Then the map ${\mathcal{O}}_{s}(U,\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}})\ni F\mapsto F(T)\in\mathcal{B}^{\rm r}(\mathcal{V})$ is ${\mathbb{R}}$-linear, and the map ${\mathcal{O}}_{s}(U)\ni f\mapsto f(T)\in\mathcal{B}^{\rm r}(\mathcal{V})$ is a unital real algebra morphism. Moreover, the following properties are true: $(1)$ For all $F\in\mathcal{O}_{s}(U,\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}}),\,f\in{\mathcal{O}}_{s}(U)$, we have $(Ff)(T)=F(T)f(T)$. $(2)$ For every polynomial $P(\zeta)=\sum_{n=0}^{m}A_{n}\zeta^{n},\,\zeta\in{\mathbb{C}}$, with $A_{n}\in\mathcal{B}^{\rm r}(\mathcal{V})$ for all $n=0,1,\ldots,m$, we have $P(T)=\sum_{n=0}^{m}A_{n}T^{n}\in\mathcal{B}^{\rm r}(\mathcal{V})$. The proof of this result is similar to that of Theorem 2 and will be omitted. ###### Remark 6 The algebra ${\mathbb{H}}$ is, in particular, a Banach ${\mathbb{H}}$-space. As already noticed, the left multiplications $L_{\bf q},\,{\bf q}\in{\mathbb{H}},$ are elements of $\mathcal{B}^{\rm r}({\mathbb{H}})$. In fact, the map ${\mathbb{H}}\ni{\bf q}\mapsto L_{\bf q}\in\mathcal{B}^{\rm r}({\mathbb{H}})$ is a injective morphism of real algebras allowing the identification of ${\mathbb{H}}$ with a subalgebra of $\mathcal{B}^{\rm r}({\mathbb{H}})$. Let $\Omega\subset{\mathbb{H}}$ be a spectrally saturated open set, and let $U=\mathfrak{S}(\Omega):=\\{\lambda\in{\mathbb{C}},\exists{\bf q}\in\Omega,\lambda\in\sigma({\bf q})\\}$, which is open and conjugate symmetric (see [21]). Denotig by $f_{\mathbb{H}}$ the function $\Omega\ni{\bf q}\mapsto f({\bf q}),{\bf q}\in\Omega$, for every $f\in\mathcal{O}_{s}(U)$, we set $\mathcal{R}(\Omega):=\\{f_{\mathbb{H}};f\in\mathcal{O}_{s}(U)\\},$ which is a commutative real algebra. Defining the function $F_{\mathbb{H}}$ in a similar way for each $F\in\mathcal{O}_{s}(U,{\mathbb{M}})$, we set $\mathcal{R}(\Omega,{\mathbb{H}}):=\\{F_{\mathbb{H}};F\in\mathcal{O}_{s}(U,{\mathbb{M}})\\},$ which, according to the next theorem, is a right $\mathcal{R}(\Omega)$-module. The next result is an analytic functional calculus for quaternions (see [21], Theorem 5), obtained as a particular case of Theorem 4 (see also its predecessor in [5]). ###### Theorem 5 Let $\Omega\subset{\mathbb{H}}$ be a spectrally saturated open set, and let $U=\mathfrak{S}(\Omega)$. The space $\mathcal{R}(\Omega)$ is a unital commutative ${\mathbb{R}}$-algebra, the space $\mathcal{R}(\Omega,{\mathbb{H}})$ is a right $\mathcal{R}(\Omega)$-module, the map ${\mathcal{O}}_{s}(U,{\mathbb{M}})\ni F\mapsto F_{\mathbb{H}}\in\mathcal{R}(\Omega,{\mathbb{H}})$ is a right module isomorphism, and its restriction ${\mathcal{O}}_{s}(U)\ni f\mapsto f_{\mathbb{H}}\in\mathcal{R}(\Omega)$ is an ${\mathbb{R}}$-algebra isomorphism. Moreover, for every polynomial $P(\zeta)=\sum_{n=0}^{m}a_{n}\zeta^{n},\,\zeta\in{\mathbb{C}}$, with $a_{n}\in{\mathbb{H}}$ for all $n=0,1,\ldots,m$, we have $P_{\mathbb{H}}(q)=\sum_{n=0}^{m}a_{n}q^{n}\in{\mathbb{H}}$ for all $q\in{\mathbb{H}}$. Most of the assertions of Theorem 5 can be obtained directly from Theorem 4. The injectivity of the map ${\mathcal{O}}_{s}(U)\ni f\mapsto f_{\mathbb{H}}\in\mathcal{R}(\Omega)$, as well as an alternative complete proof, can be obtained as in the proof of Theorem 5 from [21]. ###### Remark 7 That Theorems 3 and 4 have practically the same proof as Theorems 1 and 2 (respectively) is due to the fact that all of them can be obtained as particular cases of more general results. Indeed, considering a unital real Banach algebra ${\mathcal{A}}$, and its complexification ${\mathcal{A}}_{\mathbb{C}}$, identifying ${\mathcal{A}}$ with a real subalgebra of ${\mathcal{A}}_{\mathbb{C}}$, for a function $F\in\mathcal{O}_{s}(U,A_{\mathbb{C}})$, where $U\subset{\mathbb{C}}$ is open and conjugate symmetric, the element $F(b)\in{\mathcal{A}}$ for each $b\in{\mathcal{A}}$ with $\sigma_{\mathbb{C}}(b)\subset U$. The assertion follows as in the proof of Theorem 1. The other results also have their counterparts. We omit the details. ###### Remark 8 The space $\mathcal{R}(\Omega,{\mathbb{H}})$ can be independently defined, and it consists of the set of all ${\mathbb{H}}$-valued functions, which are slice regular in the sense of [5], Definition 4.1.1. They are used in [5] to define a quaternionic functional calculus for quaternionic linear operators (see also [4]). Roughly speaking, given a quaternionic linear operator, each regular quaternionic-valued function defined in a neighborhood $\Omega$ of its quaternionic spectrum is associated with another quaternionic linear operator, replacing formally the quaternionic variable with that operator. This constraction is largely explained in the fourth chapter of [5]. Our Theorem 4 constructs an analytic functional calculus with functions from ${\mathcal{O}}_{s}(U,\mathcal{B}^{\rm r}(\mathcal{V})_{\mathbb{C}})$, where $U$ is a a neighborhood of the complex spectrum of a given quaternionic linear operator, leading to another quaternionic linear operator, replacing formally the complex variable with that operator. We can show that those functional calculi are equivalent. This is a consequence of the fact that the class of regular quaternionic-valued function used by the construction in [5] is isomorphic to the class of analytic functions used in our Theorem 5. The advantage of our approach is its simplicity and a stronger connection with the classical approach, using spectra defined in the complex plane, and Cauchy type kernels partially commutative. Let us give a direct argument concerning the equivalence of those analytic functional calculi. For an operator $T\in\mathcal{B}^{\rm r}(\mathcal{V})$, the so-called right $S$-resolvent is defined via the formula $S_{R}^{-1}({\bf s},T)=-(T-{\bf s}^{*})(T^{2}-2\Re({\bf s})T+\|{\bf s}\|)^{-1},\,\,{\bf s}\in\rho_{\mathbb{H}}(T)$ (4) (see [5], formula (4.27)). Fixing an element $\kappa\in\mathbb{S}$, and a spectrally saturated open set $\Omega\subset{\mathbb{H}}$, for $\Phi\in\mathcal{R}(\Omega,{\mathbb{H}})$ one sets $\Phi(T)=\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}\Phi({\bf s})d{\bf s}_{\kappa}S_{R}^{-1}({\bf s},T),$ (5) where $\Sigma\subset\Omega$ is a spectrally saturated open set containing $\sigma_{\mathbb{H}}(T)$, such that $\Sigma_{\kappa}=\\{u+v\kappa\in\Sigma;u,v\in{\mathbb{R}}\\}$ is a subset whose boundary $\partial(\Sigma_{\kappa})$ consists of a finite family of closed curves, piecewise smooth, positively oriented, and $d{\bf s}_{\kappa}=-\kappa du\wedge dv$. Formula (5) is a (right) quaternionic functional calculus, as defined in [5], Section 4.10. Because the space $\mathcal{V}_{\mathbb{C}}$ is also an ${\mathbb{H}}$-space, we may extend these formulas to the operator $T_{\mathbb{C}}\in\mathcal{B}^{\rm r}(\mathcal{V}_{\mathbb{C}})$, extending the operator $T$ to $T_{\mathbb{C}}$, and replacing $T$ by $T_{\mathbb{C}}$ in formulas (4) and (5). For the function $\Phi\in\mathcal{R}(\Omega,{\mathbb{H}})$ there exists a function $F\in{\mathcal{O}}_{s}(U,\mathcal{B}^{\rm r}(\mathcal{V}_{\mathbb{C}}))$ such that $F_{\mathbb{H}}=\Phi$. Denoting by $\Gamma_{\kappa}$ the boundary of a Cauchy domain in ${\mathbb{C}}$ containing the compact set $\cup\\{\sigma({\bf s});{\bf s}\in\overline{\Sigma_{\kappa}}\\}$, we can write $\Phi(T_{\mathbb{C}})=\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}\left(\frac{1}{2\pi i}\int_{\Gamma_{\kappa}}F(\zeta)(\zeta-{\bf s})^{-1}d\zeta\right)d{\bf s}_{\kappa}S_{R}^{-1}({\bf s},T_{\mathbb{C}})=$ $\frac{1}{2\pi i}\int_{\Gamma_{\kappa}}F(\zeta)\left(\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}(\zeta-{\bf s})^{-1}d{\bf s}_{\kappa}S_{R}^{-1}({\bf s},T_{\mathbb{C}})\right)d\zeta.$ It follows from the complex linearity of $S_{R}^{-1}({\bf s},T_{\mathbb{C}})$, and from formula (4.49) in [5], that $(\zeta-{\bf s})S_{R}^{-1}({\bf s},T_{\mathbb{C}})=S_{R}^{-1}({\bf s},T_{\mathbb{C}})(\zeta-T_{\mathbb{C}})-1,$ whence $(\zeta-{\bf s})^{-1}S_{R}^{-1}({\bf s},T_{\mathbb{C}})=S_{R}^{-1}({\bf s},T_{\mathbb{C}})(\zeta-T_{\mathbb{C}})^{-1}+(\zeta-{\bf s})^{-1}(\zeta- T_{\mathbb{C}})^{-1},$ and therefore, $\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}(\zeta-{\bf s})^{-1}d{\bf s}_{\kappa}S_{R}^{-1}({\bf s},T_{\mathbb{C}})=\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}d{\bf s}_{\kappa}S_{R}^{-1}({\bf s},T_{\mathbb{C}})(\zeta-T_{\mathbb{C}})^{-1}+$ $\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}(\zeta-{\bf s})^{-1}d{\bf s}_{\kappa}(\zeta-T_{\mathbb{C}})^{-1}=(\zeta-T_{\mathbb{C}})^{-1},$ because $\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}d{\bf s}_{\kappa}S_{R}^{-1}({\bf s},T_{\mathbb{C}})=1\,\,\,{\rm and}\,\,\,\frac{1}{2\pi}\int_{\partial(\Sigma_{\kappa})}(\zeta-{\bf s})^{-1}d{\bf s}_{\kappa}=0,$ as in Theorem 4.8.11 from [5], since the ${\mathbb{M}}$-valued function ${\bf s}\mapsto(\zeta-{\bf s})^{-1}$ is analytic in a neighborhood of the set $\overline{\Sigma_{\kappa}}\subset{\mathbb{C}}_{\kappa}$ for each $\zeta\in\Gamma_{\kappa}$, respectively. Therefore $\Phi(T_{\mathbb{C}})=\Phi(T)_{\mathbb{C}}=F(T_{\mathbb{C}})=F(T)_{\mathbb{C}}$, implying $\Phi(T)=F(T)$. ## 5 Some Examples ###### Example 2 One of the simplest Banach ${\mathbb{H}}$-space is the space ${\mathbb{H}}$ itself. As already noticed (see Remark 6), taking ${\mathcal{V}}={\mathbb{H}}$, so ${\mathcal{V}}_{\mathbb{C}}={\mathbb{M}}$, and fixing an element ${\bf q}\in{\mathbb{H}}$, we may consider the operator $L_{\bf q}\in\mathcal{B}^{\rm r}({\mathbb{H}})$, whose complex spectrum is given by $\sigma_{\mathbb{C}}(L_{\bf q})=\sigma({\bf q})=\\{\Re{\bf q}\pm i\|\Im{\bf q}\|\\}$. If $U\subset{\mathbb{C}}$ is conjugate symmetric open set containing $\sigma_{\mathbb{C}}(L_{\bf q})$, and $F\in\mathcal{O}_{s}(U,{\mathbb{M}})$, then we have $F({L_{\bf q}})=F(s_{+}({\bf q}))\iota_{+}(\mathfrak{s}_{\tilde{\bf q}})+F(s_{-}({\bf q}))\iota_{-}(\mathfrak{s}_{\tilde{\bf q}})\in{\mathbb{M}},$ (6) where $s_{\pm}({\bf q})=\Re{\bf q}\pm i\|\Im{\bf q}\|$, $\tilde{\bf q}=\Im\bf q,\,\mathfrak{s}_{\tilde{\bf q}}=\tilde{\bf q}\|\tilde{\bf q}\|^{-1}$, and $\iota_{\pm}(\mathfrak{s}_{\tilde{\bf q}})=2^{-1}(1\mp i\mathfrak{s}_{\tilde{\bf q}})$ (see [21], Remark 3). ###### Example 3 Let ${\mathfrak{X}}$ be a topological compact space, and let $C({\mathfrak{X}},{\mathbb{M}})$ be the space of ${\mathbb{M}}$-valued continuous functions on ${\mathfrak{X}}$. Then $C({\mathfrak{X}},{\mathbb{H}})$ is the real subspace of $C({\mathfrak{X}},{\mathbb{M}})$ consisting of ${\mathbb{H}}$-valued functions, which is also a Banach ${\mathbb{H}}$-space with respect to the operations $({\bf q}F)(x)={\bf q}F(x)$ and $(F{\bf q})(x)=F(x){\bf q}$ for all $F\in C({\mathfrak{X}},{\mathbb{H}})$ and $x\in{\mathfrak{X}}$. Moreover, $C({\mathfrak{X}},{\mathbb{H}})_{\mathbb{C}}=C({\mathfrak{X}},{\mathbb{H}}_{\mathbb{C}})=C({\mathfrak{X}},{\mathbb{M}})$. We fix a function $\Theta\in C({\mathfrak{X}},{\mathbb{H}})$ and define the operator $T\in\mathcal{B}(C({\mathfrak{X}},{\mathbb{H}}))$ by the relation $(TF)(x)=\Theta(x)F(x)$ for all $F\in C({\mathfrak{X}},{\mathbb{H}})$ and $x\in{\mathfrak{X}}$. Note that $(T(F{\bf q}))(x)=\Theta(x)F(x){\bf q}=((TF){\bf q})(x)$ for all $F\in C({\mathfrak{X}},{\mathbb{H}}),{\bf q}\in{\mathbb{H}}$, and $x\in{\mathfrak{X}}$. In othe words, $T\in\mathcal{B}^{\rm r}(C({\mathfrak{X}},{\mathbb{H}}))$. Note also that the operator $T$ is invertible if and only if the function $\Theta$ has no zero in ${\mathfrak{X}}$. Let us compute the $Q$-spectrum of $T$. According to Definition 1, we have $\rho_{\mathbb{H}}(T)=\\{{\bf q}\in{\mathbb{H}};(T^{2}-2\Re{\bf q}\,T+\|{\bf q}\|^{2})^{-1}\in\mathcal{B}^{\rm r}(C({\mathfrak{X}},{\mathbb{H}}))\\}.$ Consequently, ${\bf q}\in\sigma_{\mathbb{H}}(T)$ if and only if zero is in the range of the function $\tau({\bf q},x):=\Theta(x)^{2}-2\Re{\bf q}\,\Theta(x)+\|{\bf q}\|^{2},\,x\in\mathfrak{X}.$ Similarly, $\rho_{\mathbb{C}}(T)=\\{\lambda\in{\mathbb{C}};(T^{2}-2\Re\lambda\,T+\|\lambda\|^{2})^{-1}\in\mathcal{B}^{\rm r}(C({\mathfrak{X}},{\mathbb{H}}))\\},$ and so $\lambda\in\sigma_{\mathbb{C}}(T)$ if and only if zero is in the range of the function $\tau(\lambda,x):=\Theta(x)^{2}-2\Re\lambda\,\Theta(x)+|\lambda|^{2},\,x\in\mathfrak{X}.$ Looking for solutions $u+iv,u,v\in{\mathbb{R}}$, of the equation $(u-\Theta(x))^{2}+v^{2}=0$, a direct calculation shows that $u=\Re\Theta(x)$ and $v=\pm\|\Im\Theta(x)\|$. Hence $\sigma_{\mathbb{C}}(T)=\\{\Re\Theta(x)\pm i\|\Im\Theta(x)\|;x\in\mathfrak{X}\\}=\cup_{x\in\mathfrak{X}}\sigma(\Theta(x)).$ Of course, for every open conjugate symmetric subset $U\subset{\mathbb{C}}$ containing $\sigma_{\mathbb{C}}(T)$, and for every function $\Phi\in\mathcal{O}_{c}(U,\mathcal{B}(C({\mathfrak{X}},{\mathbb{M}})))$, we may construct the operator $\Phi(T)\in\mathcal{B}^{\rm r}(C({\mathfrak{X}},{\mathbb{H}}))$, using Theorem 4. ## 6 Quaternionic Joint Spectrum of Paires In many applications, it is more convenient to work with matrix quaternions rather than with abstract quaternions. Specifically, one considers the injective unital algebra morphism ${\mathbb{H}}\ni x_{1}+y_{1}{\bf j}+x_{2}{\bf k}+y_{2}{\bf l}\mapsto\left(\begin{array}[]{cc}x_{1}+iy_{1}&x_{2}+iy_{2}\\\ -x_{2}+iy_{2}&x_{1}-iy_{1}\end{array}\right)\in{\mathbb{M}}_{2},$ with $x_{1},y_{1},x_{2},y_{2}\in{\mathbb{R}},$ where ${\mathbb{M}}_{2}$ is the complex algebra of $2\times 2$-matrix, whose image, denoted by ${\mathbb{H}}_{2}$ is the real algebra of matrix quaternions. The elements of ${\mathbb{H}}_{2}$ can be also written as matrices of the form $Q({\bf z})=\left(\begin{array}[]{cc}z_{1}&z_{2}\\\ -\bar{z}_{2}&\bar{z_{1}}\end{array}\right),\,\,{\bf z}=(z_{1},z_{2})\in{\mathbb{C}}^{2}.$ A strong connection between the spectral theory of pairs of commuting operators in a complex Hilbert space and the algebra of quaternions has been firstly noticed in [17]. Another connection will be presented in this section. If ${\mathcal{V}}$ is an arbitrary vector space, we denote by ${\mathcal{V}}^{2}$ the Cartesian product ${\mathcal{V}}\times{\mathcal{V}}$. Let $\mathcal{V}$ be a real Banach space, and let ${\bf T}=(T_{1},T_{2})\in\mathcal{B(V)}^{2}$ be a pair of commuting operators. The extended pair ${\bf T}_{\mathbb{C}}=(T_{1{\mathbb{C}}},T_{2{\mathbb{C}}})\in\mathcal{B(V_{\mathbb{C}})}^{2}$ also consists of commuting operators. For simplicity, we set $Q({\bf T}_{\mathbb{C}}):=\left(\begin{array}[]{cc}T_{1{\mathbb{C}}}&T_{2{\mathbb{C}}}\\\ -T_{2{\mathbb{C}}}&T_{1{\mathbb{C}}}\end{array}\right)$ which acts on the complex Banach space $\mathcal{V}_{\mathbb{C}}^{2}$. We now define the quaternionic resolvent set and spectrum for the case of a pair of operators, inspired by the previous discussion concerning a single operator. ###### Definition 2 Let $\mathcal{V}$ be a real Banach space. For a given pair ${\bf T}=(T_{1},T_{2})\in\mathcal{B(V)}^{2}$ of commuting operators, the set of those $Q({\bf z})\in{\mathbb{H}}_{2},\,{\bf z}=(z_{1},z_{2})\in{\mathbb{C}}^{2}$, such that the operator $T_{1}^{2}+T_{2}^{2}-2\Re{z_{1}}T_{1}-2\Re{z_{2}}T_{2}+|z_{1}|^{2}+|z_{2}|^{2}$ is invertible in $\mathcal{B(V)}$ is said to be the quaternionic joint resolvent (or simply the $Q$-joint resolvent) of ${\bf T}$, and is denoted by $\rho_{\mathbb{H}}({\bf T})$. The complement $\sigma_{\mathbb{H}}({\bf T})={\mathbb{H}}_{2}\setminus\rho_{\mathbb{H}}({\bf T})$ is called the quaternionic joint spectrum (or simply the $Q$-joint spectrum) of ${\bf T}$. For every pair ${\bf T}_{\mathbb{C}}=(T_{1{\mathbb{C}}},T_{2{\mathbb{C}}})\in\mathcal{B(V_{\mathbb{C}})}^{2}$ we put ${\bf T}_{\mathbb{C}}^{c}=(T_{1{\mathbb{C}}},-T_{2{\mathbb{C}}})\in\mathcal{B(V_{\mathbb{C}})}^{2}$, and for every pair ${\bf z}=(z_{1},z_{2})\in{\mathbb{C}}^{2}$ we put ${\bf z}^{c}=(\bar{z}_{1},-z_{2})\in{\mathbb{C}}^{2}$ ###### Lemma 3 A matrix quaternion $Q({\bf z})$ $({\bf z}\in{\mathbb{C}}^{2})$ is in the set $\rho_{\mathbb{H}}({\bf T})$ if and only if the operators $Q({\bf T}_{\mathbb{C}})-Q({\bf z}),\,Q({\bf T}_{\mathbb{C}}^{c})-Q({\bf z}^{c})$ are invertible in $\mathcal{B}(\mathcal{V}_{\mathbb{C}}^{2})$. Proof The assertion follows from the equalities $\left(\begin{array}[]{cc}T_{1{\mathbb{C}}}-z_{1}&T_{2{\mathbb{C}}}-z_{2}\\\ -T_{2{\mathbb{C}}}+\bar{z}_{2}&T_{1{\mathbb{C}}}-\bar{z}_{1}\end{array}\right)\left(\begin{array}[]{cc}T_{1{\mathbb{C}}}-\bar{z}_{1}&-T_{2{\mathbb{C}}}+z_{2}\\\ T_{2{\mathbb{C}}}-\bar{z}_{2}&T_{1{\mathbb{C}}}-z_{1}\end{array}\right)=$ $\left(\begin{array}[]{cc}T_{1{\mathbb{C}}}-\bar{z}_{1}&-T_{2{\mathbb{C}}}+z_{2}\\\ T_{2{\mathbb{C}}}-\bar{z}_{2}&T_{1{\mathbb{C}}}-z_{1}\end{array}\right)\left(\begin{array}[]{cc}T_{1{\mathbb{C}}}-z_{1}&T_{2{\mathbb{C}}}-z_{2}\\\ -T_{2{\mathbb{C}}}+\bar{z}_{2}&T_{1{\mathbb{C}}}-\bar{z}_{1}\end{array}\right)=$ $[(T_{1{\mathbb{C}}}-z_{1})(T_{1{\mathbb{C}}}-\bar{z}_{1})+(T_{2{\mathbb{C}}}-z_{2})(T_{2{\mathbb{C}}}-\bar{z}_{2})]{\bf I}.$ for all ${\bf z}=(z_{1},z_{2})\in{\mathbb{C}}^{2}$, where $\bf I$ is the identity. Consequently, the operators $Q({\bf T}_{\mathbb{C}})-Q({\bf z}),\,Q({\bf T}_{\mathbb{C}}^{c})-Q({\bf z}^{c})$ are invertible in $\mathcal{B}({\mathcal{V}}_{\mathbb{C}}^{2})$ if and only if the operator $(T_{1{\mathbb{C}}}-z_{1})(T_{1{\mathbb{C}}}-\bar{z}_{1})+(T_{2{\mathbb{C}}}-z_{2})(T_{2{\mathbb{C}}}-\bar{z}_{2})$ is invertible in $\mathcal{B}(\mathcal{V}_{\mathbb{C}})$. Because we have $T_{1{\mathbb{C}}}^{2}+T_{2{\mathbb{C}}}^{2}-2\Re{z_{1}}T_{1{\mathbb{C}}}-2\Re{z_{2}}T_{2{\mathbb{C}}}+|z_{1}|^{2}+|z_{2}|^{2}=$ $[T_{1}^{2}+T_{1}^{2}-2\Re{z_{1}}T_{1}-2\Re{z_{2}}T_{2}+|z_{1}|^{2}+|z_{2}|^{2}]_{\mathbb{C}},$ the operators $Q({\bf T}_{\mathbb{C}})-Q({\bf z}),\,Q({\bf T}_{\mathbb{C}}^{c})-Q({\bf z}^{c})$ are invertible in $\mathcal{B}({\mathcal{V}}_{\mathbb{C}}^{2})$ if and only if the operator $T_{1}^{2}+T_{1}^{2}-2\Re{z_{1}}T_{1}-2\Re{z_{2}}T_{2}+|z_{1}|^{2}+|z_{2}|^{2}$ is invertible in $\mathcal{B(V)}$. Lemma 3 shows that we have the property $Q({\bf z})\in\sigma_{\mathbb{H}}({\bf T})$ if and only if $Q(z^{c})\in\sigma_{\mathbb{H}}({\bf T}^{c})$. Putting $\sigma_{{\mathbb{C}}^{2}}({\bf T}):=\\{{\bf z}\in{\mathbb{C}}^{2};Q({\bf z})\in\sigma_{\mathbb{H}}({\bf T})\\},$ the set $\sigma_{{\mathbb{C}}^{2}}({\bf T})$ has a similar property, specifically $\bf z\in\sigma_{{\mathbb{C}}^{2}}({\bf T})$ if and only if $\bf z^{c}\in\sigma_{{\mathbb{C}}^{2}}({\bf T}^{c})$. As in the quaternionic case, the set $\sigma_{{\mathbb{C}}^{2}}({\bf T})$ looks like a ”complex border“ of the set $\sigma_{\mathbb{H}}({\bf T})$. ###### Remark 9 For the extended pair ${\bf T}_{\mathbb{C}}=(T_{1{\mathbb{C}}},T_{2{\mathbb{C}}})\in{B(V_{\mathbb{C}})}^{2}$ of the commuting pair ${\bf T}=(T_{1},T_{2})\in\mathcal{B(V)}$ there is an interesting connexion with the joint spectral theory of J. L. Taylor (see [15, 16]; see also [19]). Namely, if the operator $T_{1{\mathbb{C}}}^{2}+T_{2{\mathbb{C}}}^{2}-2\Re{z_{1}}T_{1{\mathbb{C}}}-2\Re{z_{2}}T_{2{\mathbb{C}}}+|z_{1}|^{2}+|z_{2}|^{2}$ is invertible, then the point ${\bf z}=(z_{1},z_{2})$ belongs to the joint resolvent of ${\bf T}_{\mathbb{C}}$. Indeed, setting $R_{j}({\bf T}_{\mathbb{C}},{\bf z})=(T_{j{\mathbb{C}}}-\bar{z}_{j})(T_{1{\mathbb{C}}}^{2}+T_{2{\mathbb{C}}}^{2}-2\Re{z_{1}}T_{1{\mathbb{C}}}-2\Re{z_{2}}T_{2{\mathbb{C}}}+|z_{1}|^{2}+|z_{2}|^{2})^{-1},$ $q=Q({\bf z})$ for $j=1,2$, we clearly have $(T_{1{\mathbb{C}}}-z_{1})R_{1}({\bf T}_{\mathbb{C}},{\bf z})+(T_{2{\mathbb{C}}}-z_{2})R_{2}({\bf T}_{\mathbb{C}},{\bf z})={\bf I},$ which, according to [15], implies that ${\bf z}$ is in the joint resolvent of ${\bf T}_{\mathbb{C}}$. A similar argument shows that, in this case the point ${\bf z}^{c}$ belongs to the joint resolvent of ${\bf T}_{\mathbb{C}}^{c}$. In addition, if $\sigma(T_{\mathbb{C}})$ designates the Taylor spectrum of $T_{\mathbb{C}}$, we have the inclusion $\sigma(T_{\mathbb{C}})\subset\sigma_{{\mathbb{C}}^{2}}({\bf T})$. In particular, for every complex-valued function $f$ analytic in a neighborhood of $\sigma_{{\mathbb{C}}^{2}}({\bf T})$, the operator $f(\bf T_{\mathbb{C}})$ can be computed via Taylor’s analytic functional calculus. In fact, we have a Martinelli type formula for the analytic functional calculus: ###### Theorem 6 Let $\mathcal{V}$ be a real Banach space, let ${\bf T}=(T_{1},T_{2})\in\mathcal{B(V)}^{2}$ be a pair of commuting operators, let $U\subset{\mathbb{C}}^{2}$ be an open set, let $D\subset U$ be a bounded domain containing $\sigma_{{\mathbb{C}}^{2}}({\bf T})$, with piecewise-smooth boundary $\Sigma$, and let $f\in\mathcal{O}(U)$. Then we have $f({\bf T}_{\mathbb{C}})=\frac{1}{(2\pi i)^{2}}\int_{\Sigma}f({\bf z}))L({\bf z,T_{\mathbb{C}}})^{-2}(\bar{z}_{1}-T_{1{\mathbb{C}}})d\bar{z}_{2}-(\bar{z}_{2}-T_{2{\mathbb{C}}})d\bar{z}_{1}]dz_{1}dz_{2},$ where $L({\bf z,T_{\mathbb{C}}})=T_{1{\mathbb{C}}}^{2}+T_{2{\mathbb{C}}}^{2}-2\Re{z_{1}}T_{1{\mathbb{C}}}-2\Re{z_{2}}T_{2{\mathbb{C}}}+|z_{1}|^{2}+|z_{2}|^{2}.$ Proof. Theorem III.9.9 from [19] implies that the map $\mathcal{O}(U)\ni f\mapsto f({\bf T}_{\mathbb{C}})\in\mathcal{B(V_{\mathbb{C}})}$, defined in terms of Taylor’s analytic functional calculus, is unital, linear, multiplicative, and ordinary complex polynomials in ${\bf z}$ are transformed into polynomials in ${\bf T}_{\mathbb{C}}$ by simple substitution, where $\mathcal{O}(U)$ is the algebra of all analytic functions in the open set $U\subset{\mathbb{C}}^{2}$, provided $U\supset\sigma({\bf T}_{\mathbb{C}})$. The only thing to prove is that, when $U\supset\sigma_{{\mathbb{C}}^{2}}({\bf T})$, Taylor’s functional calculus is given by the stated (canonical) formula. In order to do that, we use an argument from the proof of Theorem III.8.1 in [19], to make explicit the integral III(9.2) from [19] (see also [12]). We consider the exterior algebra $\Lambda[e_{1},e_{2},\bar{\xi_{1}},\bar{\xi_{2}},\mathcal{O}(U)\otimes\mathcal{V}_{\mathbb{C}}]=\Lambda[e_{1},e_{2},\bar{\xi_{1}},\bar{\xi_{2}}]\otimes\mathcal{O}(U)\otimes\mathcal{V}_{\mathbb{C}},$ where the indeterminates $e_{1},e_{2}$ are to be associated with the pair ${\bf T}_{\mathbb{C}}$, we put $\bar{\xi_{j}}=d\bar{z}_{j},\,j=1,2$, and consider the operators $\delta=(z_{1}-T_{1{\mathbb{C}}})\otimes e_{1}+(z_{2}-T_{2{\mathbb{C}}})\otimes e_{2},\,\bar{\partial}=(\partial/\partial\bar{z_{1}})\otimes\bar{\xi_{1}}+(\partial/\partial\bar{z_{2}})\otimes\bar{\xi_{2}}$, acting naturally on this exterior algebra, via the calculus with exterior forms. To simplify the computation, we omit the symbol $\otimes$, and the exterior product will be denoted simply par juxtaposition. We fix the exterior form $\eta=\eta_{2}=fye_{1}e_{2}$ for some $f\in\mathcal{O}(U)$ and $y\in\mathcal{X}_{\mathbb{C}}$, which clearly satisfy the equation $(\delta+\bar{\partial})\eta=0$, and look for a solution $\theta$ of the equation $(\delta+\bar{\partial})\theta=\eta$. We write $\theta=\theta_{0}+\theta_{1}$, where $\theta_{0},\theta_{1}$ are of degree $0$ and $1$ in $e_{1},e_{2}$, respectively. Then the equation $(\delta+\bar{\partial})\theta=\eta$ can be written under the form $\delta\theta_{1}=\eta,\,\delta\theta_{0}=-\bar{\partial}\theta_{1}$, and $\bar{\partial}\theta_{0}=0$. Note that $\theta_{1}=fL({\bf z,T_{\mathbb{C}}})^{-1}[(\bar{z}_{1}-T_{1{\mathbb{C}}})ye_{2}-(\bar{z}_{2}-T_{2{\mathbb{C}}})]ye_{1}$ is visibly a solution of the equation $\delta\theta_{1}=\eta$. Further, we have $\bar{\partial}\theta_{1}=fL({\bf z,T_{\mathbb{C}}})^{-2}[(z_{1}-T_{1{\mathbb{C}}})(\bar{z}_{2}-T_{2{\mathbb{C}}})y\bar{\xi}_{1}e_{1}-(z_{1}-T_{1{\mathbb{C}}})(\bar{z}_{1}-T_{1{\mathbb{C}}})y\bar{\xi}_{2}e_{1}+$ $(z_{2}-T_{2{\mathbb{C}}})(\bar{z}_{2}-T_{2{\mathbb{C}}})y\bar{\xi}_{1}e_{2}-(z_{2}-T_{2{\mathbb{C}}})(\bar{z}_{1}-T_{1{\mathbb{C}}})y\bar{\xi}_{2}e_{2}]=$ $\delta[fL({\bf z,T_{\mathbb{C}}})^{-2}(\bar{z}_{1}-T_{1{\mathbb{C}}})y\bar{\xi}_{2}-fL({\bf z,T_{\mathbb{C}}})^{-2}(\bar{z}_{2}-T_{2{\mathbb{C}}})y\bar{\xi}_{1}],$ so we may define $\theta_{0}=-fL({\bf z,T_{\mathbb{C}}})^{-2}(\bar{z}_{1}-T_{1{\mathbb{C}}})y\bar{\xi}_{2}+fL({\bf z,T_{\mathbb{C}}})^{-2}(\bar{z}_{2}-T_{2{\mathbb{C}}})y\bar{\xi}_{1}.$ Formula III(8.5) from [19] shows that $f({\bf T}_{\mathbb{C}})y=-\frac{1}{(2\pi i)^{2}}\int_{U}\bar{\partial}(\phi\theta_{0})dz_{1}dz_{2}=$ $\frac{1}{(2\pi i)^{2}}\int_{\Sigma}f({\bf z}))L({\bf z,T_{\mathbb{C}}})^{-2}[(\bar{z}_{1}-T_{1{\mathbb{C}}})yd\bar{z}_{2}-(\bar{z}_{2}-T_{2{\mathbb{C}}})yd\bar{z}_{1}]dz_{1}dz_{2},$ for all $y\in\mathcal{X}_{\mathbb{C}}$, via Stokes’s formula, where $\phi$ is a smooth function such that $\phi=0$ in a neighborhood of $\sigma_{{\mathbb{C}}^{2}}({\bf T})$, $\phi=1$ on $\Sigma$ and the support of $1-\phi$ is compact. ###### Remark 10 (1) We may extend the previous functional calculus to $\mathcal{B(V}_{\mathbb{C}})$-valued analytic functions, setting, for such a function $F$ and with the notation from above, $F({\bf T}_{\mathbb{C}})=\frac{1}{(2\pi i)^{2}}\int_{\Sigma}F({\bf z}))L({\bf z,T_{\mathbb{C}}})^{-2}(\bar{z}_{1}-T_{1{\mathbb{C}}})d\bar{z}_{2}-(\bar{z}_{2}-T_{2{\mathbb{C}}})d\bar{z}_{1}]dz_{1}dz_{2}.$ In particular, if $F({\bf z})=\sum_{j,k\geq 0}A_{jk{\mathbb{C}}}z_{1}^{j}z_{2}^{k}$, with $A_{j,k}\in\mathcal{B(V)}$, where the series is convergent in neighborhood of $\sigma_{{\mathbb{C}}^{2}}({\bf T})$, we obtain $F({\bf T}):=F({\bf T}_{\mathbb{C}})|\mathcal{V}=\sum_{j,k\geq 0}A_{jk}T_{1}^{j}T_{2}^{k}\in\mathcal{B(V)}.$ (2) The connexion of the spectral theory of pairs with the algebra of quaternions is even stronger in the case of complex Hilbert spaces. Specifically, if $\mathcal{H}$ is a complex Hilbert space and ${\bf V}=(V_{1},V_{2})$ is a commuting pair of bounded linear operators on $\mathcal{H}$, a point ${\bf z}=(z_{1},z_{2})\in{\mathbb{C}}^{2}$ is in the joint resolvent of ${\bf V}$ if and only if the operator $Q({\bf V})-Q({\bf z})$ is invertible in $\mathcal{H}^{2}$, where $Q({\bf V})=\left(\begin{array}[]{cc}V_{1}&V_{2}\\\ -V_{2}^{*}&V_{1}^{*}\end{array}\right).$ (see [17] for details). In this case, there is also a Martinelli type formula which can be used to construct the associated analytic functional calculus (see [18],[19]). An approach to such a construction in Banach spaces, by using a so-called splitting joint spectrum, can be found in [14]. ## References * [1] A. G. Baskakov and A. S. Zagorskii: Spectral Theory of Linear Relations on Real Banach Spaces, Mathematical Notes (Russian: Matematicheskie Zametki), 2007, Vol. 81, No. 1, pp. 15-27. * [2] S. Bochner: Analytic and meromorphic continuation by means of Green’s formula, Ann. of Math. (2) , 44 : 4 (1943) pp. 652-673. * [3] J. L. Brenner: Matrices of quaternions, Pacific J. Math. 1 (1951), 329-335. * [4] F. Colombo, J. Gantner, D. P. Kimsey: Spectral Theory on the S-Spectrum for Quaternionic Operators, Birkhäuser, 2018. * [5] F. Colombo, I. Sabadini and D. C. Struppa: Noncommutative Functional Calculus, Theory and Applications of Slice Hyperholomorphic Functions: Progress in Mathematics, Vol. 28 Birkhäuser/Springer Basel AG, Basel, 2011. * [6] N. Dunford and J. T. Schwartz: Linear Operators, Part I: General Theory, Interscience Publishers, New York, London, 1958. * [7] G. Gentili and D. C. Struppa: A new theory of regular functions of a quaternionic variable, Advances in Mathematics 216 (2007) 279-301. * [8] R. Ghiloni , V. Moretti and A. Perotti: Continuous slice functional calculus in quaternionic Hilbert spaces, Rev. Math. Phys. 25 (2013), no. 4, 1350006, 83 p. * [9] L. Ingelstam: Real Banach algebras. Ark. Mat. 5 (1964), 239–270 (1964). * [10] I. Kaplansky: Normed algebras, Duke. Math. J. 16, 399-418 (1949). * [11] S. H. Kulkarni: Representations of a Class of Real $B^{*}$-Algebras as Algebras of Quaternion-Valued Functions, Proceedings of the American Mathematical Society, Vol. 116, No. 1 (1992), 61-66. * [12] R. Levi: Notes on the Taylor joint spectrum of commuting operators. Spectral theory (Warsaw, 1977), 321–332, Banach Center Publ., 8, PWN, Warsaw, 1982. * [13] E. Martinelli: Alcuni teoremi integrali per le funzioni analitiche di più variabili complesse, Accad. Ital. Mem. Cl. Sci. fis. mat. nat. 9 (1938), 269-283. * [14] V. Müller and V. Kordula: Vasilescu-Martinelli formula for operators in Banach spaces, Studia Math. 113 (1995), no. 2, 127-139. * [15] J. L. Taylor: A joint spectrum for several commuting operators. J. Functional Anal. 6 1970 172-191. * [16] J. L. Taylor: The analytic functional calculus for several commuting operators, Acta Math. 125 (1970), 1-38. * [17] F.-H. Vasilescu: On pairs of commuting operators, Studia Math. 62 (1978), 203-207. * [18] F.-H. Vasilescu: A Martinelli type formula for the analytic functional calculus, Rev. Roumaine Math. Pures Appl. 23 (1978), no. 10, 1587-1605. * [19] F.-H. Vasilescu: Analytic functional calculus and spectral decompositions, D. Reidel Publishing Co., Dordrecht and Editura Academiei R. S. R., Bucharest, 1982. * [20] F.-H. Vasilescu: Analytic Functional Calculus in Quaternionic Framework, http://arxiv.org/abs/1902.03850 * [21] F.-H. Vasilescu: Quaternionic Regularity via Analytic Functional Calculus, Integral Equations and Operator Theory, DOI: 10.1007/s00020-020-2574-7

2024-09-04T02:54:59.172215

2003.05265

arxiv-papers

# Transient chaos under coordinate transformations in relativistic systems D. S. Fernández Á. G. López J. M. Seoane M. A. F. Sanjuán Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain ###### Abstract We use the Hénon-Heiles system as a paradigmatic model for chaotic scattering to study the Lorentz factor effects on its transient chaotic dynamics. In particular, we focus on how time dilation occurs within the scattering region by measuring the time in a clock attached to the particle. We observe that the several events of time dilation that the particle undergoes exhibit sensitivity to initial conditions. However, the structure of the singularities appearing in the escape time function remains invariant under coordinate transformations. This occurs because the singularities are closely related to the chaotic saddle. We then demonstrate using a Cantor-like set approach that the fractal dimension of the escape time function is relativistic invariant. In order to verify this result, we compute by means of the uncertainty dimension algorithm the fractal dimensions of the escape time functions as measured with inertial and comoving with the particle frames. We conclude that, from a mathematical point of view, chaotic transient phenomena are equally predictable in any reference frame and that transient chaos is coordinate invariant. ###### pacs: 05.45.Ac,05.45.Df,05.45.Pq ## I Introduction Chaotic scattering in open Hamiltonian systems is a fundamental part of the theoretical study of dynamical systems. There are many applications such as the interaction between the solar wind and the magnetosphere tail seoane2013 , the simulation in several dimensions of the molecular dynamics lin2013 , the modeling of chaotic advection of particles in fluid mechanics daitche2014 , or the analysis of the escaping mechanism from a star cluster or a galaxy zotos2017 ; navarro2019 , to name a few. A scattering phenomenon is a process in which a particle travels freely from a remote region and encounters an obstacle, often described in terms of a potential, which affects its evolution. Finally, the particle leaves the interaction region and continues its journey freely. This interaction is typically nonlinear, possibly leading the particle to perform transient chaotic dynamics, i.e., chaotic dynamics with a finite lifetime lai2010 ; grebogi1983 . Scattering processes are commonly studied by means of the scattering functions, which relate the particle states at the beginning of its evolution once the interaction with the potential has already taken place. Thus, nonlinear interactions can make these functions exhibit self-similar arrangements of singularities, which hinder the system predictability aguirre2009 . Transient chaos is a manifestation of the presence in phase space of a chaotic set called non- attracting chaotic set, also called chaotic saddle ott1993 . This phenomenon can be found in a wide variety of situations tel2015 , as for example the dynamics of decision making, the doubly transient chaos of undriven autonomous mechanical systems or even in the sedimentation of volcanic ash. There have been numerous efforts to characterize chaos in relativistic systems in an observer-independent manner hobill1994 . It has been rigorously demonstrated that the sign of the Lyapunov exponents is invariant under coordinate transformations that satisfy four minimal conditions motter2003 . More specifically, such conditions consider that a valid coordinate transformation has to leave the system autonomous, its phase space bounded, the invariant measure normalizable and the domain of the new time parameter infinite motter2003 . As a consequence, chaos is a property of relativistic systems independent of the choice of the coordinate system in which they are described. In other words, homoclinic and heteroclinic tangles cannot be untangled by means of coordinate transformations. We shall utilize the Lorentz transformations along this paper, which satisfy this set of conditions motter2009 . Although we utilize a Hamiltonian system in its open regime, from the point of view of Lyapunov exponents the phase space can be considered bounded because of the presence of the chaotic saddle. This set is located in a finite region of the system’s phase space and contains all the non-escaping orbits in the hyperbolic regime. Hence, the Lyapunov exponents are well- defined because these trajectories stay in the saddle forever. On the other hand, concerning the computation of the escape time function, we shall only consider along this work the finite part of the phase space where the escaping orbits remain bounded, and similarly from the point of view of the finite-time Lyapunov exponents the phase space can be considered bounded as well vallejo2003 . Despite the fact that the sign of the Lyapunov exponents is invariant, the precise values of these exponents, which indicate “how chaotic” a dynamical system is, are noninvariant. Therefore, this lack of invariance leaves some room to explore how coordinate transformations affect the unpredictability in dynamical systems with transient chaos. In the present work we analyze the structure of singularities of the scattering functions under a valid coordinate transformation. In particular, we compute the fractal dimension of the escape time function as measured in an inertial reference frame and another non-inertial reference frame comoving with the particle, respectively. We then characterize the system unpredictability by calculating this fractal dimension, since it enables to infer the dimension of the chaotic saddle aguirre2001 . Indeed, this purely geometrical method has been proposed as an independent-observer procedure to determine whether the system behaves chaotically motter2001 . Relevant works have been devoted to analyze the relationship between relativity and chaos in recent decades barrow1982 ; chernikov1989 ; ni2012 . More recently, the Lorentz factor effects on the dynamical properties of the system have also been studied in relativistic chaotic scattering bernal2017 ; bernal2018 . In this paper, we focus on how changes of the reference frame affect typical phenomena of chaotic scattering. We describe the model in Sec. II, which consists of a relativistic version of the Hénon-Heiles system. Two well-known scattering functions are explored in Sec. III, such as the exit through which the particle escapes and its escape time. In Sec. IV, we demonstrate the fractal dimension invariance under a coordinate transformation by using a Cantor-like set approach. Subsequently, we quantify the unpredictability of the escape times and analyze the effect of such a reference frame modification. We conclude with a discussion of the main results and findings of the present work in Sec. V. ## II Model description Figure 1: (a) The three-dimensional representation of the Hénon-Heiles potential $V(x,y)=\frac{1}{2}(x^{2}+y^{2})+x^{2}y-\frac{1}{3}y^{3}$. (b) The isopotential curves in the physical space show that the Hénon-Heiles system is open and has triangular symmetry. If the energy of the particle is higher than a threshold value, related to the potential saddle points, there exist unbounded orbits. Following these trajectories the particle leaves the scattering region through any of the three exits. The Hénon-Heiles system was proposed in 1964 to study the existence of a third integral of motion in galactic models with axial symmetry henon1964 . We consider a single particle whose total mechanical energy can be denoted as $E_{N}$ in the Newtonian approximation. This energy is conserved along the trajectory described by the particle, which is launched from the interior of the potential well, within a finite region of the phase space called the scattering region. We have utilized a dimensionless form of the Hénon-Heiles system, so that the potential is written as $V(x,y)=\frac{1}{2}(x^{2}+y^{2})+x^{2}y-\frac{1}{3}y^{3},$ (1) where $x$ and $y$ are the spatial coordinates. When the energy is above a threshold value, the potential well exhibits three exits due to its triangular symmetry in the physical space, i.e., the plane $(x,y)$, as visualized in Fig. 1. We call Exit 1 the exit located at the top $(y\to+\infty)$, Exit 2 the one located downwards to the left $(x\to-\infty,y\to-\infty)$ and Exit 3 the one at the right $(x\to+\infty,y\to-\infty)$. One of the characteristics of open Hamiltonian systems with escapes is the existence of highly unstable periodic orbits known as Lyapunov orbits contopoulos1990 , which are placed near the saddle points. In fact, when a trajectory crosses through a Lyapunov orbit, it escapes to infinity and never returns back to the scattering region. Furthermore, we recall that the energy of the particle determines also the dynamical regime. We can distinguish two open regimes in which escapes are allowed. On the one hand, in the nonhyperbolic regime the KAM tori coexist with the chaotic saddle and the phase space exhibits regions where dynamics is regular and also chaotic sideris2006 , whereas the chaotic saddle rules the dynamics in the hyperbolic regime, making it completely chaotic. When the speed of the particle is comparable to the speed of light, the relativistic effects have to be taken into account ohanian2001 . In the present work we consider a particle which interacts in the limit of weak external fields, and therefore we deal with a special relativistic version of the Hénon-Heiles system, whose dynamics is governed by the conservative Hamiltonian lan2011 ; chanda2018 ; kovacs2011 ; calura1997 $H=c\sqrt{c^{2}+p^{2}+q^{2}}+V(x,y),$ (2) where $c$ is the value of the speed of light, and $p$ and $q$ are the momentum coordinates. On the other hand, the Lorentz factor is defined as $\gamma=\frac{1}{\sqrt{1-\frac{\textbf{v}^{2}}{c^{2}}}}=\frac{1}{\sqrt{1-\beta^{2}}},$ (3) where v is the velocity vector of the particle and $\beta=|\textbf{v}|/c$ the ratio between the speed of the particle and the speed of light. The Lorentz factor $\gamma$ and $\beta$ are two equivalent ways to express how large is the speed of the particle compared to the speed of light. These two factors vary in the ranges $\gamma\in[1,+\infty)$ and $\beta\in[0,1)$, respectively. For convenience, we shall use $\beta$ as a parameter along this work. Hamilton’s canonical equations can be derived from Eq. (2), yielding the equations of motion $\displaystyle\dot{x}=$ $\displaystyle\frac{\partial H}{\partial p}=\frac{p}{\gamma},$ $\displaystyle\dot{p}=-\frac{\partial H}{\partial x}=-x-2xy,$ (4) $\displaystyle\dot{y}=$ $\displaystyle\frac{\partial H}{\partial q}=\frac{q}{\gamma},$ $\displaystyle\dot{q}=-\frac{\partial H}{\partial y}=y^{2}-x^{2}-y,$ where the Lorentz factor can be alternatively written in the momentum- dependent form as $\gamma=\frac{1}{c}\sqrt{c^{2}+p^{2}+q^{2}}$. Although the complete phase space is four-dimensional, the conservative Hamiltonian constrains the dynamics to a three-dimensional manifold of the phase space, known as the energy shell. Some recent works aim at isolating the effects of the variation of the Lorentz factor $\gamma$ (or $\beta$ equivalently) from the remaining variables of the system bernal2017 ; bernal2018 . In order to accomplish this, they modify the initial value of $\beta$ and use it as the only parameter of the dynamical system. Since $\beta$ is a quantity that depends on $|\textbf{v}|$ and $c$, they choose to vary the numerical value of $c$. Needless to say, the value of the speed of light $c$ remains constant during the particle trajectory. The fundamental reason for deciding to increase the kinetic energy of the system by reducing the numerical value of the speed of light is simply as follows. If we keep the Hénon-Heiles potential constant and increase the speed of the particle to values close to the speed of light, the potential will be in a much lower energy regime compared to the kinetic energy of the particle. Therefore, the potential becomes negligible and the interaction between them becomes irrelevant. Consequently, each time we select a value of the speed of light we are scaling the system, and hence the ratio of the kinetic energy and the potential as well. The sequence of potential wells with different values of $\beta$ represents potential wells with the Hénon-Heiles morphology, but at different scales in which the interaction of a relativistic particle is not trivial. In this way, the effects of the Lorentz factor on the dynamics are isolated from the other system variables, because the Lorentz factor is the only parameter that differentiates all these scaled systems. We then consider the same initial value of the particle speed $|\textbf{v}_{0}|$ in every simulation with a different value of $\beta$, launching the particle from the minimum potential, which is located at $(x_{0},y_{0})=(0,0)$ and where the potential energy is null. We have arbitrarily chosen $|\textbf{v}_{0}|\approx 0.5831$ (as in bernal2017 ; bernal2018 ), which corresponds to the open nonhyperbolic regime with energy $E_{N}=0.17$, close to the escape energy in the Newtonian approximation. Thus, we analyze how the relativistic parameter $\beta$, as its value increases, affects the dynamical properties starting from the nonhyperbolic regime. The numerical value of $c$ varies, as shown in Fig. 2, and for instance if the simulation is carried out for a small $\beta$, where $|\textbf{v}_{0}|\ll c$, the initial speed of the particle only represents a very low percentage of the speed of light. In this case, we recover the Newtonian approximation and the classical version of the Hénon-Heiles system. On the contrary, if the simulation takes place with a value of $\beta$ near one, the speed of the particle represents a high percentage of the speed of light and the relativistic effects on the dynamics become more intense. Numerical computations reveal that the KAM tori are mostly destroyed at $\beta\approx 0.4$, and hence the dynamics is hyperbolic for higher values of $\beta$ bernal2018 . If some small tori survive, they certainly do not rule the system overall dynamics. As we focus on the hyperbolic regime, the simulations are run for values of $\beta\in[0.5,0.99]$ and by means of a fixed step fourth-order Runge-Kutta method press1992 . We recall that the initial values of the momentum $(p_{0},q_{0})$ depend on the chosen initial value of $\beta$, and therefore this computational technique (to vary the value of $\beta$ fixing $|\textbf{v}_{0}|$) is an ideal method to increase the particle kinetic energy to the relativistic regime. For example, a particle trapped in the KAM tori can escape if the initial value of $\beta$ is high enough, as shown in Fig. 2. Figure 2: The evolution of a particle launched within the scattering region from the same initial condition for different values of $\beta$. (a) For a very low $\beta$ (Newtonian approximation), the particle is trapped in the KAM tori and describes a bounded trajectory. (b) The value of $\beta$ is large enough to destroy the KAM tori and the particle leaves the scattering region following a trajectory typical of transient chaos. (c) Finally, a larger value of $\beta$ than in (b) makes the particle escape faster. ## III Escape times in inertial and non-inertial frames The scattering functions enable us to represent the relation between input and output dynamical states of the particle, i.e., how the interaction of the particle with the potential takes place. The potential of Hénon-Heiles leads the particle to describe chaotic trajectories before converging to a specific exit, which makes the scattering functions exhibit a fractal structure. In order to verify the sensitivity of the system to exits and escape times, we launch particles from the potential minimum slightly varying the shooting angle $\theta$ that is formed by the initial velocity vector and the positive $x$-axis, as shown in Fig. 3(a). The maximum value of the kinetic energy is reached at the potential minimum, as the system is conservative. We define the value of the Lorentz factor associated with this maximum kinetic energy as the critical Lorentz factor $\gamma_{c}(\beta)=\frac{1}{\sqrt{1-\beta^{2}}}.$ (5) We emphasize that the initial Lorentz factor of every particle is the critical Lorentz factor, since every trajectory is initialized from the potential minimum in this work. We shall monitor the Lorentz factor of the particle along its trajectory and use the critical Lorentz factor as the criterion of whether the particle has escaped or not. This escape criterion is based on the fact that the value of the kinetic energy remains bounded while the particle evolves chaotically within the potential well, bouncing back and forth against the potential barriers before escaping. The Lorentz factor value then varies between the unity and the critical value inside the scattering region, i.e., $\gamma(t)\in[1,\gamma_{c}]$. Eventually, the particle leaves the scattering region and the value of its Lorentz factor breaks out towards infinity, because its kinetic energy does not remain bounded anymore. In order to prevent this asymptotic behavior of the Lorentz factor, it is convenient to set that the escape happens at the time $t_{e}$ when $\gamma(t_{e})>\gamma_{c}$. In this manner, we define the scattering region as the part of the physical space where the dynamics is bounded. This escape criterion is computationally affordable and useful to implement in any Hamiltonian system without knowing specific information about the exits. In addition, it includes all the escapes that take place when the Lyapunov orbit criterion is considered. Figure 3: (a) Each of the exits is identified with a different color, such that Exit 1 (red), Exit 2 (green) and finally Exit 3 (blue). In order to avoid redundant results due to the triangular symmetry of the well, we only let the particle evolve from the angular region $\theta_{0}\in\left[\pi/2,5\pi/6\right]$ (black dashed lines). (b) The scattering function of the exits $(2000\times 2000)$ given the parameter map $(\beta\in[0.5,0.99],\theta_{0}\in[\pi/2,5\pi/6])$ in the hyperbolic regime. A particle launched with $\theta=\pi/2$ escapes directly towards the Exit 1 for every value of $\beta$ as shown in Fig. 3(b), whereas if it is launched with $\theta=5\pi/6$ the particle bounces against the potential barrier placed between Exit 1 and Exit 2 and escapes through the Exit 3. The whole structure of exits in between is apparently fractal. Nonetheless, the exit function becomes smoother when the value of $\beta$ increases, but it is never completely smooth. On the other hand, we recall that the chaotic saddle is an observer-independent set of points formed by the intersection of the stable and unstable manifolds. Concretely, the stable manifold of an open Hamiltonian system is defined as the boundary between the exit basins ott1993 . If a particle starts from a point arbitrarily close to the stable manifold it will spend an infinite time in converging to an exit, i.e., it never escapes. The unstable manifold is the set along which particles lying infinitesimally close to the chaotic saddle will eventually leave the scattering region in the course of time tel2015 . The escape time can be easily defined as the time the particle spends evolving inside the scattering region before escaping to infinity. In nonrelativistic systems, the particular clock in which the time is measured is irrelevant since time is absolute. However, here we consider two time quantities: the time $t$ that is measured by an inertial reference frame at rest and the proper time $\tau$ as measured by a non-inertial reference frame comoving with the particle. This proper time is simply the time measured by a clock attached to the particle. As is well known, an uniformly moving clock runs slower by a factor $\sqrt{1-\beta^{2}}$ in comparison to another identically constructed and synchronized clock at rest in an inertial frame. Therefore, we assume that at any instant of time the clock of the accelerating particle advances at the same rate as an inertial clock that momentarily had the same velocity barton1999 . In this manner, given an infinitesimal time interval $dt$, the particle clock will measure a time interval $d\tau=\frac{dt}{\gamma(t)},$ (6) where $\gamma(t)$ is the particle Lorentz factor at the instant of time $t$. Since the Lorentz factor is greater than the unity, the proper time interval always obeys that $d\tau\leq dt$, which is just the mathematical statement of the twin paradox. When the particle velocities are very close to the speed of light, the time dilation phenomenon takes place so that the time of the particle clock runs more slowly in comparison to clocks at rest in the potential. In the context of special relativity, it is important to bear in mind that it is assumed that the potential does not affect the clocks rate. In other words, all the clocks placed at rest in any point of the potential are ticking at the same rate along this work. Without loss of generality, Eq. 6 can be expressed as an integral in the form $\tau_{e}=\int_{0}^{t_{e}}\frac{dt}{\gamma(t)},$ (7) where the final time of the integration interval is the escape time in the inertial frame. We shall solve this integral using the Simpson’s rule jeffreys1988 . Since each evolution of the Lorentz factor is unique because each particle describes a distinct chaotic trajectory, every particle clock measures a different proper time at any instant of time $t$. Nonetheless, as the dynamics is bounded in the same energetic conditions given a value of $\beta$, the Lorentz factor of all trajectories is similar on average at any instant of time $t$. For this reason, we assume that there exists an average value of the Lorentz factor along the particle trajectory, and estimate it as the arithmetic mean between the maximum and minimum values of the bounded Lorentz factor inside the scattering region, i.e., $\bar{\gamma}(\beta)=\frac{1+\gamma_{c}}{2}=\frac{1+\sqrt{1-\beta^{2}}}{2\sqrt{1-\beta^{2}}}.$ (8) Using this definition to Eq. 7, we can define an average time dilation in the form $\bar{\tau}_{e}\equiv t_{e}/\bar{\gamma}$. This value should only be regarded as an approximation, which shall prove of great usefulness to interpret the numerical results obtained ahead. Accordingly, the difference between both the average escape time and the time $t_{e}$ is also approximately linear on average. In this manner, we can also define the magnitude $\delta\bar{t}_{e}\equiv t_{e}-\bar{\tau}_{e}=\frac{1-\sqrt{1-\beta^{2}}}{1+\sqrt{1-\beta^{2}}}t_{e}.$ (9) We emphasize that this value is again just an approximation representing the average behavior of the system, which disregards the fluctuations of the Lorentz factor. It reproduces qualitatively the behavior when the dynamics is bounded in the well, as shown in Fig. 4(a). Figure 4: (a) The Lorentz factor evolution $\gamma(t)$ of three different trajectories: a fast escape (yellow) and two typical transient chaotic trajectories (red and blue). The dashed guideline represents the Lorentz factor value of $\bar{\gamma}$ (black), corresponding to $\beta=0.75$. The time differences $\delta t(t)$ along these trajectories is also shown. (b) The scattering function of escape times $t_{e}$ in logarithmic scale given the parameter map $(\beta\in[0.5,0.99],\theta_{0}\in[\pi/2,5\pi/6])$. The two black dashed lines corresponds to the subfigures (c) and (d), which show the scattering function of escape time $t_{e}(\theta_{0})$ (blue) and $\tau_{e}(\theta_{0})$ (red) for $\beta=0.5$ and $\beta=0.8$, respectively. (e, f) The time difference function $\delta t_{e}(\theta_{0})$ (black) for the same values of $\beta$. The escape time function is similar to the exit function, as shown in Fig. 4(b); the longest escape times are located close to the the boundary of the exit regions, i.e., the mentioned stable manifold, because these trajectories spend long transient times before escaping. In this manner, the structure of singularities is again associated to the stable manifold, equally that the exit function. This is an evidence that the fractality of the escape time function must be an observer-independent feature, since the exit through which the particle escapes does not depend on the considered clock. Indeed, we observe that the escape proper time function exhibits a similar structure of singularities because of the approximated linear relation described by $\bar{\tau}_{e}$ (see Figs. 4(c) and 4(d)). Despite being almost identical structures, the dilation time phenomenon always makes $\tau_{e}(\theta_{0})<t_{e}(\theta_{0})$. Importantly, the time difference function $\delta t_{e}(\theta_{0})$ also preserves the fractal structure as shown in Figs. 4(e) and 4(f). This occurs because sensitivity to initial conditions is translated into sensitivity to time dilation phenomena. The longer the time the particle spends in the well, the more travels from the center to the potential barriers and back. If we think of each of these travels as an example of a twin paradox journey, we get an increasing time dilation for particles that spend more time in the well. Since these times are sensitive to modifications in the initial conditions, so are time dilation effects. We could then introduce what might be called the triplet paradox. In this case an additional third sibling leaves the planet and comes back to the starting point having a different age than their two other siblings, because of the sensitivity to initial conditions. This phenomenon in particular illustrates how chaotic dynamics affects typical relativistic phenomena. ## IV Invariant fractal dimension and persistence of transient chaos The chaotic saddle and the stable manifold are self-similar fractal sets when the underlying dynamics is hyperbolic ott1993 . This fact is reflected in the peaks structure of the escape time functions, which is present at any scale of initial conditions. In this sense, the escape time functions share with the Cantor set some properties with regard to their singularities, and therefore to their fractal dimensions. It is possible to study the fractal dimensions of the escape time functions in terms of a Cantor-like set lau1991 ; seoane2007 . In this manner, we can build a Cantor-like set to schematically represent the escape of particles launched from different initial conditions $\theta_{0}$. We consider that a certain fraction $\eta_{t}$ of particles escapes from the scattering region when a minimal characteristic time $t_{0}$ has elapsed. If these particles were launched from initial conditions centered in the original interval, two identical segments are created; the trajectories that began in those segments do not escape at least by a time $t_{0}$. Similarly, a same fraction of particles $\eta_{t}$ from the two surviving segments escapes by a time $2t_{0}$. If we continue this iterative procedure for $3t_{0}$, $4t_{0}$ and so on, we obtain a Cantor-like set of Lebesgue measure zero with associated fractal dimension $d_{t}$ that can be computed as $d_{t}=\frac{\ln 2}{\ln 2-\ln\left(1-\eta_{t}\right)}.$ (10) Similarly, if the escape times are measured by a non-inertial reference frame comoving with a particle, a fraction of particles $\eta_{\tau}$ escapes every time $\tau_{0}$, and therefore the associated fractal dimension can be defined as $d_{\tau}$. The behavior is governed by Poisson statistics in the hyperbolic regime. Therefore, the average number of particles that escape follow an exponential decay law. More specifically, the number of particles remaining in the scattering region according to an inertial reference frame at rest in the potential is given by $N(t)=N_{0}e^{-\sigma t}.$ (11) We note that this decay is homogeneous in an inertial reference frame, whereas according to an observer describing the decay in a non-inertial reference frame comoving with a particle, we get the decay law $\tilde{N}(\tau)\equiv N(t(\tau))=N_{0}e^{-\sigma\int_{0}^{\tau}\gamma(t(\tau^{\prime}))d\tau^{\prime}},$ (12) where we have substituted the equality $t=\int_{0}^{\tau}\gamma(t(\tau^{\prime}))d\tau^{\prime}$ from solving the Eq. (6). In other words, for an accelerated observer the decay is still Poissonian, but inhomogeneous. Nevertheless, if we disregard the fluctuations in the Lorentz factor, an homogeneous statistics can be nicely approximated once again, by defining the average constant rate $\bar{\sigma}_{\tau}\equiv\sigma\bar{\gamma}$. We recall that $\gamma(t)$ is the Lorentz factor along the trajectory of a certain particle, and therefore $\tilde{N}(\tau)$ here describes the number of particles remaining in the scattering region according to the accelerated frame co-moving with such a particle. This particle must be sufficiently close to the chaotic saddle in order to remain trapped in the well a sufficiently long time so as to render useful statistics, by counting a high number of escaping test bodies. Now we calculate, without loss of generality, the fraction of particles that escape during an iteration according to this reference frame as $\eta_{\tau}=\frac{\tilde{N}(\tau_{0})-\tilde{N}(\tau_{0}^{\prime})}{\tilde{N}(\tau_{0})}=\frac{N(t_{0})-N(2t_{0})}{N(t_{0})}=\eta_{t},$ (13) where $\tau_{0}^{\prime}$ is the proper time observed by the accelerated body when the clocks at rest in the potential mark $2t_{0}$. In this manner, we obtain that the fraction of escaping particles is invariant under reference frame transformations, because there exists an unequivocal relation between the times $t$ and $\tau$ given by $\gamma(t)$. From this result we derive that the fractal dimension of the Cantor-like set associated with the escape times function is invariant under coordinate transformations, $d_{t}=d_{\tau}$. This equality holds for every particle clock evolving in the well, as long as it stays long enough. On the other hand, this result is in consonance with the Cantor-like set nature, because its fractal dimension does not depend on how much time an iteration lasts. In order to compute the fractal dimensions associated with these scattering functions, we make use of the uncertainty dimension algorithm lau1991 ; grebogi1983_2 and the shooting method previously described. We launch a particle from the potential minimum with a random shooting angle $\theta_{0}$ in the interval $[\pi/2,5\pi/6]$ and measure the escape times $t_{e}(\theta_{0})$ and $\tau_{e}(\theta_{0})$, and the exit $e(\theta_{0})$ through it escapes. Then, we carry out again the same procedure from a slightly different shooting angle $\theta_{0}+\epsilon$, where $\epsilon$ can be considered a small perturbation, and calculate the quantities $t_{e}(\theta_{0}+\epsilon)$, $\tau_{e}(\theta_{0}+\epsilon)$ and $e(\theta_{0}+\epsilon)$. We then say that an initial condition $\theta_{0}$ is uncertain in measuring, e.g., the escape time $t_{e}$, if the difference between the escape times, $|t_{e}(\theta_{0})-t_{e}(\theta_{0}+\epsilon)|$, is higher than a given time. This time is usually associated with the integration step $h$ of the numerical method, which is the resolution of an inertial clock. Conveniently, we set this criterion of uncertain initial conditions as $3h/2$, i.e., the half between the step and two times the step of the integrator, for any clock. The reason for it is that the time differences according to a particle clock are the result of a computation by means of Eq. (7). Therefore, an initial condition $\theta_{0}$ is uncertain in measuring the escape time $t_{e}$ if $\Delta t_{e}(\theta_{0})=|t_{e}(\theta_{0})-t_{e}(\theta_{0}+\epsilon)|>3h/2.$ (14) Similarly, an initial condition $\theta_{0}$ is uncertain in measuring the escape time $\tau_{e}$ if $\Delta\tau_{e}(\theta_{0})=|\tau_{e}(\theta_{0})-\tau_{e}(\theta_{0}+\epsilon)|>3h/2.$ (15) Finally, an initial condition is uncertain with respect to the exit through which the particle escapes if $e(\theta_{0})\neq e(\theta_{0}+\epsilon)$. We generally expect that the time differences holds $\Delta\tau_{e}(\theta_{0})<\Delta t_{e}(\theta_{0})$, since we have defined previously that $\bar{\tau}_{e}\equiv t_{e}/\bar{\gamma}$. Thus, given the same criterion $3h/2$ in both clocks, there will be some uncertain initial conditions $\theta_{0}$ in the inertial clock $\left(\Delta t_{e}(\theta_{0})>3h/2\right)$ that become certain in the particle clock $\left(\Delta\tau_{e}(\theta_{0})<3h/2\right)$. We show a scheme in Fig. 5(a) to clarify this physical effect on the escape times unpredictability. It is easy to see that this effect is caused by the limited resolution of the hypothetical clocks, and becomes more intense for high values of $\beta$ because it is proportional to the Lorentz factor. Figure 5: (a) A scheme to visualize the physical effect of a reference frame modification on the unpredictability of the escape times, where $h=0.005$. (b) Fractal dimensions according to exits $d_{e}$ (green), escape time $d_{t}$ (blue) and escape proper time $d_{\tau}$ (red) with standard deviations computed by the uncertainty dimension algorithm versus twenty five equally spaced values of $\beta\in[0.5,0.98]$. The fraction of uncertain initial conditions behaves as $f(\epsilon)\sim\epsilon^{1-d},$ (16) where $d$ is the value of the fractal dimension, which enables us to quantify the unpredictability in foreseeing the particle final dynamical state. In particular, $d=0$ ($d=1$) implies minimum (maximum) unpredictability of the system lau1991 . All the cases in between, $0<d<1$, imply also unpredictability, and the closer to the unity the value of the fractal dimension is, the more unpredictable the system is. According to our scattering functions, it is expected that the values of their fractal dimensions decrease as the value of $\beta$ increase, since these functions become smoother. Taking decimal logarithms in Eq. (16), we obtain $\log_{10}\frac{f(\epsilon)}{\epsilon}\sim-d\log_{10}\epsilon.$ (17) This formula allows us to compute the fractal dimension of the scattering functions from the slope of the linear relation, which obeys a representation $\log_{10}f(\epsilon)/\epsilon$ versus $\log_{10}\epsilon$. We use an adequate range of angular perturbations according to our shooting method and the established criterion of uncertain initial conditions, concretely, $\log_{10}\epsilon\in[-6,-1]$. The computed fractal dimensions always hold $d_{e}<d_{t},d_{\tau}$ as shown in Fig. 5(b). This occurs because it is generally more predictable to determine the exit through which the particle escapes than exactly its escape time when the clocks resolution is small. Therefore, there is a greater number of uncertain conditions concerning escape times than in relation to exits. The former ones are located outside and over the stable manifold, whereas the uncertain conditions regarding exits can only be located on the stable manifold by definition. We obtain computationally $d_{t}\approx d_{\tau}$ for almost every value of $\beta$. Nonetheless, the physical effect explained above causes a small difference between the computed fractal dimensions regarding escape times, implying $d_{\tau}<d_{t}$ in a very energetic regime. From a mathematical point of view, if we consider a infinitely small clock resolution, i.e., $h\to 0$, uncertain initial conditions in any clock will be only the ones whose associated escape time differences are also infinitely small. Such uncertain conditions will be located on the stable manifold. In that case, the geometric and observer-independent nature of the fractality caused by the chaotic saddle is reflected into the values of the fractal dimensions. It is expected that in this limit the equality $d_{e}=d_{t}=d_{\tau}$ holds. This equality extends the very important statement that relativistic _chaos_ is coordinate invariant to _transient chaos_ as well. The result provided in motter2003 showing that the signs of the Lyapunov exponents of a chaotic dynamical system are invariant under coordinate transformations can be perfectly extended to transient chaotic dynamics. For this purpose, it is only required to consider a chaotic trajectory on the chaotic saddle, which meets the necessary four conditions described in motter2003 . Since the sign of the Lyapunov exponents of a trajectory on the chaotic saddle are also invariant, it is therefore evident that the existence of transient chaotic dynamics can not be avoided by considering suitable changes of the reference frame. We believe that this analytical result is at the basis of the results arising from all the numerical explorations performed in the previous sections. ## V Conclusions Despite the fact that the Hénon-Heiles system has been widely studied as a paradigmatic open Hamiltonian system, we have added a convenient definition of its scattering region. In this manner, the scattering region can be defined as the part of the physical space where the particle dynamics is bounded, and therefore a particle escapes when its kinetic energy is greater than the kinetic energy value at the potential minimum. Since relativistic chaos has been demonstrated as coordinate invariant, we have been focused on the special relativistic version of the Hénon-Heiles system to extend this occurrence to transient chaos. We have then analyzed the Lorentz factor effects on the system dynamics, concretely, how the time dilation phenomenon affects the scattering function structure. The exit and the escape time functions exhibit a fractal structure of singularities as a consequence of the presence of the chaotic saddle. Since the origin of the escape time singularities is geometric, the fractality of the escape time function must be independent of the observer. We conclude that the time dilation phenomenon does not affect the typical structure of the singularities of the escape times, and interestingly this phenomenon occurs chaotically. The escape time function as measured in any clock is closely related to a Cantor-like set of Lebesgue measure zero, since it is a self-similar set in the hyperbolic regime. This feature allows us to demonstrate that the fractal dimension of the escape time function is relativistic invariant. The key point of the demonstration is that, knowing the evolution of the Lorentz factor, there exists an unequivocal relation between the transformed times. In order to verify this result computationally, we have used the uncertainty dimension algorithm. Furthermore, we have pointed out that the system is more likely to be predictable in a reference frame comoving with the particle if a limited clock resolution is considered, even though from a mathematical point of view the predictability of the system is independent of the reference frame. The main conclusion of the present work is that transient chaos is coordinate invariant from a theoretical point of view. This statement extends the universality of occurrence of chaos and fractals under coordinate transformations to the realm of transient chaotic phenomena as well. ## ACKNOWLEDGMENTS We acknowledge interesting discussions with Prof. Hans C. Ohanian. This work was supported by the Spanish State Research Agency (AEI) and the European Regional Development Fund (ERDF) under Project No. FIS2016-76883-P. ## References * (1) J. M. Seoane and M. A. F. Sanjuán, Rep. Prog. 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2003.05288

arxiv-papers

11footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS> # The Brownian Motion in an Ideal Quantum Qas Chi-Chun Zhou Ping Zhang and Wu-Sheng Dai ###### Abstract A Brownian particle in an ideal quantum gas is considered. The mean square displacement (MSD) is derived. The Bose-Einstein or Fermi-Dirac distribution, other than the Maxwell-Boltzmann distribution, provides a different stochastic force compared with the classical Brownian motion. The MSD, which depends on the thermal wavelength and the density of medium particles, reflects the quantum effect on the Brownian particle explicitly. The result shows that the MSD in an ideal Bose gas is shorter than that in a Fermi gas. The behavior of the quantum Brownian particle recovers the classical Brownian particle as the temperature raises. At low temperatures, the quantum effect becomes obvious. For example, there is a random motion of the Brownian particle due to the fermionic exchange interaction even the temperature is near the absolute zero. ## 1 Introduction The Brownian motion is first observed by Robert Brown in 1827 and then explained by Einstein (1905), Smoluchowski (1905), and Langevin (1908) in the early 20th century [1]. The early theory of the Brownian motion not only provides an evidence for the atomistic hypothesis of matter [2], but also builds a bridge between the microscopic dynamics and the macroscopic observable phenomena [2]. The classical understanding of the Brownian motion is quite well established. However, there is an assumption in the early theory of the Brownian motion that the medium particle obeys the Maxwell-Boltzmann distribution. The behavior of a Brownian particle in an ideal quantum gas draws some attentions because to study the motion of a Brownian particle in an quantum system is now within reach of experimental tests. For example, an electron in a black body radiation [3]. In such systems, the quantum exchange interaction, which always leads to real difficulty in mechanics and statistical mechanics [4, 5, 6, 7], exists and causes the medium particle obeying the Bose-Einstein or Fermi-Dirac distribution. It is difficult to make exact or even detailed dynamical calculations [8, 1], since a different stochastic force is provided by the Bose-Einstein or Fermi-Dirac distribution. At high-temperature and low- density, the classical theory serves as good approximation. In this paper, we give an explicit expression of the mean square displacement (MSD) of a Brownian particle in an ideal quantum gas using, e.g., the virial expansion. Comparison with the classical Brownian motion, a correction for the MSD, which depends on the thermal wavelength and the density of medium particles, is deduced. The result shows that the MSD in an ideal Bose gas is shorter than that in a Fermi gas. The behavior of the quantum Brownian particle recovers the classical Brownian particle as the temperature raises. At low temperature, the quantum effect becomes obvious. For example, there is a random motion of the Brownian particle due to the fermionic exchange interaction even the temperature is near the absolute zero. The early studies of the Brownian motion inspired many prominent developments in various areas such as physics, mathematics, financial markets, and biology. In physics, exact solutions of Brownian particles in different cases, such as in a constant field of force [1] and in a harmonically potential field [1], are given. The Brownian motion with a time dependent diffusion coefficient is studied in Ref. [9]. The boundary problem of Brownian motions is studied in Refs. [10, 11]. The anomalous diffusion process, frequently described by the scaled Brownian motion, is studied in Refs. [12, 13]. The Kramers-Klein equation considers the Brownian particle that is in an general field of force [1]. The generalized Langevin equations and the master equation for the quantum Brownian motion are studied in Refs. [14, 15]. In mathematics, the rigorous interpretation of Brownian motions based on concepts of random walks, martingales, and stochastic processes is given [16, 8, 1]. In financial markets, the theory of the Brownian motion is used to describe the movement of the price of stocks and options [8, 1, 17, 18, 19, 20]. The application of the fractional Brownian motion, which is a generalized Brownian motion, in financial markets is studied in Refs. [21, 22, 23]. Moreover, the Brownian motion plays a central and fundamental role in studies of soft matter and biophysics [8], e.g., active Brownian motions, which can be used to describe the motion of swarms of animals in fluid, are studied in Refs. [24, 25, 26, 27, 28, 8]. Among many quantities, the MSD, which is measurable, describes the Brownian motion intuitively. There are studies focus on the MSD related problems. For examples, the relation between the MSD and the time interval can be generally written as $\left\langle x_{t}^{2}\right\rangle\sim t^{\alpha}$ [9]. One distinguishes the subdiffusion with $0<\alpha<1$ and the superdiffusion with $\alpha>1$ [29, 30, 31]. The relation between the MSD and the time interval of the so called ultraslow Brownian motions is $\left\langle x_{t}^{2}\right\rangle\sim\left(\ln t\right)^{\gamma}$ [32]. There are different approaches to build a quantum analog of the Brownian motion [33, 34, 35, 36]. For examples, the method of the path integral is used to study the quantum Brownian motion [35]. The approach of a quantum analog or quantum generalization of the Langevin equation and the master equation, e.g., the quantum master equation [3] and the quantum Langevin equation [37] is used to build a quantum Brownian motion. Among them, the method of quantum dynamical semigroups [38] is prominent. They point it out that the quantum equation should be casted into the Lindblad form [38, 39]. A completely positive master equation describing quantum dissipation for a Brownian particle is derived in Ref. [39]. This paper is organized as follows. In Sec. 2, for the sake of completeness, we derive the brownian motion from the perspective of the particle distribution in an ideal Boltzmann gas. In Sec. 3, we derive the MSD of a Brownian particle in an ideal quantum gas. High-temperature and low- temperature expansions are given. The $d$-dimensional case is considered. The conclusion and outlook are given in Sec. 4. Some details of the calculation is given in the Appendix. ## 2 A Brownian particle in an ideal classical gas: the Brownian motion In this section, we consider a Brownian particle in an ideal classical gas. For the sake of completeness, we derive, in detail, the Brownian motion from the perspective of the particle distribution. A brief review on the Langevin equation. For a Brownian particle with mass $M$, the dynamic equation is given by Paul Langevin [40] $\displaystyle dv$ $\displaystyle=-\frac{\gamma}{M}vdt+\frac{1}{M}F_{t}dt,$ (2.1) $\displaystyle dx$ $\displaystyle=vdt,$ (2.2) where $\gamma=6\pi\eta r$ with $\eta$ the viscous coefficient and $r$ the radius of the medium particles. $F_{t}$ is the stochastic force generated by numerous collisions of the medium particle. It is reasonable to make the assumption that $F_{t}$ is isotropic, i.e., $\left\langle F_{t}\right\rangle=0.$ (2.3) If the collision of the medium particle is uncorrelated; that is, for $t\neq s$, $F_{t}$ is independent of $F_{s}$: $\left\langle F_{s}F_{t}\right\rangle\propto\delta\left(s-t\right),$ (2.4) then, the solution of Eqs. (2.1) and (2.2) is [40] $\displaystyle v_{t}$ $\displaystyle=v_{0}\exp\left(-\frac{\gamma}{M}t\right)+\frac{1}{M}\int_{0}^{t}\exp\left[-\frac{\gamma}{M}\left(t-s\right)\right]F_{s}ds,$ (2.5) $\displaystyle x_{t}$ $\displaystyle=x_{0}+\frac{M}{\gamma}v_{0}\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]+\frac{1}{\gamma}\int_{0}^{t}\left\\{1-\exp\left[-\frac{\gamma}{M}\left(t-s\right)\right]\right\\}F_{s}ds,$ (2.6) where $x_{0}$ and $v_{0}$ are the initial position and velocity. The stochastic force determined by the Maxwell-Boltzmann distribution and the MSD. In an ideal classical gas, the gas particle obeys the Maxwell-Boltzmann distribution [41]. The number of particles possessing energy within $\varepsilon$ to $\varepsilon+d\varepsilon$, denoted by $\tilde{a}_{\varepsilon}$, is proportional to $e^{-\beta\varepsilon}$ [41], i.e., $\tilde{a}_{\varepsilon}=\omega_{\varepsilon}e^{-\beta\varepsilon}d\varepsilon,$ (2.7) where $\omega_{\varepsilon}$ is the degeneracy of the energy $\varepsilon$ and $\beta=\left(kT\right)^{-1}$ with $k$ the Boltzmann constant $T$ the temperature [41]. A collision of the medium particle with energy $\varepsilon$ gives a force of magnitude proportional to $\sqrt{2m\varepsilon}$, which is the momentum of the particle. We have $F=\rho\sqrt{2m\varepsilon},$ (2.8) where $\rho$ is a coefficient and $m$ is the mass of the medium particle. Thus, the probability of the Brownian particle subjected to a stochastic force with magnitude within $F$ to $F+dF$ is $P\left(F\right)dF=\sqrt{\frac{\beta}{2\pi m\rho^{2}}}\exp\left(-\frac{F^{2}\beta}{2m\rho^{2}}\right)dF.$ (2.9) In an ideal classical gas, there is no inter-particle interactions among medium particles, thus, for $t\neq s$, the force $F_{s}$ and $F_{t}$ are independent. Substituting Eq. (2.9) into Eq. (2.4) gives $\left\langle F_{s}F_{t}\right\rangle=\frac{m\rho^{2}}{\beta}\delta\left(s-t\right).$ (2.10) By using Eqs (2.6), (2.9), and (2.10), a direct calculation of the MSD gives $\displaystyle\left\langle\left(x_{t}-x_{0}\right)^{2}\right\rangle$ $\displaystyle=\frac{M^{2}}{\gamma^{2}}\left(v_{0}^{2}-\frac{1}{2m\gamma}\frac{m\rho^{2}}{\beta}\right)\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]^{2}$ $\displaystyle+\frac{1}{\gamma^{2}}\frac{m\rho^{2}}{\beta}\left\\{t-\frac{M}{\gamma}\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]\right\\}.$ (2.11) For a large-scale time, $t\gg 1$, Eq. (2.11) recovers $\left\langle x_{t}^{2}\right\rangle=\frac{kT}{3\pi\eta r}t,$ (2.12) where $\rho=\sqrt{12\pi\eta r/m}$ and $x_{0}$ is chosen to be $0$ without lose of generality. Eq. (2.12) is the famous Einstein’s long-time result of the MSD. The motion of a brownian particle in an ideal classical gas is the Brownian motion. ## 3 The MSD of a Brownian particle in an ideal quantum gas In this section, we give the MSD of a Brownian particle in an ideal quantum gas. High-temperature and low-temperature expansions explain the quantum effect intuitively. ### 3.1 The stochastic force determined by the Bose-Einstein or Fermi-Dirac distribution In an ideal quantum gas, the gas particle obeys Bose-Einstein or Fermi-Dirac distribution other than the Maxwell-Boltzmann distribution. The stochastic force is different from that in an ideal classical gas. In this section, we discuss the properties of the stochastic force in an ideal quantum gas. In an ideal quantum gas, the number of particles possessing energy within $\varepsilon$ to $\varepsilon+d\varepsilon$, denoted by $a_{\varepsilon}$, is $a_{\varepsilon}=\frac{\omega_{\varepsilon}}{\exp\left(\beta\varepsilon+\alpha\right)+g}d\varepsilon,$ (3.1) where $\alpha$ is defined by $z=e^{-\alpha}$ with $z$ the fugacity [41]. For Bose cases, $g=-1$, and for Fermi cases, $g=1$. Thus, the probability of the Brownian particle subjected to a stochastic force with magnitude within $F$ to $F+dF$ is $p\left(F\right)dF=\sqrt{\frac{\beta}{2\rho^{2}m\pi}}\frac{1}{h_{1/2}\left(z\right)}\frac{1}{\exp\left[\beta F^{2}/\left(2\rho^{2}m\right)+\alpha\right]+g}dF,$ (3.2) where we $h_{\nu}\left(x\right)$ equals Bose-Einstein integral $g_{\nu}\left(x\right)$ in Bose cases [41] and Fermi-Dirac integral $f_{\nu}\left(x\right)$ [41] in Fermi cases. In an ideal quantum gas, the stochastic force is also isotropic, that is, Eq. (2.3) holds. However, the collision, due to the overlapping of the wave package, can be correlated; that is, $\left\langle F_{s}F_{t}\right\rangle$ is no longer a delta function but a function of $s-t$ with a peak at $s=t$. However, as the ratio of the thermal wavelength and the average distance between the medium particles decreases, $\left\langle F_{s}F_{t}\right\rangle$, can be well approximated by a delta function: $\left\langle F_{s}F_{t}\right\rangle\sim\frac{m\rho^{2}}{\beta}\frac{h_{3/2}\left(z\right)}{h_{1/2}\left(z\right)}\delta\left(s-t\right),$ (3.3) for $n\lambda\ll 1$, where $\lambda=h/\sqrt{2\pi mkT}$ is the thermal wavelength and $n$ is the density of the medium particle. In this paper, we consider the case that the ratio of the thermal wavelength and the average distance between the medium particles is small. ### 3.2 The MSD For $n\lambda\ll 1$, by using Eqs. (3.2), (2.3), and (2.6), a direct calculation of MSD gives $\displaystyle\left\langle x_{t}^{2}\right\rangle$ $\displaystyle=\frac{M^{2}}{\gamma^{2}}\left\\{v_{0}^{2}-\frac{1}{2m\gamma}\frac{m\rho^{2}}{\beta}\frac{h_{3/2}\left(z\right)}{h_{1/2}\left(z\right)}\right\\}\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]^{2}$ $\displaystyle+\frac{1}{\gamma^{2}}\frac{m\rho^{2}}{\beta}\frac{h_{3/2}\left(z\right)}{h_{1/2}\left(z\right)}\left\\{t-\frac{M}{\gamma}\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]\right\\}.$ (3.4) For a large-scale time, $t\gg 1$, Eq. (3.4) recovers $\left\langle x_{t}^{2}\right\rangle=\frac{kT}{3\pi\eta r}t\frac{h_{3/2}\left(z\right)}{h_{1/2}\left(z\right)},$ (3.5) where $h_{3/2}\left(z\right)/h_{1/2}\left(z\right)$ is a correction for the MSD due to the Bose-Einstein or Fermi-Dirac distribution, a result of the quantum exchange interaction among gases particles. ### 3.3 High-temperature and low-temperature expansions In order to compare with Eq. (2.12) intuitively, we give high-temperature and low-temperature expansions of Eq. (3.5) by using the state equation of ideal Bose or Fermi gases [40, 41] $\displaystyle p$ $\displaystyle=\frac{kT}{\lambda}h_{3/2}\left(z\right),$ (3.6) $\displaystyle\frac{N}{V}$ $\displaystyle=\frac{1}{\lambda}h_{1/2}\left(z\right).$ (3.7) The high-temperature expansion. At high temperatures, the virial expansion of Eqs. (3.6) and (3.7) directly gives [40, 41] $\frac{pV}{N}\sim kT\left[1+ga_{1}\left(T\right)n\lambda+...\right],$ (3.8) where $a_{1}\left(T\right)$ is the first virial coefficient [40]. For a $1$-dimensional ideal Bose or Fermi gas, $a_{1}\left(T\right)=0.353553$ [41]. Substituting Eqs. (3.6) and (3.7) into Eq. (3.8) gives $\frac{h_{3/2}\left(z\right)}{h_{1/2}\left(z\right)}\sim\left[1+ga_{1}\left(T\right)n\lambda+...\right].$ (3.9) Substituting Eq. (3.9) into Eq. (3.5) gives the MSD at high temperatures: $\left\langle x_{t}^{2}\right\rangle=\frac{kT}{3\pi\eta r}t\left[1+ga_{1}\left(T\right)n\lambda+\ldots\right].$ (3.10) The result, Eq. (3.10), shows that the MSD in an ideal Bose gas is shorter than that in a Fermi gas. Since $\lambda$ decreases as $T$ raises, the behavior of the quantum Brownian particle returns the classical Brownian particle as the temperature raises. The low-temperature expansion for Fermi cases. At low temperatures, for Fermi cases, $g=-1$, $\frac{h_{3/2}\left(z\right)}{h_{1/2}\left(z\right)}=\frac{f_{3/2}\left(z\right)}{f_{1/2}\left(z\right)}.$ (3.11) The expansion of the Fermi-Dirac integral at large $z$ gives [41] $f_{\nu}\left(e^{\xi}\right)=\frac{\xi^{\nu}}{\Gamma\left(1+\nu\right)}\left\\{1+2\nu\sum_{j=1,3,5,...}\left[\left(\nu-1\right)....\left(\nu-j\right)\left(1-2^{-j}\right)\frac{\zeta\left(j+1\right)}{\xi^{j+1}}\right]\right\\}.$ (3.12) Keeping only the first two terms in Eq. (3.12) gives $f_{\nu}\left(z\right)=\frac{\left(\ln z\right)^{\nu}}{\Gamma\left(1+\nu\right)}+2\nu\left(\nu-1\right)\frac{1}{2}\frac{\zeta\left(2\right)}{\left(\ln z\right)^{2}}.$ (3.13) Substituting Eq. (3.13) into Eqs. (3.6) and (3.7) gives $\displaystyle p$ $\displaystyle=\frac{kT}{\lambda}\frac{\left(\ln z\right)^{3/2}}{\Gamma\left(5/2\right)}\left[1+\frac{3}{4}\frac{\zeta\left(2\right)}{\left(\ln z\right)^{2}}\right],$ (3.14) $\displaystyle\frac{N}{V}$ $\displaystyle=\frac{1}{\lambda}\frac{\left(\ln z\right)^{1/2}}{\Gamma\left(5/2\right)}\left[1-\frac{1}{4}\frac{\zeta\left(2\right)}{\left(\ln z\right)^{2}}\right].$ (3.15) The fugacity $z$ can be solved from Eq. (3.15): $\ln z\sim\frac{\epsilon_{f}}{kT}\left[1+\frac{1}{2}\zeta\left(2\right)\left(\frac{kT}{\epsilon_{f}}\right)^{2}\right],$ (3.16) where $\epsilon_{f}=\lambda^{2}kT\left[\frac{1}{2}\Gamma\left(\frac{3}{2}\right)n\right]^{2}$ is the Fermi energy [41]. By substituting Eq. (3.13) into Eq. (3.11) with fugacity $z$ given by Eq. (3.16), we have $\frac{f_{3/2}\left(z\right)}{f_{1/2}\left(z\right)}=\frac{\Gamma\left(3/2\right)}{\Gamma\left(5/2\right)}\frac{\epsilon_{f}}{kT}\left\\{1+\left[\frac{\zeta\left(2\right)}{2}+\zeta\left(2\right)\right]\left(\frac{kT}{\epsilon_{f}}\right)^{2}\right\\}.$ (3.17) Substituting Eq. (3.17) into Eq. (3.5) gives the MSD of Fermi cases at low temperatures: $\left\langle x_{t}^{2}\right\rangle\sim\frac{1}{3\pi\eta r}t\frac{\Gamma\left(3/2\right)}{\Gamma\left(5/2\right)}\epsilon_{f}\left[1+\frac{3}{2}\zeta\left(2\right)\left(\frac{kT}{\epsilon_{f}}\right)^{2}\right].$ (3.18) The first term of Eq. (3.18) is independent of the temperature $T$, which means that there is a random motion of the Brownian particle due to the fermionic exchange interaction even the temperature is near the absolute zero. It is a result of Pauli exclusion principle [41]. The low-temperature expansion for Bose cases. At low temperatures, for Bose cases, $g=1$, $\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}=\frac{g_{1+d/2}\left(z\right)}{g_{d/2}\left(z\right)}.$ (3.19) Expanding $g_{\nu}\left(z\right)$ around $z=1$ gives [41] $g_{\nu}\left(z\right)=\frac{\Gamma\left(1-\nu\right)}{\left(-\ln z\right)^{1-\nu}}+\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}}{j!}\zeta\left(\nu-j\right)\left(-\ln z\right)^{j}.$ (3.20) Substituting Eq. (3.20) into Eqs. (3.6) and (3.7) gives $\displaystyle p$ $\displaystyle=\frac{1}{\lambda^{d}}\Gamma\left(-\frac{1}{2}\right)\left(-\ln z\right)^{1/2}+\zeta\left(\frac{3}{2}\right)-\zeta\left(\frac{1}{2}\right)\left(-\ln z\right),$ (3.21) $\displaystyle\frac{N}{V}$ $\displaystyle=\frac{1}{\lambda^{d}}\frac{\Gamma\left(1/2\right)}{\left(-\ln z\right)^{1/2}}+\zeta\left(\frac{1}{2}\right)-\zeta\left(-\frac{1}{2}\right)\left(-\ln z\right).$ (3.22) The fugacity can be solved from Eq. (3.22): $\ln z=-\frac{\pi}{n^{2}\lambda^{2}}.$ (3.23) By substituting Eq. (3.20) into Eq. (3.19) with fugacity $z$ given by Eq. (3.23), we have $\frac{g_{3/2}\left(z\right)}{g_{1/2}\left(z\right)}=\frac{\zeta\left(3/2\right)}{\sqrt{n^{2}\lambda^{2}}}-\left(2+\frac{\zeta\left(3/2\right)\zeta\left(1/2\right)}{\pi}\right)\frac{\pi}{n^{2}\lambda^{2}}.$ (3.24) Substituting Eq. (3.24) into Eq. (3.5) gives the MSD of Bose cases at low temperatures: $\displaystyle\left\langle x_{t}^{2}\right\rangle$ $\displaystyle\sim\frac{kT}{3\pi\eta r}t\left[\zeta\left(\frac{3}{2}\right)\frac{1}{n\lambda}-\left(2\pi+\zeta\left(\frac{3}{2}\right)\zeta\left(\frac{1}{2}\right)\right)\frac{1}{n^{2}\lambda^{2}}\right]$ (3.25) $\displaystyle\sim\frac{1}{3\pi\eta r}\zeta\left(\frac{3}{2}\right)\frac{\sqrt{2\pi m}}{h}\frac{1}{n}\left(kT\right)^{3/2}t.$ (3.26) The MSD is proportional to $T^{3/2}$ and is reversely proportional to the density of particle, which is also different from that of the Brownian motion. ### 3.4 The $d$-dimensional case In this section, a similar procedure gives the MSD of a Brownian particle in a $d$-dimensional space. For the sake of clarity, we list the result. The detail of the calculation can be found in the Appendix. The MSD. The MSD for a Brownian particle in an ideal quantum gas in a $d$-dimensional space is $\left\langle\mathbf{x}_{t}^{2}\right\rangle=\frac{kTd}{3\pi\eta r}t\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}.$ (3.27) The high-temperature expansion. At high temperatures, the MSD, Eq.(3.27), becomes $\left\langle\mathbf{x}_{t}^{2}\right\rangle=\frac{kTd}{3\pi\eta r}t\left[1+ga_{1}\left(T\right)n\lambda^{d}+\ldots\right],$ (3.28) where $a_{1}\left(T\right)=\frac{1}{2^{1+d/2}}$ and is the first virial coefficient of ideal Bose or Fermi gases in a $d$-dimensional space [40, 41]. The low-temperature expansion for Fermi cases. At low temperatures, for Fermi cases, the MSD, Eq. (3.27), becomes $\left\langle\mathbf{x}_{t}^{2}\right\rangle\sim\frac{d}{3\pi\eta r}\frac{\Gamma\left(1+d/2\right)}{\Gamma\left(2+d/2\right)}t\epsilon_{f}\left[1+\left(\frac{d}{2}+1\right)\zeta\left(2\right)\left(\frac{kT}{\epsilon_{f}}\right)^{2}\right],$ (3.29) where $\epsilon_{f}$ is the Fermi energy in a $d$-dimensional space and $\epsilon_{f}=\lambda^{2}kT\left[\frac{1}{2}\Gamma\left(1+\frac{d}{2}\right)n\right]^{2/d}$ [40, 41]. The low-temperature expansion for Bose cases with $d=2$. The low-temperature expansion for a Bose gas at any dimension higher than $2$ is not given in this section, because the Bose-Einstein condensation (BEC) occurs. Here, we only consider the $2$-dimensional case. At low temperatures, for Bose cases, the MSD, Eq. (3.27), becomes $\left\langle\mathbf{x}_{t}^{2}\right\rangle=\frac{2kT}{3\pi\eta r}t\left[-\exp\left(-n\lambda^{2}\right)-\frac{\exp\left(-n\lambda^{2}\right)}{n\lambda^{2}}+\frac{\pi^{2}}{6n\lambda^{2}}\right].$ (3.30) ## 4 Conclusions and outlooks The difficulty in the calculation of the behavior of a Brownian particle in an ideal quantum gas directly comes from the stochastic force caused by the Bose- Einstein and Fermi-Dirac distribution other than the Maxwell-Boltzmann distribution. Comparison with the classical Brownian motion, on one hand, the distribution of the stochastic force is different; on the other hand, the collision, due to the overlapping of the wave package, could be correlated, that is, $\left\langle F_{s}F_{t}\right\rangle$ is no longer a delta function but a function of $s-t$ with a peak at $s=t$. Thus, it is difficult to make exact or even detailed dynamical calculations [8, 1]. In this paper, we consider the motion of a Brownian particle in an ideal quantum gas. We give an explicit expression of the MSD, which depends on the thermal wavelength and the density of medium particles. High-temperature and low-temperature expansions explain the quantum effect intuitively. For examples, the MSD in an ideal Bose gas is shorter than that in a Ferm gas. There is a random motion of the Brownian particle due to the fermionic exchange interaction even the temperature is near the absolute zero. The result in this work can be verified by experiment test. ## 5 Acknowledgments We are very indebted to Dr G Zeitrauman for his encouragement. This work is supported in part by NSF of China under Grant No. 11575125 and No. 11675119. ## 6 Appendix In the appendix, we give the detail of the calculation of Eqs (3.27), (3.28), (3.29), and (3.30). The detail of the calculate for the MSD, Eq. (3.27), of a Brownian particle in a $d$-dimensional space. The Langevin equation in $d$-dimensional is $\displaystyle M\frac{d\mathbf{v}}{dt}$ $\displaystyle=-\gamma\mathbf{v}+\mathbf{F}_{t},$ (6.1) $\displaystyle\frac{d\mathbf{x}}{dt}$ $\displaystyle=\mathbf{v.}$ (6.2) In a $d$-dimensional space, the stochastic force $\mathbf{F}_{t}$ is isotropic: $\left\langle\mathbf{F}\right\rangle=0.$ (6.3) For different time $t$ and $s$, $\mathbf{F}_{s}$ and $\mathbf{F}_{t}$ are almost independent when the ratio of the thermal wavelength and the average distance between the medium particles is small, that is, $\left\langle\mathbf{F}_{s}\cdot\mathbf{F}_{t}\right\rangle\sim\delta\left(s-t\right)$ (6.4) holds for $n\lambda^{d}\ll 1$. The solution of Eqs. (6.1) and (6.2) is $\displaystyle\mathbf{v}_{t}$ $\displaystyle=\mathbf{v}_{0}e^{-\frac{\gamma}{M}t}+\frac{1}{M}\int_{0}^{t}\exp\left[-\frac{\gamma}{M}\left(t-s\right)\right]\mathbf{F}_{s}ds,$ (6.5) $\displaystyle\mathbf{x}_{t}$ $\displaystyle=\mathbf{x}_{0}+\frac{M}{\gamma}v_{0}\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]+\frac{1}{\gamma}\int_{0}^{t}\left\\{1-\exp\left[-\frac{\gamma}{M}\left(t-s\right)\right]\right\\}\mathbf{F}_{s}ds.$ (6.6) In the $d$-dimensional case, the number of particle possessing momentum within $\mathbf{P}$ to $\mathbf{P+}d\mathbf{P}$, denoted by $a\left(\mathbf{P}\right)$, is [40, 41] $a\left(\mathbf{P}\right)=\frac{1}{\exp\left[\beta\left(p_{x^{1}}^{2}+p_{x^{2}}^{2}+...p_{x^{d}}^{2}\right)/\left(2m\right)+\alpha\right]+g}.$ (6.7) The force given by a collision of a particle with momentum $\mathbf{P}$ is proportional to $\mathbf{P}$, $\mathbf{F=}\rho\mathbf{P}$. Thus the probability of the stochastic force with magnitude within $\left|\mathbf{F}\right|$ to $\left|\mathbf{F+}d\mathbf{F}\right|$ is $P\left(\mathbf{F}\right)d\mathbf{F}=\left(\sqrt{\frac{\beta}{2\pi m\rho^{2}}}\right)^{d}\frac{1}{h_{d/2}\left(z\right)}\frac{1}{\exp\left[\beta\left|\mathbf{F}\right|^{2}/\left(2m\rho^{2}\right)+\alpha\right]+g}.$ (6.8) Substituting Eq. (6.8) into Eq. (6.4) gives $\left\langle\mathbf{F}_{s}\cdot\mathbf{F}_{t}\right\rangle\sim\frac{dm\rho^{2}}{\beta}\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}\delta\left(s-t\right).$ (6.9) By using Eqs. (6.6), (6.8), and (6.9), a direct calculation of MSD gives $\displaystyle\left\langle\mathbf{x}_{t}^{2}\right\rangle$ $\displaystyle=\frac{M^{2}}{\gamma^{2}}\left[\mathbf{v}_{0}^{2}-\frac{d}{2m\gamma}\frac{m\rho^{2}}{\beta}\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}\right]\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]^{2}$ $\displaystyle+\frac{1}{\gamma^{2}}\frac{dm\rho^{2}}{\beta}\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}\left[t-\frac{M}{\gamma}\left[1-\exp\left(-\frac{\gamma}{M}t\right)\right]\right].$ (6.10) where $\mathbf{x}_{0}$ is chosen to be the origin. For a large-scale time, $t\gg 1$, Eq. (6.10) recovers Eq. (3.27). The high-temperature expansion. For the $d$-dimensional case, the state equation of an ideal quantum gas is [40, 41] $\displaystyle p$ $\displaystyle=\frac{kT}{\lambda^{d}}h_{1+d/2}\left(z\right),$ (6.11) $\displaystyle\frac{N}{V}$ $\displaystyle=\frac{1}{\lambda^{d}}h_{d/2}\left(z\right).$ (6.12) The virial expansion [40, 41] directly gives $\frac{pV}{N}\sim kT\left[1+ga_{1}\left(T\right)n\lambda^{d}+...\right]$ (6.13) Substituting Eqs. (6.11) and (6.12) into Eq. (6.13) gives $\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}\sim\left[1+ga_{1}\left(T\right)n\lambda^{d}+...\right].$ (6.14) Substituting Eq. (6.14) into Eq. (3.27) gives Eq. (3.28). The low-temperature expansion for Fermi cases. For Fermi cases, $g=-1$, $\frac{h_{1+d/2}\left(z\right)}{h_{d/2}\left(z\right)}=\frac{f_{1+d/2}\left(z\right)}{f_{d/2}\left(z\right)}.$ (6.15) By the expansion of the Fermi-Dirac integral at large $z$, we have [41] $f_{\nu}\left(e^{\xi}\right)=\frac{\xi^{\nu}}{\Gamma\left(1+\nu\right)}\left\\{1+2\nu\sum_{j=1,3,5,...}\left[\left(\nu-1\right)....\left(\nu-j\right)\left(1-2^{-j}\right)\frac{\zeta\left(j+1\right)}{\xi^{j+1}}\right]\right\\}.$ (6.16) Keeping only the first two terms gives $f_{\nu}\left(z\right)=\frac{\left(\ln z\right)^{\nu}}{\Gamma\left(1+\nu\right)}+2\nu\left(\nu-1\right)\frac{1}{2}\frac{\zeta\left(2\right)}{\left(\ln z\right)^{2}}.$ (6.17) Substituting Eq. (6.17) into Eqs. (6.11) and (6.12) gives $\displaystyle\frac{p}{kT}$ $\displaystyle=\frac{1}{\lambda^{d}}\frac{\left(\ln z\right)^{1+d/2}}{\Gamma\left(2+d/2\right)}\left[1+\frac{d}{2}\left(1+\frac{d}{2}\right)\frac{\zeta\left(2\right)}{\left(\ln z\right)^{2}}\right],$ (6.18) $\displaystyle N$ $\displaystyle=\frac{\Omega}{\lambda^{d}}\frac{\left(\ln z\right)^{d/2}}{\Gamma\left(1+d/2\right)}\left[1+\frac{d}{2}\left(\frac{d}{2}-1\right)\frac{\zeta\left(2\right)}{\left(\ln z\right)^{2}}\right],$ (6.19) where $\Omega$ is the volume. The fugacity can be solved from Eq (6.19): $\ln z\sim\frac{\epsilon_{f}}{kT}\left[1-\zeta\left(2\right)\left(\frac{d}{2}-1\right)\left(\frac{kT}{\epsilon_{f}}\right)^{2}\right],$ (6.20) where $\epsilon_{f}=\lambda^{2}kT\left[\frac{1}{2}\Gamma\left(1+\frac{d}{2}\right)n\right]^{2/d}$ is the Fermi energy. By substituting Eq. (6.17) into Eq. (6.15) with fugacity $z$ given by Eq. (6.20), we have $\frac{f_{1+d/2}\left(z\right)}{f_{d/2}\left(z\right)}=\frac{\Gamma\left(1+\frac{d}{2}\right)}{\Gamma\left(2+\frac{d}{2}\right)}\frac{\epsilon_{f}}{kT}\left\\{1+\left[\frac{d\zeta\left(2\right)}{2}+\zeta\left(2\right)\right]\left(\frac{kT}{\epsilon_{f}}\right)^{2}\right\\}.$ (6.21) Substituting Eq. (6.21) into Eq. (3.27) gives Eq. (3.29). The low-temperature expansion for Bose cases in the $2$-dimensional space. For Bose cases, $g=1$, $\frac{h_{2}\left(z\right)}{h_{1}\left(z\right)}=\frac{g_{2}\left(z\right)}{g_{1}\left(z\right)},$ (6.22) where $d=2$. For $d=2$, $g_{1}\left(z\right)=-\ln\left(1-z\right).$ (6.23) Substituting Eq. (6.23) into Eq. (6.12) gives $\frac{N}{V}=-\frac{1}{\lambda^{2}}\ln\left(1-z\right).$ (6.24) Then, the fugacity can be solved from Eq (6.24): $z=1-\exp\left(-n\lambda^{2}\right).$ (6.25) Expanding $g_{2}\left(z\right)$ around $z=1$ gives $\displaystyle g_{2}\left(z\right)$ $\displaystyle=\frac{\pi^{2}}{6}-\left(1-z\right)-\frac{\left(1-z\right)^{2}}{2^{2}}-\frac{\left(1-z\right)^{3}}{3^{2}}-\ldots$ $\displaystyle+\left(1-z\right)\ln\left(1-z\right)+\frac{\left(1-z\right)^{2}}{2}\ln\left(1-z\right)+\frac{\left(1-z\right)^{3}}{3}\ln\left(1-z\right)+\ldots$ (6.26) Substituting Eqs. (6.26) and (6.23) with fugacity given in Eq. (6.25) into Eq. (6.22) gives $\frac{g_{2}\left(z\right)}{g_{1}\left(z\right)}=-\exp\left(-n\lambda^{2}\right)-\frac{\exp\left(-n\lambda^{2}\right)}{n\lambda^{2}}+\frac{\pi^{2}}{6n\lambda^{2}},$ (6.27) Substituting Eq. (6.27) into Eq. (3.27) gives Eq. (3.30). ## Acknowledgments We are very indebted to Dr G. Zeitrauman for his encouragement. 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2024-09-04T02:54:59.199255

2003.05336

arxiv-papers

# On Tracking Java Methods with Git Mechanisms Yoshiki Higo<EMAIL_ADDRESS>Shinpei Hayashi<EMAIL_ADDRESS>Shinji Kusumoto<EMAIL_ADDRESS>Graduate School of Information Science and Technology, Osaka University, Yamadaoka 1–5, Suita, Osaka 565–0871, Japan School of Computing, Tokyo Institute of Technology, Ookayama 2–12–1–W8–71, Ookayama, Meguro-ku, Tokyo 152–8550, Japan ###### Abstract Method-level historical information is useful in various research on mining software repositories such as fault-prone module detection or evolutionary coupling identification. An existing technique named Historage converts a Git repository of a Java project to a finer-grained one. In a finer-grained repository, each Java method exists as a single file. Treating Java methods as files has an advantage, which is that Java methods can be tracked with Git mechanisms. The biggest benefit of tracking methods with Git mechanisms is that it can easily connect with any other tools and techniques build on Git infrastructure. However, Historage’s tracking has an issue of accuracy, especially on small methods. More concretely, in the case that a small method is renamed or moved to another class, Historage has a limited capability to track the method. In this paper, we propose a new technique, FinerGit, to improve the trackability of Java methods with Git mechanisms. We implement FinerGit as a system and apply it to 182 open source software projects, which include 1,768K methods in total. The experimental results show that our tool has a higher capability of tracking methods in the case that methods are renamed or moved to other classes. ###### keywords: Mining software repositories , Source code analysis , Tracking Java methods ††journal: Journal of Systems and Software ## 1 Introduction One feature of version control systems is the ability to know file-level change information. Thus, it is easy to identify which files were changed in given commits or counting changes for files in a given repository. However, many approaches in mining software repositories (in short, MSR) require information on finer-grained units such as Java methods or C functions. If we want to count changes for Java methods, we need to parse source files to identify method positions and then we need to match method positions with changed code positions to identify which methods were changed. To conduct finer-grained analyses, developers have to implement code/scripts. Besides, incorrect analysis results will be obtained if the implemented code/scripts include bugs. Hata et al. proposed a technique, Historage, which enables Java methods to be tracked with Git mechanisms [1]. Historage takes a Git repository of a Java project as its input, and it outputs another Git repository in which each method gets extracted as a file. Treating Java methods as files realizes that developers/practitioners can obtain method-level historical information only by executing Git commands such as git-log. (a) Git repository. (b) Historage repository. Figure 1: Differences between Git and Historage repositories. Figure 1 shows a simple model of Git and Historage repositories. In the Git repository, file Person.java is managed. We can see that Person.java was changed in two commits c100 and c101. Information for the changes on Person.java can be retrieved by executing git-log. However, if we want to know which methods were changed in the two commits, we have to parse Person.java to obtain the positions of the methods and then we have to match method positions with the positions of the changed code in the two commits. On the other hand, in the Historage repository, each method exists as a file. Thus, just executing git-log is sufficient to know in which commits the two methods were changed. The command identifies that getLength() in Person.java was changed in commit c100 and setLength(int) was changed in c101. However, Historage has a limited capability of tracking methods in the case that methods are renamed or moved to other classes. We explain the issue with Figure 2, which shows refactorings on file Person.java in Figure 1. The refactorings include the following four changes. LABEL:Rename_Class: Person $\rightarrow$ Engineer LABEL:Rename_Field: length $\rightarrow$ height LABEL:Rename_Method (Getter): getLength $\rightarrow$ getHeight LABEL:Rename_Method (Setter): setLength $\rightarrow$ setHeight (a) Git. (b) Historage. Figure 2: Trackability differences between Git and Historage repositories. In the case of the changes in Figure 2LABEL:sub@fig:GitTrackingModel, the Git rename detection function can identify that file Person.java was renamed to Engineer.java because the two files sufficiently share the identical lines. On the other hand, in the Historage repository, files of Java methods get much smaller than their original file as shown in Figure 2LABEL:sub@fig:HistorageTrackingModel. Thus, the ratio of the changed lines against all the lines gets higher, which makes the Git function not work well. Hata et al. addressed that changing the threshold for the Git rename function is a way to realize a better method tracking [1]. They recommend using 30% instead of 60%, which is a default value of Git. However, we consider that only using a lower threshold may produce incorrect tracking results. For example, if we use 30% instead of 60%, the Git rename function can identify that Engineer/getHeight() is a renamed file of Person/getLength(). However, at the same time, Person/getLength() can be tracked wrongly from Engineer/setHeight(int) because their similarity is 1/3, which is higher than 30%. Tracking method accurately is essential. If not, MSR approaches using historical data gets affected. Hora et al. reported that between 10 and 21% of changes at the method level in 15 large Java systems were untracked in the context of refactoring detection [2]. They also found that 37% of the top-25% most changed entities (classes and methods) have at least one untracked change in their histories. By assessing two MSR approaches, they detected that their results could be improved when untracked changes were resolved. In this paper, we propose a new technique named FinerGit to improve the trackability of Java methods. Several research areas benefit from FinerGit. FinerGit is useful for studies in the context of assessing bug introducing changes [3, 4, 5] or detecting code authorship [6, 7]. More broadly, any study that compares two versions of methods can be benefited, for example, API evolution detection [8, 9], code warning prioritization [10, 11], and many other. Figure 3: Tracking files with our technique. The main contributions of this paper are the followings. * 1. We raise an issue on method trackability in Historage. * 2. We propose a new technique, FinerGit, to increase method trackability with Git mechanisms. * 3. We provide a software tool based on FinerGit. The tool is open to the public on GitHub 111https://github.com/kusumotolab/FinerGit. The tool is sufficiently fast even for huge repositories, as shown in the evaluation. * 4. We show the experimental results on the tracking results of 182 open source software (OSS) projects. These experiments have two aspects. First, they clarify the advantage of FinerGit with an existing technique, Historage. Second, they are the first attempt of large-scale empirical studies for the tracking results of method-level repositories. The remainder of this paper is organized as follows: in Section 2, we explain our research goal and our key idea to achieve the goal; in Section 3, we propose our new technique named FinerGit on the top of the key idea; Section 4 describes an implementation of FinerGit; then, we report the evaluation results with the implementation in Section 5; we also describe threats to validity on the experiments in Section 7; related work is introduced in Section 8; lastly, we conclude this paper in Section 9. ## 2 Basic Approach At present, there are various techniques of tracking source code entities [12, 13, 14, 15]. Those techniques utilize many types of information such as text similarities, data dependencies, and call dependencies. On the other hand, in this research, we utilize only line-based text similarity to track Java methods. The reason is that our research goal is realizing accurate method tracking with Git mechanisms. The biggest benefit of tracking methods with Git mechanisms is that it can easily connect with any other tools and techniques built on Git infrastructure. For example, the following analyses can be easily performed by using the basic commands provided by Git. * 1. We can know how many times each method was changed in the past by git-log. * 2. We can know how many developers changed a specified method in the past by collecting author names of the commits in which the method was changed. Git performs file comparisons by using hash values. If the size of a line is equal to or shorter than 64 bytes, a hash value is calculated from the entire line. If the size of a line is longer than 64 bytes, the line is chunked by 64 bytes, and a hash value is calculated from each chunk. Thus, even if just a single token in a given line (which is shorter than 64 bytes) has been changed, Git regards that the entire line has been changed. Method-level tracking with Git mechanisms can be realized by treating each method as a single file (a _method file_ hereafter). Based on this idea, Hata et al. developed technique named Historage [16]. However, as explained with Figure 2, simple extraction as files are inadequate for small methods. In this research, we propose a file format that each line includes only a single token. By using this format, each hash is calculated from a single token. In Figure 2LABEL:sub@fig:HistorageTrackingModel, Git regards that the two red lines of methods getLength and setLength were changed, though only the method name and the field name were changed in methods. As a result, the ratio of unchanged lines becomes 1/3, which is less than 60% of Git’s default value so that the method is not tracked with Git mechanisms. We state two restrictions for the techniques to improve method tracking with Git mechanisms as follows. * 1. Since the file tracking mechanism in Git is based on line-based text similarity, the characteristics of methods to be used in comparison must be represented as a sequence of text lines. Based on this restriction, complex comparison techniques of file contents such as tf/idf are not applicable. * 2. Since the contents of method files are visible and are utilized by developers, they should follow a representation of source code in an understandable way by users. Users may apply git-diff command to a method file to see how a method was modified, and the obtained difference should represent the difference of method contents in this case. Based on this restriction, converting method contents to a sequence of computed numeric values used only for a comparison purpose is not suitable. Figure 3 shows how the changes in Figure 2LABEL:sub@fig:HistorageTrackingModel are treated in FinerGit. The file changing mechanism in this technique satisfies the above restrictions. The ratio of unchanged lines becomes 8/10 for getLength and 11/15 for setLength. Both values are higher than 60%, so that both methods are tracked with Git mechanisms. ## 3 Proposed Technique Herein, we explain our proposed technique named FinerGit to realize a better method tracking with Git mechanisms. FinerGit is designed on the top of the basic approach explained in Section 2. FinerGit consists of (1) naming convention and (2) two heuristics. ### 3.1 Naming Convention In FinerGit, a file name for a Java method includes the following information: * 1. a class name including the method, * 2. access modifiers of the method, * 3. a return type of the method, * 4. a name of the method, and * 5. a list of parameter types of the method. For example, the file name for method setLength in Figure 2 becomes as follows. `Person#public_void_setLength(int).mjava` Extension .mjava means that this is a method file and the file includes source code of a Java method. Including the above information in the file name reflects code changes around a given method as follows. * 1. If the name of the class including the given method is changed, the file name of the given method gets changed, but its contents are not changed. * 2. If another method in the class including the given method is changed, neither file name nor contents of the given method are changed. * 3. If the signature of the given method is changed, the file name of the given method gets changed and its contents are also slightly changed since the contents include the tokens of the method signature. * 4. If the contents of the given method are changed, the file name of the given method does not get changed while its contents get changed. We can track methods with Git mechanisms in any of the above cases if either of them occurs alone. However, if a signature of a method is changed and its contents are also changed broadly, it is difficult to track the method. (a) Historage. (b) w/o Heuristic-1. (c) w/ Heuristic-1. Figure 4: Tracking files w/o and w/ Heuristic-1. ### 3.2 Introducing Heuristics It is not difficult to imagine that Git tracks wrong methods with FinerGit because each line has only a single token and such lines will coincidentally match with many other lines. Thus, we introduce two heuristics to reduce such coincidental matches of unrelated lines. Heuristic-1: Classifying brackets, parentheses, and semicolons of termination characters in detail. Heuristic-2: Removing tokens existing in all methods from the targets of similarity calculation. #### 3.2.1 Heuristic-1 Some termination characters such as brackets, parentheses, and semicolons are omnipresent in Java source code. Such termination characters are used as a part of various program elements. For example, brackets (“{” and “}”) are used to initialize arrays in addition to code blocks such as if-statements and for- statements. Thus, if just a bracket is placed on a line, brackets of different roles are coincidentally matched with each other. Such accidental matchings make the similarity between deleted and added methods inappropriately higher. To prevent such accidental matchings, we classify termination characters in detail. More concretely, we add a token explanation to each line. Token explanations prevent accidental matchings of different-role characters from being matched. In this heuristic, semicolons, brackets, and parentheses are classified into 18, 21, and 20 categories, respectively. Figure 4 shows how Heuristic-1 affects method tracking. Figure 4LABEL:sub@fig:Heuristic1Historage is a method file that Historage outputs. The deleted method includes an if-statement for checking whether variable a is null or not. The added method includes a while-statement for adding variable b to variable total repeatedly. Those are different methods, which means a lower similarity between them is better. In the case of Historage, the last line of the if-statement coincidentally matches with the last line of the while- statement so that the similarity between them becomes 1/3 (=33%). In the case of FinerGit without Heuristic-1, the parentheses and the brackets of the if- statement coincidentally matches with ones of the while-statement. Moreover, the semicolon of the return-statement coincidentally matches with the one of the expression-statement. As a result, the similarity between them becomes 5/12 (=42%). If we introduce Heuristic-1 to this example, the parentheses, the brackets, and the semicolons get unmatched. Thus, the similarity between them becomes 0/12 (=0%). (a) w/o Heuristic-2. (b) w/ Heuristic-2. Figure 5: Tracking files w/ and w/o Heuristic-2. #### 3.2.2 Heuristic-2 The parentheses for parameters and the brackets for method bodies are omnipresent in compilable Java methods. The fact means that at least the four tokens always match between any Java methods. Thus, the similarity between non-related methods gets inappropriately higher. If methods include many tokens, the impact of the four tokens is negligible. However, if methods are small such as getters and setters, the impact of the four tokens become serious. Consequently, we decided not to put the four tokens into files for methods. By removing the four tokens, we prevent the similarity of two non- related methods from getting higher inappropriately. Figure 5 shows how Heuristic-2 affects tracking. This example shows a similarity calculation between getLength (before refactoring) and setHeight (after refactoring) in Figure 2. A lower similarity between the two methods is better because they are different methods. In the case that we calculate a similarity without Heuristic-2, the similarity becomes 5/10 (=50%). However, in the case that we adopt Heuristic-2, the similarity becomes 1/6 (=17%) because the four tokens are ignored. ## 4 Implementation We have implemented a tool based on FinerGit. Our tool is open to the public in GitHub, and anyone can use it freely. Our tool takes a Git repository of a Java project, and it outputs another Git repository where each Java method gets extracted as a file. In FinerGit repositories, method files have extension .mjava. By executing git-log command with option \--follow for .mjava files, we can get their histories. The name of a method file includes the information of the signature of the method and the class name including the method so that the file name occasionally becomes very long. Very long file names are not compatible with widely-used operating systems. For example, in the case of Windows 10, the absolute path of a file must not exceed 260 characters. If a file name violates the restriction, its file cannot be accessed with Windows’ file manager and some other problems occur. In the case of Linux and MacOS, a file name (not a file path) must not exceed 255 characters. For practical use in such widely-used operating systems, if a file name becomes longer than the restriction of operating systems, our tool cuts the file name in the middle and then it appends a hash value that is calculated from the entire file name. This manipulation can shorten the file name while keeping its identity. There are three types of comments in Java source code: line comments, block comments, and Javadoc comments. Line and block comments are removed from .mjava files while Javadoc comments are included in .mjava files as they are in .java files. This means that a Javadoc comment exists in the header part of .mjava file if its original method has it. Our tool also has a function to extract each field in Java source code as a single file. Files for fields have extension .fjava. A field declaration includes multiple tokens such as field name, field type, modifiers, initializations, and annotations. Thus, fields can be tracked as well as methods by placing a single token on a line. A file name for a Java field include the following information: * 1. a class including the field, * 2. access modifiers of the field, * 3. a type of the field, and * 4. a name of the field. For example, the file name for field length in Figure 2 becomes as follows. `Person#private_int_length.fjava` Including the above information in the file name reflects code changes around a given field as follows. * 1. If the name of class including the given field is changed, the file name of the given method gets changed, but its contents are not changed. * 2. If another method or field in the class including the given field is changed, neither file name nor the contents of the given method are changed. * 3. If the access modifiers, type, or name of the field is changed, the file name of the given field gets changed and its contents are also changed. * 4. If the annotations and/or initializations of the field are changed, the file name of the given field does not get changed while its contents get changed. In Historage repository, a file path of a method includes its signature information. Historage makes a directory for each Java class. Methods included in a class are placed in its corresponding directory. On the other hand, our technique places files of Java methods in the same directory of their original Java files. A reason why FinerGit does not make new directories for Java classes is that the conversion time of Historage is long and making a large number of directories in the conversion process is a factor of taking a long time. Both FinerGit and Historage make a large number of files because each Java method is extracted as a single file, but our technique does not make new directories for Java classes. In both FinerGit and Historage, file name collisions for extracted files do not occur as long as their source code is compilable. ## 5 Evaluation We evaluated FinerGit by comparing it with Historage [1]. We did not use the published version of Historage implementation222https://github.com/niyaton/kenja but we added Historage’s functionality to our tool. By using the same implementation for FinerGit and Historage, we can avoid different tracking results due to the differences in implementation details. For example, original Historage makes directories for each Java class while our Historage implementation outputs files of Java methods in the same directory as their original files. The file name convention of our Historage implementation is the same as FinerGit. Thus, in this way, we can evaluate how much method trackability with Git mechanisms gets improved by FinerGit. We selected 182 Java projects in GitHub as our evaluation targets. In the process of our target selection, we used Borges dataset [17]. This dataset includes 2,279 popular projects in GitHub. Firstly, we extracted 202 projects that are labeled as “Java projects”. Borges et al. classified the projects in the dataset into six categories: Application software, System software, Web libraries and frameworks, Non-web libraries and frameworks, Software tools, and Documentation. Secondly, we extracted 185 projects that are other than Documentation projects because they are repositories with documentation, tutorials, source code examples, etc. (e.g., java-design- patterns333https://github.com/iluwatar/java-design-patterns). Documentation projects are outside of the scope of this evaluation. Then, we cloned the 185 repositories to our local storage on March 4th 2019. Unfortunately, we found that three of the 185 projects did not include .java file. The three projects (google/iosched, afollestad/material-dialogs, and googlesamples/android- topeka) are Kotlin projects. Finally, we removed the three projects from the 185 projects. Figure 6: Project size. Figure 6 shows the distributions of the number of commits and LOC of the target projects. The two largest repositories in the targets are platform_frameworks_base444https://github.com/aosp- mirror/platform_frameworks_base and intellij- community555https://github.com/JetBrains/intellij-community. The two repositories include approximate 380K and 240K commits, and their latest revisions consist of about 3.7M and 5.0M LOC, respectively. We generated FinerGit repositories and Historage ones from the 182 target projects. Herein, FinerGit repositories have the file format of including a single token per line with the two heuristics while Historage repositories have the same line format as the original repositories. We have evaluated FinerGit from the five viewpoints: * 1. tracking accuracy, * 2. heuristics impacts, * 3. project-level tracking results, * 4. method-size-level tracking results, and * 5. execution time. Hereafter in this section, we report the results in detail. ### 5.1 Tracking Accuracy It is not realistic to manually check whether FinerGit generates correct tracking results for each method in the target projects. Thus, we make an oracle for a method for each target project with the following procedure. 1. 1. A method was randomly selected from each target project. In total, 182 methods were selected. 2. 2. Each of the methods in FinerGit repositories was tracked with the following command. `> git log --follow -U15 -M20% -C20% -p` ` -- `path/to/method`.mjava` With the above command, a specified file is tracked even if the file was renamed. If there is a file that has a 20% or more similarity, Git regards that file renaming or copying occurred. 3. 3. The tracking results were examined, and oracles of renaming and copying history were made by two of the authors independently. Each author spent several hours on this task. The two authors made different oracles for 34 out of the 182 methods. 4. 4. The two authors discussed the 34 methods so that they obtain consensus for them. After a two-hour discussion, they got consensus oracles for the 34 methods. With the above procedure, we obtained consensus oracles of tracking results for the 182 methods. Finally, we obtained the resulting oracle set consisting of 426 renaming/copying changes for the 182 methods in total. Next, we track the methods in FinerGit’s repositories and Historage’s ones with different thresholds. We used the following command to count how many times Git found renaming and copying with a specified threshold. `> git log --follow --oneline -M`t` -C`t` -p` ` -- `path/to/method`.mjava` ` | grep -e "^rename from\|^copy from"` ` | wc -l` In the above command, t is the threshold that Git regards given two files have a renaming or copying relationship. We tracked the target methods with 13 different thresholds (i.e., 20%, 25%, 30%, $\ldots$, 80%). If tracking results for a method include a higher number of renaming/copying than its oracle, we regard renaming/copying in the over-tracking part as false positives. If tracking results for a method include a lower number of renaming/copying than its oracle, we regard renaming/copying that are not detected as false negatives. We calculated precision, recall, and F-measure for each threshold by summing up the number of false positives and false negatives of all the methods. Figure 7: Precision, recall, and F-measure values. Figure 7 shows how precision, recall, and F-measure changes according to given thresholds. The graphs of Historage and FinerGit have the following features. * 1. Precision of Historage is very high. Historage has 93.01% of precision even in the case of threshold 20%. * 2. Recall of Historage is low. Historage has only 57.04% of recall in the case of threshold 20%. * 3. FinerGit has high precision in the case of high thresholds, but precision gets rapidly decreased for lower thresholds. * 4. FinerGit has higher recall than Historage for all the thresholds. The recall differences between FinerGit and Historage get bigger for lower thresholds. Historage has a low possibility to track wrong methods while it often misses renaming and copying. On the other hand, in FinerGit repositories, precision gets decreased for lower thresholds while recall improves much. The highest F-measure on FinerGit is 84.52% on threshold 50% while the highest F-measure on Historage is 70.72% and 70.23% on thresholds 20% and 25%, respectively. ### 5.2 Heuristics Impacts (a) Precision. (b) Recall. (c) F-measure. (d) Rename count. Figure 8: Precision, recall, F-measure, and rename count when heuristics 1 and 2 are on and/or off. To reveal how each heuristic impacts on method tracking, we measured precision, recall, and F-measure and we also counted found renames for the following four types of fine-grained repositories. The target methods are the same as Subsection 5.1. Herein, rename count means the sum of found renames for all the target methods in a type of repositories. H1 OFF, H2 OFF: neither heuristics are applied to. H1 ON, H2 OFF: only Heuristic-1 is applied to. H1 OFF, H2 ON: only Heuristic-2 is applied to. H1 ON, H2 ON: both heuristics are applied to. This is the same repository as what we used in Subsection 5.1. Figure 8 shows the results. Applying only Heuristic-1 makes it possible to find more renaming so that precision gets decreased while recall gets increased. On the other hand, applying only Heuristic-2 slightly shorten method tracking. As a result, precision gets increased while recall gets decreased. The reasons why applying Heuristic-1 and Heuristic-2 have opposite impacts on method tracking are as follows. * 1. Applying Heuristic-1 reduces similarities between methods. How much the similarities are decreased depends on the contents on methods. Thus, a different method can be tracked at a commit compared to the case that Heuristic-1 is not applied to. * 2. Applying Heuristic-2 reduces similarities between all methods. Unlike Heuristic-1, Heuristic-2 does not make a different method tracked. Thus, Heuristic-2 just shortens method tracking. Table 1 shows the maximum F-measure for each type of finer-grained repositories. In this table, the maximum F-measure is the greatest F-measure in all data. All types have almost the same maximum values. This table also shows the maximum recall when we track methods with over 95% precision. These results show that more method renames are found with keeping 95% precision by applying both heuristics. Table 1: Maximum F-measure and Maximum Recall Repository type | Max F-measure (thr.) | Max Recall (thr.) ---|---|--- H1 OFF, | H2 OFF | 82.63% (40%) | 58.45% (55%) H1 ON, | H2 OFF | 83.77% (55%) | 56.81% (65%) H1 OFF, | H2 ON | 83.26% (35%) | 60.09% (50%) H1 ON, | H2 ON | 84.52% (50%) | 68.78% (55%) ### 5.3 Project-Level Tracking Results (a) Ratio of different tracking results. (b) Average change counts. Figure 9: Project-level comparisons. (a) shows the ratio of methods whose tracking results are different between FinerGit and Historage for each project. (b) shows the average of change counts for all the methods for each project. In this evaluation, we measured the ratio of methods whose tracking results are different between the two tools for each project. We compare how much the number of detected renames is different from FinerGit and Historage under the same precision. As shown in the previous subsection, the two tools have different precision values for different thresholds. To realize a fair comparison, we decided to select different thresholds for FinerGit and Historage that satisfy the following condition: method tracking results with the thresholds have the same precision values and the precision values are as high as possible. Thus, we used threshold 55% for FinerGit and 25% for Historage. The precision of FinerGit on threshold 55% is 95.73%, and Historage on threshold 25% is 96.60%. Those precision values are almost the same and high enough. Figure 9 shows the comparison results. In Figure 9LABEL:sub@fig:ProjectLevelComparison:Ratio, the blue boxplot shows the ratio of methods for which FinerGit found more renames than Historage per project and the red boxplot shows the opposite one. FinerGit found more renames for 22.71% methods on average while the ratio of methods that Historage found more renames than FinerGit is only 5.26%. In Figure 9LABEL:sub@fig:ProjectLevelComparison:Count, the blue boxplot shows the average number of changes identified by FinerGit for all methods of each project. The red one shows the average number of changes identified by Historage. The median values of those boxplots are 3.67 and 2.86, respectively. These results mean that FinerGit can find more renames for all the methods on average. Next, we show that the tracking improvement by FinerGit is effective via the following two ways: * 1. considering the fact that some methods were never changed after their initial creation, and * 2. conducting statistical testing for the tracking results. #### 5.3.1 Considering Never-Changed Methods In software development, some methods are never changed after their initial creation. If the 182 target projects include many never-changed methods, it is quite natural that the comparison results between FinerGit and Historage are not so different from each other. Thus, we investigate how many never-changed methods are included in the projects. It is not realistic to manually collect real never-changed methods. In this experiment, we decided to regard methods that both FinerGit and Historage were not able to detect any changes as never- changed methods. Figure 10 shows the relationship between the ratio of never-changed methods and the ratio of methods for which FinerGit found more renames than Historage. The 25 percentile, the median, and the 75 percentile of never-changed methods are 6.88%, 15.27%, and 26.50%, respectively. The figure indicates that the more never-changed methods there are, the fewer methods FinerGit found more renames for. Figure 11 shows the same figures as Figure 9LABEL:sub@fig:ProjectLevelComparison:Ratio only for the projects that include 50% or more never-changed methods. As shown in Figure 11LABEL:sub@fig:ProjectLevel50on, the differences between FinerGit and Historage are small because the majority of their methods is never-changed. Figure 11LABEL:sub@fig:ProjectLevel50off shows the differences after we removed never-changed methods from the projects. We can see that the differences between the two tools get much larger. MSR approaches are naturally applied to methods that have change histories. Never-changed methods are exempt from MSR approaches. We also investigated how many methods only FinerGit or Historage found at least a change for. The former number is 97,629 and the latter one is 35,553. They are 5.52% and 2.01% of all methods, respectively. Finding changes for more methods means that various MSR approaches requiring past changes can be applied more broadly. Figure 10: Relationships between the ratio of methods for which FinerGit found more renames than Historage and the ratio of never-changed methods. (a) w/ never-changed methods. (b) w/o never-changed methods. Figure 11: The ratio of methods whose tracking results are different between FinerGit and Historage for projects where 50% or more methods are never- changed ones. #### 5.3.2 Conducting Statistical Testing We applied Paired Wilcoxson’s signed ranked test to the comparison results between FinerGit and Historage shown in Figure 9. The test showed that the comparison results include significant differences regarding both aspects of the ratio ($p$-value $<$ 0.001) and average change counts ($p$-value $<$ 0.001). We also applied Cliff’s Delta to the comparison results to see the effect size. The resulting values were computed as 0.712 for the ratio and 0.221 for the average change counts, which revealed a _large_ and a _small_ effect size of the improvement achieved by using FinerGit, respectively. Consequently, we can say that FinerGit significantly improves tracking Java methods compared to Historage. ### 5.4 Method-Size-Level Tracking Results We also conducted comparisons based on method size. In this comparison, we made several method groups based on their size. Then, we compared the tracking results for each group. Figure 12 shows the comparison results. We can see that there are 1,036K methods whose LOC is in the range between 1 and 5. Herein, the LOC was computed using the original format, not the single-token- per-line one. FinerGit generated longer tracking results for 26.21% of the 1,036K methods. Our research motivation was improving the trackability for small methods, but surprisingly FinerGit improved the trackability for methods of any size. This figure also shows the average rename counts that were found by FinerGit and Historage. We can see that FinerGit found more renames for methods of any size than Historage. Interestingly, more renames tend to be found for larger methods by both tools. Consequently, we conclude that the method tracking capability of FinerGit is higher than Historage. (a) Ratio of methods for which FinerGit or Historage found more renames than the other tool. (b) Average renames that were found by FinerGit or Historage. Figure 12: Comparison based on method size. ### 5.5 Execution Time Figure 13: Execution time of FinerGit. We measured the time that FinerGit reconstructed the repositories of the target projects on MacBook Pro666CPU: 2.7GHz quad-core Intel Core i7, memory size: 16 GBytes. Figure 13 shows the measurement results. This figure shows that FinerGit is scalable enough for large repositories. In the longest case, FinerGit took 4,209 seconds to reconstruct the repository of intellij- community, which includes more than 240K commits. Of course, this execution time can be shorter if a higher specification computer is used777We also measured execution time with our workstation whose CPU is 3.6GHz octet-core Intel Core i9 and memory size is 32 GBytes. The execution time was approximately 22% of MacBook Pro’s one.. Figure 13 includes the regression line for all the data. The regression line shows that FinerGit takes around 100 seconds to process each 10K commits for large repositories. ## 6 Comparisons with Other Techniques We also compared FinerGit with two other techniques, AURA and RefactoringMiner (RMiner). The first comparison target is AURA, which is a technique that takes two versions of Java source code and generates mappings of methods between them [15]. AURA performs call dependency and text similarity analyses to generate mappings. The second comparison target is RMiner, which is a technique that detects refactorings from commit history [18]. RMiner’s refactoring detection is based on an AST-based statement matching algorithm. RMiner defines different rules for different refactoring patterns. RMiner checks if matching results of two ASTs before and after changes in a given commit follow any of the rules. We conducted this comparison on the development history of JHotDraw between releases 5.2 and 5.3. This development history is one of the evaluation targets in AURA’s literature [15]. Releases 5.2 and 5.3 include 1,519 and 1,981 methods, respectively. There are 19 commits between releases 5.2 and 5.3. ### 6.1 AURA Table 2: Refactorings detected by RMiner Refactoring pattern | # of detected instances ---|--- LABEL:Change_Parameter_Type | 56 LABEL:Change_Return_Type | 10 LABEL:Move_Method | 3 LABEL:Rename_Method | 44 LABEL:Rename_Parameter | 45 Total | 158 We made FinerGit’s repository and tracked the 1,981 methods with 20% threshold with the command shown in Subsection 5.1. The tracking results of 185 methods included renaming and the total number of renaming was 241. Two of the authors independently examined the tracking results to make oracles. Each author spent several hours on this task. The two authors make different oracles for 18 out of the 185 methods. The authors had a discussion on the 18 methods to obtain consensus for them. After a one-hour discussion, they got consensus oracles for the 18 methods. Our consensus oracle includes 161 renamings on 124 methods. Next, we tracked the 1,981 methods with 50% threshold, which is the best F-measure threshold in the evaluation in Subsection 5.1. As a result, we obtained 161 renamings on 124 methods. By comparing the tracking results of 50% threshold with the consensus oracle, We calculated two kinds of precision and recall: one was calculated based on renaming instances; the other was calculated based on methods whose tracking results included at least one renaming in the consensus oracle. * 1. From the viewpoint of renaming instances, precision and recall were 91.30% and 83.52%, respectively. * 2. From the viewpoint of methods including renames, precision and recall were 86.29% and 83.59%, respectively. According to AURA’s literature [15], AURA generated mappings for 97 rules888A rule is a mapping group of multiple methods. and its precision was 92.38%. By comparing those results, we conclude that FinerGit generated mappings for more methods with slightly-lower precision. AURA utilizes text similarity and call dependency to generate mappings while FinerGit utilizes only text similarity. On the other hand, AURA takes only two versions of source code to generate mappings while FinerGit utilizes all commits to track methods. Those are the reason why the precision values of the two tools were not so different. ### 6.2 RefactoringMiner We performed RMiner 999RMiner is available at https://github.com/tsantalis/RefactoringMiner. We used the latest version of the tool at 17th November, 2019. The commit ID is 4bb0e11550b781b61ce1c382a58ea182a2f46944. on the commit history of JHotDraw between release 5.2 and 5.3. RMiner has a capability of detecting 38 types of refactoring patterns and the following five refactoring patterns correspond to renamings that FinerGit detects: LABEL:Change_Parameter_Type, LABEL:Change_Return_Type, LABEL:Move_Method, LABEL:Rename_Method, and LABEL:Rename_Parameter. RMiner detected 158 refactoring instances of the five patterns. The detail numbers of refactorings detected by RMiner are shown in Table 2. We compared the 158 refactorings with the 161 renamings detected by FinerGit with 50% threshold. The number of common instances was 65, which was 41.14% of RMiner’s refactorings and 40.37% of FinerGit’s renamings. Table 3: Precision and Recall of RMiner in literature [18] Refactoring pattern | Precision | Recall ---|---|--- LABEL:Move_Method | 95.17% | 76.36% LABEL:Rename_Method | 97.78% | 83.28% The FinerGit evaluation in Subsection 6.1 shows that FinerGit’s tracking accuracy on JHotDraw is high (precision and recall are 91.30% and 83.52%, respectively in 50% threshold). Table 3 shows precision and recall of RMiner for each refactoring pattern in literature [18]101010LABEL:Change_Parameter_Type, LABEL:Change_Return_Type, and LABEL:Rename_Parameter were not investigated in the literature because those refactoring patterns have been recently supported by RMiner.. According to this table, precision and recall of RMiner are also high. However, the common instances between FinerGit and RMiner do not occupy a large portion of all instances detected by either of the techniques. We manually investigated renames and refactorings that had been detected only either of the techniques and found that the results faithfully reflected their different inheritances. There were two major cases of renames that were detected only by FinerGit. * 1. New parameters were added to methods or return types of methods were changed according to the changes in method’s bodies. Those changes were not refactorings but functional enhancements. * 2. Access modifiers (public, protected, and private) were added/removed/changed. Such changes were refactorings; however they were not supported by RMiner. On the other hand, refactorings that were detected only by RMiner had changed a large part of method’s bodies. Thus, line similarities of method’s bodies between such refactorings become low, which leaded to fail to be detected as a renaming by FinerGit. Herein, we compared FinerGit with RMiner; however their purposes are different from each other. The FinerGit’s purpose is tracking Java methods with high accuracy. No matter what kinds of changes are made, FinerGit is able to track methods if a line similarity of the method’s bodies between a change is higher than a given threshold. On the other hand, the purpose of RMiner is detecting refactorings in a commit history. No matter how unsimilar between method’s bodies are between a refactoring, RMiner is able to detect the refactoring if the refactoring is supported by RMiner. ## 7 Threats to Validity In the experiment, we used 182 Java projects, and we investigated on tracking results on 1,768K methods in total. Those numbers of projects and methods are large enough so that we expect that the same results are obtained if we conduct another experiment on different Java projects. To measure precision, recall, and F-measure of method tracking by FinerGit and Historage, we manually constructed oracle for 182 methods. Firstly, two of the authors made oracle for all the 182 methods independently, and then they discussed for which they made different oracle. This process of making oracle is designed to avoid making mistakes and to reduce subjective view on constructing oracle as much as possible. One more thing about oracle is that, essentially, oracle should be made independently from tracking results of FinerGit and Historage. However, constructing oracle with a fully-manual work is extraordinarily difficult even for a small number of methods. Consequently, in the experiment, we firstly obtained high-recall tracking results with an enough low threshold, and then, we checked how many false positives were included in the tracking results. We consider that this construction process does not ensure 100%-correct oracle but high enough for comparing different techniques. In other word, we made oracle of reasonable quality with a realistic time cost. In the manual investigation, we checked surrounding 15 lines (as shown in Subsection 5.1) of changes in commits to judge whether method tracking by FinerGit was correct or not. The number 15 came from our experiences with FinerGit because we had checked tracking results of FinerGit before conducting the experiment in this paper. In the experiment, we discussed the comparison results by focusing on whether FinerGit had found more renaming and copying for Java methods than Historage. However, we also need to see the fact that there were some cases that short tracking results by FinerGit were better than long tracking results by Historage. Such cases mean that FinerGit was able to avoid tracking methods incorrectly. We investigated some of such cases, and then we found that the reason why Historage found a higher number of renames is due to the existences of coincidentally matched lines as shown in Figure 4LABEL:sub@fig:Heuristic1Historage. ## 8 Related Work The research that is most related to this paper is of course Historage [1]. Historage is useful in research on mining software repositories because researchers can obtain Java method histories without implementing code/scripts by themselves. Historage has been used in many research before now. * 1. Hata et al. researched predicting fault-prone Java methods by using method histories obtained with Historage [19]. Their experimental results showed that the method-level prediction outperformed package-level and file-level predictions from the viewpoint of efforts for finding bugs. * 2. Hata et al. also used Historage to infer restructuring operations on the logical structure of Java source code [16]. * 3. Fujiwara et al. developed a hosting service of Historage repositories, Kataribe111111http://sdlab.naist.jp/kataribe/ [20]. Kataribe enables researchers/practitioners to browse method histories on the web, and they can clone Historage repositories in Kataribe into their local storages if they want to conduct further analyses. * 4. Tantithamthavorn et al. investigated the impact of granularity levels (class- level and function-level) on a feature location technique [21]. The results indicated that function-level feature location technique outperforms class- level feature location technique. Moreover, function-level feature location technique also required seven times less effort than class-level feature location technique to localize the first relevant source code entity. * 5. Kashiwabara et al. proposed a technique to recommend appropriate verbs for a method name of a given method so that developers can use various verbs consistently [22]. Their technique recommends candidate verbs by using association rules extracted from existing methods. They extracted renamed methods from repositories of target projects using Historage. * 6. Oliveira et al. presented an approach to analyze the conceptual cohesion of the source code associated with co-changed clusters of fine-grained entities [23]. They obtained change histories of Java methods with Historage. By using the change histories, they identified a set of methods that were frequently changed together. * 7. Yamamori et al. proposed to use two types of logical couplings of Java methods for recommending code changes [24]. The first type is logical couplings that are extracted from code repositories. They used Historage and Kataribe to obtain logical couplings of Java methods. The second type is logical couplings that are extracted from interaction data. They used a dataset that had been collected by Mylyn [25]. Their experimental results showed that there was a significant improvement in the efficiency of the change recommendation process. * 8. Yuzuki et al. conducted an empirical study to investigate how often change conflicts happen in large projects and how they are resolved [26]. In their empirical study, they used Historage to conduct method-level analysis. As a result, they found that 44% of conflicts were caused by changing concurrently the same positions of methods, 48% is by deleting methods, and 8% is by renaming methods. They also found that 99% of the conflicts were resolved by adopting one method directly. * 9. Suzuki et al. investigated relationships between method names and their implementation features [27]. They showed that focusing on the gap between method names and their implementation features is useful to predict fault- prone methods. They used Historage to collect change histories of Java methods in the investigation. All the above research can be conducted with FinerGit instead of Historage. Moreover, the experimental results may change if FinerGit is used because there is a significant difference in the tracking results between FinerGit and Historage. We are not the first research group that has used single-token-per-line format for Git repositories. To the best of our knowledge, the study by German et al. was the first attempt to follow this approach [28]. They proposed to rearrange source files with single-token-per-line for enabling fine-grained git-blame. By using their technique, we can see the person who changed last for each token of the source code. They showed that blame-by-token reports the correct commit that adds a given source code token between 94.5% and 99.2% of the times, while the traditional approach of blame-by-line reports the correct commit that adds a given token between 74.8% and 90.9%. German developed a system cregit 121212https://github.com/cregit/ based on their proposed technique. cregit has being used in Linux development community131313https://cregit.linuxsources.org/. cregit does not extract Java methods as files, which is a difference between cregit and FinerGit. Heuristic-1, which is described in Subsection 3.2, is refining symbols in source code. On the one hand, symbol refinements are often performed in the process of code clone detection techniques. In the context of clone detection, some symbols are replaced with special ones prior to the matching process. For example, in CCFinder [29] and NICAD [30], which are representative code clone detection techniques, all variables and literals are replaced with a specific wildcard symbol. The purpose of replacements is to detect syntactically- similar code as code clones as much as possible. Such replacements can realize that the matching process ignores differences in variables or literals. On the other hand, in the context of FinerGit, we do not want to ignore differences in variables or literals. If we ignore such differences, the similarity between non-related methods can rise accidentally, which leads FinerGit to make wrong method tracking. The purpose of our Heuristic-1 is to calculate lower similarity values between non-related methods. There are many research studies of program element matching other than Historage [13]. Lozano et al. and Saha et al. implemented method tracking techniques since they need to track method-level clones in their experiments [31, 32]. Their method-level tracking techniques are line-based comparisons and their comparisons compute numerical similarity values by comparing lines as texts. Thus, in the case that only a small part of a line is changed, the similarity between a before-change line and its after-change line should be high while a simple line-based comparison like diff regards that a before- change line is completely different from its after-change line. However, their comparisons are still line-based ones, which include some flaws compared to token-based ones. * 1. In the cases that the first token of the line is moved to the previous line or the last token of the line is moved to the next line (e.g., left bracket (“{”) is moved to the next line due to format change), their line-based techniques regard that multiple lines have been changed while our technique regards that no lines have been changed. * 2. The same changes have different impacts on lines of different length. For example, variable abc is changed to def in a 10-character line, the similarity becomes 7/10 while the same change occur in a 40-character line, the similarity becomes 37/40. Godfrey and Zou detected merging and splitting source code entities such as files and functions. They extended origin analysis [33] to track source code entities. They utilize various information for entities such as entity names, caller/callee relationship, and code metrics values. Wu et al. proposed a technique to identify change rules for one-replaced-by-many and many-replaced- by-one methods [15]. Their approach is a hybrid one, which means that it uses two kinds of data: caller/callee relationship and text similarity. Kim et al. proposed a technique to track functions even if their names get changed [14]. Their technique computes function similarities between given two methods. They introduced eight similarity factors such as complexity metrics and clone existences to determine if a function is renamed from another function. Dig et al. proposed a technique to detect refactorings performed during component evolution [12]. Their technique can track methods even if refactorings change their names. Their detection algorithm uses a combination of a fast syntactic analysis to detect refactoring candidates and a more expensive semantic analysis to refine the results. There are many other approaches for identifying refactorings, and many of them support refactorings that changes method names/signatures such as LABEL:Rename_Method and LABEL:Parameterize_Method pattern [34, 35, 36, 37, 18, 38, 39]. The advantage of the proposed technique against the above approach should be the ease to use because it utilizes Git mechanisms to track methods. A researcher/practitioner who wants method evolution data does not have to learn how to use new tools. ## 9 Conclusion In this paper, we firstly discuss Historage, which is proposed in literature [16]. Historage is a tool that converts a Git repository to a finer-grained one. In the finer-grained repository, each Java method exists as a single file. Thus, we can track Java method with Git commands such as git-log. However, tracking small methods with Git mechanisms does not work well because small methods do not have good chemistry with the Git rename detection function. Thus, we proposed a new technique that puts only a single token of Java methods per line. In other words, in our technique, each line includes only a single token. We also derived two heuristics to reduce incorrect tracking. We implemented a software tool based on the proposed technique. We applied our tool and Historage to 182 repositories of Java OSS projects to compare the two tools. The 182 repositories include 1,768K methods in total, which are the targets our comparisons. We found that FinerGit scored 84.52% as maximum F-measure while Historage scored 70.23%. We also confirmed that the proposed technique worked well for methods of any size in spite that our research motivation was to realize better tracking for small methods. 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2024-09-04T02:54:59.212489

2003.05342

arxiv-papers

# Dynamics of Chiral Cosmology Andronikos Paliathanasis<EMAIL_ADDRESS>Institute of Systems Science, Durban University of Technology, Durban 4000, Republic of South Africa Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia 5090000, Chile ###### Abstract We perform a detailed analysis for the dynamics of Chiral cosmology in a spatially flat Friedmann-Lemaître-Robertson-Walker universe with a mixed potential term. The stationary points are categorized in four families. Previous results in the literature are recovered while new phases in the cosmological evolution are found. From our analysis we find nine different cosmological solutions, the eight describe scaling solutions, where the one is that of a pressureless fluid, while only one de Sitter solution is recovered. Cosmology; Scalar field; Chiral Cosmology; Stability; Dark energy; Dynamics ###### pacs: 98.80.-k, 95.35.+d, 95.36.+x ## I Introduction A detailed analysis of the recent cosmological observations dataacc1 ; dataacc2 ; data1 ; data2 ; Hinshaw:2012aka ; Ade:2015xua indicates that the universe has gone through two acceleration phases during its evolution. In particular into a late-time acceleration phase which is attributed to dark energy, and into an early acceleration phase known as inflation. Inflation was proposed four decades ago guth in order to explain why in large scales the universe appears isotropic and homogeneous. The inflationary era is described by a scalar field known as the inflaton which when dominates drives the dynamics of the universe such that the observations are explained. In addition, scalar fields have been used to describe the recent acceleration epoch of the universe, that is, they have been applied as a source of the dark energy Ratra . In scalar field theory the gravitational field equations remain of second-order with extra degrees of freedom as many as the scalar fields and corresponding conservation equations hor1 ; hor2 ; hor3 . These extra degrees of freedom can attribute the geometrodynamical degrees of freedom provided by invariants to the modification of Einstein-Hilbert action in the context of modified/alternative theories of gravity mod1 ; mod2 ; mod3 . The simplest scalar field theory proposed in the literature is the quintessence model Ratra . Quintessence is described by a minimally coupled scalar field $\phi\left(x^{\kappa}\right)~{}$with a potential function $V\left(\phi^{\kappa}\right)$. The scalar field satisfies the weak energy condition, i.e. $\rho\geq 0,~{}\rho+p\geq 0,$ while the equation of state parameters $w_{Q}=\frac{p}{\rho}$ is bounded as $\left|w_{Q}\right|\leq 1$. For some power-law quintessence models, the gravitational field equations provide finite-time singularities during inflation leading to chaotic dynamics sing ; page . On the other hand, for some kind of potentials the quintessence can describe the late-time acceleration udm . In the cosmological scenario of a Friedmann-Lemaître-Robertson-Walker universe (FLRW) exact and analytic solutions of the field equations for different potentials are presented in jdbnew ; muslinov ; ellis ; barrow1 ; newref2 ; ref001 ; ref002 and references therein. Results of similar analysis on the dynamics of quintessence models are summarized in the recent review gen01 . Other scalar field models which have been proposed in the literature are: phantom fields, Galileon, scalar tensor, multi-scalar field models and others ph1 ; ph2 ; ph3 ; ph4 ; ph5 ; ph7 ; ph8 ; ph9 ; ph10 ; ph11 . Multi-scalar field models have been used to provide alternative models for the description of inflation hy1 ; hy2 ; hy3 , such as hybrid inflation, double inflation, $\alpha$-attractors hy4 ; atr1 ; atr3 and as alternative dark energy models. Multi-scalar field models which have drawn the attention of cosmologists are, the quintom model and the Chiral model. A common feature of these two theories is that they are described by two-scalar fields, namely $\phi\left(x^{\kappa}\right)$ and $\psi\left(x^{\kappa}\right)$. For the quintom model, one of the two fields is quintessence while the second scalar field is phantom which means that the energy density of the field can be negative. One of the main characteristics of quintom cosmology is that the parameter for the equation of state for the effective cosmological fluid can cross the phantom divide line more than once qq1 ; qq2 . The general dynamics of quintom cosmology is presented in qq3 . In Chiral theory, the two scalar fields have a mixed kinetic term. The two scalar fields are defined on a two-dimensional space of constant nonvanishing curvature atr6 ; atr7 . That model is inspired by the non-linear sigma cosmological model sigm0 . Chiral cosmology is linked with the $\alpha-$attractor models atr3 . Exact solutions and for specific cases the dynamics of Chiral cosmology were studied before in andimakis , while analytic solutions in Chiral cosmology are presented in 2sfand . In the latter reference, it was found that pressureless fluid is provided by the model, consequently, the model can also be seen as an alternative model for the description of the dark sector of the universe. Last but not least scaling attractors in Chiral theory were studied in andimakis ; per1 . In this piece of work we are interested in the evolution of the dynamics for the gravitational field equations of Chiral cosmology in a spatially flat FLRW background space. We consider a general scenario where an interaction term for the two scalar fields exists in the potential term $V\left(\phi,\psi\right)$ of the two fields, that is, $V_{,\phi\psi}\neq 0$. Specifically, we determine the stationary points of the cosmological equations and we study the stability of these points. Each stationary point describes a solution in the cosmological evolution. Such an analysis is important in order to understand the general behaviour of the model and to infer about its viability. This approach has been applied in various gravitational theories with important results for the viability of specific theories of gravity, see for instance dyn1 ; dyn2 ; dyn3 ; dyn4 ; dyn5 ; dyn6 ; dyn7 ; dyn8 ; dyn9 and references therein. From such an analysis we can conclude about for which eras of the cosmological history can be provided by the specific theory, we refer the reader in the discussion of dyn1 . The plan of the paper is as follows. In Section II we present the model of our consideration which is that of Chiral cosmology in a spatially flat FLRW spacetime with a mixed potential term. We write the field equations which are of second-order. By using the energy density and pressure variables we observe that the interaction of the two fields depends on the pressure term. In Section III, we rewrite the field equations by using dimensionless variables in the $H-$normalization. We find an algebraic-differential dynamical system consists of one algebraic constraint and six first-order ordinary differential equations. We consider a specific form for the potential in order to reduce dynamical system the system by one-dimension; and with the use of the constraint equation we end with a four-dimensional system. The main results of this work are presented in Section IV. We find the stationary points of the field equations which form four different families. The stationary points of family A are those of quintessence, in family B only the kinetic part of the second scalar field contributes to the cosmological solutions. On the other hand, the points of family C are those where only the dynamic part of the second field contributes. Furthermore, for the cosmological solutions at the points of family D all the components of the second field contributes to the cosmological fluid. For all the stationary points we determine the physical properties which describe the corresponding exact solutions, as also we determine the stability conditions. An application of this analysis is presented in Section V with some numerical results. Moreover, for completeness of our study we present an analytic solution of the field equations by using previous results of the literature, from where we can verify the main results of this work. In Section VII we discuss the additional stationary points when matter source is included in the cosmological model. Finally, in Section VIII we draw our conclusions. ## II Chiral cosmology We consider the gravitational Action Integral to be 2sfand $S=\int\sqrt{-g}dx^{4}R-\int\sqrt{-g}dx^{4}\left(\frac{1}{2}g^{\mu\nu}H_{AB}\left(\Phi^{C}\right)\nabla_{\mu}\Phi^{A}\nabla_{\nu}\Phi^{B}+V\left(\Phi^{C}\right)\right)$ (1) where $\Phi^{A}=\left(\phi\left(x^{\mu}\right),\psi\left(x^{\mu}\right)\right)$, $H_{AB}\left(\Phi^{C}\right)$ is a second rank tensor which defines the kinetic energy of the scalar fields, while $V\left(\Phi^{C}\right)$ is the potential. The Action Integral (1) describes a interacting two-scalar field cosmological model where the interaction follows by the potential $V\left(\Phi^{C}\right)=V\left(\phi,\psi\right),$ and the kinetic part. In this work we assume that $H_{AB}\left(\Phi^{C}\right)$ is diagonal and admits at least one isometry such that (1) $S=\int\sqrt{-g}dx^{4}R-\int\sqrt{-g}dx^{4}\left(\frac{1}{2}g^{\mu\nu}\left(\phi_{;\mu}\phi_{;\nu}+M\left(\phi\right)\psi_{;\mu}\psi_{;\mu}\right)+V\left(\Phi^{C}\right)\right)$ (2) where $M\left(\phi\right)_{,\phi}\neq 0$ and $M\left(\phi\right)\neq M_{0}\phi^{2}$. In the latter two cases, $H_{AB}\left(\Phi^{C}\right)$ describe a two-dimensional flat space and if it is of Lorentzian signature then it describes the quintom model. Functional of forms of $M\left(\phi\right)$ where $H_{AB}\left(\Phi^{C}\right)$ is a maximally symmetric space of constant curvature $R_{0}$, are given by the second-order differential equation $2M_{,\phi\phi}M-\left(M_{,\phi}\right)^{2}+2M^{2}R_{0}=0.$ (3) A solution of the latter equation is $M\left(\phi\right)=M_{0}e^{\kappa\phi}$, which can be seen as the general case since new fields can be defined under coordinate transformations to rewrite the form of $H_{AB}\left(\Phi^{C}\right)$. This is the case of Chiral model that we study in this work. Variation with respect to the metric tensor of (1) provides the gravitational field equations $G_{\mu\nu}=H_{AB}\left(\Phi^{C}\right)\nabla_{\mu}\Phi^{A}\nabla_{\nu}\Phi^{B}-g_{\mu\nu}\left(\frac{1}{2}g^{\mu\nu}H_{AB}\left(\Phi^{C}\right)\nabla_{\mu}\Phi^{A}\nabla_{\nu}\Phi^{B}+V\left(\Phi^{C}\right)\right),$ (4) while variation with respect to the fields $\Phi^{A}$ give the Klein-Gordon vector-equation $g^{\mu\nu}\left(\nabla_{\mu}\left(H_{~{}B}^{A}\left(\Phi^{C}\right)\nabla_{\nu}\Phi^{B}\right)\right)+H_{~{}B}^{A}\left(\Phi^{C}\right)\frac{\partial V\left(\Phi^{C}\right)}{\partial\Phi^{B}}=0.$ (5) According to the cosmological principle, the universe in large scales is isotropic and homogeneous described by the spatially flat FLRW spacetime with line element $ds^{2}=-dt^{2}+a^{2}\left(t\right)\left(dx^{2}+dy^{2}+dz^{2}\right).$ (6) where $a\left(t\right)$ denotes the scale factor and the Hubble function is defined as $H\left(t\right)=\frac{\dot{a}}{a}$. For the line element (6) and the second-rank tensor $H_{AB}\left(\Phi^{C}\right)$ of our consideration the field equations are written as follows $3H^{2}=\frac{1}{2}\left(\dot{\phi}^{2}+M\left(\phi\right)\dot{\psi}^{2}\right)+V\left(\phi\right)+M\left(\phi\right)U\left(\psi\right),$ (7) $2\dot{H}+3H^{2}=-\left(\frac{1}{2}\left(\dot{\phi}^{2}+M\left(\phi\right)\dot{\psi}^{2}\right)-V\left(\phi\right)-M\left(\phi\right)U\left(\psi\right)\right),$ (8) $\ddot{\phi}+3H\dot{\phi}-\frac{1}{2}M_{,\phi}\dot{\psi}^{2}+V_{,\phi}\left(\phi\right)+M_{,\phi}U\left(\psi\right)=0,$ (9) $\ddot{\psi}+3H\dot{\psi}+\frac{M_{,\phi}}{M}\dot{\phi}\dot{\psi}+U_{,\psi}=0.$ (10) where we replaced $V\left(\phi,\psi\right)=V\left(\phi\right)+M\left(\phi\right)U\left(\psi\right)$ and we have assumed that the fields $\phi,\psi$ inherit the symmetries of the FLRW space such that $\phi\left(x^{\mu}\right)=\phi\left(t\right)$ and $\psi\left(x^{\mu}\right)=\psi\left(t\right)$. At this point we remark that the field equations (8)-(10) can be produced by the variation principle of the point-like Lagrangian $\mathcal{L}\left(a,\dot{a},\phi,\dot{\phi},\psi,\dot{\psi}\right)=-3a\dot{a}^{2}+\frac{1}{2}a^{3}\left(\dot{\phi}^{2}+M\left(\phi\right)\dot{\psi}^{2}\right)-a^{3}\left(V\left(\phi\right)+M\left(\phi\right)U\left(\psi\right)\right),$ (11) while equation (7) can be seen as the Hamiltonian constraint of the time- independent Lagrangian (11). An equivalent way to write the field equations (7), (8) is by defining the quantities $\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V\left(\phi\right)~{},~{}p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V\left(\phi\right),$ (12) $\rho_{\psi}=\left(\frac{1}{2}\dot{\psi}^{2}+U\left(\psi\right)\right)M\left(\phi\right)~{},~{}p_{\psi}=\left(\frac{1}{2}\dot{\psi}^{2}-U\left(\psi\right)\right)M\left(\phi\right),$ (13) that is, $3H^{2}=\rho_{\phi}+\rho_{\psi},$ (14) $2\dot{H}+3H^{2}=-\left(p_{\phi}+p_{\psi}\right),$ (15) $\dot{\rho}_{\phi}+3H\left(\rho_{\phi}+p_{\phi}\right)=\dot{\phi}\frac{\partial}{\partial\phi}p_{\psi},$ (16) $\dot{\rho}_{\psi}+3H\left(\rho_{\psi}+p_{\psi}\right)=-\dot{\phi}\frac{\partial}{\partial\phi}p_{\psi}.$ (17) The latter equations give us an interesting observation, since we can write the interacting functions of the two fields. The interaction models, with interaction between dark matter and dark energy have been proposed as an potential mechanism to explain the cosmic coincidence problem and provide a varying cosmological constant. Some interaction models which have been studied before in the literature are presented in Amendola:2006dg ; Pavon:2007gt ; Chimento:2009hj ; Arevalo:2011hh ; an001 ; an002 while some cosmological constraints on interacting models can be found in in1 ; in2 ; in3 ; in4 . ## III Dimensionless variables We consider the dimensionless variables in the $H$-normalization cop $\dot{\phi}=\sqrt{6}xH~{},~{}V\left(\phi\right)=3y^{2}H^{2}~{},~{}\dot{\psi}=\frac{\sqrt{6}}{\sqrt{M\left(\phi\right)}}zH~{},~{}U\left(\psi\right)=\frac{3}{M\left(\phi\right)}u^{2}H^{2}$ (18) or $x=\frac{\dot{\phi}}{\sqrt{6}H}~{},~{}y^{2}=\frac{V\left(\phi\right)}{3H^{2}}~{},~{}z=\frac{\sqrt{M\left(\phi\right)}\dot{\psi}}{\sqrt{6}H}~{},~{}u^{2}=\frac{M\left(\phi\right)U\left(\psi\right)}{3H^{2}},$ (19) where the field equations become $\displaystyle\frac{dx}{d\tau}$ $\displaystyle=\frac{3}{2}x\left(x^{2}-\left(1+u^{2}+y^{2}-z^{2}\right)\right)-\frac{\sqrt{6}}{2}\left(\lambda y^{2}+\kappa\left(u^{2}-z^{2}\right)\right),$ (20) $\displaystyle\frac{dy}{d\tau}$ $\displaystyle=\frac{3}{2}y\left(1+x^{2}+z^{2}-y^{2}-u^{2}\right)+\frac{\sqrt{6}}{2}\lambda xy,$ (21) $\displaystyle\frac{dz}{d\tau}$ $\displaystyle=\frac{3}{2}z\left(z^{2}-\left(1+u^{2}+y^{2}-x^{2}\right)\right)-\frac{\sqrt{6}}{2}\left(\kappa xz+\mu u^{2}\right),$ (22) $\displaystyle\frac{du}{d\tau}$ $\displaystyle=\frac{3}{2}u\left(1+x^{2}+z^{2}-y^{2}-u^{2}\right)+\frac{\sqrt{6}}{2}u\left(\kappa x+\mu z\right),$ (23) $\displaystyle\frac{d\mu}{d\tau}$ $\displaystyle=\sqrt{\frac{3}{2}}\mu\left(2\mu z\bar{\Gamma}\left(\mu,\lambda\right)-\kappa x-2\mu z\right),$ (24) $\displaystyle\frac{d\lambda}{d\tau}$ $\displaystyle=\sqrt{6}\lambda^{2}x\left(\Gamma\left(\lambda\right)-1\right),$ (25) in which $\tau=\ln a,~{}\lambda\left(\phi\right)=\frac{V_{,\phi}}{V}~{},~{}\kappa\left(\lambda\right)=\frac{M_{,\phi}}{M}~{},~{}\mu\left(\phi,\psi\right)=\frac{1}{\sqrt{M\left(\phi\right)}}\frac{U_{,\psi}}{U},~{}$ (26) and functions $\Gamma\left(\lambda\right),~{}\bar{\Gamma}\left(\mu,\lambda\right)$ are defined as $\Gamma\left(\lambda\right)=\frac{V_{,\phi\phi}V}{\left(V_{,\phi}\right)^{2}}~{},~{}\bar{\Gamma}\left(\mu,\lambda\right)=\frac{U_{,\psi\psi}U}{\left(U_{,\psi}\right)^{2}},$ (27) while the constraint equation is $1-x^{2}-y^{2}-z^{2}-u^{2}=0.$ (28) The equation of state parameter for the effective cosmological fluid $w_{tot},~{}$is given in terms of the dimensionless parameters as follows $w_{tot}=-1-\frac{2}{3}\frac{\dot{H}}{H^{2}}=x^{2}+z^{2}-y^{2}-u^{2}$ (29) while we define the variables $\Omega_{\phi}=x^{2}+y^{2}~{},~{}\Omega_{\psi}=z^{2}+u^{2},$ (30) with equation of state parameters $w_{\phi}=-1+\frac{2x^{2}}{x^{2}+y^{2}}~{},~{}w_{\psi}=-1+\frac{2z^{2}}{z^{2}+u^{2}}.$ (31) At this point it is important to mention that since the two fields interact that is not the unique definition of the physical variables $\Omega_{\phi}$ and $\Omega_{\psi}$, $w_{\phi}$ and $w_{\psi}$. Moreover, from the constraint equation (28) it follows that the stationary points are on the surface of a four-dimensional unitary sphere, while the field equations remain invariant under the transformations $\left\\{y,u\right\\}\rightarrow\left(-y,-u\right)$, that is, the variables $\left\\{x,y,z,u\right\\}$ take values in the following regions $\left|x\right|\leq 1~{},~{}\left|z\right|\leq 1~{},~{}0\leq y\leq 1~{}\ $and $0\leq u\leq 1$. For the arbitrary functions $V\left(\phi\right),$ $U\left(\psi\right)$ and $M\left(\phi\right)$, there are six dependent, namely $\left\\{x,y,z,u,\lambda,\mu\right\\}$, where in general $\kappa=\kappa\left(\lambda\right)$, however the dimension of the system can be reduced by one, if we apply the constraint condition (28). In the following Section, we determine the stationary points for the cases where $M\left(\phi\right)=M_{0}e^{\kappa\phi},~{}\ V\left(\phi\right)=V_{0}e^{\lambda\phi},~{}$ and $U\left(\psi\right)=U_{0}\psi^{\frac{1}{\sigma}}$. Consequently, we calculate $\Gamma\left(\lambda\right)=1$ and $\bar{\Gamma}\left(\mu,\lambda\right)=1-\sigma$ and $\kappa=const$. Therefore, $\frac{d\lambda}{d\tau}=0$ is satisfied identically and the dimension of the dynamical system is reduced by one. Therefore we end with the dynamical system (20)-(24) with constraint (28). We remark that in Chiral model, the kinetic parts of the two fields are defined on a two-dimensional space of constant curvature. ## IV Dynamical behaviour The stationary points of the dynamical system have coordinates which make the rhs of equations (20)-(24) vanish. We categorize the stationary points into four families. Family A, are the points with coordinates $\left(x_{A},y_{A},z_{A},u_{A},\mu_{A}\right)=\left(x_{A},y_{A},0,0,0\right)$ and correspond to the points of the minimally coupled scalar field cosmology cop . The points with coordinates $\left(x_{B},y_{B},z_{B},u_{B},\mu_{B}\right)=\left(x_{B},y_{B},z_{B},0,\mu_{B}\right)$ and $z_{B}\neq 0$ define the points of Family B. These points describe physical solutions without any contribution of the potential $U\left(\psi\right)$ to the energy density of the total fluid source, but only when $\mu_{B}=0$ there is not any contribution of potential $U\left(\psi\right)$ to the dynamics. When $\mu_{B}=0$, the stationary points are those found before in andimakis . Points of family $C$ have coordinates $\left(x_{C},y_{C},z_{C},u_{C},\mu_{C}\right)=\left(x_{C},y_{C},0,u_{C},\mu_{C}\right),~{}u_{C}\neq 0$ which describe exact solutions with no contribution of the kinetic part of the scalar fields $\psi$. Finally, the points of family D have coordinates of the form $\left(x_{C},y_{C},z_{C},u_{C},\mu_{C}\right)~{}$with$~{}z_{D}u_{D}\neq 0$. Let $P$ be a stationary point of the dynamical system (20)-(24), that is,$~{}\dot{q}^{A}=f^{A}\left(q^{B}\right)$, where $f^{A}\left(P\right)=0$. In order to study the stability properties of the critical point $P$, we write the linearized system which is $\delta\dot{x}^{A}=J_{B}^{A}\delta x^{B}~{}$where $J_{B}^{A}$ is the Jacobian matrix at the point $P$, i.e.$~{}J_{B}^{A}=\frac{\partial f^{A}\left(P\right)}{\partial x^{B}}$. The eigenvalues $\mathbf{e}\left(P\right)$ of the Jacobian matrix determine the stability of the station point. When all the eigenvalues have negative real part then point $P$ is an attractor and the exact solution at the point is stable, otherwise the exact solution at the critical point is unstable and point $P$ is a source, when all the eigenvalues have a positive real part, or $P$ is a saddle point. ### IV.1 Family A There are three stationary points which describe cosmological solutions without any contribution of the second field $\psi$. The points have coordinates cop $A_{1}^{\pm}=\left(\pm 1,0,0,0,0\right)~{},~{}A_{2}=\left(-\frac{\lambda}{\sqrt{6}},\sqrt{1-\frac{\lambda^{2}}{6}},0,0,0\right).$ (32) Points $A_{1}^{\pm}$ describe universes dominated by the kinetic part of the scalar field $\phi,~{}$that is by the term $\frac{1}{2}\dot{\phi}^{2}$. The physical quantities are derived $\left(w_{tot}\left(A_{1}^{\pm}\right),w_{\phi}\left(A_{1}^{\pm}\right),w_{\psi}\left(A_{1}^{\pm}\right),\Omega_{\phi}\left(A_{1}^{\pm}\right),\Omega_{\psi}\left(A_{1}^{\pm}\right)\right)=\left(1,1,\nexists,1,0\right).$ Point $A_{2}$ is physically accepted when $\left|\lambda\right|<\sqrt{6},$ the physical quantities are calculated $\left(w_{tot}\left(A_{2}\right),w_{\phi}\left(A_{2}\right),w_{\psi}\left(A_{2}\right),\Omega_{\phi}\left(A_{2}\right),\Omega_{\psi}\left(A_{2}\right)\right)=\left(-1+\frac{\lambda^{2}}{3},-1+\frac{\lambda^{2}}{3},\nexists,1,0\right).$ Therefore, point $A_{2}$ describes a scaling solution. The latter solution is that of an accelerated universe when $\left|\lambda\right|<\sqrt{2}$. In the case of quintessence scalar field cosmology, points $A_{1}^{\pm}$ are always unstable, while $A_{2}$ is the unique attractor of the dynamical system when $\left|\lambda\right|<\sqrt{3}$. However, for the model of our analysis the stability conditions are different. In order to conclude for the stability of the stationary points we determine the eigenvalues of the linearized dynamical system (20)-(24) around to the stationary points. For the points $A_{1}^{\pm}$ it follows $\displaystyle e_{1}\left(A_{1}^{\pm}\right)$ $\displaystyle=3,$ $\displaystyle e_{2}\left(A_{1}^{\pm}\right)$ $\displaystyle=\frac{1}{2}\left(6\pm\sqrt{6}\lambda\right),$ $\displaystyle~{}e_{3}\left(A_{1}^{\pm}\right)$ $\displaystyle=\frac{1}{2}\left(6\pm\sqrt{6}\kappa\right),$ $\displaystyle~{}e_{4}\left(A_{1}^{\pm}\right)$ $\displaystyle=\mp\sqrt{\frac{3}{2}}\kappa,~{}$ $\displaystyle e_{5}\left(A_{1}^{\pm}\right)$ $\displaystyle=\mp\sqrt{\frac{3}{2}}\kappa,$ from where we conclude that points $A_{1}^{\pm}~{}$are saddle points, while the solutions at points $A_{1}^{\pm}$ are always unstable’ because at least one of the eigenvalues is always positive, i.e. eigenvalue $e_{1}\left(A_{1}^{\pm}\right)>0$. For the stationary point $A_{2}$ the eigenvalues are derived $\displaystyle e_{1}\left(A_{2}\right)$ $\displaystyle=\frac{1}{2}\left(\lambda^{2}-6\right),$ $\displaystyle e_{2}\left(A_{2}\right)$ $\displaystyle=\lambda^{2}-3,$ $\displaystyle~{}e_{3}\left(A_{2}\right)$ $\displaystyle=\frac{1}{2}\kappa\lambda,$ $\displaystyle~{}e_{4}\left(A_{2}\right)$ $\displaystyle=\frac{1}{2}\left(\lambda^{2}-\kappa\lambda\right),~{}$ $\displaystyle e_{5}\left(A_{2}\right)$ $\displaystyle=\frac{1}{2}\left(\lambda^{2}-6+\kappa\lambda\right),$ that is, the exact solution at point $A_{2}$ is always unstable. However. from the two eigenvalues $e_{1}\left(A_{2}\right),~{}e_{2}\left(A_{2}\right)$ we can infer that in the surface $\left\\{x,y\right\\}$ of the phase space the stationary point $A_{2}$ acts like an attractor for $\left|\lambda\right|<\sqrt{3}$, which however becomes a saddle point for the higher-dimensional phase space. We remark that we determined the stability of the stationary points without using the constant equation and reducing the dynamical system by one- dimension. However, by replacing $z^{2}=1-x^{2}-y^{2}-u^{2}$ in the (20)-(24) we end with a four-dimensional system, from where we find the same results, that is, the exact solutions at the points $A_{1}^{\pm}$ and $A_{2}$ are always unstable. ### IV.2 Family B For $z_{B}\neq 0$ and $u_{B}=0,$ we found four stationary points which are $\displaystyle B_{1}^{\pm}$ $\displaystyle=\left(-\frac{\sqrt{6}}{\kappa+\lambda},\sqrt{\frac{\kappa}{\kappa+\lambda}},\pm\sqrt{\frac{\lambda^{2}+\kappa\lambda-6}{\left(\kappa+\lambda\right)^{2}}},0,0\right),$ (33) $\displaystyle B_{2}^{\pm}$ $\displaystyle=\left(-\frac{\sqrt{6}}{\kappa+\lambda},\sqrt{\frac{\kappa}{\kappa+\lambda}},\pm\sqrt{\frac{\lambda^{2}+\kappa\lambda-6}{\left(\kappa+\lambda\right)^{2}}},0,\sqrt{\frac{3}{2}}\frac{\kappa}{\sqrt{\left(\lambda^{2}+\kappa\lambda-6\right)}}\right),$ (34) which are real and are physically accepted when $\left\\{\kappa>0,\lambda>\sqrt{6}\right\\}$ or $\left\\{0<\lambda\leq\sqrt{6},~{}\kappa>\frac{6-\lambda^{2}}{\lambda}\right\\}$ or $\left\\{\lambda<-\sqrt{6},\kappa<0\right\\}$ or $\left\\{-\sqrt{6}<\lambda<0,\kappa<\frac{6-\lambda^{2}}{\lambda}\right\\}$. The latter region plots are presented in Fig. 1. Figure 1: Region plot in the space $\left\\{\lambda,\kappa\right\\}$ where points $\mathbf{B=}\left(B_{1}^{\pm},B_{2}^{\pm}\right)$ are real. The stationary points have the same physical properties, that is, the points describe universes with the same physical properties, where the physical quantities have the following values $w_{tot}\left(\mathbf{B}\right)=1-\frac{2\kappa}{\kappa+\lambda}~{},~{}w_{\phi}\left(\mathbf{B}\right)=-1+\frac{12}{6+\kappa\left(\kappa+\lambda\right)}~{},~{}w_{\psi}\left(\mathbf{B}\right)=1~{},$ (35) $\Omega_{\phi}\left(\mathbf{B}\right)=1-\Omega_{\psi}\left(\mathbf{B}\right)~{},~{}\Omega_{\psi}\left(\mathbf{B}\right)=\left|\frac{\lambda\left(\kappa+\lambda\right)-6}{\left(\kappa+\lambda\right)^{2}}\right|.$ (36) From $w_{tot}\left(\mathbf{B}\right)$ it follows that the points describe scaling solutions and the de Sitter universe is recovered only when $\lambda=0$, which is excluded because for $\lambda=0$, the stationary points are not real. We continue by studying the stability of the stationary points. In Fig. 2, we present counter plots for the physical parameters $w_{tot}\left(\mathbf{B}\right),~{}w_{\phi}\left(\mathbf{B}\right)\,$ and $\Omega_{\psi}\left(\mathbf{B}\right)$ in the space of variables $\left\\{\lambda,\kappa\right\\}$. Figure 2: Qualitative evolution of the physical variables $w_{tot}\left(\mathbf{B}\right),~{}w_{\phi}\left(\mathbf{B}\right)\,$ and $\Omega_{\psi}\left(\mathbf{B}\right)$ of the exact solutions at the critical points $\mathbf{B=}\left(B_{1}^{\pm},B_{2}^{\pm}\right)$ for various values of the free variables $\left\\{\lambda,\kappa\right\\}$. For the stationary points $B_{1}^{\pm}$ two of the five eigenvalues are expressed as $e_{1}\left(B_{1}^{\pm}\right)=3\frac{\kappa}{\kappa+\lambda},~{}e_{2}\left(B_{1}^{\pm}\right)=-3\frac{\kappa-\lambda}{\kappa+\lambda},$ from where we observe that $e_{1}\left(B_{1}^{\pm}\right)>0$ in order for the points to be real, consequently the exact solutions at the stationary points $B_{1}^{\pm}$ are unstable. We use the constraint $z^{2}=1-x^{2}-y^{2}-u^{2}$ such that the dynamical system is reduced by one-dimension. Thus, for the new four-dimensional system the eigenvalues of the linearized system around points $B_{1}^{\pm}$ are found $\displaystyle e_{1}\left(B_{1}^{\pm}\right)$ $\displaystyle=3\frac{\kappa}{\kappa+\lambda},~{}e_{2}\left(B_{1}^{\pm}\right)=-3\frac{\kappa-\lambda}{\kappa+\lambda},$ $\displaystyle e_{3}\left(B_{1}^{\pm}\right)$ $\displaystyle=-\frac{3\kappa+i\sqrt{3\kappa\left(4\lambda^{3}+8\kappa\lambda^{2}+4\left(\kappa^{2}-6\right)\lambda-27\kappa\right)}}{2\left(\kappa+\lambda\right)},$ $\displaystyle e_{4}\left(B_{1}^{\pm}\right)$ $\displaystyle=-\frac{3\kappa-i\sqrt{3\kappa\left(4\lambda^{3}+8\kappa\lambda^{2}+4\left(\kappa^{2}-6\right)\lambda-27\kappa\right)}}{2\left(\kappa+\lambda\right)},$ from where we conclude again that the exact scaling solutions at points $B_{1}^{\pm}$ are unstable. In particular points Similarly, the eigenvalues of the linearized system around the points $B_{2}^{\pm}$ are calculated $\displaystyle e_{1}\left(B_{2}^{\pm}\right)$ $\displaystyle=-3\frac{\kappa}{\kappa+\lambda},~{}e_{2}\left(B_{2}^{\pm}\right)=-3\frac{2\sigma\left(\kappa-\lambda\right)-\kappa}{2\sigma\left(\kappa+\lambda\right)},$ $\displaystyle e_{3}\left(B_{2}^{\pm}\right)$ $\displaystyle=e_{3}\left(B_{1}^{\pm}\right),~{}e_{4}\left(B_{2}^{\pm}\right)=e_{3}\left(B_{1}^{\pm}\right),$ Hence, we infer that the stationary points $B_{2}^{\pm}$ are attractors, and the exact solutions at the points are stable when the free parameters $\left\\{\lambda,\kappa,\sigma\right\\}$ are constraints as follows $\lambda\leq-\sqrt{6}:\left\\{\kappa<\lambda,\sigma<0,\sigma>\frac{\kappa}{2\left(\kappa-\lambda\right)}\right\\}\cup\left\\{\kappa=\lambda,\sigma<0\right\\}\cup\left\\{\lambda<\kappa<0,\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma<0\right\\},$ $-\sqrt{6}<\lambda<-\sqrt{3}:\left\\{\kappa<\lambda,\sigma<0,\sigma>\frac{\kappa}{2\left(\kappa-\lambda\right)}\right\\}\cup\left\\{\kappa=\lambda,\sigma<0\right\\}\cup\left\\{\lambda<\kappa<\frac{6-\lambda^{2}}{\lambda},\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma<0\right\\},$ $\lambda=-\sqrt{3}:\left\\{\kappa<-\sqrt{3},~{}\sigma<0\right\\}\cup\left\\{\kappa<-\sqrt{3},~{}\frac{\kappa}{2\left(\sqrt{3}+\kappa\right)}<\sigma\right\\},$ $-\sqrt{3}<\lambda<0:\left\\{\kappa<\frac{6-\lambda^{2}}{\lambda},\sigma<0\right\\}\cup\left\\{\kappa<\frac{6-\lambda^{2}}{\lambda},\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma\right\\},$ $0<\lambda<\sqrt{3}:\left\\{\kappa>\frac{6-\lambda^{2}}{\lambda},\sigma<0\right\\}\cup\left\\{\kappa>\frac{6-\lambda^{2}}{\lambda},\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma\right\\},$ $\lambda=\sqrt{3}:\left\\{\kappa<\sqrt{3},~{}\sigma<0\right\\}\cup\left\\{\kappa<-\sqrt{3},~{}-\frac{\kappa}{2\left(\sqrt{3}-\kappa\right)}<\sigma\right\\},$ $\sqrt{3}<\lambda<\sqrt{6}:\left\\{\frac{6-\lambda^{2}}{\lambda}<\kappa<\lambda,\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma<0\right\\}\cup\left\\{\kappa\geq\lambda,\sigma<0\right\\}\cup\left\\{\kappa>\lambda,\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma\right\\},$ $\lambda\geq\sqrt{6}:\left\\{0<\kappa<\lambda,\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma<0\right\\}\cup\left\\{\kappa\geq\lambda,\sigma<0\right\\}\cup\left\\{\kappa>\lambda,\frac{\kappa}{2\left(\kappa-\lambda\right)}<\sigma\right\\}.$ In Figs. 3 and 4 we plot the regions where the stationary points $B_{2}^{\pm}$ are attractors and the exact solutions on the stationary points points are stable. Figure 3: Region plot in the space of variabels $\left\\{\kappa,\lambda,\sigma\right\\}$ where the points $B_{2}^{\pm}$ are attractors. Figure 4: Region plots in the the planes $\kappa-\sigma,~{}\lambda-\sigma$ and $\lambda-\kappa$ where points $B_{2}^{\pm}$ are attractors. Left figures present the region in the plane $\kappa-\sigma$ for $\lambda=-2$ and $\lambda=2$; middle figures present the region in the plane $\lambda-\sigma$, for $\kappa=-2$ and $\kappa-2$ while right figures are in the plane for $\lambda-\kappa$ for $\sigma=-1$ and $\sigma=1$. ### IV.3 Family C The stationary points of Family C are two and they have coordinates $\displaystyle C_{1}$ $\displaystyle=\left(-\frac{\kappa}{\sqrt{6}},0,0,\sqrt{1-\frac{\kappa^{2}}{6}},0\right),$ (37) $\displaystyle C_{2}$ $\displaystyle=\left(0,\sqrt{\frac{\kappa}{\kappa-\lambda}},0,\sqrt{\frac{\lambda}{\lambda-\kappa}},0\right).$ (38) Point $C_{1}$ is real when $\left|\kappa\right|\leq\sqrt{6}$ and the physical quantities of the exact solution at the point are $\left(w_{tot}\left(C_{1}\right),w_{\phi}\left(C_{1}\right),w_{\psi}\left(C_{1}\right),\Omega_{\phi}\left(C_{1}\right),\Omega_{\psi}\left(C_{1}\right)\right)=\left(-1+\frac{\kappa^{2}}{3},1,-1,\frac{\kappa^{2}}{6},1-\frac{\kappa^{2}}{6}\right).$ (39) Thus, stationary point $C_{1}$ describes a scaling solution. The scaling solution describes an accelerated universe when $\left|\kappa\right|<\sqrt{2}$. Furthermore, the exact solution at the stationary point $C_{2}$ describes a de Sitter universe, where the two scalar fields mimic the cosmological constant, the physical quantities are $\left(w_{tot}\left(C_{2}\right),w_{\phi}\left(C_{2}\right),w_{\psi}\left(C_{2}\right),\Omega_{\phi}\left(C_{2}\right),\Omega_{\psi}\left(C_{2}\right)\right)=\left(-1,-1,-1,\frac{\kappa}{\kappa-\lambda},\frac{\lambda}{\lambda-\kappa}\right).$ (40) Point $C_{2}$ is real and physically accepted when $\lambda\kappa<0$, i.e. $\left\\{\lambda<0,\kappa>0\right\\}$ or $\left\\{\lambda>0,\kappa<0\right\\}$. The linearized four-dimensional system around the stationary point $C_{1}$ admits the eigenvalues $\displaystyle e_{1}\left(C_{1}\right)$ $\displaystyle=\frac{\kappa^{2}}{2},$ $\displaystyle e_{2}\left(C_{1}\right)$ $\displaystyle=-\frac{1}{2}\left(6-\kappa^{2}\right)$ $\displaystyle e_{3}\left(C_{1}\right)$ $\displaystyle=2\left(\kappa^{2}-3\right)$ $\displaystyle e_{4}\left(C_{1}\right)$ $\displaystyle=\frac{1}{2}\kappa\left(\kappa-\lambda\right)$ from where we infer that the exact solution at the stationary point is always unstable. Specifically, point $C_{1}$ is a saddle point. For the stationary point $C_{2}$, we find that one of the eigenvalues of the linearized system around $C_{2}$ is zero. That eigenvalue corresponds to the linearize equation (24). As far as concerns the other three eigenvalues we plot numerically their values and we find that they have negative real parts for all the range of parameters $\left\\{\lambda,\kappa\right\\}$ where the point exists. In Fig. 5 we plot the real parts of the three nonzero eigenvalues of the linearized system. Therefore, we infer that the there exists a four-dimensional stable submanifold around the stationary point. However, because of the eigenvalues has zero real part the center manifold theorem (CMT) should be applied. For simplicity on our calculations we apply the CMT for the five dimensional system. We find that the variables with nonzero real part on their eigenvalues, that is, variables $\left\\{x,y,z,u\right\\}$, according to the CMT theorem are approximated as functions of variable $\mu$ as follows $\displaystyle x$ $\displaystyle=x_{00}\mu^{2}+x_{10}\mu^{3}+x_{20}\mu^{4}+O\left(\mu^{5}\right)~{},~{}y=y_{00}\mu^{2}+y_{10}\mu^{3}+y_{20}\mu^{4}+O\left(\mu^{5}\right),~{}$ $\displaystyle z$ $\displaystyle=z_{00}\mu^{2}+z_{10}\mu^{3}+z_{20}\mu^{4}+O\left(\mu^{5}\right)~{},~{}u=u_{00}\mu^{2}+u_{10}\mu^{3}+u_{20}\mu^{4}+O\left(\mu^{5}\right)$ where $\left\\{x_{00},y_{00},z_{00},u_{00}\right\\}=\left(0,0,z_{00},0\right)$; $x_{10}=-\frac{z_{00}}{\kappa},~{}y_{10}=\sqrt{\frac{3}{2}}\frac{z_{00}}{\sqrt{\kappa^{3}\left(\kappa-\lambda\right)}},~{}$etc. Hence, the fifth equation, i.e. equation (20) is written $\frac{d\mu}{d\tau}=\alpha\mu^{4}+a_{1}\mu^{5}+O\left(\mu^{6}\right)~{}$where $\alpha=\frac{\sqrt{6}\left(\kappa\lambda-2\left(\kappa\lambda+3\right)\sigma\right)}{2\kappa\lambda+6}z_{00}-\frac{6\kappa\left(\sqrt{\lambda\left(\lambda-\kappa\right)}\right)}{2\kappa\lambda+6}u_{10}$. Therefore, the point is always unstable for $a\neq 0$, however from the coefficient term $a_{1}\mu^{5}$ we find that the point can be stable. Figure 5: Qualitative evolution for the real parts of the nonzero eigenvalues of the linearized system around the stationary point $C_{2}$. ### IV.4 Family D The fourth family of stationary points is consists of the following six stationary points $D_{1}^{\pm}=\left(-\sqrt{\frac{3}{2}}\frac{1}{\kappa},0,\pm\frac{\sqrt{\kappa^{2}-3}}{\sqrt{2}\kappa},\frac{1}{\sqrt{2}},0\right),$ (41) $D_{2}^{\pm}=\left(x_{D_{2}},0,\pm z_{D_{2}},\sqrt{1-\left(x_{D_{2}}\right)^{2}-\left(z_{D_{2}}\right)^{2}},\mu_{D_{2}}\right),$ (42) $D_{3}^{\pm}=\left(x_{D_{3}},0,\pm z_{D3},\sqrt{1-\left(x_{D_{3}}\right)^{2}-\left(z_{D_{3}}\right)^{2}},\mu_{D_{3}}\right),$ (43) with $\displaystyle x_{D_{2}}$ $\displaystyle=-\frac{\kappa^{2}(2\sigma-1)+\sqrt{-4\kappa^{4}\sigma+\kappa^{4}+4\left(\kappa^{2}-3\right)^{2}\sigma^{2}}+6\sigma}{\sqrt{6}\kappa(4\sigma-1)},$ $\displaystyle z_{D_{2}}$ $\displaystyle=\frac{\sqrt{-\kappa^{4}(1-2\sigma)^{2}+6\kappa^{2}\sigma\left(8\sigma^{2}-2\sigma+1\right)-\sqrt{-4\kappa^{4}\sigma+\kappa^{4}+4\left(\kappa^{2}-3\right)^{2}\sigma^{2}}\left(\kappa^{2}(2\sigma-1)+24\sigma^{2}\right)-144\sigma^{3}}}{2\sqrt{3}\kappa\sqrt{\sigma}(4\sigma-1)},$ $\displaystyle\mu_{D_{2}}$ $\displaystyle=z_{D_{2}}\frac{\sqrt{6}\left(\kappa^{2}(1-2\sigma)^{2}+2\sigma\left(\sqrt{-4\kappa^{4}\sigma+\kappa^{4}+4\left(\kappa^{2}-3\right)^{2}\sigma^{2}}-6\sigma\right)\right)}{\kappa^{2}(1-2\sigma)^{2}-24\sigma^{2}},$ $\displaystyle x_{D_{3}}$ $\displaystyle=\frac{\kappa^{2}(1-2\sigma)+\sqrt{-4\kappa^{4}\sigma+\kappa^{4}+4\left(\kappa^{2}-3\right)^{2}\sigma^{2}}-6\sigma}{\sqrt{6}\kappa(4\sigma-1)},$ $\displaystyle z_{D_{3}}$ $\displaystyle=\frac{\sqrt{-\kappa^{4}(1-2\sigma)^{2}+6\kappa^{2}\sigma\left(8\sigma^{2}-2\sigma+1\right)+\sqrt{-4\kappa^{4}\sigma+\kappa^{4}+4\left(\kappa^{2}-3\right)^{2}\sigma^{2}}\left(\kappa^{2}(2\sigma-1)+24\sigma^{2}\right)-144\sigma^{3}}}{2\sqrt{3}\kappa\sqrt{\sigma}(4\sigma-1)},$ $\displaystyle\mu_{D_{3}}$ $\displaystyle=z_{D_{3}}\frac{\sqrt{6}\left(2\sigma\left(\sqrt{-4\kappa^{4}\sigma+\kappa^{4}+4\left(\kappa^{2}-3\right)^{2}\sigma^{2}}+6\sigma\right)-\kappa^{2}(1-2\sigma)^{2}\right)}{\kappa^{2}(1-2\sigma)^{2}-24\sigma^{2}}\text{. }$ Points $D_{1}^{\pm}$ describe a scaling solution where the effective fluid is pressureless, that is, it describes a dust fluid source and the scale factor is $a\left(t\right)=a_{0}t^{\frac{2}{3}}$. The physical parameters of the exact solution at points $D_{1}^{\pm}$ are $w_{tot}\left(D_{1}^{\pm}\right)=0~{},~{}w_{\phi}\left(D_{1}^{\pm}\right)=1~{},~{}w_{\psi}\left(D_{1}^{\pm}\right)=\frac{3}{3-2\kappa^{2}}~{},$ (44) $\Omega_{\phi}\left(D_{1}^{\pm}\right)=\frac{3}{2\kappa^{2}}~{},~{}\Omega_{\psi}\left(D_{1}^{\pm}\right)=1-\frac{3}{2\kappa^{2}}.$ (45) Remark that points $D_{1}^{\pm}$ are real when $\left|\kappa\right|>\sqrt{3}$. The eigenvalues of the four-dimensional linearized system around the stationary points $D_{1}^{\pm}$ are derived $\displaystyle e_{1}\left(D_{1}^{\pm}\right)$ $\displaystyle=\frac{3}{2}$ $\displaystyle e_{2}\left(D_{1}^{\pm}\right)$ $\displaystyle=\frac{3}{2}\left(\kappa-\lambda\right)$ $\displaystyle e_{3}\left(D_{1}^{\pm}\right)$ $\displaystyle=-\frac{3+\sqrt{3\left(51-16\kappa^{2}\right)}}{4}$ $\displaystyle e_{4}\left(D_{1}^{\pm}\right)$ $\displaystyle=-\frac{3-\sqrt{3\left(51-16\kappa^{2}\right)}}{4}$ from where we infer that the stationary points $D_{1}^{\pm}$ are always unstable. Points $D_{1}^{\pm}$ are saddle points. Points $D_{2}^{\pm}$ are real and physically accepted when $\left\\{\sigma\in\left(0,\frac{1}{4}\right)\cup\left(\frac{1}{4},\frac{1}{2}\right),\kappa>\frac{2\sqrt{6}\sigma}{2\sigma-1}\right\\}\cup\left\\{\frac{2\sqrt{6}\sigma}{1-2\sigma}<\kappa<-\sqrt{6}\sqrt{\frac{2\sigma^{2}+\sigma\sqrt{4\sigma-1}}{\left(1-2\sigma\right)^{2}}},\sigma>\frac{1}{2}\right\\}$ and $\left\\{\kappa<0,\sigma\,<0\right\\}$ as they are presented in Fig. 6. The exact solution at the stationary points describe a scaling solution with values of the equation of state parameter $w_{tot}\left(\kappa,\sigma\right)$ as they presented in Fig. 6. For the linearized four-dimensional system one of the eigenvalues is $e_{1}\left(D_{2}^{\pm}\right)=\frac{A\left(\kappa,\sigma\right)(2\kappa\sigma-\kappa-2\lambda\sigma)}{4\kappa\sigma(4\sigma-1)\left(2\kappa^{2}\sigma-\kappa^{2}+24\sigma^{2}\right)},$ where $\displaystyle A\left(\kappa,\sigma\right)$ $\displaystyle=4\kappa^{4}\sigma^{2}-4\kappa^{4}\sigma+\kappa^{4}+48\kappa^{2}\sigma^{3}-12\kappa^{2}\sigma^{2}-6\kappa^{2}\sigma$ $\displaystyle+\sqrt{\left(2\kappa^{2}\sigma-\kappa^{2}+24\sigma^{2}\right)^{2}\left(4\kappa^{4}\sigma^{2}-4\kappa^{4}\sigma+\kappa^{4}-24\kappa^{2}\sigma^{2}+36\sigma^{2}\right)}+144\sigma^{3}.$ The other three eigenvalues are only functions of $\kappa,\sigma$, that is $e_{2,3,4}\left(D_{2}^{\pm}\right)=e_{2,3,4}\left(\kappa,\sigma\right)$. Numerically, we find that there are not any values of $\left\\{\kappa,\sigma\right\\}$ where the points $D_{2}^{\pm}$ are defined, such that all the eigenvalues have real part negative, consequently, the stationary points are always sources and the exact solutions at the stationary points $D_{2}^{\pm}$ are always unstable. Figure 6: Left figure: Region plot in the space $\left\\{\kappa,\sigma\right\\}$ where points $D_{2}^{\pm}$ are real and physical accepted. Right Figure: Contour plot of the equation of state parameter for the effective fluid $w_{tot}\left(\kappa,\sigma\right)$ at the critical points $D_{2}^{\pm}$. Figure 7: Left figure: Region plot in the space $\left\\{\kappa,\sigma\right\\}$ where points $D_{3}^{\pm}$ are real and physical accepted. Right Figure: Contour plot of the equation of state parameter for the effective fluid $w_{tot}\left(\kappa,\sigma\right)$ at the critical points $D_{3}^{\pm}$. Stationary points $D_{3}^{\pm}$ have similar physical properties with points $D_{2}^{\pm}$, indeed they describe scaling solutions only. The points are real and physically accepted in the region $\left\\{\sigma>\frac{1}{2},\kappa<-\sqrt{\frac{6\sigma}{\sqrt{4\sigma-1}-2\sigma}}\right\\}$. In Fig. 7 we present the region in the space $\left\\{\sigma,\kappa\right\\}$ where the points are defined as also the counter plot of the equation of state parameter for the effective fluid source which describes the exact solution at the points $D_{3}^{\pm}$. In a similar way with points $D_{2}^{\pm}$ we find that there is not any range in the space $\left\\{\kappa,\sigma\right\\}$ where the points are attractors. Consequently, the stationary points $D_{3}^{\pm}$ are sources. The main physical results of the stationary points are summarized in Table 1. Table 1: The physical propreties of the stationary models in chiral cosmology Point | Contribution of $\phi$ | Contribution of $\psi$ | Scaling/de Sitter | Possible $w_{tot}<-\frac{1}{3}$ | Stability ---|---|---|---|---|--- $A_{1}$ | Yes only kinetic part | No | Scaling | No | Unstable $A_{2}$ | Yes | No | Scaling | Yes | Unstable $B_{1}^{\pm}$ | Yes | Yes only kinetic part | Scaling | Yes | Unstable $B_{2}^{\pm}$ | Yes | Yes only kinetic part | Scaling | Yes | Can be Stable $C_{1}$ | Yes only kinetic | Yes only potential | Scaling | Yes | Unstable $C_{2}$ | Yes only potential | Yes only potential | de Sitter $\left(w_{tot}=-1\right)$ | Always | CMT $D_{1}^{\pm}$ | Yes | Yes | Scaling $\left(w_{tot}=0\right)$ | No | Unstable $D_{2}^{\pm}$ | Yes | Yes | Scaling | Yes | Unstable $D_{3}^{\pm}$ | Yes | Yes | Scaling | Yes | Unstable ## V Application $\left(\kappa,\sigma\right)=\left(2,\frac{1}{2}\right)$ Consider now the case where $\kappa=2$ and $\sigma=\frac{1}{2}$, while $\lambda$ is an arbitrary constant. For that consideration, the stationary points of the dynamical system (20)-(24) have the following coordinates $\displaystyle\bar{A}_{1}^{\pm}$ $\displaystyle=\left(\pm 1,0,0,0,0\right),~{}$ $\displaystyle\bar{A}_{2}$ $\displaystyle=\left(-\frac{\lambda}{\sqrt{6}},\sqrt{1-\frac{\lambda^{2}}{6}},0,0,0\right),$ $\displaystyle\bar{B}_{1}^{\pm}$ $\displaystyle=\left(-\frac{\sqrt{6}}{\lambda+2},\sqrt{\frac{2}{\lambda+2}},\pm\sqrt{\frac{\left(\lambda+1\right)^{2}-7}{\left(\lambda+2\right)^{2}}},0,0\right),~{}$ $\displaystyle\bar{B}_{2}^{\pm}$ $\displaystyle=\left(-\frac{\sqrt{6}}{\lambda+2},\sqrt{\frac{2}{\lambda+2}},\sqrt{\frac{\left(\lambda+1\right)^{2}-7}{\left(\lambda+2\right)^{2}}},0,2\sqrt{\frac{6}{\left(\lambda+1\right)^{2}-7}}\right),$ $\displaystyle\bar{C}_{1}$ $\displaystyle=\left(-\sqrt{\frac{2}{3}},0,0,\frac{1}{\sqrt{3}},0\right),~{}$ $\displaystyle\bar{C}_{2}$ $\displaystyle=\left(0,\left(1-\frac{\lambda}{2}\right)^{-1},0,\sqrt{\frac{\lambda}{\lambda-2}}\right),$ $\displaystyle\bar{D}_{1}^{\pm}$ $\displaystyle=\left(-\frac{1}{2}\sqrt{\frac{3}{2}},0,\frac{1}{2\sqrt{2}},\frac{1}{\sqrt{2}},0\right).$ Points $\bar{A}_{1}^{\pm},~{}\bar{A}_{2}$ are sources and since they do not depend on the parameters $\kappa,\sigma$ their physical properties are the same as before. Recall that point $\bar{A}_{2}$ is real for $\left|\lambda\right|<\sqrt{6}$. Stationary points $\mathbf{B}=\left(\bar{B}_{1}^{\pm},\bar{B}_{2}^{\pm}\right)$ exist when $\lambda>\sqrt{7}-1$. The physical parameters at the points are simplified as follows $w_{tot}\left(\mathbf{B}\right)=\frac{\lambda-2}{\lambda+2}~{},~{}w_{\phi}\left(\mathbf{B}\right)=-1+\frac{6}{5\lambda}~{},~{}w_{\psi}\left(\mathbf{B}\right)=1~{},$ (46) $\Omega_{\phi}\left(\mathbf{B}\right)=\frac{2\left(\lambda+5\right)}{\left(\lambda+2\right)^{2}}~{},~{}\Omega_{\psi}\left(\mathbf{B}\right)=1-\frac{2\left(\lambda+5\right)}{\left(\lambda+2\right)^{2}}.$ (47) The exact solutions at points $\bar{B}_{1}^{\pm}$ are always unstable. However, for points $\bar{B}_{2}^{\pm}$ we find that $e_{2}\left(\bar{B}_{2}^{\pm}\right)>0$ for $\lambda>\sqrt{7}-1$ which means that points $\bar{B}_{2}^{\pm}$ are sources. The parameter for the equation of state $w_{tot}\left(\mathbf{B}\right)$ is constraint as $\frac{\sqrt{7}-3}{\sqrt{7+1}}<w_{tot}\left(B\right)<1$, while for $\lambda=2$, $w_{tot}\left(\mathbf{B}\right)=0$ the exact solutions have the scale factor $a\left(t\right)=a_{0}t^{\frac{2}{3}}$, while for $\lambda=4$, $w_{tot}\left(\mathbf{B}\right)=\frac{1}{3}$, that is $a\left(t\right)=a_{0}t^{\frac{1}{2}}$. Furthermore, stationary point $\bar{C}_{1}$ is a source and describes the radiation epoch, $w_{tot}\left(\bar{C}_{1}\right)=\frac{1}{3}$, on the other hand, at point $\bar{C}_{2}$ the exact solution is that of de Sitter universe, the point is real for $\lambda<0\mathbf{.}$Finally, points $\bar{D}_{1}^{\pm}$ points describe the unstable scaling solutions which describe the matter dominated era, that is, $w_{tot}\left(\bar{D}_{1}^{\pm}\right)=0$. In Figs. 8 and 9, the evolution of the physical variables $\left\\{w_{tot},w_{\phi},w_{\psi},\Omega_{\phi},\Omega_{\psi}\right\\}$ is presented for the specific model for $\lambda=-4$ and $\lambda=-2$ and for different initial conditions for the integration of the dynamical system (20)-(24). Recall that the de Sitter point $\bar{C}_{2}$ is a source; however, it admits a four-dimensional stable manifold when $\mu\rightarrow 0$. We observe that in the de Sitter point the physical parameters $\Omega_{\phi},~{}\Omega_{\psi}$ are not zero which means that the all the parts of the potential $V\left(\phi,\psi\right)$ contributes to the cosmological fluid. The initial conditions have been considered such that to describe a wide range of solutions and different behaviour. The large number of stationary points is observed from the behaviour of $w_{tot}$, which has various maxima before reach the de Sitter point. Similarly from the diagram of $\left\\{\Omega_{\phi},~{}\Omega_{\psi}\right\\}$, we observe that there is a alternation between the domination of the two fields. Figure 8: Evolution of the physical variables $\left\\{w_{tot},w_{\phi},w_{\psi},\Omega_{\phi},\Omega_{\psi}\right\\}$ for numerical solutions of the field equations with $\kappa=2,~{}\sigma=\frac{1}{2}$ and $\lambda=-4$. The plots are for different initial conditions $\left(x\left(0\right),y\left(0\right),z\left(0\right),u\left(0\right),\mu\left(0\right)\right)$ where $\mu\left(0\right)$ has been chosen to be near to zero, such that the de Sitter point $\bar{C}_{2}$ to be an attractor. Figure 9: Evolution of the physical variables $\left\\{w_{tot},w_{\phi},w_{\psi},\Omega_{\phi},\Omega_{\psi}\right\\}$ for numerical solutions of the field equations with $\kappa=2,~{}\sigma=\frac{1}{2}$ and $\lambda=-2$. The plots are for different initial conditions $\left(x\left(0\right),y\left(0\right),z\left(0\right),u\left(0\right),\mu\left(0\right)\right)$ where $\mu\left(0\right)$ has been chosen to be near to zero, such that the de Sitter point $\bar{C}_{2}$ to be an attractor. Consider now the cosmographic parameters $q,~{}j$ and $s$ which are defined as cowei $q\left(x,y,z,u,\mu;\lambda,\kappa,\sigma\right)=-1-\frac{\dot{H}}{H^{2}}$ (48) $j\left(x,y,z,u,\mu;\lambda,\kappa,\sigma\right)=\frac{\ddot{H}}{H^{3}}-3q-2$ (49) $s\left(x,y,z,u,\mu;\lambda,\kappa,\sigma\right)=\frac{H^{\left(3\right)}}{H^{4}}+4j+3q(q+4)+6$ (50) In Fig. 10 we present the evolution of the cosmographic parameters for the application we considered in this example as also for additional values of the free parameters $\left\\{\lambda,\kappa,\sigma\right\\}$, while all the plots are for the same initial conditions. Here we present the qualitative evolution of these parameters, however the cosmographic parameters as also the free parameters of the theory can be constrained by the observations cob1 . Figure 10: Qualitative evolution of the cosmographic parameters $\left\\{q,j,s\right\\}$ for various values of the free parameters $\left\\{\lambda,\kappa,\sigma\right\\}$. The plots of the first row are for $\left\\{\lambda,\kappa,\sigma\right\\}=\left\\{-4,\pm 2,\frac{1}{2}\right\\}$ while the plots of the second row are for $\left\\{\lambda,\kappa,\sigma\right\\}=\left\\{-2,\pm 2,\frac{1}{2}\right\\}$. From the figure we observe that in order the future attractor to be a de Sitter point then $\kappa>0.$ In the following section we continue our analysis by presenting analytic solutions for the model of our study. ## VI Analytic solution We consider the point-like Lagrangian $\mathcal{L}\left(a,\dot{a},\phi,\dot{\phi},\psi,\dot{\psi}\right)=-3a\dot{a}^{2}+\frac{1}{2}a^{3}\left(\dot{\phi}^{2}+e^{\kappa\phi}\dot{\psi}^{2}\right)-a^{3}\left(V_{0}e^{\lambda\phi}+U_{0}\psi^{\frac{1}{\sigma}}e^{\kappa\phi}\right).$ (51) Analytic solutions of form of Lagrangian (51) were presented before in 2sfand . By using the results and the analysis of 2sfand we present an analytic solutions for specific values of the parameters $\left\\{\lambda,\kappa,\sigma\right\\}$ in order to support the results of the previous section. Specifically for the free variables we select $\left(\lambda,\kappa,\sigma\right)=\left(-\frac{\sqrt{6}}{2},-\frac{\sqrt{6}}{2},\frac{1}{2}\right)$. These values are not random. In particular, from the results of 2sfand it follows that for these specific values the field equations admit conservation laws and they form a Liouville integrable dynamical system, such that the field equations can be solved by quadratures. In order to simplify the field equations and write the analytic solution by using closed-form functions, we apply the point transformation $a=\left(xz-\frac{3}{8}y^{2}\right)^{\frac{1}{3}}~{},~{}\phi=-2\sqrt{\frac{2}{3}}\ln\left(\frac{x}{\sqrt{\left(xz-\frac{3}{8}y^{2}\right)}}\right)~{},~{}\psi=\frac{y}{x}$ (52) such that Lagrangian (51) is written as $\mathcal{L}\left(x,\dot{x},y,\dot{y},z,\dot{z}\right)=-\frac{4}{3}\dot{x}\dot{z}-V_{0}x^{2}+\frac{1}{2}\dot{y}^{2}-U_{0}y^{2}.$ (53) In the new coordinates the field equations are $\ddot{x}=0~{},~{}\ddot{y}+2U_{0}y=0~{},~{}\ddot{z}-\frac{3}{2}V_{0}x=0,$ (54) with constraint equation $-\frac{4}{3}\dot{x}\dot{z}+V_{0}x^{2}+\frac{1}{2}\dot{y}^{2}+U_{0}y^{2}=0.$ (55) Easily, we find the exact solution $x=x_{1}t+x_{0}~{},~{}z=\frac{1}{4}V_{0}x_{1}t^{3}+\frac{3}{4}V_{0}x_{0}t^{2}+z_{1}t+z_{0}~{},$ (56) $y\left(t\right)=y_{1}\cos\left(\sqrt{2U_{0}}t\right)+y_{2}\sin\left(\sqrt{2U_{0}}t\right)$ (57) with constraint condition $V_{0}x_{0}^{2}-\frac{4}{3}x_{1}z_{1}+U_{0}\left(y_{1}^{2}+y_{2}^{2}\right)$. For $x_{0}=z_{0}=y_{1}=0$, the scale factor is written $a\left(t\right)=\left(\frac{x_{1}}{4}V_{0}t^{4}+x_{1}z_{1}t^{2}-\frac{3}{8}\left(y_{2}\right)^{2}\sin\left(\sqrt{2U_{0}}t\right)\right)^{\frac{1}{3}}.$ It is easy to observe that the present analytic solution does not provide any de Sitter point. That is in agreement with the result of the previous section, since for $\lambda=\kappa$, the de Sitter point $C_{2}$ does not exist. For more general solutions with expansion eras and de Sitter phases we refer the reader to 2sfand . ## VII With a matter source Let us assume now the presence of an additional pressureless matter source in field equations with energy density $\rho_{m}$ and let us discuss the existence of additional stationary points. For a pressureless fluid source the dimensionless field equations (20)-(25) remain the same, while the constraint equation (28) becomes $\Omega_{m}=1-x^{2}-y^{2}-z^{2}-u^{2}$ (58) where $\Omega_{m}=\frac{\rho_{m}}{3H^{2}}$, and $0\leq\Omega_{m}\leq 1$. For this model, the stationary points found before exist and give $\Omega_{m}=0$, while when $\Omega_{m}\neq 0$ the additional points exist $E_{1}=\left(-\sqrt{\frac{3}{2}}\frac{1}{\lambda},\sqrt{\frac{3}{2}}\frac{1}{\lambda},0,0,0\right)~{},~{}E_{2}=\left(-\sqrt{\frac{3}{2}}\frac{1}{\kappa},0,0,\sqrt{\frac{3}{2}}\frac{1}{\kappa},0\right)$ (59) $E_{3}=\left(-\sqrt{\frac{3}{2}}\frac{1}{\kappa},0,z,\sqrt{\frac{3}{2}+\kappa^{2}z^{2}}\frac{1}{\kappa},0\right)$ (60) Point $E_{1}$ is physically accepted when $\left|\lambda\right|>\sqrt{\frac{3}{2}}$ and describes the tracking solution with $\Omega_{m}\left(E_{1}\right)=1-\frac{3}{\lambda^{2}}$ where the field $\phi$ mimics the ideal gas $\rho_{m},~{}$that is, $w_{\phi}\left(E_{1}\right)=0,$ while the second field $\psi$ does not contribute, i.e. $z\left(E_{1}\right)=u\left(E_{1}\right)=0$. For $E_{2}$ we find $\left(w_{tot}\left(E_{2}\right),w_{\phi}\left(E_{2}\right),w_{\psi}\left(E_{2}\right),\Omega_{\phi}\left(E_{2}\right),\Omega_{\psi}\left(E_{2}\right)\right)=\left(0,1,-1,\frac{3}{2\kappa^{2}},\frac{3}{2\kappa^{2}}\right)$, which means that it is another tracking tracking solution with $\Omega_{m}=1-\frac{3}{\kappa^{2}}$; the point is physically accepted when $\left|\kappa\right|\geq\sqrt{\frac{3}{2}}$. $E_{3}$ does not describe one point, but a family of points on the surface $u\left(z\right)=$ $\sqrt{\frac{3}{2}+\kappa^{2}z^{2}}$ , for $x\left(E_{3}\right)=-\sqrt{\frac{3}{2}}\frac{1}{\kappa},$ $y\left(E_{3}\right)=\mu\left(E_{3}\right)=0$. It describes a tracking solution, that is $w_{tot}\left(E_{3}\right)=0$, with physical parameters $\left(w_{tot}\left(E_{3}\right),w_{\phi}\left(E_{3}\right),w_{\psi}\left(E_{3}\right),\Omega_{\phi}\left(E_{3}\right),\Omega_{\psi}\left(E_{3}\right)\right)=\left(0,1,-\frac{3}{4+3\kappa^{2}z^{2}},\frac{3}{2\kappa^{2}},2z^{2}+\frac{3}{2\kappa^{2}}\right),$ (61) while the point is physically accepted when $\left|\kappa\right|\geq\sqrt{\frac{3}{2}}$ and $\left|z\right|\leq\frac{1}{2}\sqrt{2-\frac{3}{\kappa^{2}}}$. When $z\left(E_{3}\right)=0$, then $E_{3}$ reduces to $E_{2}$. What it is important, to mention is that the stability analysis for all the previous points changes, since we made use of the constraint equation (28). ## VIII Conclusions In this work we performed a detailed study of the dynamics for a two scalar field model with a mixed potential term known as Chiral model. The purpose of our analysis was to study the cosmological evolution of that specific model as also the cosmological viability of the model and which epochs of the cosmological evolution can be described by the Chiral model. For the scalar field potential we assumed that it is of the form $V\left(\phi,\psi\right)=V_{0}e^{\lambda\phi}+U_{0}\psi^{\frac{1}{\sigma}}e^{\kappa\phi}$. For this consideration and without assuming the existence of additional matter source, we found four families of stationary points which provide nine different cosmological solutions. Eight of the cosmological solutions are scaling solutions which describe spacetimes with a a perfect fluid with a constant equation of state parameter $w\left(P\right)$. One of the scaling solutions describes a universe with a stiff matter, $w\left(P\right)=1$, another scaling solution correspond to a universe with a pressureless fluid source, $w\left(P\right)=0$, while for the rest six scaling solutions $w\left(P\right)=w\left(P,\lambda,\kappa,\sigma\right)$, which can describe accelerated eras for for specific values of the free parameters $\left\\{\lambda,\kappa,\sigma\right\\}$. Moreover, the ninth exact cosmological solution which was found from the analysis of the stationary points describes a de Sitter universe. As far as the stability of the exact solutions at the stationary points is concerned, seven of the points are always unstable. while only the set of the points $B_{2}^{\pm}~{}$can be stable. Point $C_{2}$ which describes the de Sitter universe, has one eigenvalue negative while the rest of the eigenvalues are always negative. 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2024-09-04T02:54:59.224615

2003.05370

arxiv-papers

# Dividing the Ontology Alignment Task with Semantic Embeddings and Logic-based Modules††Accepted to the 24th European Conference on Artificial Intelligence (ECAI 2020) Ernesto Jiménez-Ruiz City, University of London, UK, email: ernesto.jimenez- <EMAIL_ADDRESS>and Department of Informatics, University of Oslo, NorwaySection for Artificial Intelligence and Decision Support, Medical University of Vienna, Vienna, AustriaDepartment of Computer Science, University of Oxford, UKMiami University, Oxford, OH 45056, United States Asan Agibetov Section for Artificial Intelligence and Decision Support, Medical University of Vienna, Vienna, AustriaDepartment of Computer Science, University of Oxford, UKMiami University, Oxford, OH 45056, United States Jiaoyan Chen Department of Computer Science, University of Oxford, UKMiami University, Oxford, OH 45056, United States Matthias Samwald3 Miami University, Oxford, OH 45056, United States Valerie Cross Miami University, Oxford, OH 45056, United States ###### Abstract Large ontologies still pose serious challenges to state-of-the-art ontology alignment systems. In this paper we present an approach that combines a neural embedding model and logic-based modules to accurately divide an input ontology matching task into smaller and more tractable matching (sub)tasks. We have conducted a comprehensive evaluation using the datasets of the Ontology Alignment Evaluation Initiative. The results are encouraging and suggest that the proposed method is adequate in practice and can be integrated within the workflow of systems unable to cope with very large ontologies. ## 1 Introduction The problem of (semi-)automatically computing an alignment between independently developed ontologies has been extensively studied in the last years. As a result, a number of sophisticated ontology alignment systems currently exist [44, 15].222Ontology matching surveys and approaches: http://ontologymatching.org/ The Ontology Alignment Evaluation Initiative333OAEI evaluation campaigns: http://oaei.ontologymatching.org/ (OAEI) [3, 4] has played a key role in the benchmarking of these systems by facilitating their comparison on the same basis and the reproducibility of the results. The OAEI includes different tracks organised by different research groups. Each track contains one or more matching tasks involving small-size (e.g., conference), medium-size (e.g., anatomy), large (e.g., phenotype) or very large (e.g., largebio) ontologies. Some tracks only involve matching at the terminological level (e.g., concepts and properties) while other tracks also expect an alignment at the assertional level (i.e., instance data). Large ontologies still pose serious challenges to ontology alignment systems. For example, several systems participating in the _largebio track_ were unable to complete the largest tasks during the latest OAEI campaigns.444Largebio track: http://www.cs.ox.ac.uk/isg/projects/SEALS/oaei/ These systems typically use advanced alignment methods and are able to cope with small and medium size ontologies with competitive results, but fail to complete large tasks in a given time frame or with the available resources such as memory. There have been several efforts in the literature to divide the ontology alignment task (e.g., [20, 22]). These approaches, however, have not been successfully evaluated with very large ontologies, failing to scale or producing partitions of the ontologies leading to information loss [42]. In this paper we propose a novel method to accurately divide the matching task into several independent, smaller and manageable (sub)tasks, so as to scale systems that cannot cope with very large ontologies.555A preliminary version of this work has been published in arXiv [25] and in the Ontology Matching workshop [26]. Unlike state-of-the-art approaches, our method: (i) preserves the coverage of the relevant ontology alignments while keeping manageable matching subtasks; (ii) provides a formal notion of matching subtask and semantic context; (iii) uses neural embeddings to compute an accurate division by learning semantic similarities between words and ontology entities according to the ontology alignment task at hand; (iv) computes self- contained (logical) modules to guarantee the inclusion of the (semantically) relevant information required by an alignment system; and (v) has been successfully evaluated with very large ontologies. ## 2 Preliminaries A _mapping_ (also called match) between entities666In this work we accept any input ontology in the OWL 2 language [18]. We refer to (OWL 2) concepts, properties and individuals as entities. of two ontologies $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ is typically represented as a 4-tuple $\langle e_{1},\allowbreak e_{2},\allowbreak r,\allowbreak c\rangle$ where $e_{1}$ and $e_{2}$ are entities of $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$, respectively; $r$ is a semantic relation, typically one of $\\{\sqsubseteq,\sqsupseteq,\equiv\\}$; and $c$ is a confidence value, usually, a real number within the interval $\left(0,1\right]$. For simplicity, we refer to a mapping as a pair $\langle e_{1},\allowbreak e_{2}\rangle$. An ontology _alignment_ is a set of mappings $\mathcal{M}$ between two ontologies $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. An ontology _matching task_ $\mathcal{M}\mathcal{T}$ is composed of a pair of ontologies $\mathcal{O}_{1}$ (typically called source) and $\mathcal{O}_{2}$ (typically called target) and possibly an associated _reference alignment_ $\mathcal{M}^{RA}$. The objective of a matching task is to discover an overlapping of $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ in the form of an alignment $\mathcal{M}$. The _size_ or _search space_ of a matching task is typically bound to the size of the Cartesian product between the entities of the input ontologies: $\lvert Sig(\mathcal{O}_{1})\rvert\times\lvert Sig(\mathcal{O}_{2})\rvert$, where $Sig(\mathcal{O})$ denotes the signature (i.e., entities) of $\mathcal{O}$ and $\lvert\cdot\lvert$ denotes the size of a set. An ontology _matching system_ is a program that, given as input a matching task $\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$, generates an ontology alignment $\mathcal{M}^{S}$.777Typically automatic, although there are systems that also allow human interaction [32]. The standard evaluation measures for an alignment $\mathcal{M}^{S}$ are _precision_ (P), _recall_ (R) and _f-measure_ (F) computed against a reference alignment $\mathcal{M}^{RA}$ as follows: $P=\frac{\lvert\mathcal{M}^{S}\cap\mathcal{M}^{RA}\rvert}{\lvert\mathcal{M}^{S}\rvert},~{}R=\frac{\lvert\mathcal{M}^{S}\cap\mathcal{M}^{RA}\rvert}{\lvert\mathcal{M}^{RA}\rvert},~{}F=2\cdot\frac{P\cdot R}{P+R}$ (1) Figure 1: Pipeline to divide a given matching task $\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$. ### 2.1 Problem definition and quality measures We denote _division_ of an ontology matching task $\mathcal{M}\mathcal{T}$, composed by the ontologies $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$, as the process of finding $n$ matching subtasks $\mathcal{M}\mathcal{T}_{i}=\langle\mathcal{O}_{1}^{i},\mathcal{O}_{2}^{i}\rangle$ (with $i$=$1$,…,$n$), where $\mathcal{O}_{1}^{i}\subset\mathcal{O}_{1}$ and $\mathcal{O}_{2}^{i}\subset\mathcal{O}_{2}$. Size of the division. The size of each matching subtask is smaller than the original task and thus reduces the search space. Let $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}=\\{\mathcal{M}\mathcal{T}_{1},\ldots,\mathcal{M}\mathcal{T}_{n}\\}$ be the division of a matching task $\mathcal{M}\mathcal{T}$ into $n$ subtasks. The _size ratio_ of the subtasks $\mathcal{M}\mathcal{T}_{i}$ and $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ with respect to the original matching task size is computed as follows: $\mathsf{SizeRatio}(\mathcal{M}\mathcal{T}_{i},\mathcal{M}\mathcal{T})=\frac{\lvert Sig(\mathcal{O}_{1}^{i})\rvert\times\lvert Sig(\mathcal{O}_{2}^{i})\rvert}{\lvert Sig(\mathcal{O}_{1})\rvert\times\lvert Sig(\mathcal{O}_{2})\rvert}$ (2) $\mathsf{SizeRatio}(\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n},\mathcal{M}\mathcal{T})=\sum_{i=1}^{n}\mathsf{SizeRatio}(\mathcal{M}\mathcal{T}_{i},\mathcal{M}\mathcal{T})$ (3) The ratio $\mathsf{SizeRatio}(\mathcal{M}\mathcal{T}_{i},\mathcal{M}\mathcal{T})$ is less than $1.0$ while the aggregation $\sum_{i=1}^{n}\mathsf{SizeRatio}(\mathcal{M}\mathcal{T}_{i},\mathcal{M}\mathcal{T})$, being $n$ the number of matching subtasks, can be greater than $1.0$ as matching subtasks depend on the division technique and may overlap. Alignment coverage. The division of the matching task aims at preserving the target outcomes of the original matching task. The _coverage_ is calculated with respect to a relevant alignment $\mathcal{M}$, possibly the reference alignment $\mathcal{M}^{RA}$ of the matching task if it exists, and indicates whether that alignment can still be (potentially) discovered with the matching subtasks. The formal notion of coverage is given in Definitions 1 and 2. ###### Definition 1 (Coverage of a matching task) Let $\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$ be a matching task and $\mathcal{M}$ an alignment. We say that a mapping $m=\langle e_{1},\allowbreak e_{2}\rangle\in\mathcal{M}$ is covered by the matching task if $e_{1}\in Sig(\mathcal{O}_{1})$ and $e_{2}\in Sig(\mathcal{O}_{2})$. The coverage of $\mathcal{M}\mathcal{T}$ w.r.t. $\mathcal{M}$ (denoted as $\mathsf{Coverage}(\mathcal{M}\mathcal{T},\mathcal{M})$) represents the set of mappings $\mathcal{M}^{\prime}\subseteq\mathcal{M}$ covered by $\mathcal{M}\mathcal{T}$. ###### Definition 2 (Coverage of the matching task division) Let $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}=\\{\mathcal{M}\mathcal{T}_{1},\ldots,\mathcal{M}\mathcal{T}_{n}\\}$ be the result of dividing a matching task $\mathcal{M}\mathcal{T}$ and $\mathcal{M}$ an alignment. We say that a mapping $m\in\mathcal{M}$ is covered by $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ if $m$ is at least covered by one of the matching subtask $\mathcal{M}\mathcal{T}_{i}$ (with $i$=$1$,…,$n$) as in Definition 1. The coverage of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ w.r.t. $\mathcal{M}$ (denoted as $\mathsf{Coverage}(\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n},\mathcal{M})$) represents the set of mappings $\mathcal{M}^{\prime}\subseteq\mathcal{M}$ covered by $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$. The coverage is given as a ratio with respect to the (covered) alignment: $\mathsf{CoverageRatio}(\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n},\mathcal{M})=\frac{\lvert\mathsf{Coverage}(\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n},\mathcal{M})\rvert}{\lvert\mathcal{M}\rvert}$ (4) ## 3 Methods In this section we present our approach to compute a division $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}=\\{\mathcal{M}\mathcal{T}_{1},\ldots,\mathcal{M}\mathcal{T}_{n}\\}$ given a matching task $\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$ and the number of target subtasks $n$. We rely on locality ontology modules to extract self-contained modules of the input ontologies. The module extraction and task division is tailored to the ontology alignment task at hand by embedding the contextual semantics of a (combined) inverted index of the ontologies in the matching task. Figure 1 shows an overview of our approach. (i) The ontologies $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ are indexed using the lexical index LexI (see Section 3.2); (ii) LexI is divided into clusters based on the semantic embeddings of its entries (see Section 3.4); (iii) entries in those clusters derive potential mapping sets (see Section 3.3); and (iv) the context of these mapping sets lead to matching subtasks (see Sections 3.1 and 3.3). Next, we elaborate on the methods behind these steps. ### 3.1 Locality modules and context Logic-based module extraction techniques compute ontology fragments that capture the meaning of an input signature (e.g., set of entities) with respect to a given ontology. That is, a module contains the context (i.e., sets of _semantically related_ entities) of the input signature. In this paper we rely on bottom-locality modules [13, 29], which will be referred to as locality- modules or simply as modules. These modules include the ontology axioms required to describe the entities in the signature. Locality-modules compute self-contained ontologies and are tailored to tasks that require reusing a fragment of an ontology. Please refer to [13, 29] for further details. Locality-modules play an key role in our approach as they provide the context for the entities in a given mapping or set of mappings as formally presented in Definition 3. ###### Definition 3 (Context of a mapping and an alignment) Let $m=\langle e_{1},\allowbreak e_{2}\rangle$ be a mapping between two ontologies $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. We define the context of $m$ (denoted as $\mathsf{Context}(m,\mathcal{O}_{1},\mathcal{O}_{2})$) as a pair of locality modules $\mathcal{O}_{1}^{\prime}\subseteq\mathcal{O}_{1}$ and $\mathcal{O}_{2}^{\prime}\subseteq\mathcal{O}_{2}$, where $\mathcal{O}_{1}^{\prime}$ and $\mathcal{O}_{2}^{\prime}$ include the semantically related entities to $e_{1}$ and $e_{2}$, respectively. Similarly, the _context_ for an alignment $\mathcal{M}$ between two ontologies $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ is denoted as $\mathsf{Context}(\mathcal{M},\mathcal{O}_{1},\mathcal{O}_{2})=\langle\mathcal{O}_{1}^{\prime},\mathcal{O}_{2}^{\prime}\rangle$, where $\mathcal{O}_{1}^{\prime}$ and $\mathcal{O}_{2}^{\prime}$ are modules including the semantically related entities for the entities $e_{1}\in Sig(\mathcal{O}_{1})$ and $e_{2}\in Sig(\mathcal{O}_{2})$ in each mapping $m=\langle e_{1},\allowbreak e_{2}\rangle\in\mathcal{M}$. Intuitively, as the context of an alignment (i.e., $\mathsf{Context}(\mathcal{M},\mathcal{O}_{1},\mathcal{O}_{2})=\langle\mathcal{O}_{1}^{\prime},\mathcal{O}_{2}^{\prime}\rangle$) semantically characterises the entities involved in that alignment, a matching task $\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$ can be reduced to the task $\mathcal{M}\mathcal{T}^{\mathcal{M}}_{\mathcal{O}_{1}\text{-}\mathcal{O}_{2}}=\langle\mathcal{O}_{1}^{\prime},\mathcal{O}_{2}^{\prime}\rangle$ without information loss in terms of finding $\mathcal{M}$ (i.e., $\mathsf{Coverage}(\mathcal{M}\mathcal{T}^{\mathcal{M}}_{\mathcal{O}_{1}\text{-}\mathcal{O}_{2}},\mathcal{M})=\mathcal{M}$). For example, in the small OAEI _largebio_ tasks [3, 4] systems are given the context of the reference alignment as a (reduced) matching task (e.g., $\mathcal{M}\mathcal{T}^{RA}_{\text{fma- nci}}=\mathsf{Context}(\mathcal{M}^{RA}_{\text{fma-nci}},\ \mathcal{O}_{\text{FMA}},\mathcal{O}_{\text{NCI}})=\langle\mathcal{O}_{\text{FMA}}^{\prime},\mathcal{O}_{\text{NCI}}^{\prime}\rangle$), instead of the whole FMA and NCI ontologies. Table 1: Inverted lexical index LexI. For readability, index values have been split into elements of $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. ‘-’ indicates that the ontology does not contain entities for that entry. | # | Index key | Index value ---|---|--- Entities $\mathcal{O}_{1}$ | Entities $\mathcal{O}_{2}$ 1 | $\\{$ disorder $\\}$ | $\mathcal{O}_{1}$:Disorder_of_pregnancy, $\mathcal{O}_{1}$:Disorder_of_stomach | $\mathcal{O}_{2}$:Pregnancy_Disorder 2 | $\\{$ disorder, pregnancy $\\}$ | $\mathcal{O}_{1}$:Disorder_of_pregnancy | $\mathcal{O}_{2}$:Pregnancy_Disorder 3 | $\\{$ carcinoma, basaloid $\\}$ | $\mathcal{O}_{1}$:Basaloid_carcinoma | $\mathcal{O}_{2}$:Basaloid_Carcinoma, $\mathcal{O}_{2}$:Basaloid_Lung_Carcinoma 4 | $\\{$ follicul, thyroid, carcinom $\\}$ | $\mathcal{O}_{1}$:Follicular_thyroid_carcinoma | $\mathcal{O}_{2}$:Follicular_Thyroid_carcinoma 5 | $\\{$ hamate, lunate $\\}$ | $\mathcal{O}_{1}$:Lunate_facet_of_hamate | - ### 3.2 Indexing the ontology vocabulary We rely on a semantic inverted index (we will refer to this index as LexI). This index maps sets of words to the entities where these words appear. LexI encodes the labels of all entities of the input ontologies $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$, including their lexical variations (e.g., preferred labels, synonyms), in the form of _key-value_ pairs where the key is a set of words and the value is a set of entities such that the set of words of the key appears in (one of) the entity labels. Similar indexes are commonly used in information retrieval applications [11], Entity Resolution systems [40], and also exploited in ontology alignment systems (e.g., LogMap [27], ServOMap [14] and AML [16]) to reduce the search space and enhance the matching process. Table 1 shows a few example entries of LexI for two input ontologies. LexI is created as follows. (i) Each label associated to an ontology entity is split into a set of words; for example, the label “Lunate facet of hamate” is split into the set {“lunate”, “facet”, “of”, “hamate”}. (ii) Stop-words are removed from the set of words. (iii) Stemming techniques are applied to each word (i.e., {“lunat”, “facet”, “hamat”}). (iv) Combinations of subsets of words also serve as keys in LexI; for example, {“lunat”, “facet”}, {“hamat”, “lunat”} and so on.888In order to avoid a combinatorial blow-up, the number of computed subsets of words is limited. (v) Entities leading to the same (sub)set of words are associated to the same key in LexI, for example {“disorder”} is associated with three entities. Finally, (vi) entries in LexI pointing to entities of only one ontology or associated to a number of entities larger than $\alpha$ are not considered.999In the experiments we used $\alpha=60$. Note that a single entity label may lead to several entries in LexI, and each entry in LexI points to one or more entities. ### 3.3 Covering matching subtasks Each entry (i.e., a _key-value_ pair) in LexI is a source of candidate mappings. For instance, the example in Table 1 suggests that there is a candidate mapping $m=\langle\mathsf{\mathcal{O}_{1}\negthickspace:\negthickspace Disorder\\_of\\_stomach},\allowbreak\mathsf{\mathcal{O}_{2}\negthickspace:\negthickspace Pregnancy\\_disorder}\rangle$ since these entities are associated to the _{ “disorder”}_ entry in LexI. These mappings are not necessarily correct but will link lexically-related entities, that is, those entities sharing at least one word among their labels (e.g., “disorder”). Given a subset of entries or rows of LexI (i.e., $l\subseteq\textsf{LexI}$), the function $\mathsf{Mappings}(l)=\mathcal{M}^{l}$ provides the set of mappings derived from $l$. We refer to the set of all (potential) mappings suggested by LexI (i.e., $\mathsf{Mappings}(\textsf{LexI})$) as $\mathcal{M}^{\textsf{LexI}}$. $\mathcal{M}^{\textsf{LexI}}$ represents a manageable subset of the Cartesian product between the entities of the input ontologies. For example, LexI suggest around $2\times 10^{4}$ potential mappings for the matching task $\mathcal{M}\mathcal{T}_{\text{fma- nci}}=\langle\mathcal{O}_{\text{FMA}},\mathcal{O}_{\text{NCI}}\rangle$, while the Cartesian product between $\mathcal{O}_{\text{FMA}}$ and $\mathcal{O}_{\text{NCI}}$ involves more than $5\times 10^{9}$ mappings. Since standard ontology alignment systems rarely discover mappings outside $\mathcal{M}^{\textsf{LexI}}$, the context of $\mathcal{M}^{\textsf{LexI}}$ (recall Definition 3) can be seen as a reduced matching task $\mathcal{M}\mathcal{T}^{\textsf{LexI}}=\mathsf{Context}(\mathcal{M}^{\textsf{LexI}},\mathcal{O}_{1},\mathcal{O}_{2})=\langle\mathcal{O}_{1}^{\textsf{LexI}},\mathcal{O}_{2}^{\textsf{LexI}}\rangle$ of the original task $\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$. However, the modules $\mathcal{O}_{1}^{\textsf{LexI}}$ and $\mathcal{O}_{2}^{\textsf{LexI}}$, although smaller than $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$, can still be challenging for many ontology matching systems. A solution is to divide or cluster the entries in LexI to lead to several tasks involving smaller ontology modules. ###### Definition 4 (Matching subtasks from LexI) Let $\mathcal{M}\mathcal{T}=\langle\mathcal{O}_{1},\mathcal{O}_{2}\rangle$ be a matching task, _LexI_ the inverted index of the ontologies $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$, and $\\{l_{1},\ldots,l_{n}\\}$ a set of $n$ clusters of entries in _LexI_. We denote the set of matching subtasks from _LexI_ as $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}=\\{\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{1},\ldots,\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{n}\\}$ where each cluster $l_{i}$ leads to the matching subtask $\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{i}=\langle\mathcal{O}_{1}^{i},\mathcal{O}_{2}^{i}\rangle$, such that $\mathsf{Mappings}(l_{i})=\mathcal{M}^{\textsf{LexI}}_{i}$ is the set of mappings suggested by the _LexI_ entries in $l_{i}$ (i.e., _key-value_ pairs) and $\mathcal{O}_{1}^{i}$ and $\mathcal{O}_{2}^{i}$ represent the context of $\mathcal{M}^{\textsf{LexI}}_{i}$ w.r.t. $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. Quality of the matching subtasks. The matching subtasks in Definition 4 rely on LexI and the notion of context, thus it is expected that the tasks in $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ will cover most of the mappings $\mathcal{M}^{S}$ that a matching system can compute, that is $\mathsf{CoverageRatio}(\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n},\mathcal{M}^{S})$ will be close to $1.0$. Furthermore, the use of locality modules to compute the context guarantees the extraction of matching subtasks that are suitable to ontology alignment systems in terms of preservation of the logical properties of the given signature. Intuitively each cluster of LexI will lead to a smaller matching task $\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{i}$ (with respect to both $\mathcal{M}\mathcal{T}^{\textsf{LexI}}$ and $\mathcal{M}\mathcal{T}$) in terms of search space. Hence $\mathsf{SizeRatio}(\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{i},\mathcal{M}\mathcal{T})$ will be smaller than $1.0$. The overall aggregation of ratios (cf. Equation 3) depends on the clustering strategy of the entries in LexI and it is also expected to be smaller than $1.0$. Reducing the search space in each matching subtask $\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{i}$ has the potential of enabling the evaluation of systems that cannot cope with the original matching task $\mathcal{M}\mathcal{T}$ in a given time-frame or with (limited) computational resources. Table 2: Matching tasks. AMA: Adult Mouse Anatomy. DOID: Human Disease Ontology. FMA: Foundational Model of Anatomy. HPO: Human Phenotype Ontology. MP: Mammalian Phenotype. NCI: National Cancer Institute Thesaurus. NCIA: Anatomy fragment of NCI. ORDO: Orphanet Rare Disease Ontology. SNOMED CT: Systematized Nomenclature of Medicine – Clinical Terms. Phenotype ontologies downloaded from BioPortal. For all tracks we use the consensus with vote=3 as system mappings $\mathcal{M}^{S}$. The Phenotype track does not have a gold standard so a consensus alignment with vote=2 is used as reference. OAEI track | Source of $\mathcal{M}^{RA}$ | Source of $\mathcal{M}^{S}$ | Task | Ontology | Version | Size (classes) ---|---|---|---|---|---|--- Anatomy | Manually created [10] | Consensus (vote=3) | AMA-NCIA | AMA | v.2007 | 2,744 NCIA | v.2007 | 3,304 Largebio | UMLS-Metathesaurus [28] | Consensus (vote=3) | FMA-NCI | FMA | v.2.0 | 78,989 FMA-SNOMED | NCI | v.08.05d | 66,724 SNOMED-NCI | SNOMED CT | v.2009 | 306,591 Phenotype | Consensus alignment (vote=2) [21] | Consensus (vote=3) | HPO-MP | HPO | v.2016 | 11,786 MP | v.2016 | 11,721 DOID-ORDO | DOID | v.2016 | 9,248 ORDO | v.2016 | 12,936 ### 3.4 Semantic embeddings We use a _semantic embedding_ approach to identify, given $n$, a set of clusters of entries $\\{l_{1},\ldots,l_{n}\\}$ from LexI. As in Definition 4, these clusters lead to the set of matching subtasks $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}=\\{\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{1},\ldots,\mathcal{M}\mathcal{T}^{\textsf{LexI}}_{n}\\}$. The _semantic embeddings_ aim at representing into the same (vector) space the features about the relationships among words and ontology entities that occur in LexI. Hence, words and entities that belong to similar semantic contexts will typically have similar vector representations. Embedding model. Our approach currently relies on the StarSpace toolkit101010StarSpace: https://github.com/facebookresearch/StarSpace and its neural embedding model [49] to learn _embeddings_ for the words and ontology entities in LexI. We adopt the _TagSpace_ [48] training setting of StarSpace. Applied to our setting, StarSpace learns associations between a set of words (i.e., keys in LexI) and a set of relevant ontology entities (i.e., values in LexI). The StarSpace model is trained by assigning a $d$-dimensional vector to each of the relevant features (e.g., the individual words and the ontology entities in LexI). Ultimately, the look-up matrix (the matrix of embeddings - latent vectors) is learned by minimising the loss function in Equation 5. $\\!\sum_{\begin{subarray}{c}(w,e)\in E^{+},\\\ e^{-}\in E^{-}\end{subarray}}L^{batch}(sim(\bm{v}_{w},\bm{v}_{e}),sim(\bm{v}_{w},\bm{v}_{e_{1}^{-}}),\ldots,\\\ sim(\bm{v}_{w},\bm{v}_{e_{j}^{-}}))$ (5) In this loss function we compare positive samples with negative samples. Hence we need to indicate the generator of positive pairs $(w,e)\in E^{+}$ (in our setting those are _word-entity_ pairs from LexI) and the generator of negative entries $e^{-}\in E^{-}$ (in our case we sample from the list of entities in the values of LexI). StarSpace follows the strategy by Mikolov et al. [36] and selects a random subset of $j$ negative examples for each batch update. Note that we tailor the generators to the alignment task by sampling from LexI. The similarity function $sim$ operates on $d$-dimensional vectors (e.g., $\bm{v}_{w}$, $\bm{v}_{e}$ and $\bm{v}_{e}^{-}$), in our case we use the standard dot product in Euclidean space. Clustering strategy. The semantic embedding of each entry $\varepsilon=(K,V)\in$ LexI is calculated by concatenating (i) the mean vector representation of the vectors associated to each word in the key $K$, with (ii) the mean vector of the vectors of the ontology entities in the value $V$, as in Equation 6, where $\oplus$ represents the concatenation of two vectors, $\bm{v}_{w}$ and $\bm{v}_{e}$ represents $d$-dimensional vector embeddings learnt by StarSpace, and $\bm{v}_{\varepsilon}$ is a ($2*d$)-dimension vector. $\bm{v}_{\varepsilon}=\frac{1}{|K|}\sum_{w\in K}\bm{v}_{w}\oplus\frac{1}{|V|}\sum_{e\in V}\bm{v}_{e}$ (6) Based on the embeddings $\bm{v}_{\varepsilon}$ we then perform standard clustering with the K-means algorithm to obtain the clusters of LexI entries $\\{l_{1},\ldots,l_{n}\\}$. For example, following our approach, in the example of Table 1 entries in rows $1$ and $2$ (respectively $3$ and $4$) would belong to the same cluster. Suitability of the embedding model. Although we could have followed other embedding strategies, we advocated to learn new entity embeddings with StarSpace for the following reasons: (i) ontologies, particularly in the biomedical domain, may bring specialised vocabulary that is not fully covered by precomputed word embeddings; (ii) to embed not only words but also concepts of both ontologies; and (iii) to obtain embeddings tailored to the ontology alignment task (i.e., to learn similarities among words and concepts dependant on the task). StarSpace provides the required functionalities to embed the semantics of LexI and identify accurate clusters. Precise clusters will lead to smaller matching tasks, and thus, to a reduced global size of the computed division of the matching task $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ (cf. Equation 3). ## 4 Evaluation In this section we provide empirical evidence to support the suitability of the proposed method to divide the ontology alignment task. We rely on the datasets of the Ontology Alignment Evaluation Initiative (OAEI) [3, 4], more specifically, on the matching tasks provided in the _anatomy_ , _largebio_ and _phenotype_ tracks. Table 2 provides an overview of these OAEI tasks and the related ontologies and mapping sets. The methods have been implemented in Java111111Java codes: https://github.com/ernestojimenezruiz/logmap-matcher and Python121212Python codes: https://github.com/plumdeq/neuro-onto-part (neural embedding strategy), tested on a Ubuntu Laptop with an Intel Core i9-8950HK<EMAIL_ADDRESS>and allocating up to $25Gb$ of RAM. Datasets, matching subtasks, computed mappings and other supporting resources are available in the _Zenodo_ repository [24]. For all of our experiments we used the following StarSpace hyperparameters: -trainMode 0 -similarity dot --epoch 100 --dim 64. (a) $\mathsf{CoverageRatio}$ of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ over $\mathcal{M}^{RA}$ (b) $\mathsf{CoverageRatio}$ of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ over $\mathcal{M}^{S}$ (c) $\mathsf{SizeRatio}$ of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ (d) Module sizes of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ for FMA-NCI Figure 2: Quality measures of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ with respect to the number of matching subtasks $n$. ### 4.1 Adequacy of the division approach We have evaluated the adequacy of our division strategy in terms of coverage (as in Equation 4) and size (as in Equation 3) of the resulting division $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ for each of the matching task in Table 2. Coverage ratio. Figures 2(a) and 2(b) shows the coverage of the different divisions $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ with respect to the reference alignment and system computed mappings, respectively. As system mappings we have used the consensus alignment with vote=3, that is, mappings that have been voted by at least $3$ systems in the last OAEI campaigns. The overall coverage results are encouraging: (i) the divisions $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ cover over $94\%$ of the reference alignments for all tasks, with the exception of the SNOMED-NCI case where coverage ranges from $0.94$ to $0.90$; (ii) when considering system mappings, the coverage for all divisions is over $0.98$ with the exception of AMA-NCIA, where it ranges from $0.956$ to $0.974$; (iii) increasing the number of divisions $n$ tends to slightly decrease the coverage in some of the test cases, this is an expected behaviour as the computed divisions include different semantic contexts (i.e., locality modules) and some relevant entities may fall out the division; finally (iv) as shown in [42], the results in terms of coverage of state-of-the-art partitioning methods (e.g., [22, 20]) are very low for the OAEI _largebio_ track ($0.76$, $0.59$ and $0.67$ as the best results for FMA-NCI, FMA-SNOMED and SNOMED-NCI, respectively), thus, making the obtained results even more valuable. Size ratio. The results in terms of the size (i.e., search space) of the selected divisions $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ are presented in Figure 2(c). The search space is improved with respect to the original $\mathcal{M}\mathcal{T}$ for all the cases, getting as low as $5\%$ of the original matching task size for the FMA-NCI and FMA-SNOMED cases. The gain in the reduction of the search space gets relatively stable after a given division size; this result is expected since the context provided by locality modules ensures modules with the necessary semantically related entities. The scatter plot in Figure 2(d) visualise the size of the source modules against the size of the target modules for the FMA-NCI matching subtasks with divisions of size $n\in\\{5,20,50,100\\}$. For instance, the (blue) circles represent points $\big{(}\lvert Sig(\mathcal{O}_{1}^{i})\rvert,\lvert Sig(\mathcal{O}_{2}^{i})\rvert\big{)}$ being $\mathcal{O}_{1}^{i}$ and $\mathcal{O}_{2}^{i}$ the source and target modules (with $i$=$1$,…,$5$) in the matching subtasks of $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{5}$. It can be noted that, on average, the size of source and target modules decreases as the size of the division increases. For example, the largest task in $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{20}$ is represented in point $(6754,9168)$, while the largest task in $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{100}$ is represented in point $(2657,11842)$. Table 3: Evaluation of systems that failed to complete OAEI tasks in the 2015-2018 campaigns. Times reported in seconds (s). Tool | Task | Matching | Performance measures | Computation times (s) ---|---|---|---|--- subtasks | P | R | F | Min | Max | Total MAMBA (v.2015) | AMA-NCIA | 5 | 0.870 | 0.624 | 0.727 | 73 | 785 | 1,981 10 | 0.885 | 0.623 | 0.731 | 41 | 379 | 1,608 50 | 0.897 | 0.623 | 0.735 | 8 | 154 | 1,377 FCA-Map (v.2016) | FMA-NCI | 20 | 0.656 | 0.874 | 0.749 | 39 | 340 | 2,934 50 | 0.625 | 0.875 | 0.729 | 19 | 222 | 3,213 FMA-SNOMED | 50 | 0.599 | 0.251 | 0.354 | 6 | 280 | 3,455 100 | 0.569 | 0.253 | 0.350 | 5 | 191 | 3,028 SNOMED-NCI | 150 | 0.704 | 0.629 | 0.664 | 5 | 547 | 16,822 | 200 | 0.696 | 0.630 | 0.661 | 5 | 395 | 16,874 SANOM (v.2017) | FMA-NCI | 20 | 0.475 | 0.720 | 0.572 | 40 | 1,467 | 9,374 50 | 0.466 | 0.726 | 0.568 | 15 | 728 | 7,069 FMA-SNOMED | 100 | 0.145 | 0.210 | 0.172 | 3 | 1,044 | 13,073 150 | 0.143 | 0.209 | 0.170 | 3 | 799 | 10,814 POMap++ (v.2018) | FMA-NCI | 20 | 0.697 | 0.732 | 0.714 | 24 | 850 | 5,448 50 | 0.701 | 0.748 | 0.724 | 11 | 388 | 4,041 FMA-SNOMED | 50 | 0.520 | 0.209 | 0.298 | 4 | 439 | 5,879 100 | 0.522 | 0.209 | 0.298 | 3 | 327 | 4,408 ALOD2vec (v.2018) | FMA-NCI | 20 | 0.697 | 0.813 | 0.751 | 115 | 2,141 | 13,592 50 | 0.698 | 0.813 | 0.751 | 48 | 933 | 12,162 FMA-SNOMED | 100 | 0.702 | 0.183 | 0.29 | 9 | 858 | 12,688 150 | 0.708 | 0.183 | 0.291 | 7 | 581 | 10,449 Computation times. The time to compute the divisions of the matching task is tied to the number of locality modules to extract, which can be computed in polynomial time relative to the size of the input ontology [13]. The creation of LexI does not add an important overhead, while the training of the neural embedding model ranges from $21s$ in AMA-NCI to $224s$ in SNOMED-NCI. Overall, for example, the required time to compute the division with $100$ matching subtasks ranges from $23s$ (AMA-NCIA) to approx. $600s$ (SNOMED-NCI). ### 4.2 Evaluation of OAEI systems In this section we show that the division of the alignment task enables systems that, given some computational constraints, were unable to complete an OAEI task. We have selected the following five systems from the latest OAEI campaigns, which include novel alignment techniques but failed to scale to very large matching tasks: MAMBA (v.2015) [35], FCA-Map (v.2016) [52], SANOM (v.2017) [37], ALOD2vec (v.2018) [43] and POMap++ (v.2018) [30]. MAMBA failed to complete the anatomy track, while FCA-Map, SANOM, ALOD2vec and POMap++ could not complete the largest tasks in the largebio track. MAMBA and SANOM threw an out-of-memory exception with $25Gb$, whereas FCA-Map, ALOD2vec and POMap++ did not complete the tasks within a $6$ hours time-frame. We have used the SEALS infrastructure to conduct the evaluation [3, 4]. Table 3 shows the obtained results in terms of precision, recall, f-measure, and computation times (time for the easiest and the hardest task, and total time for all tasks) over different divisions $\mathcal{D}_{\mathcal{M}\mathcal{T}}^{n}$ computed using our strategy. For example, FCA-Map was run over divisions with 20 and 50 matching subtasks (i.e., $n\in\\{20,50\\}$) in the FMA-NCI case. Note that for each matching subtask a system generates a partial alignment $\mathcal{M}^{S}_{i}$, the final alignment for the (original) matching task is computed as the union of all partial alignments ($\mathcal{M}^{S}=\bigcup_{i=1}^{n}\mathcal{M}^{S}_{i}$). The results are encouraging and can be summarised as follows: 1. ) We enabled several systems to produce results even for the largest OAEI test case (e.g., FCA-Map with SNOMED-NCI). 2. ) The computation times are also very good falling under the $6$ hours time frame, specially given that the (independent) matching subtasks have been run sequentially without parallelization. 3. ) The size of the divisions, with the exception of FCA-Map, is beneficial in terms of total computation time. 4. ) The increase of number of matching subtasks is positive or neutral for MAMBA, POMap++ and ALOD2vec in terms of f-measure, while it is slightly reduced for FCA-Map and SANOM. 5. ) Global f-measure results are lower than top OAEI systems; nevertheless, since the above systems could not be evaluated without the divisions, these results are obtained without any fine-tuning of their parameters. 6. ) The computation times of the hardest tasks, as $n$ increases, is also reduced. This has a positive impact in the monitoring of alignment systems as the hardest task is completed in a reasonable time. ## 5 Related work Partitioning and blocking. Partitioning and modularization techniques have been extensively used within the Semantic Web to improve the efficiency when solving the task at hand (e.g., visualization [45, 1], reuse [29], debugging [47], classification [7]). Partitioning or blocking has also been widely used to reduce the complexity of the ontology alignment task [16]. In the literature there are two major categories of partitioning techniques, namely: _independent_ and _dependent_. Independent techniques typically use only the structure of the ontologies and are not concerned about the ontology alignment task when performing the partitioning. Whereas dependent partitioning methods rely on both the structure of the ontology and the ontology alignment task at hand. Although our approach does not compute (non-overlapping) partitions of the ontologies, it can be considered a dependent technique. Prominent examples of ontology alignment systems including partitioning techniques are Falcon-AO [22], GOMMA [19], COMA++ [5] and TaxoMap [20]. Falcon-AO, GOMMA and COMA++ perform independent partitioning where the clusters of the source and target ontologies are independently extracted. Then pairs of similar clusters (i.e., matching subtasks) are aligned using standard techniques. TaxoMap [20] implements a dependent technique where the partitioning is combined with the matching process. TaxoMap proposes two methods, namely: PAP (partition, anchor, partition) and APP (anchor, partition, partition). The main difference of these methods is the order of extraction of (preliminary) anchors to discover pairs of partitions to be matched (i.e., matching subtasks). SeeCOnt [2] presents a seeding-based clustering technique to discover independent clusters in the input ontologies. Their approach has been evaluated with the Falcon-AO system by replacing its native PBM (Partition-based Block Matching) module [23]. Laadhar et al. [30] have recently integrated within the system POMap++ a hierarchical agglomerative clustering algorithm to divide an ontology into a set of partitions. The above approaches, although presented interesting ideas, did not provide guarantees about the size and coverage of the discovered partitions or divisions. Furthermore, they have not been successfully evaluated on very large ontologies. On the one hand, as reported by Pereira et al. [42] the results in terms of coverage of the PBM method of Falcon-OA, and the PAP and APP methods of TaxoMap are very low for the OAEI largebio track. On the other hand, as discussed in Section 4, POMap++ fails to scale with the largest largebio tasks. Note that the recent work in [31] has borrowed from our workshop paper [26] the quality measures presented in Section 2.1. They obtain competitive coverage results for medium size ontologies; however, their approach, as in POMap++, does not scale for large ontologies. Blocking techniques are also extensively used in Entity Resolution (see [40] for a survey). Although related, the problem of blocking in ontologies is different as the logically related axioms for a seed signature play an important role when computing the blocks. Our dependent approach, unlike traditional partitioning and blocking methods, computes overlapping self-contained modules (i.e., locality modules [13]). Locality modules guarantee the extraction of all semantically related entities for a given signature. This capability enhances the coverage results and enables the inclusion of the (semantically) relevant information required by an alignment system. It is worth mentioning that the need of self-contained and covering modules, although not thoroughly studied, was also highlighted in a preliminary work by Paulheim [41]. Embedding and clustering. Recently, machine learning techniques such as semantic embedding [12] have been investigated for ontology alignment. They often first learn vector representations of the entities and then predict the alignment [9, 51, 46]. However, most of them focus on alignment of ontology individuals (i.e., ABox) without considering the ontology concepts and axioms at the terminological level (i.e., TBox). Nkisi-Orji et al. [39] predicts the alignment between ontology concepts with Random Forest, but incorporates the embeddings of words alone, without any other semantic components like in our work. Furthermore, these approaches focus on predicting the alignment, while our work aims at boosting an existing alignment system. Our framework could potentially be adopted in systems like in [39] if facing scalability problems for large ontologies. Another piece of related work is the clustering of semantic components, using the canopy clustering algorithm [33] where objects are grouped into canopies and each object can be a member of multiple canopies. For example, Wu et al. [50] first extracted canopies (i.e., mentions) from a knowledge base, and then grouped the entities accordingly so as to finding out the entities with the same semantics (i.e., canonicalization). As we focus on a totally different task – ontology alignment, the context that can be used, such as the embeddings for the words and ontology entities in LexI, is different from these works, which leads to a different clustering method. ## 6 Conclusions and future work We have developed a novel framework to split the ontology alignment task into several matching subtasks based on a semantic inverted index, locality modules, and a neural embedding model. We have performed a comprehensive evaluation which suggests that the obtained divisions are suitable in practice in terms of both coverage and size. The division of the matching task allowed us to obtain results for five systems which failed to complete these tasks in the past. We have focused on systems failing to complete a task, but a suitable adoption and integration of the presented framework within the pipeline of any ontology alignment system has the potential to improve the results in terms of computation times. Opportunities. Reducing the ontology matching task into smaller and more manageable tasks may also bring opportunities to enhance (i) user interaction [32], (ii) reasoning and repair [34], (iii) benchmarking and monitoring [3, 4], and (iv) parallelization. The computed independent matching subtasks can potentially be run in parallel in evaluation platforms like the HOBBIT [38]. The current evaluation was conducted sequentially as (i) the SEALS instance only allows running one task at a time, and (ii) the evaluated systems were not designed to run several tasks in parallel; for instance, we managed to run MAMBA outside SEALS, but it relies on a MySQL database and raised a concurrent access exception. Impact on the f-measure. As shown in Section 4.2, the impact of the number of divisions on the f-measure depends on the evaluated systems. In the near future we aim at conducting an extensive evaluation of our framework over OAEI systems able to deal with the largest tasks in order to obtain more insights about the impact on the f-measure. In [25] we reported a preliminary evaluation where YAM-Bio [6] and AML [17] kept similar f-measure values, while LogMap [27] had a reduction in f-measure, as the number of divisions increased. Number of subdivisions. Currently our strategy requires the size of the number of matching subtasks or divisions as input. The (required) matching subtasks may be known before hand if, for example, the matching tasks are to be run in parallel in a number of available CPUs. For the cases where the resources are limited or where a matching system is known to cope with small ontologies, we plan to design an algorithm to estimate the number of divisions so that the size of the matching subtasks in the computed divisions is appropriate to the system and resource constraints. Dealing with a limited or large lexicon. The construction of LexI shares a limitation with state-of-the-art systems when the input ontologies are lexically disparate or in different languages. In such cases, LexI can be enriched with general-purpose lexicons (e.g., WordNet), more specialised background knowledge (e.g., UMLS Metathesaurus) or with translated labels using online services (e.g., Google). On the other hand, a large lexicon may also have an important impact in the computation times. Our conducted evaluation shows, however, that we can cope with very large ontologies with a rich lexicon (e.g., NCI Thesaurus). Notion of context. Locality-based modules are typically much smaller than the whole ontology and they have led to very good results in terms of size and coverage. We plan, however, to study different notions of _context_ of an alignment (e.g., the tailored modules proposed in [8]) to further improve the results in terms of size while keeping the same level of coverage. This work was supported by the SIRIUS Centre for Scalable Data Access (Norges forskningsråd), the AIDA project (Alan Turing Institute), Samsung Research UK, Siemens AG, and the EPSRC projects AnaLOG, OASIS and UK FIRES. We would also like to thank the anonymous reviewers that helped us improve this work. ## References * [1] Asan Agibetov, Giuseppe Patanè, and Michela Spagnuolo, ‘Grontocrawler: Graph-Based Ontology Exploration’, in STAG, (2015). * [2] Alsayed Algergawy, Samira Babalou, Mohammad J. Kargar, and S. 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2024-09-04T02:54:59.237774

2003.05398

arxiv-papers

# Stratification of polymer-colloid mixtures via fast nonequilibrium evaporation Kyoungmun Lee<EMAIL_ADDRESS>Siyoung Q. Choi<EMAIL_ADDRESS>Department of Chemical and Biomolecular Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Korea ###### Abstract In drying liquid films of polymer-colloid mixtures, the stratification in which polymers are placed on top of larger colloids is studied. It is often presumed that the formation of segregated polymer-colloid layers is solely due to the proportion in size at fast evaporation as in binary colloid mixtures. By comparing experiments with a theoretical model, we found that the transition in viscosity near the drying interface was another important parameter for controlling the formation of stratified layers in polymer- colloid mixtures. At high evaporation rates, increased polymer concentrations near the surface lead to a phase transition from semidilute to concentrated regime, in which colloidal particles are kinetically arrested. Stratification only occurs if the formation of a stratified layer precedes the evolution to the concentrated regime near the drying interfaces. Otherwise, the colloids will be trapped by the polymers in the concentrated regime before forming a segregated layer. Also, no stratification is observed if the initial polymer concentration is too low to form a sufficiently high polymer concentration gradient within a short period of time. Our findings are relevant for developing solution-cast polymer composite for painting, antifouling and antireflective coatings. ††preprint: APS/123-QED ## I Introduction Solution-cast polymer composite films composed of polymer matrices containing colloidal particles have been widely studied for many applications, including paints [1], coatings [2,3], and cosmetics [4,5] because they provide highly improved macroscopic properties relative to the pure polymer [6], through a simple manufacturing process. The enhanced properties of the dried films are largely dependent on the spatial distribution of the polymer and colloid [7-10]. In particular, stratified layers consisting of a polymer layer on a colloidal layer have exhibited highly improved antifouling performance [11,12], and photoactive properties [13]. Several previous studies have demonstrated ways of controlling the segregated layers of polymer-colloid mixtures in an equilibrium state [14-16]. However, relatively little is known about how polymer-colloid mixtures can be stratified during the simple, fast and inexpensive nonequilibrium solvent evaporation process. Although solvent casting is one of the simplest manufacturing methods, from coffee ring stains [17] to many industrial applications [1-5], the inherent nonequilibrium nature of drying has made it difficult to clarify the underlying mechanism. As a solvent evaporates, the spatial distribution of the solutes in liquid films is determined by two competing factors: diffusion [18] and receding drying interfaces. Solutes tend to distribute uniformly in drying films with a diffusion constant _D_ , while the nonuniform concentration gradient is developed by the downward velocity of the interface _$v_{ev}$_. Which of the two phenomena dominates can be quantified by the dimensionless _Péclet_ number _Pe =_ _$v_{ev}$__$z_{0}/D$_ , where _$z_{0}$_ is the initial film thickness. If _Pe_ $>$ 1, the solutes cannot diffuse uniformly within the time of evaporation, and they accumulate near the top of the film. On the other hand, the drying film shows almost uniform distribution if _Pe_ $<$ 1. In binary colloid mixtures, it was recently shown that stratifications with smaller colloids placed on large colloids can be realized if _Pe_ is larger than 1 [19-22]. This occurs when the concentration gradient of both the large and smaller particles increases near the liquid/air interface. Fortini _et al._ [20] proposed that the inverted stratification was caused by an imbalance in the osmotic pressure between the larger and smaller colloids. Zhou _et al._ [21] suggested that the stratification phenomenon could be explained quantitatively using a diffusion model, with cross-interaction between the colloids. Sear and Warren [22] argued that diffusiophoretic motion induced by the concentration gradient of the smaller components can exclude the larger colloids from the drying interfaces. In a way similar to binary colloid mixtures, it has been proposed that a polymer-colloid mixture can yield the same stratified layers if the _Pe_ of both the polymer and colloid are larger than 1 [23,24]. However, these results have only been demonstrated by simulation and modeling studies, and few experimental studies have been made on polymer-colloid stratification. Although polymers and colloids can show similar behaviors at very dilute concentrations [24,25], they might behave much differently at the high concentrations that any drying solutions must experience for the complete drying [26,27]. The obvious difference is viscosity. It rapidly increases at relatively low concentrations in the polymer solution, slowing the motions of the species [27-29]. In contrast, the viscosity of the colloidal suspension increases relatively slowly [30]. Thus, the growth in viscosity near the interface, which can kinetically arrest larger colloids [31-33], needs to be considered differently for polymer and colloidal systems, but no appropriate studies have been performed yet. In this work, we experimentally show that the formation of stratified layers, where a small polymer layer is placed on larger colloids, can be predicted using two competing time scales: the time at which the colloid begins to stratify (_$t_{s}^{*}$_) and the time the colloid is arrested by the transitions of viscosity near the interface (_$t_{c}^{*}$_). We consider that the colloid starts to be arrested near the drying interfaces when the polymer concentration reaches a concentrated regime where the polymer chains are densely packed [29]. The stratification can be observed only if _$t_{s}^{*}$_ precedes _$t_{c}^{*}$_ , or _$t_{c}^{*}$_ /_$t_{s}^{*}$_ $>$ 1\. Otherwise, the viscosity near the drying interface rapidly grows within a very short time and the colloids are kinetically trapped before a sufficient downward velocity away from the surface of large colloids is generated. In addition, when the initial polymer concentration is too low, no stratification can also occur because the concentration gradient of the polymer, or the additional migration velocity of the larger colloid, is not enough until the evaporation ends. For the predictive analysis of _$t_{s}^{*}$_ and _$t_{c}^{*}$_ , we propose a simple model modified from the previous work [22]. We observed quite excellent agreement in the final film morphology of the model prediction and experimental studies. Our comprehensive study predicts the spatial distribution of polymers and colloids in the final dried film, based on the experimental system and drying conditions. ## II Result and discussion ### II.1 Structure of dried films of polymer-colloid Mixtures of aqueous polystyrene (PS) suspension with a mean diameter _$d_{c}$_ = 1 _$\mu$__m_ , and poly(ethylene glycol) (PEG) or poly(vinyl alcohol) (PVA) were used as a model system for stratification. The molecular weights of the polymers with PEG _$M_{n}$_ (number average molecular weight) 6,000 gmol-1, PEG _$M_{n}$_ 20,000 gmol-1, PVA _MW_ 6,000 gmol-1, and PVA _$M_{w}$_ (weight average molecular weight) 13,000-23,000 gmol-1 (PVA _$M_{w}$_ 18,000) were chosen for radius of colloid (_$R_{colloid}$_) _$\gg$_ radius of polymer (_$R_{polymer}$_). Before drying, the film solutions contained an initial volume fraction of _$\phi_{i,p}$_ = 0.01 or 0.04 for the polymer and _$\phi_{i,c}$_ = 0.67 _$\phi_{i,p}$_ for the colloid, respectively. The mixture solutions were deposited on glass substrates as _$z_{0}$_ = 1.25 mm. The evaporation was performed at ambient temperature and a relative humidity of 23 %, resulting in an initial polymer _Péclet_ number _$Pe_{i,p}$_ $>$ 1 (See Supplemental Material). All of the experimental systems are summarized in Table I. When the evaporation was completed, the final film morphologies were analyzed with the help of scanning electronic microscopy (SEM) and ImageJ analysis. Table 1: Various polymer-colloid systems that were tested. Colloid was fixed as PS to exclude gravitational effect during drying ($\rho_{PS}$ $\approx$ $\rho_{water}$). A total of 8 systems were experimentally performed. | | | | | | | | _$Pe_{i,p}$_ ---|---|---|---|---|---|---|---|--- Colloid | Polymer | | _$R_{g}$_ 111See Supplemental Material (nm) | _$\phi_{i,p}$_ | _$\phi_{i,p}:\phi_{i,c}$_ | _$h_{0}$_ (mm) | Relative humidity | _$\phi_{i,p}$_ 0.01 | _$\phi_{i,p}$_ 0.04 PS (r = 500 nm) | PEG | _$M_{n}$_ 6,000 | 3.6 | 0.01 | 3 : 2 | 1.25 | 23 % | 4 | 7 | | _$M_{n}$_ 20,000 | 7.4 | or | | | | 9 | 22 | PVA | _MW_ 6,000 | 3.5 | 0.04 | | | | 4 | 9 | | _$M_{w}$_ 18,000 | 6.8 | | | | | 8 | 24 After complete drying, the polymers were enriched at the top of the films in PEG _$M_{n}$_ 6,000 gmol-1 (_$\phi_{i,p}$_ = 0.04) [Fig. 1(a)] and PVA _MW_ 6,000 gmol-1 (_$\phi_{i,p}$_ = 0.04) [Fig. 1(c)] while other 6 dried films in Fig. 1(b), 1(d) and Fig. 2(a) - 2(d) were not segregated, but randomly distributed. Although the stratified layers in Fig. 1(a) and Fig. 1(c) also showed different degrees of stratification, there was a clear boundary between the stratified layers [Fig. 1(e)] and nonstratified layers [Fig. 1(f), Fig. 2(e), and Fig. 2(f)]. Figure 1: Cross sectional SEM images of dried films of polymer-colloid mixtures (_$\phi_{i,p}=0.04$_ , _$\phi_{i,p}:\phi_{i,c}=3:2$_). The upper row shows various polymer-colloid distributions according to the polymer types and molecular weights (a) PEG _$M_{n}$_ 6,000, (b) PEG _$M_{n}$_ 20,000, (c) PVA _MW_ 6,000, (d) PVA _$M_{w}$_ 18,000. The yellow lines represent boundary of stratified layers. If there is no clear boundary, nothing is denoted. The lower rows are estimated relative volume fraction of polymer _$\phi_{p}$_ (red circles) and colloid _$\phi_{c}$_ (blue triangles) of two representatives: (e) PEG _$M_{n}$_ 6,000 and (f) PEG _$M_{n}$_ 20,000. The colloidal volume fractions were obtained from SEM images through the ImageJ analysis. The remained volume fraction was considered as polymer volume fraction _$\phi_{p}=1-\phi_{c}$_. Figure 2: SEM images of dried films formed from polymer-colloid mixtures (_$\phi_{i,p}=0.01$_ , _$\phi_{i,p}:\phi_{i,c}=3:2$_). Distributions of polymer and colloid are shown through the upper row depending on the polymer types and molecular weights (a) PEG _$M_{n}$_ 6,000, (b) PEG _$M_{n}$_ 20,000, (c) PVA _MW_ 6,000, (d) PVA _$M_{w}$_ 18,000. There was no clear stratified layer in all four images. The volume fractions of polymer _$\phi_{p}$_ (red circles) and colloid _$\phi_{c}$_ (blue triangles) of the two dried films were obtained from SEM image analysis: (e) PEG _$M_{n}$_ 6,000 and (f) PEG _$M_{n}$_ 20,000. The volume fractions of colloids are estimated by ImageJ analysis, and the polymer volume fraction was determined by _$\phi_{p}=1-\phi_{c}$_. ### II.2 Modified theoretical model of dynamic stratification As the solvent evaporated at _Pe_ $>$ 1 for both polymer and colloid, the descending air/water interface _$z_{interface}$_ compressed the polymer and colloid, and they accumulated near the drying interface. From previous studies [22,34], the transition of the polymer concentration in drying film _$\phi_{p}$_(z,t) can be written as $\displaystyle\phi_{p}(z,t^{*})\approx\phi_{i,p}(1+Pe_{p}t^{*}exp{\bf[}-\frac{|z-z_{interface}|}{D_{p}/v_{ev}}{\bf]}),$ (1) $\displaystyle z_{interface}(t^{*})=z_{0}-v_{ev}t=(1-t^{*})z_{0}$ (2) if _Péclet_ number of polymer _$Pe_{p}$_ $\gg$ 1, where _$t^{*}=tv_{ev}/z_{0}$_ (0 $\leq$ _$t^{*}$_ $\leq$ 1) is the dimensionless time. Here, _$Pe_{p}$_ and diffusion coefficient of polymer _$D_{p}$_ can be expressed as a function of drying time when _$Pe_{p}$_ and _$D_{p}$_ vary slowly. Since the viscosity growth derived from the increased polymer concentration can be accompanied by the kinetic arrest of the colloidal particles, _$t_{c}^{*}$_ can be determined by the time when the volume fraction of polymer reaches the concentrated regime _$\phi_{p}=\phi_{p}^{**}$_. We consider that the colloidal particles at the drying interface (_$z=z_{interface}$_) are kinetically arrested when the polymer fraction reaches _$\phi_{p}^{**}$_ at _$z=z_{interface}-r_{colloid}$_ $\displaystyle\phi_{p}(z_{interface}-r_{colloid},t_{c}^{*})=\phi_{p}^{**}.$ (3) Meanwhile, increasing the concentration gradients of the small polymers can also create the diffusiophoretic drift velocity of larger colloids _$v_{diffusiophoresis}$_ [35,36] $\displaystyle v_{diffusiophoresis}=-\frac{9}{4}D_{p}\nabla\phi_{p}$ (4) under the condition of _$R_{colloid}$_ $\gg$ _$R_{polymer}$_. From the simple 1D diffusion model, the polymer concentration gradient at the interface is _$\nabla\phi_{p}=-v_{ev}\phi_{interface}/D_{p}$_ [37]. This gives the diffusiophoretic velocity of interfacial colloids with the combination of _$\phi_{interface}=\phi_{i,p}(1+Pe_{p}t^{*})$_ originating from Eq. (1) at _$z=z_{interface}$_ , $\displaystyle v_{colloid,interface}\approx\frac{9}{4}v_{ev}\phi_{i,p}(1+Pe_{p}t^{*}).$ (5) The time at which the colloid begins to stratify during the evaporation process (_$t_{s}^{*}$_) is determined by comparing _$v_{colloid,interface}$_ and _$v_{ev}$_. Near the time when evaporation begins, the gradient of polymer concentration is not too large and _$v_{colloid,interface}$_ does not overcome _$v_{ev}$_. At this state, both the polymer and colloid simply accumulate at the drying interface. If the concentration gradient of the polymer is large enough for the formation of a higher colloidal diffusiophoretic velocity, however, _$v_{colloid,interface}$_ is larger than _$v_{ev}$_ and it starts to create stratified layers in the drying film. We consider the time _$t_{s}^{*}$_ when _$v_{colloid,interface}=v_{ev}$_ , resulting in $\displaystyle v_{colloid,interface}(t_{s}^{*})=v_{ev}.$ (6) The final morphologies of the drying polymer-colloid mixtures are determined by the two competing time scales _$t_{s}^{*}$_ and _$t_{c}^{*}$_. There are three regimes for the predictive analysis of the stratification of polymer- colloid mixtures. The first is _$t_{c}^{*}/t_{s}^{*}$_ $>$ 1, where the downward motion of the colloidal particles appears before _$\phi_{p}(z_{interface}-r_{colloid},t_{c}^{*})=\phi_{p}^{**}$_. The second is _$t_{c}^{*}/t_{s}^{*}$_ $<$ 1, where the polymer volume fraction reaches _$\phi_{p}^{**}$_ before the evolution of _$v_{colloid,interface}(t_{s}^{*})=v_{ev}$_. The third is _$t_{s}^{*}\approx 1$_ , where _$t_{s}^{*}$_ reaches to the time at which evaporation ends (_$t^{*}=1$_), even though _$t_{s}^{*}$_ precedes _$t_{c}^{*}$_. ### II.3 Comparison of experimental results and theoretical model As described above, the prediction for the polymer-colloid stratification can be estimated using the competition between _$t_{c}^{*}$_ and _$t_{s}^{*}$_. For the time dependent volume fraction of the polymer in the drying films, evaporation rates were determined by measuring mass reduction (Fig. SM3). To calculate the time dependent (or concentration dependent) polymer diffusion coefficient, the average volume fractions of polymer in the drying film were used as _$D_{p}$_ (See Supplemental Material). The transition volume fraction of semidilute entangled _$\phi_{e}$_ to concentrated regime _$\phi^{**}$_ in good solvent were determined by the specific viscosity _$\eta_{sp}$_ slope transition [27,28,38] in Fig. 3. From the slope transition of semidilute unentangled (_$\eta_{sp}$_ $\sim$ _$\phi_{p}^{1.3}$_) to semidilute entangled (_$\eta_{sp}$_ $\sim$ _$\phi_{p}^{3.9}$_), _$\phi_{e}$_ of the polymer in good solvent was measured. Similarly, _$\phi^{**}$_ can be estimated using the slope transition point between the semidilute entangled regime (_$\eta_{sp}$_ $\sim$ _$\phi_{p}^{3.9}$_) and the concentrated regime (_$\eta_{sp}$_ $\sim$ _$\phi_{p}^{\alpha}$_ , where $\alpha$ $>$ 3.9). Figure 3: Specific viscosity of four polymer solutions as a function of polymer volume fraction. Polymer volume fraction where it goes to concentrated regime _$\phi^{**}$_ is estimated by the slope transition point from 3.9 to larger than 3.9. In case of PEG _$M_{n}$_ 6,000, _$\phi^{**}$_ is considered as max solubility ($\approx$ 630 mg/ml at 20oC). As the PEG _$M_{n}$_ 6,000 solution goes to higher than max solubility, it shows abrupt increment of specific viscosity (empty red triangle). In drying films of polymer-colloid mixtures, the final film morphology can be predicted using the three regimes in the (_$t_{s}^{*}$_ , _$t_{c}^{*}$_) plane. Regime 1 with _$t_{c}^{*}$_ /_$t_{s}^{*}$_ $>$ 1 indicates clearly stratified layers in the dried films. Regime 2 represents nonsegregated layers, because _$t_{c}^{*}$_ appears before _$t_{s}^{*}$_. Regime 3 also shows nonstratified layers in the final morphology of the complete dried polymer-colloids mixtures, since _$t_{s}^{*}$_ appears very close to 1 (_$t_{s}^{*}$_ $\approx$ 1). The theoretical predictions based on Eq. (3), Eq. (6) and the experimental stratification results from 8 different systems are presented in Fig. 4. There is quite excellent agreement between the model prediction and experimental results except for the PVA _MW_ 6,000 (_$\phi_{i,p}=0.04$_) system, which also appears to be closest to _$t_{c}^{*}/t_{s}^{*}$_ = 1. This might be due to the air/water interfacial activity of PVA _MW_ 6,000 (Fig. SM4), which can make faster _$t_{s}^{*}$_ under real drying conditions, but it cannot bring _$t_{c}^{*}$_ forward since _$t_{c}^{*}$_ is related to the _$z=z_{interface}-r_{colloid}$_ , not _$z=z_{interface}$_. To reduce the interfacial activity effect of PVA _MW_ 6,000 (_$\phi_{i,p}=0.04$_) on stratification, we moved the point to deviate from _$t_{c}^{*}/t_{s}^{*}$_ $=$ 1 in our theoretical model by changing _$v_{ev}$_. As it deviates from _$t_{c}^{*}/t_{s}^{*}=1$_ , the theoretical prediction becomes consistent with the experimental result for PVA _MW_ 6,000 (_$\phi_{i,p}=0.04$_) (Fig. 5). Figure 4: State diagram on the (_$t_{s}^{*}$_ ,_$t_{c}^{*}$_) plane. The dotted line corresponds to _$t_{c}^{*}/t_{s}^{*}=1$_. Theoretical predictions of 8 different systems are denoted as symbols in the diagram, and the experimental results are represented by colors. Blue indicates regime 1 (_$t_{c}^{*}/t_{s}^{*}$_ $>$ 1) where stratified layer expected and red shows regime 2 (_$t_{c}^{*}/t_{s}^{*}$_ $<$ 1). Orange designated regime 3 (_$t_{s}^{*}\approx 1$_) (Fig. SM5). The green indicates the intermediate state where stratified layer is observed in experiments while it belongs to regime 2 in model prediction. All data points show overall agreement with one exception, filled green triangle, which also appears close to the _$t_{c}^{*}/t_{s}^{*}=1$_. Figure 5: State diagram of PVA _MW_ 6,000 (_$\phi_{i,p}$_ = 0.04) on the (_$t_{s}^{*}$_ ,_$t_{c}^{*}$_) plane. The dotted line corresponds to _$t_{c}^{*}/t_{s}^{*}=1$_. The filled green triangle deviated from _$t_{c}^{*}/t_{s}^{*}=1$_ in theoretical model only by increasing _$v_{ev}$_. As it stays away from _$t_{c}^{*}/t_{s}^{*}=1$_ , the intermediate stratified morphology where stratified layer is observed in experiments while it belongs to regime 2 in model prediction become consistent with model prediction. (a) SEM image of PVA _MW_ 6,000 (_$\phi_{i,p}$_ = 0.04) at fast evaporation. (b) Top of the cross-sectional SEM image (a). The evaporation rate was controlled by convective flow of air with a relative humidity of 23 % at ambient temperature. ### II.4 Conditions for polymer-colloid stratification To analyze the general conditions for polymer-colloid stratification, we represented _$t_{c}^{*}$_ and _$t_{s}^{*}$_ in another experimental parameter. As mentioned above, the polymer-on-top structure can be formed when the two conditions, both _$t_{s}^{*}$_ $<$ 1 and _$t_{c}^{*}/t_{s}^{*}$_ $>$ 1, are satisfied. From Eq. (3) and Eq. (6), _$t_{c}^{*}$_ and _$t_{s}^{*}$_ are (See Supplemental Material) $\displaystyle t_{c}^{*}\approx\frac{\frac{\phi^{**}}{\phi_{i,p}}-1}{Pe_{p}(t_{c}^{*})},$ (7) $\displaystyle t_{s}^{*}\approx\frac{\frac{4}{9}\frac{1}{\phi_{i,p}}-1}{Pe_{p}(t_{s}^{*})},$ (8) where _$Pe_{p}(t_{c}^{*})$_ and _$Pe_{p}(t_{s}^{*})$_ are _Pe_ of the polymer at dimensionless time _$t^{*}=t_{c}^{*}$_ and _$t^{*}=t_{s}^{*}$_ in respectively. The first condition for the stratification to happen, _$t_{s}^{*}$_ $<$ 1, is $\displaystyle Pe_{p}(t_{s}^{*})\phi_{i,p}>\frac{4}{9}-\phi_{i,p}.$ (9) This follows a condition for similar to that for the inverted stratification of binary colloidal mixtures [21,33]. The second requirement for stratified layers in polymer-colloid mixtures, _$t_{c}^{*}/t_{s}^{*}$_ $>$ 1, can be expressed as $\displaystyle\frac{t_{c}^{*}}{t_{s}^{*}}\approx\frac{(\frac{\phi^{**}}{\phi_{i,p}}-1)}{(\frac{4}{9}\frac{1}{\phi_{i,p}}-1)}\frac{Pe_{p}(t_{s}^{*})}{Pe_{p}(t_{c}^{*})}>1,$ (10) $\displaystyle\frac{t_{c}^{*}}{t_{s}^{*}}\approx\frac{(\frac{\phi^{**}}{\phi_{i,p}}-1)}{(\frac{4}{9}\frac{1}{\phi_{i,p}}-1)}\frac{\eta(t_{s}^{*})}{\eta(t_{c}^{*})}>1.$ (11) Since _$t_{c}^{*}$_ and _$t_{s}^{*}$_ come out when the polymer solution in the semi-dilute entangled regime (close to _$\phi_{p}=\phi^{**}$_), _$\eta(t^{*})$_ is $\displaystyle\eta(t^{*})=(\frac{1-t_{e}^{*}}{1-t^{*}})^{3.9}(\eta_{e}-\eta_{s})+\eta_{s}$ (12) from Eq. (14) of Supplemental Material, where _$t_{e}^{*}$_ is the dimensionless time when _$\eta=\eta_{e}$_ (viscosity when _$\phi_{p}=\phi_{e}$_) from Eq. (10) of Supplemental Material. If we neglect the last term in Eq. (12), $\displaystyle\frac{t_{c}^{*}}{t_{s}^{*}}\approx\frac{(\frac{\phi^{**}}{\phi_{i,p}}-1)}{(\frac{4}{9}\frac{1}{\phi_{i,p}}-1)}(\frac{1-t_{c}^{*}}{1-t_{s}^{*}})^{3.9},$ (13) $\displaystyle\frac{t_{c}^{*}}{t_{s}^{*}}(\frac{1-t_{s}^{*}}{1-t_{c}^{*}})^{3.9}\approx\frac{(\frac{\phi^{**}}{\phi_{i,p}}-1)}{(\frac{4}{9}\frac{1}{\phi_{i,p}}-1)}.$ (14) To satisfy the condition of _$t_{c}^{*}/t_{s}^{*}$_ $>$ 1 for polymer-colloid stratification, $\displaystyle\frac{(\frac{\phi^{**}}{\phi_{i,p}}-1)}{(\frac{4}{9}\frac{1}{\phi_{i,p}}-1)}>1,$ (15) $\displaystyle\phi^{**}-\phi_{i,p}>\frac{4}{9}-\phi_{i,p},$ (16) $\displaystyle\phi^{**}>\frac{4}{9}.$ (17) It is interesting to note that the predicted stratification of the polymer- colloid mixtures does not depend on the drying rate _$v_{ev}$_ , or _Pe_ , as long as _$Pe\gg 1$_. This tendency also can be seen in Fig. 6, which shows the theoretical predictions of the 8 systems above, with _$v_{ev}$_ values changed. Ignoring the data points of _$Pe_{i,p}\leq 5$_ , failing to follow the aforementioned assumption _$Pe\gg 1$_ , all the other points belong in same regime once the polymer type and initial volume fraction are determined. This is quite plausible since the increase in polymer concentration near the drying interface accelerates both _$t_{c}^{*}$_ and _$t_{s}^{*}$_ in similar order. Thus, it might be hard to create stratified layers in polymer-colloid mixtures only by varying the evaporation rate _$v_{ev}$_ , or _Pe_. Altering other properties which can increase _$t_{c}^{*}/t_{s}^{*}$_ larger than 1, such as the interfacial activity of the polymer in Fig. 5 or the gravitational velocity from the density difference in Fig. SM6, could be another solution to achieve stratified layers in polymer-colloid mixtures. Figure 6: Theoretical prediction of the stratification of 8 different systems on the (_$t_{s}^{*}$_ ,_$t_{c}^{*}$_) plane with controlled _$v_{ev}$_ (or _$Pe_{i,p}$_) (a) PEG _$M_{n}$_ 6,000, (b) PEG _$M_{n}$_ 20,000, (c) PVA _MW_ 6,000, (d) PVA _$M_{w}$_ 18,000. As _$Pe_{i,p}$_ increases, both _$t_{s}^{*}$_ and _$t_{c}^{*}$_ decrease and data points go to left bottom side on the (_$t_{s}^{*}$_ ,_$t_{c}^{*}$_) plane. Regardless of the polymer type or molecular weight, most of the data points belong in the same regime once the type of polymer and initial volume fraction are determined except the relatively slow drying rate (_$Pe_{i,p}\leq 5$_ , red circles). ## III Conclusion In summary, we demonstrated that dynamic stratification of polymer-colloid mixtures can be achieved by controlling viscosity near the drying interface, which results from increasing polymer concentration. When the polymer-colloid solution evaporates, the polymer starts to increase the solution viscosity near the air/water interface within a relatively very short time, unlike colloidal suspensions. Since the transition in viscosity due to the polymer can cause the kinetic arrest of colloidal particles, which hinders the diffusiophoretic downward motion of colloids, stratified layers are only observed if the formation of a stratified layer precedes the transition in viscosity near the liquid/air interfaces. Our model calculations for _$t_{c}^{*}$_ and _$t_{s}^{*}$_ , inspired by the previous study [22], show that the segregation of polymer-colloid mixtures can only occur under the condition of _$t_{c}^{*}/t_{s}^{*}$_ $>$ 1, unless the solute fraction of the polymer is sufficiently low. The requirement for stratification, _$t_{c}^{*}/t_{s}^{*}$_ $>$ 1, implies that the stratification of polymer-colloid mixtures may not rely on drying rate if _$Pe\gg 1$_ , since both _$t_{c}^{*}$_ and _$t_{s}^{*}$_ vary in similar order as _$v_{ev}$_ changes. Our model calculations are further supported by the consistency between the model prediction and final experimental film morphologies In more general terms, the consistent results of the experiments and model prediction may shed light on methods of controlling surface enrichment in general solution-cast polymer composites. The ability to predict morphology in a simple nonequilibrium solvent evaporation process is highly desirable for preparing materials whose surface properties are crucial to performance, such as antireflective or organic photovoltaics. 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2024-09-04T02:54:59.247362

2003.05399

arxiv-papers

Open Communications in Nonlinear Mathematical Physics ]ocnmp[ Vol.1 (2021) pp 1–References Article ††footnotetext: © The author(s). Distributed under a Creative Commons Attribution 4.0 International License Hamiltonian structures for integrable hierarchies of Lagrangian PDEs Mats Vermeeren 1 1 School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK. <EMAIL_ADDRESS> Received May 18, 2021; Accepted August 31, 2021 ###### Abstract Many integrable hierarchies of differential equations allow a variational description, called a Lagrangian multiform or a pluri-Lagrangian structure. The fundamental object in this theory is not a Lagrange function but a differential $d$-form that is integrated over arbitrary $d$-dimensional submanifolds. All such action integrals must be stationary for a field to be a solution to the pluri-Lagrangian problem. In this paper we present a procedure to obtain Hamiltonian structures from the pluri-Lagrangian formulation of an integrable hierarchy of PDEs. As a prelude, we review a similar procedure for integrable ODEs. We show that the exterior derivative of the Lagrangian $d$-form is closely related to the Poisson brackets between the corresponding Hamilton functions. In the ODE (Lagrangian 1-form) case we discuss as examples the Toda hierarchy and the Kepler problem. As examples for the PDE (Lagrangian 2-form) case we present the potential and Schwarzian Korteweg-de Vries hierarchies, as well as the Boussinesq hierarchy. ## 1 Introduction Some of the most powerful descriptions of integrable systems use the Hamiltonian formalism. In mechanics, Liouville-Arnold integrability means having as many independent Hamilton functions as the system has degrees of freedom, such that the Poisson bracket of any two of them vanishes. In the case of integrable PDEs, which have infinitely many degrees of freedom, integrability is often defined as having an infinite number of commuting Hamiltonian flows, where again each two Hamilton functions have a zero Poisson bracket. In addition, many integrable PDEs have two compatible Poisson brackets. Such a bi-Hamiltonian structure can be used to obtain a recursion operator, which in turn is an effective way to construct an integrable hierarchy of PDEs. In many cases, especially in mechanics, Hamiltonian systems have an equivalent Lagrangian description. This raises the question whether integrability can be described from a variational perspective too. Indeed, a Lagrangian theory of integrable hierarchies has been developed over the last decade or so, originating in the theory of integrable lattice equations (see for example [14], [3], [12, Chapter 12]). It is called the theory of _Lagrangian multiform_ systems, or, of _pluri-Lagrangian_ systems. The continuous version of this theory, i.e. its application to differential equations, was developed among others in [26, 28]. Recently, connections have been established between pluri-Lagrangian systems and variational symmetries [18, 19, 22] as well as Lax pairs [21]. Already in one of the earliest studies of continuous pluri-Lagrangian structures [26], the pluri-Lagrangian principle for ODEs was shown to be equivalent to the existence of commuting Hamiltonian flows (see also [24]). In addition, the property that Hamilton functions are in involution can be expressed in Lagrangian terms as closedness of the Lagrangian form. The main goal of this work is to generalize this connection between pluri-Lagrangian and Hamiltonian structures to the case of integrable PDEs. A complementary approach to connecting pluri-Lagrangian structures to Hamiltonian structures was recently taken in [6]. There, a generalisation of covariant Hamiltonian field theory is proposed, under the name _Hamiltonian multiform_ , as the Hamiltonian counterpart of Lagrangian multiform systems. This yields a Hamiltonian framework where all independent variables are on the same footing. In the present work we obtain classical Hamiltonian structures where one of the independent variables is singled out as the common space variable of all equations in a hierarchy. We begin this paper with an introduction to pluri-Lagrangian systems in Section 2. The exposition there relies mostly on examples, while proofs of the main theorems can be found in Appendix A. Then we discuss how pluri-Lagrangian systems generate Hamiltonian structures, using symplectic forms in configuration space. In Section 3 we review this for ODEs (Lagrangian 1-form systems) and in Section 4 we present the case of $(1+1)$-dimensional PDEs (Lagrangian 2-form systems). In each section, we illustrate the results by examples. ## 2 Pluri-Lagrangian systems A hierarchy of commuting differential equations can be embedded in a higher- dimensional space of independent variables, where each equation has its own time variable. All equations share the same space variables (if any) and have the same configuration manifold $Q$. We use coordinates $t_{1},t_{2},\ldots,t_{N}$ in the _multi-time_ $M=\mathbb{R}^{N}$. In the case of a hierarchy of $(1+1)$-dimensional PDEs, the first of these coordinates is a common space coordinate, $t_{1}=x$, and we assume that for each $i\geq 2$ there is a PDE in the hierarchy expressing the $t_{i}$-derivative of a field $u:M\rightarrow Q$ in terms of $u$ and its $x$-derivatives. Then the field $u$ is determined on the whole multi-time $M$ if initial values are prescribed on the $x$-axis. In the case of ODEs, we assume that there is a differential equation for each of the time directions. Then initial conditions at one point in multi-time suffice to determine a solution. We view a field $u:M\rightarrow Q$ as a smooth section of the trivial bundle $M\times Q$, which has coordinates $(t_{1},\ldots,t_{N},u)$. The extension of this bundle containing all partial derivatives of $u$ is called the _infinite jet bundle_ and denoted by $M\times J^{\infty}$. Given a field $u$, we call the corresponding section $\llbracket u\rrbracket=(u,u_{t_{i}},u_{t_{i}t_{j}},\ldots)$ of the infinite jet bundle the _infinite jet prolongation_ of $u$. (See e.g. [1] or [17, Sec. 3.5].) In the pluri-Lagrangian context, the Lagrange function is replaced by a jet- dependent $d$-form. More precisely we consider a fiber-preserving map $\textstyle\mathcal{L}:M\times J^{\infty}\rightarrow\bigwedge^{d}(T^{*}M).$ Since a field $u:M\rightarrow Q$ defines a section of the infinite jet bundle, $\mathcal{L}$ associates to it a section of $\bigwedge^{d}(T^{*}M)$, that is, a $d$-form $\mathcal{L}\llbracket u\rrbracket$. We use the square brackets $\llbracket u\rrbracket$ to denote dependence on the infinite jet prolongation of $u$. We take $d=1$ if we are dealing with ODEs and $d=2$ if we are dealing with $(1+1)$-dimensional PDEs. Higher-dimensional PDEs would correspond to $d>2$, but are not considered in the present work. (An example of a Lagrangian $3$-form system, the KP hierarchy, can be found in [22].) We write $\mathcal{L}\llbracket u\rrbracket=\sum_{i}\mathcal{L}_{i}\llbracket u\rrbracket\,\mbox{\rm d}t_{i}$ for 1-forms and $\mathcal{L}\llbracket u\rrbracket=\sum_{i,j}\mathcal{L}_{ij}\llbracket u\rrbracket\,\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}$ for 2-forms. ###### Definition 2.1. A field $u:M\rightarrow Q$ is a solution to the _pluri-Lagrangian problem_ for the jet-dependent $d$-form $\mathcal{L}$, if for every $d$-dimensional submanifold $\Gamma\subset M$ the action $\int_{\Gamma}\mathcal{L}\llbracket u\rrbracket$ is critical with respect to variations of the field $u$, i.e. $\frac{\mbox{\rm d}}{\mbox{\rm d}\varepsilon}\int_{\Gamma}\mathcal{L}\llbracket u+\epsilon v\rrbracket\bigg{|}_{\varepsilon=0}=0$ for all variations $v:M\rightarrow Q$ such that $v$ and all its partial derivatives are zero on $\partial\Gamma$. Some authors include in the definition that the Lagrangian $d$-form must be closed when evaluated on solutions. That is equivalent to requiring that the action is not just critical on every $d$-submanifold, but also takes the same value on every $d$-submanifold (with the same boundary and topology). In this perspective, one can take variations of the submanifold $\Gamma$ as well as of the fields. We choose not to include the closedness in our definition, because slightly weaker property can be obtained as a consequence Definition 2.1 (see Proposition A.2 in the Appendix). Most of the authors that include closedness in the definition use the term “Lagrangian multiform” (e.g. [14, 12, 32, 33, 22]), whereas those that do not tend to use “pluri-Lagrangian” (e.g. [4, 5, 27]). Whether or not it is included in the definition, closedness of the Lagrangian $d$-form is an important property. As we will see in Sections 3.4 and 4.4, it is the Lagrangian counterpart to vanishing Poisson brackets between Hamilton functions. Clearly the pluri-Lagrangian principle is stronger than the usual variational principle for the individual coefficients $\mathcal{L}_{i}$ or $\mathcal{L}_{ij}$ of the Lagrangian form. Hence the classical Euler-Lagrange equations are only a part of the system equations characterizing a solution to the pluri-Lagrangian problem. This system, which we call the _multi-time Euler-Lagrange equations_ , was derived in [28] for Lagrangian 1- and 2-forms by approximating an arbitrary given curve or surface $\Gamma$ by _stepped_ curves or surfaces, which are piecewise flat with tangent spaces spanned by coordinate directions. In Appendix A we give a more intrinsic proof that the multi-time Euler-Lagrange equations imply criticality in the pluri-Lagrangian sense. Yet another proof can be found in [23]. In order to write the multi-time Euler-Lagrange equations in a convenient form, we will use the multi-index notation for (mixed) partial derivatives. Let $I$ be an $N$-index, i.e. a $N$-tuple of non-negative integers. We denote by $u_{I}$ the mixed partial derivative of $u:\mathbb{R}^{N}\rightarrow Q$, where the number of derivatives with respect to each $t_{i}$ is given by the entries of $I$. Note that if $I=(0,\ldots,0)$, then $u_{I}=u$. We will often denote a multi-index suggestively by a string of $t_{i}$-variables, but it should be noted that this representation is not always unique. For example, $t_{1}=(1,0,\ldots,0),\qquad t_{N}=(0,\ldots,0,1),\qquad t_{1}t_{2}=t_{2}t_{1}=(1,1,0,\ldots,0).$ In this notation we will also make use of exponents to compactify the expressions, for example $t_{2}^{3}=t_{2}t_{2}t_{2}=(0,3,0,\ldots,0).$ The notation $It_{j}$ should be interpreted as concatenation in the string representation, hence it denotes the multi-index obtained from $I$ by increasing the $j$-th entry by one. Finally, if the $j$-th entry of $I$ is nonzero we say that $I$ contains $t_{j}$, and write $I\ni t_{j}$. ### 2.1 Lagrangian 1-forms ###### Theorem 2.2 ([28]). Consider the Lagrangian 1-form $\mathcal{L}\llbracket u\rrbracket=\sum_{j=1}^{N}\mathcal{L}_{j}\llbracket u\rrbracket\,\mbox{\rm d}t_{j},$ depending on an arbitrary number of derivatives of $u$. A field $u$ is critical in the pluri-Lagrangian sense if and only if it satisfies the multi- time Euler-Lagrange equations $\displaystyle\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{I}}}=0$ $\displaystyle\forall I\not\ni t_{j},$ (1) $\displaystyle\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{It_{j}}}}-\frac{\delta_{1}{\mathcal{L}_{1}}}{\delta{u_{It_{1}}}}=0$ $\displaystyle\forall I,$ (2) for all indices $j\in\\{1,\ldots,N\\}$, where $\frac{\delta_{j}{}}{\delta{u_{I}}}$ denotes the variational derivative in the direction of $t_{j}$ with respect to $u_{I}$, $\frac{\delta_{j}{}}{\delta{u_{I}}}=\frac{\partial{}}{\partial{u_{I}}}-\partial_{j}\frac{\partial{}}{\partial{u_{It_{j}}}}+\partial_{j}^{2}\frac{\partial{}}{\partial{u_{It_{j}t_{j}}}}-\cdots,$ and $\partial_{j}=\frac{\mbox{\rm d}}{\mbox{\rm d}t_{j}}$. Note the derivative $\partial_{j}$ equals the total derivative $\sum_{I}u_{It_{j}}\frac{\partial{}}{\partial{u_{I}}}$ if it is applied to a function $f\llbracket u\rrbracket$ that only depends on $t_{j}$ through $u$. Using the total derivative has the advantage that calculations can be done on an algebraic level, where the $u_{I}$ are formal symbols that do not necessarily have an analytic interpretation as a derivative. ###### Example 2.3. The Toda lattice describes $N$ particles on a line with an exponential nearest-neighbor interaction. We denote the displacement from equilibrium of the particles by $u=(q^{\scriptscriptstyle[1]},\ldots,q^{\scriptscriptstyle[N]})$. We impose either periodic boundary conditions (formally $q^{\scriptscriptstyle[0]}=q^{\scriptscriptstyle[N]}$ and $q^{\scriptscriptstyle[N+1]}=q^{\scriptscriptstyle[1]}$) or open-ended boundary conditions (formally $q^{\scriptscriptstyle[0]}=\infty$ and $q^{\scriptscriptstyle[N+1]}=-\infty$). We will use $q^{\scriptscriptstyle[k]}_{j}$ as shorthand notation for the derivative $q^{\scriptscriptstyle[k]}_{t_{j}}=\frac{\mbox{\rm d}q^{\scriptscriptstyle[k]}}{\mbox{\rm d}t_{j}}$. Consider the hierarchy consisting of the Newtonian equation for the Toda lattice, $q^{\scriptscriptstyle[k]}_{11}=\exp\\!\left(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}\right)-\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right),\\\ $ (3) along with its variational symmetries, $\begin{split}q^{\scriptscriptstyle[k]}_{2}&=\left(q^{\scriptscriptstyle[k]}_{1}\right)^{2}+\exp\\!\left(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}\right)+\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right),\\\ q^{\scriptscriptstyle[k]}_{3}&=\left(q^{\scriptscriptstyle[k]}_{1}\right)^{3}+q^{\scriptscriptstyle[k+1]}_{1}\exp\\!\left(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}\right)+q^{\scriptscriptstyle[k-1]}_{1}\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right)\\\ &\quad+2q^{\scriptscriptstyle[k]}_{1}\left(\exp\\!\left(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}\right)+\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right)\right),\\\ &\mathmakebox[\widthof{{}={}}][c]{\vdots}\end{split}$ (4) The hierarchy (3)–(4) has a Lagrangian 1-form with coefficients $\displaystyle\mathcal{L}_{1}$ $\displaystyle=\sum_{k}\left(\frac{1}{2}\left(q^{\scriptscriptstyle[k]}_{1}\right)^{2}-\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right)\right),$ $\displaystyle\mathcal{L}_{2}$ $\displaystyle=\sum_{k}\left(q^{\scriptscriptstyle[k]}_{1}q^{\scriptscriptstyle[k]}_{2}-\frac{1}{3}\left(q^{\scriptscriptstyle[k]}_{1}\right)^{3}-\left(q^{\scriptscriptstyle[k]}_{1}+q^{\scriptscriptstyle[k-1]}_{1}\right)\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right)\right),$ $\displaystyle\mathcal{L}_{3}$ $\displaystyle=\sum_{k}\bigg{(}-\frac{1}{4}\left(q^{\scriptscriptstyle[k]}_{1}\right)^{4}-\left(\left(q^{\scriptscriptstyle[k+1]}_{1}\right)^{2}+q^{\scriptscriptstyle[k+1]}_{1}q^{\scriptscriptstyle[k]}_{1}+\left(q^{\scriptscriptstyle[k]}_{1}\right)^{2}\right)\exp\\!\left(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}\right)$ $\displaystyle\hskip 42.67912pt+q^{\scriptscriptstyle[k]}_{1}q^{\scriptscriptstyle[k]}_{3}-\exp\\!\left(q^{\scriptscriptstyle[k+2]}-q^{\scriptscriptstyle[k]}\right)-\frac{1}{2}\exp\\!\left(2(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]})\right)\bigg{)},$ $\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$ See [18, 29] for constructions of this pluri-Lagrangian structure. The classical Euler-Lagrange equations of these Lagrangian coefficients are $\displaystyle\frac{\delta_{1}{\mathcal{L}_{1}}}{\delta{q^{\scriptscriptstyle[k]}}}=0\quad$ $\displaystyle\Leftrightarrow\quad q^{\scriptscriptstyle[k]}_{11}=e^{q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}}-e^{q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}},$ $\displaystyle\frac{\delta_{2}{\mathcal{L}_{2}}}{\delta{q^{\scriptscriptstyle[k]}}}=0\quad$ $\displaystyle\Leftrightarrow\quad q^{\scriptscriptstyle[k]}_{12}=\left(q^{\scriptscriptstyle[k]}_{1}+q^{\scriptscriptstyle[k+1]}_{1}\right)e^{q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}}-\left(q^{\scriptscriptstyle[k-1]}_{1}+q^{\scriptscriptstyle[k]}_{1}\right)e^{q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}},$ $\displaystyle\mathmakebox[\widthof{{}\Rightarrow{}}][c]{\vdots}$ We recover Equation (3), but for the other equations of the hierarchy we only get a differentiated form. However, we do get their evolutionary form, as in Equation (4), from the multi-time Euler-Lagrange equations $\frac{\delta_{2}{\mathcal{L}_{2}}}{\delta{q^{\scriptscriptstyle[k]}_{1}}}=0,\qquad\frac{\delta_{3}{\mathcal{L}_{3}}}{\delta{q^{\scriptscriptstyle[k]}_{1}}}=0,\qquad\cdots.$ The multi-time Euler-Lagrange equations of type (2) are trivially satisfied in this case: $\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{q^{\scriptscriptstyle[k]}_{j}}}=q^{\scriptscriptstyle[k]}_{1}$ for all $j$. ### 2.2 Lagrangian 2-forms ###### Theorem 2.4 ([28]). Consider the Lagrangian 2-form $\mathcal{L}\llbracket u\rrbracket=\sum_{i<j}\mathcal{L}_{ij}\llbracket u\rrbracket\,\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j},$ depending on an arbitrary number of derivatives of $u$. A field $u$ is critical in the pluri-Lagrangian sense if and only if it satisfies the multi- time Euler-Lagrange equations $\displaystyle\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{I}}}=0$ $\displaystyle\forall I\not\ni t_{i},t_{j},$ (5) $\displaystyle\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{j}}}}-\frac{\delta_{ik}{\mathcal{L}_{ik}}}{\delta{u_{It_{k}}}}=0$ $\displaystyle\forall I\not\ni t_{i},$ (6) $\displaystyle\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}t_{j}}}}+\frac{\delta_{jk}{\mathcal{L}_{jk}}}{\delta{u_{It_{j}t_{k}}}}+\frac{\delta_{ki}{\mathcal{L}_{ki}}}{\delta{u_{It_{k}t_{i}}}}=0$ $\displaystyle\forall I,$ (7) for all triples $(i,j,k)$ of distinct indices, where $\frac{\delta_{ij}{}}{\delta{u_{I}}}=\sum_{\alpha,\beta=0}^{\infty}(-1)^{\alpha+\beta}\partial_{i}^{\alpha}\partial_{j}^{\beta}\frac{\partial{}}{\partial{u_{It_{i}^{\alpha}t_{j}^{\beta}}}}.$ ###### Example 2.5. A Lagrangian 2-form for the potential KdV hierarchy was first given in [28]. It is instructive to look at just two of the equations embedded in $\mathbb{R}^{3}$. Then the Lagrangian 2-form has three coefficients, $\mathcal{L}=\mathcal{L}_{12}\,\mbox{\rm d}t_{1}\wedge\mbox{\rm d}t_{2}+\mathcal{L}_{13}\,\mbox{\rm d}t_{1}\wedge\mbox{\rm d}t_{3}+\mathcal{L}_{23}\,\mbox{\rm d}t_{2}\wedge\mbox{\rm d}t_{3},$ where $t_{1}$ is viewed as the space variable. We can take $\displaystyle\mathcal{L}_{12}$ $\displaystyle=-u_{1}^{3}-\frac{1}{2}u_{1}u_{111}+\frac{1}{2}u_{1}u_{2},$ $\displaystyle\mathcal{L}_{13}$ $\displaystyle=-\frac{5}{2}u_{1}^{4}+5u_{1}u_{11}^{2}-\frac{1}{2}u_{111}^{2}+\frac{1}{2}u_{1}u_{3},$ where $u_{i}$ is a shorthand notation for the partial derivative $u_{t_{i}}$, and similar notations are used for higher derivatives. These are the classical Lagrangians of the potential KdV hierarchy. However, their classical Euler- Lagrange equations give the hierarchy only in a differentiated form, $\displaystyle u_{12}$ $\displaystyle=6u_{1}u_{11}+u_{1111},$ $\displaystyle u_{13}$ $\displaystyle=30u_{1}^{2}u_{11}+20u_{11}u_{111}+10u_{1}u_{1111}+u_{111111}.$ The Lagrangian 2-form also contains a coefficient $\displaystyle\mathcal{L}_{23}$ $\displaystyle=3u_{1}^{5}-\frac{15}{2}u_{1}^{2}u_{11}^{2}+10u_{1}^{3}u_{111}-5u_{1}^{3}u_{2}+\frac{7}{2}u_{11}^{2}u_{111}+3u_{1}u_{111}^{2}-6u_{1}u_{11}u_{1111}$ $\displaystyle\quad+\frac{3}{2}u_{1}^{2}u_{11111}+10u_{1}u_{11}u_{12}-\frac{5}{2}u_{11}^{2}u_{2}-5u_{1}u_{111}u_{2}+\frac{3}{2}u_{1}^{2}u_{3}-\frac{1}{2}u_{1111}^{2}$ $\displaystyle\quad+\frac{1}{2}u_{111}u_{11111}-u_{111}u_{112}+\frac{1}{2}u_{1}u_{113}+u_{1111}u_{12}-\frac{1}{2}u_{11}u_{13}-\frac{1}{2}u_{11111}u_{2}$ $\displaystyle\quad+\frac{1}{2}u_{111}u_{3}$ which does not have a classical interpretation, but contributes meaningfully in the pluri-Lagrangian formalism. In particular, the multi-time Euler- Lagrange equations $\frac{\delta_{12}{\mathcal{L}_{12}}}{\delta{u_{1}}}+\frac{\delta_{23}{\mathcal{L}_{23}}}{\delta{u_{3}}}=0\qquad\text{and}\qquad\frac{\delta_{13}{\mathcal{L}_{13}}}{\delta{u_{1}}}-\frac{\delta_{23}{\mathcal{L}_{23}}}{\delta{u_{3}}}=0$ yield the potential KdV equations in their evolutionary form, $\displaystyle u_{2}$ $\displaystyle=3u_{1}^{2}+u_{111},$ $\displaystyle u_{3}$ $\displaystyle=10u_{1}^{3}+5u_{11}^{2}+10u_{1}u_{111}+u_{11111}.$ All other multi-time Euler-Lagrange equations are consequences of these evolutionary equations. This example can be extended to contain an arbitrary number of equations from the potential KdV hierarchy. The coefficients $\mathcal{L}_{1j}$ will be Lagrangians of the individual equations, whereas the $\mathcal{L}_{ij}$ for $i,j>1$ do not appear in the traditional Lagrangian picture. ###### Example 2.6. The Boussinesq equation $u_{22}=12u_{1}u_{11}-3u_{1111}$ (8) is of second order in its time $t_{2}$, but the higher equations of its hierarchy are of first order in their respective times, beginning with $u_{3}=-6u_{1}u_{2}+3u_{112}.$ (9) A Lagrangian 2-form for this system has coefficients $\displaystyle\mathcal{L}_{12}$ $\displaystyle=\frac{1}{2}u_{2}^{2}-2u_{1}^{3}-\frac{3}{2}u_{11}^{2},$ $\displaystyle\mathcal{L}_{13}$ $\displaystyle=u_{2}u_{3}+6u_{1}^{4}+27u_{1}u_{11}^{2}-6uu_{12}u_{2}+\frac{9}{2}u_{111}^{2}+\frac{3}{2}u_{12}^{2},$ $\displaystyle\mathcal{L}_{23}$ $\displaystyle=24u_{1}^{3}u_{2}+18u_{1}u_{11}u_{12}+9u_{11}^{2}u_{2}-18u_{1}u_{111}u_{2}-2u_{2}^{3}-6uu_{2}u_{22}$ $\displaystyle\quad+6u_{1}^{2}u_{3}+9u_{111}u_{112}+3u_{11}u_{13}+3u_{12}u_{22}-3u_{111}u_{3}.$ They can be found in [30] with a different scaling of $\mathcal{L}$ and a different numbering of the time variables. Equation (8) is equivalent to the Euler-Lagrange equation $\frac{\delta_{12}{\mathcal{L}_{12}}}{\delta{u}}=0$ and Equation (9) to $\frac{\delta_{13}{\mathcal{L}_{13}}}{\delta{u_{2}}}=0.$ All other multi-time Euler-Lagrange equations are differential consequences of Equations (8) and (9). As in the previous example, it is possible to extend this 2-form to represent an arbitrary number of equations from the hierarchy. Further examples of pluri-Lagrangian 2-form systems can be found in [21, 22, 29, 30]. ## 3 Hamiltonian structure of Lagrangian 1-form systems A connection between Lagrangian 1-form systems and Hamiltonian or symplectic systems was found in [26], both in the continuous and the discrete case. Here we specialize that result to the common case where one coefficient of the Lagrangian 1-form is a mechanical Lagrangian and all others are linear in their respective time-derivatives. We formulate explicitly the underlying symplectic structures, which will provide guidance for the case of Lagrangian 2-form systems. Since some of the coefficients of the Lagrangian form will be linear in velocities, it is helpful to first have a look at the Hamiltonian formulation for Lagrangians of this type, independent of a pluri-Lagrangian structure. ### 3.1 Lagrangians that are linear in velocities Let the configuration space be a finite-dimensional real vector space $Q=\mathbb{R}^{N}$ and consider a Lagrangian $\mathcal{L}:TQ\rightarrow\mathbb{R}$ of the form $\mathcal{L}(q,q_{t})=p(q)^{T}q_{t}-V(q),$ (10) where $\det\left(\frac{\partial{p}}{\partial{q}}-\left(\frac{\partial{p}}{\partial{q}}\right)^{T}\right)\neq 0.$ (11) Note that $p$ denotes a function of the position $q$; later on we will use $\pi$ to denote the momentum as an element of cotangent space. If $Q$ is a manifold, the arguments of this subsection will still apply if there exists local coordinates in which the Lagrangian is of the form (10). The Euler- Lagrange equations are first order ODEs: $\dot{q}=\left(\left(\frac{\partial{p}}{\partial{q}}\right)^{T}-\frac{\partial{p}}{\partial{q}}\right)^{-1}\nabla V,$ (12) where $\nabla V=\left(\frac{\partial{V}}{\partial{q}}\right)^{T}$ is the gradient of $V$. Note that Equation (11) implies that $Q$ is even-dimensional, hence $Q$ admits a (local) symplectic structure. Instead of a symplectic form on $T^{*}Q$, the Lagrangian system preserves a symplectic form on $Q$ itself [2, 20]: $\displaystyle\omega=\sum_{i}-\mbox{\rm d}p_{i}(q)\wedge\mbox{\rm d}q_{i}$ $\displaystyle=\sum_{i,j}-\frac{\partial{p_{i}}}{\partial{q_{j}}}\,\mbox{\rm d}q_{j}\wedge\mbox{\rm d}q_{i}$ (13) $\displaystyle=\sum_{i<j}\left(\frac{\partial{p_{i}}}{\partial{q_{j}}}-\frac{\partial{p_{j}}}{\partial{q_{i}}}\right)\mbox{\rm d}q_{i}\wedge\mbox{\rm d}q_{j},$ which is non-degenerate by virtue of Equation (11). ###### Proposition 3.1. The Euler-Lagrange equation (12) of the Lagrangian (10) corresponds to a Hamiltonian vector field with respect to the symplectic structure $\omega$, with Hamilton function $V$. * Proof. The Hamiltonian vector field $X=\sum_{i}X_{i}\frac{\partial{}}{\partial{q_{i}}}$ of the Hamilton function $V$ with respect to $\omega$ satisfies $\iota_{X}\omega=\mbox{\rm d}V,$ where $\iota_{X}\omega=\sum_{i}\sum_{j\neq i}\left(\frac{\partial{p_{j}}}{\partial{q_{i}}}-\frac{\partial{p_{i}}}{\partial{q_{j}}}\right)X_{j}\,\mbox{\rm d}q_{i}$ and $\mbox{\rm d}V=\sum_{i}\frac{\partial{V}}{\partial{q_{i}}}\,\mbox{\rm d}q_{i}.$ Hence $X=\left(\left(\frac{\partial{p}}{\partial{q}}\right)^{T}-\frac{\partial{p}}{\partial{q}}\right)^{-1}\nabla V,$ which is the vector field corresponding to the Euler-Lagrange equation (12). ∎ ### 3.2 From pluri-Lagrangian to Hamiltonian systems On a finite-dimensional real vector space $Q$, consider a Lagrangian 1-form $\mathcal{L}=\sum_{i}\mathcal{L}_{i}\,\mbox{\rm d}t_{i}$ consisting of a mechanical Lagrangian $\mathcal{L}_{1}(q,q_{1})=\frac{1}{2}|q_{1}|^{2}-V_{1}(q),$ (14) where $|q_{1}|^{2}=q_{1}^{T}q_{1}$, and additional coefficients of the form $\mathcal{L}_{i}(q,q_{1},q_{i})=q_{1}^{T}q_{i}-V_{i}(q,q_{1})\qquad\text{for }i\geq 2,$ (15) where the indices of $q$ denote partial derivatives, $q_{i}=q_{t_{i}}=\frac{\mbox{\rm d}q}{\mbox{\rm d}t_{i}}$, whereas the indices of $\mathcal{L}$ and $V$ are labels. We have chosen the Lagrangian coefficients such that they share a common momentum $p=q_{1}$, which is forced upon us by the multi-time Euler-Lagrange equation (2). Note that for each $i$, the coefficient $\mathcal{L}_{i}$ contains derivatives of $q$ with respect to $t_{1}$ and $t_{i}$ only. Many Lagrangian 1-forms are of this form, including the Toda hierarchy, presented in Example 2.3. The nontrivial multi-time Euler-Lagrange equations are $\frac{\delta_{1}{\mathcal{L}_{1}}}{\delta{q}}=0\quad\Leftrightarrow\quad q_{11}=-\frac{\partial{V_{1}}}{\partial{q}},$ and $\frac{\delta_{i}{\mathcal{L}_{i}}}{\delta{q_{1}}}=0\quad\Leftrightarrow\quad q_{i}=\frac{\partial{V_{i}}}{\partial{q_{1}}}\qquad\qquad\text{for }i\geq 2,$ with the additional condition that $\frac{\delta_{i}{\mathcal{L}_{i}}}{\delta{q}}=0\quad\Leftrightarrow\quad q_{1i}+\frac{\partial{V_{i}}}{\partial{q}}=0.$ Hence the multi-time Euler-Lagrange equations are overdetermined. Only for particular choices of $V_{i}$ will the last equation be a differential consequence of the other multi-time Euler-Lagrange equations. The existence of suitable $V_{i}$ for a given hierarchy could be taken as a definition of its integrability. Note that there is no multi-time Euler-Lagrange equation involving the variational derivative $\frac{\delta_{1}{\mathcal{L}_{i}}}{\delta{q}}=\frac{\partial{V_{i}}}{\partial{q}}-\frac{\mbox{\rm d}}{\mbox{\rm d}t_{1}}\frac{\partial{V_{i}}}{\partial{q_{1}}}$ because of the mismatch between the time direction $t_{1}$ in which the variational derivative acts and the index $i$ of the Lagrangian coefficient. The multi-time Euler-Lagrange equations of the type $\frac{\delta_{i}{\mathcal{L}_{i}}}{\delta{q_{i}}}=\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{q_{j}}}$ all reduce to the trivial equation $q_{1}=q_{1}$, expressing the fact that all $\mathcal{L}_{i}$ yield the same momentum. Since $\mathcal{L}_{1}$ is regular, $\det\left(\frac{\partial{{}^{2}\mathcal{L}_{1}}}{\partial{q_{1}^{2}}}\right)\neq 0$, we can find a canonical Hamiltonian for the first equation by Legendre transformation, $H_{1}(q,\pi)=\frac{1}{2}|\pi|^{2}+V_{1}(q),$ where we use $\pi$ to denote the cotangent space coordinate and $|\pi|^{2}=\pi^{T}\pi$. For $i\geq 2$ we consider $r=q_{1}$ as a second dependent variable. In other words, we double the dimension of the configuration space, which is now has coordinates $(q,r)=(q,q_{1})$. The Lagrangians $\mathcal{L}_{i}(q,r,q_{i},r_{i})=rq_{i}-V_{i}(q,r)$ are linear in velocities. We have $p(q,r)=r$, hence the symplectic form (13) is $\omega=\mbox{\rm d}r\wedge\mbox{\rm d}q.$ This is the canonical symplectic form, with the momentum replaced by $r=q_{1}$. Hence we can consider $r$ as momentum, thus identifying the extended configuration space spanned by $q$ and $r$ with the phase space $T^{*}Q$. Applying Proposition 3.1, we arrive at the following result: ###### Theorem 3.2. The multi-time Euler-Lagrange equations of a 1-form with coefficients (14)–(15) are equivalent, under the identification $\pi=q_{1}$, to a system of Hamiltonian equations with respect to the canonical symplectic form $\omega=\mbox{\rm d}\pi\wedge\mbox{\rm d}q$, with Hamilton functions $H_{1}(q,\pi)=\frac{1}{2}|\pi|^{2}+V_{1}(q)\qquad\text{and}\qquad H_{i}(q,\pi)=V_{i}(q,\pi)\quad\text{for }i\geq 2$ ###### Example 3.3. From the Lagrangian 1-form for the Toda lattice given in Example 2.3 we find $\displaystyle H_{1}$ $\displaystyle=\sum_{k}\left(\frac{1}{2}\left(\pi^{[k]}\right)^{2}+\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right)\right),$ $\displaystyle H_{2}$ $\displaystyle=\sum_{k}\left(\frac{1}{3}\left(\pi^{[k]}\right)^{3}+\left(\pi^{[k]}+\pi^{[k-1]}\right)\exp\\!\left(q^{\scriptscriptstyle[k]}-q^{\scriptscriptstyle[k-1]}\right)\right),$ $\displaystyle H_{3}$ $\displaystyle=\sum_{k}\bigg{(}\frac{1}{4}\left(\pi^{[k]}\right)^{4}+\left(\left(\pi^{[k+1]}\right)^{2}+\pi^{[k+1]}\pi^{[k]}+\left(\pi^{[k]}\right)^{2}\right)\exp\\!\left(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]}\right)$ $\displaystyle\hskip 42.67912pt+\exp\\!\left(q^{\scriptscriptstyle[k+2]}-q^{\scriptscriptstyle[k]}\right)+\frac{1}{2}\exp\\!\left(2(q^{\scriptscriptstyle[k+1]}-q^{\scriptscriptstyle[k]})\right)\bigg{)},$ $\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$ We have limited the discussion in this section to the case where $\mathcal{L}_{1}$ is quadratic in the velocity. There are some interesting examples that do not fall into this category, like the Volterra lattice, which has a Lagrangian linear in velocities, and the relativistic Toda lattice, which has a Lagrangian with a more complicated dependence on velocities (see e.g. [25] and the references therein). The discussion above can be adapted to other types of Lagrangian 1-forms if one of its coefficients $\mathcal{L}_{i}$ has an invertible Legendre transform, or if they are collectively Legendre- transformable as described in [26]. ### 3.3 From Hamiltonian to Pluri-Lagrangian systems The procedure from Section 3.2 can be reversed to construct a Lagrangian 1-form from a number of Hamiltonians. ###### Theorem 3.4. Consider Hamilton functions $H_{i}:T^{*}Q\rightarrow\mathbb{R}$, with $H_{1}(q,\pi)=\frac{1}{2}|\pi|^{2}+V_{1}(q)$. Then the multi-time Euler- Lagrange equations of the Lagrangian 1-form $\sum_{i}\mathcal{L}_{i}\,\mbox{\rm d}t_{i}$ with $\displaystyle\mathcal{L}_{1}$ $\displaystyle=\frac{1}{2}|q_{1}|^{2}-V_{1}(q)$ $\displaystyle\mathcal{L}_{i}$ $\displaystyle=q_{1}q_{i}-H_{i}(q,q_{1})\qquad\text{for }i\geq 2$ are equivalent to the Hamiltonian equations under the identification $\pi=q_{1}$. * Proof. Identifying $\pi=q_{1}$, the multi-time Euler-Lagrange equations of the type (1) are $\displaystyle\frac{\delta_{1}{\mathcal{L}_{1}}}{\delta{q}}$ $\displaystyle=0\quad\Leftrightarrow\quad q_{11}=-\frac{\partial{V_{1}(q)}}{\partial{q}},$ $\displaystyle\frac{\delta_{i}{\mathcal{L}_{i}}}{\delta{q_{1}}}$ $\displaystyle=0\quad\Leftrightarrow\quad q_{i}=\frac{\partial{H_{i}(q,\pi)}}{\partial{p}},$ $\displaystyle\frac{\delta_{i}{\mathcal{L}_{i}}}{\delta{q}}$ $\displaystyle=0\quad\Leftrightarrow\quad\pi_{i}=-\frac{\partial{H_{i}(q,\pi)}}{\partial{q}}.$ The multi-time Euler-Lagrange equations of the type (2) are trivially satisfied because $\frac{\delta_{i}{\mathcal{L}_{i}}}{\delta{q_{i}}}=q_{1}$ for all $i$. ∎ Note that the statement of Theorem 3.4 does not require the Hamiltonian equations to commute, i.e. it is not imposed that the Hamiltonian vector fields $X_{H_{i}}$ associated to the Hamilton functions $H_{i}$ satisfy $[X_{H_{i}},X_{H_{j}}]=0$. However, if they do not commute then for a generic initial condition $(q_{0},\pi_{0})$ there will be no solution $(q,\pi):\mathbb{R}^{N}\rightarrow T^{*}Q$ to the equations $\displaystyle\frac{\partial{}}{\partial{t_{i}}}(q(t_{1},\ldots t_{N}),\pi(t_{1},\ldots t_{N}))=X_{H_{i}}(q(t_{1},\ldots t_{N}),\pi(t_{1},\ldots t_{N}))\qquad(i=1,\ldots,N),$ $\displaystyle(q(0,\ldots,0),\pi(0,\ldots,0))=(q_{0},\pi_{0}).$ Hence the relevance of Theorem 3.4 lies almost entirely in the case of commuting Hamiltonian equations. If they do not commute then it is an (almost) empty statement because neither the system of Hamiltonian equations nor the multi-time Euler-Lagrange equations will have solutions for generic initial data. ###### Example 3.5. The Kepler Problem, describing the motion of a point mass around a gravitational center, is one of the classic examples of a completely integrable system. It possesses Poisson-commuting Hamiltonians $H_{1},H_{2},H_{3}:T^{*}\mathbb{R}^{3}\rightarrow\mathbb{R}$ given by $\displaystyle H_{1}(q,\pi)$ $\displaystyle=\frac{1}{2}|\pi|^{2}-|q|^{-1},\quad$ the energy, Hamiltonian for the physical motion, $\displaystyle H_{2}(q,\pi)$ $\displaystyle=(q\times\pi)\cdot\mathsf{e}_{z},$ the 3rd component of the angular momentum, and $\displaystyle H_{3}(q,\pi)$ $\displaystyle=|q\times\pi|^{2},$ the squared magnitude of the angular momentum, where $q=(x,y,z)$ and $\mathsf{e}_{z}$ is the unit vector in the $z$-direction. The corresponding coefficients of the Lagrangian 1-form are $\displaystyle\mathcal{L}_{1}$ $\displaystyle=\frac{1}{2}|q_{1}|^{2}+|q|^{-1},$ $\displaystyle\mathcal{L}_{2}$ $\displaystyle=q_{1}\cdot q_{2}-(q\times q_{1})\cdot\mathsf{e}_{z},$ $\displaystyle\mathcal{L}_{3}$ $\displaystyle=q_{1}\cdot q_{3}-|q\times q_{1}|^{2}.$ The multi-time Euler-Lagrange equations are $\frac{\delta_{1}{\mathcal{L}_{1}}}{\delta{q}}=0\quad\Rightarrow\quad q_{11}=\frac{q}{|q|^{3}},$ the physical equations of motion, $\displaystyle\frac{\delta_{2}{\mathcal{L}_{2}}}{\delta{q_{1}}}=0$ $\displaystyle\quad\Rightarrow\quad q_{2}=\mathsf{e}_{z}\times q,$ $\displaystyle\frac{\delta_{2}{\mathcal{L}_{2}}}{\delta{q}}=0$ $\displaystyle\quad\Rightarrow\quad q_{12}=-q_{1}\times\mathsf{e}_{z},\ $ describing a rotation around the $z$-axis, and $\displaystyle\frac{\delta_{3}{\mathcal{L}_{3}}}{\delta{q_{1}}}=0$ $\displaystyle\quad\Rightarrow\quad q_{3}=2(q\times q_{1})\times q,$ $\displaystyle\frac{\delta_{3}{\mathcal{L}_{3}}}{\delta{q}}=0$ $\displaystyle\quad\Rightarrow\quad q_{13}=2(q\times q_{1})\times q_{1},$ describing a rotation around the angular momentum vector. ### 3.4 Closedness and involutivity In the pluri-Lagrangian theory, the exterior derivative $\mbox{\rm d}\mathcal{L}$ is constant on solutions (see Proposition A.2 in the Appendix). In many cases this constant is zero, i.e. the Lagrangian 1-form is closed on solutions. Here we relate this property to the vanishing of Poisson brackets between the Hamilton functions. ###### Proposition 3.6 ([26, Theorem 3]). Consider a Lagrangian 1-form $\mathcal{L}$ as in Section 3.2 and the corresponding Hamilton functions $H_{i}$. On solutions to the multi-time Euler-Lagrange equations, and identifying $\pi=p(q,q_{1})=\frac{\partial{\mathcal{L}_{i}}}{\partial{q_{i}}}$, there holds $\begin{split}\frac{\mbox{\rm d}\mathcal{L}_{j}}{\mbox{\rm d}t_{i}}-\frac{\mbox{\rm d}\mathcal{L}_{i}}{\mbox{\rm d}t_{j}}&=p_{j}q_{i}-p_{i}q_{j}\\\ &=\\{H_{j},H_{i}\\},\end{split}$ (16) where $\\{\cdot,\cdot\\}$ denotes the canonical Poisson bracket and $p_{j}$ and $q_{j}$ are shorthand for $\frac{\mbox{\rm d}p}{\mbox{\rm d}t_{j}}$ and $\frac{\mbox{\rm d}q}{\mbox{\rm d}t_{j}}$. * Proof. On solutions of the multi-time Euler-Lagrange equations there holds $\displaystyle\frac{\mbox{\rm d}\mathcal{L}_{j}}{\mbox{\rm d}t_{i}}$ $\displaystyle=\frac{\partial{\mathcal{L}_{j}}}{\partial{q}}q_{i}+\frac{\partial{\mathcal{L}_{j}}}{\partial{q_{1}}}q_{1i}+\frac{\partial{\mathcal{L}_{j}}}{\partial{q_{j}}}q_{ij}$ $\displaystyle=\left(\frac{\mbox{\rm d}}{\mbox{\rm d}t_{j}}\frac{\partial{\mathcal{L}_{j}}}{\partial{q}}\right)q_{i}+\frac{\partial{\mathcal{L}_{j}}}{\partial{q_{j}}}q_{ij}$ $\displaystyle=p_{j}q_{i}+pq_{ij}.$ Hence $\frac{\mbox{\rm d}\mathcal{L}_{j}}{\mbox{\rm d}t_{i}}-\frac{\mbox{\rm d}\mathcal{L}_{i}}{\mbox{\rm d}t_{j}}=p_{j}q_{i}-p_{i}q_{j}.$ (17) Alternatively, we can calculate this expression using the Hamiltonian formalism. We have $\displaystyle\frac{\mbox{\rm d}\mathcal{L}_{j}}{\mbox{\rm d}t_{i}}-\frac{\mbox{\rm d}\mathcal{L}_{i}}{\mbox{\rm d}t_{j}}$ $\displaystyle=\frac{\mbox{\rm d}}{\mbox{\rm d}t_{i}}(pq_{j}-H_{j})-\frac{\mbox{\rm d}}{\mbox{\rm d}t_{j}}(pq_{i}-H_{i})$ $\displaystyle=p_{i}q_{j}-p_{j}q_{i}+2\\{H_{j},H_{i}\\}.$ Combined with Equation (17), this implies Equation (16). ∎ As a corollary we have: ###### Theorem 3.7. The Hamiltonians $H_{i}$ from Theorem 3.2 are in involution if and only if $\mbox{\rm d}\mathcal{L}=0$ on solutions. All examples of Lagrangian 1-forms discussed so far satisfy $\mbox{\rm d}\mathcal{L}=0$ on solutions. This need not be the case. ###### Example 3.8. Let us consider a system of commuting equations that is not Liouville integrable. Fix a constant $c\neq 0$ and consider the 1-form $\mathcal{L}=\mathcal{L}_{1}\,\mbox{\rm d}t_{1}+\mathcal{L}_{2}\,\mbox{\rm d}t_{2}$ with $\mathcal{L}_{1}\llbracket r,\theta\rrbracket=\frac{1}{2}r^{2}\theta_{1}^{2}+\frac{1}{2}r_{1}^{2}-V(r)-c\theta,$ which for $c=0$ would describe a central force in the plane governed by the potential $V$, and $\mathcal{L}_{2}\llbracket r,\theta\rrbracket=r^{2}\theta_{1}(\theta_{2}-1)+r_{1}r_{2}.$ Its multi-time Euler-Lagrange equations are $\displaystyle r_{11}=-V^{\prime}(r)+r\theta_{1}^{2},$ $\displaystyle\frac{\mbox{\rm d}}{\mbox{\rm d}t_{1}}(r^{2}\theta_{1})=-c,$ $\displaystyle r_{2}=0,$ $\displaystyle\theta_{2}=1,$ and consequences thereof. Notably, we have $\frac{\mbox{\rm d}\mathcal{L}_{2}}{\mbox{\rm d}t_{1}}-\frac{\mbox{\rm d}\mathcal{L}_{1}}{\mbox{\rm d}t_{2}}=c$ on solutions, hence $\mbox{\rm d}\mathcal{L}$ is nonzero. By Theorem 3.2 the multi-time Euler-Lagrange equations are equivalent to the canonical Hamiltonian systems with $\displaystyle H_{1}(r,\theta,\pi,\sigma)$ $\displaystyle=\frac{1}{2}\frac{\sigma^{2}}{r^{2}}+\frac{1}{2}\pi^{2}+V(r)+c\theta$ $\displaystyle H_{2}(r,\theta,\pi,\sigma)$ $\displaystyle=\sigma,$ where $\pi$ and $\sigma$ are the conjugate momenta to $r$ and $\theta$. The Hamiltonians are not in involution, but rather $\\{H_{2},H_{1}\\}=c=\frac{\mbox{\rm d}\mathcal{L}_{2}}{\mbox{\rm d}t_{1}}-\frac{\mbox{\rm d}\mathcal{L}_{1}}{\mbox{\rm d}t_{2}}.$ ## 4 Hamiltonian structure of Lagrangian 2-form systems In order to generalize the results from Section 3 to the case of 2-forms, we need to carefully examine the relevant geometric structure. A useful tool for this is the variational bicomplex, which is also used in Appendix A to study the multi-time Euler-Lagrange equations. ### 4.1 The variational bicomplex To facilitate the variational calculus in the pluri-Lagrangian setting, it is useful to consider the variation operator $\delta$ as an exterior derivative, acting in the fiber $J^{\infty}$ of the infinite jet bundle. We call $\delta$ the _vertical exterior derivative_ and d, which acts in the base manifold $M$, the _horizontal exterior derivative_. Together they provide a double grading of the space $\Omega(M\times J^{\infty})$ of differential forms on the jet bundle. The space of _$(a,b)$ -forms_ is generated by those $(a+b)$-forms structured as $f\llbracket u\rrbracket\,\delta u_{I_{1}}\wedge\ldots\wedge\delta u_{I_{a}}\wedge\mbox{\rm d}t_{j_{1}}\ldots\wedge\mbox{\rm d}t_{j_{b}}.$ We denote the space of $(a,b)$-forms by $\Omega^{(a,b)}\subset\Omega^{a+b}(M\times J^{\infty})$. We call elements of $\Omega^{(0,b)}$ horizontal forms and elements of $\Omega^{(a,0)}$ vertical forms. The Lagrangian is a horizontal $d$-form, $\mathcal{L}\in\Omega^{(0,d)}$. The horizontal and vertical exterior derivatives are characterized by the anti-derivation property, $\displaystyle\mbox{\rm d}\left(\omega_{1}^{p_{1},q_{1}}\wedge\omega_{2}^{p_{2},q_{2}}\right)$ $\displaystyle=\mbox{\rm d}\omega_{1}^{p_{1},q_{1}}\wedge\omega_{2}^{p_{2},q_{2}}+(-1)^{p_{1}+q_{1}}\,\omega_{1}^{p_{1},q_{1}}\wedge\mbox{\rm d}\omega_{2}^{p_{2},q_{2}},$ $\displaystyle\delta\left(\omega_{1}^{p_{1},q_{1}}\wedge\omega_{2}^{p_{2},q_{2}}\right)$ $\displaystyle=\delta\omega_{1}^{p_{1},q_{1}}\wedge\omega_{2}^{p_{2},q_{2}}+(-1)^{p_{1}+q_{1}}\,\omega_{1}^{p_{1},q_{1}}\wedge\delta\omega_{2}^{p_{2},q_{2}},$ where the upper indices denote the type of the forms, and by the way they act on $(0,0)$-forms, and basic $(1,0)$ and $(0,1)$-forms: $\displaystyle\mbox{\rm d}f\llbracket u\rrbracket$ $\displaystyle=\sum_{j}\partial_{j}f\llbracket u\rrbracket\,\mbox{\rm d}t_{j},$ $\displaystyle\delta f\llbracket u\rrbracket$ $\displaystyle=\sum_{I}\frac{\partial{f\llbracket u\rrbracket}}{\partial{u_{I}}}\delta u_{I},$ $\displaystyle\mbox{\rm d}(\delta u_{I})$ $\displaystyle=-\sum_{j}\delta u_{Ij}\wedge\mbox{\rm d}t_{j},\hskip 56.9055pt$ $\displaystyle\delta(\delta u_{I})$ $\displaystyle=0,$ $\displaystyle\mbox{\rm d}(\mbox{\rm d}t_{j})$ $\displaystyle=0,$ $\displaystyle\delta(\mbox{\rm d}t_{j})$ $\displaystyle=0.$ One can verify that $\mbox{\rm d}+\delta:\Omega^{a+b}\rightarrow\Omega^{a+b+1}$ is the usual exterior derivative and that $\delta^{2}=\mbox{\rm d}^{2}=\delta\mbox{\rm d}+\mbox{\rm d}\delta=0.$ Time-derivatives $\partial_{j}$ act on vertical forms as $\partial_{j}(\delta u_{I})=\delta u_{Ij}$, on horizontal forms as $\partial_{j}(\mbox{\rm d}t_{k})=0$, and obey the Leibniz rule with respect to the wedge product. As a simple but important example, note that $\mbox{\rm d}(f\llbracket u\rrbracket\,\delta u_{I})=\sum_{j=1}^{N}\partial_{j}f\llbracket u\rrbracket\,\mbox{\rm d}t_{j}\wedge\delta u_{I}-f\llbracket u\rrbracket\,\delta u_{It_{j}}\wedge\mbox{\rm d}t_{j}=\sum_{j=1}^{N}-\partial_{j}(f\llbracket u\rrbracket\,\delta u_{I})\wedge\mbox{\rm d}t_{j}.$ The spaces $\Omega^{(a,b)}$, for $a\geq 0$ and $0\leq b\leq N$, related to each other by the maps d and $\delta$, are collectively known as the _variational bicomplex_ [8, Chapter 19]. A slightly different version of the variational bicomplex, using contact 1-forms instead of vertical forms, is presented in [1]. We will not discuss the rich algebraic structure of the variational bicomplex here. For a horizontal $(0,d)$-form $\mathcal{L}\llbracket u\rrbracket$, the variational principle $\delta\int_{\Gamma}\mathcal{L}\llbracket u\rrbracket=\delta\int_{\Gamma}\sum_{i_{1}<\ldots<i_{d}}\mathcal{L}_{i_{1},\ldots,i_{d}}\llbracket u\rrbracket\,\mbox{\rm d}t_{i_{1}}\wedge\ldots\wedge\mbox{\rm d}t_{i_{d}}=0$ can be understood as follows. Every vertical vector field $V=v(t_{1},\ldots,t_{a})\frac{\partial}{\partial u}$, such that its _prolongation_ $\operatorname{pr}V=\sum_{I}v_{I}\frac{\partial}{\partial u_{I}}$ vanishes on the boundary $\partial\Gamma$, must satisfy $\int_{\Gamma}\iota_{\operatorname{pr}V}\delta\mathcal{L}=\int_{\Gamma}\sum_{i_{1}<\ldots<i_{d}}\iota_{\operatorname{pr}V}(\delta\mathcal{L}_{i_{1},\ldots,i_{d}}\llbracket u\rrbracket)\,\mbox{\rm d}t_{i_{1}}\wedge\ldots\wedge\mbox{\rm d}t_{i_{d}}=0.$ Note that the integrand is a horizontal form, so the integration takes place on $\Gamma\subset M$, independent of the bundle structure. ### 4.2 The space of functionals and its pre-symplectic structure In the rest of our discussion, we will single out the variable $t_{1}=x$ and view it as the space variable, as opposed to the time variables $t_{2},\ldots,t_{N}$. For ease of presentation we will limit the discussion here to real scalar fields, but it is easily extended to complex or vector- valued fields. We consider functions $u:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto u(x)$ as fields at a fixed time. Let $J^{\infty}$ be the fiber of the corresponding infinite jet bundle, where the prolongation of $u$ has coordinates $[u]=(u,u_{x},u_{xx},\ldots)$. Consider the space of functions of the infinite jet of $u$, $\mathcal{V}=\left\\{v:J^{\infty}\rightarrow\mathbb{R}\right\\}.$ Note that the domain $J^{\infty}$ is the fiber of the jet bundle, hence the elements $v\in\mathcal{V}$ depend on $x$ only through $u$. We will be dealing with integrals $\int v\,\mbox{\rm d}x$ of elements $v\in\mathcal{V}$. In order to avoid convergence questions, we understand the symbol $\int v\,\mbox{\rm d}x$ as a _formal integral_ , defined as the equivalence class of $v$ modulo space-derivatives. In other words, we consider the space of functionals $\mathcal{F}=\mathcal{V}\big{/}\partial_{x}\\!\mathcal{V},$ where $\partial_{x}=\frac{\mbox{\rm d}}{\mbox{\rm d}x}=\sum_{I}u_{Ix}\frac{\partial{}}{\partial{u_{I}}}.$ The variation of an element of $\mathcal{F}$ is computed as $\delta\int v\,\mbox{\rm d}x=\int\frac{\delta{v}}{\delta{u}}\,\delta u\wedge\mbox{\rm d}x,$ (18) where $\frac{\delta{}}{\delta{u}}=\sum_{\alpha=0}^{\infty}(-1)^{\alpha}\partial_{x}^{\alpha}\frac{\partial{}}{\partial{u_{x^{\alpha}}}}.$ Equation (18) is independent of the choice of representative $v\in\mathcal{V}$ because the variational derivative of a full $x$-derivative is zero. Since $\mathcal{V}$ is a linear space, its tangent spaces can be identified with $\mathcal{V}$ itself. In turn, every $v\in\mathcal{V}$ can be identified with a vector field $v\frac{\partial}{\partial u}$. We will define Hamiltonian vector fields in terms of $\mathcal{F}$-valued forms on $\mathcal{V}$. An $\mathcal{F}$-valued 1-form $\theta$ can be represented as the integral of a $(1,1)$-form in the variational bicomplex, $\theta=\int\sum_{k}a_{k}[u]\,\delta u_{x^{k}}\wedge\mbox{\rm d}x$ and defines a map $\mathcal{V}\rightarrow\mathcal{F}:v\mapsto\iota_{v}\theta=\int\sum_{k}a_{k}[u]\,\partial_{x}^{k}v[u]\,\mbox{\rm d}x.$ This amounts to pairing the 1-form with the infinite jet prolongation of the vector field $v\frac{\partial}{\partial u}$. Note that $\mathcal{F}$-valued forms are defined modulo $x$-derivatives: $\int\partial_{x}\theta\wedge\mbox{\rm d}x=0$ because its pairing with any vector field in $\mathcal{V}$ will yield a full $x$-derivative, which represents the zero functional in $\mathcal{F}$. Hence the space of $\mathcal{F}$-valued 1-forms is $\Omega^{(1,1)}/\partial_{x}\Omega^{(1,1)}$. An $\mathcal{F}$-valued 2-form $\omega=\int\sum_{k,\ell}a_{k,\ell}[u]\,\delta u_{x^{k}}\wedge\delta u_{x^{\ell}}\wedge\mbox{\rm d}x$ defines a skew-symmetric map $\mathcal{V}\times\mathcal{V}\rightarrow\mathcal{F}:(v,w)\mapsto\iota_{w}\iota_{v}\omega=\int\sum_{k,\ell}a_{k,\ell}[u]\left(\partial_{x}^{k}v[u]\,\partial_{x}^{\ell}w[u]-\partial_{x}^{k}w[u]\,\partial_{x}^{\ell}v[u]\right)\mbox{\rm d}x$ as well as a map from vector fields to $\mathcal{F}$-valued 1-forms $\mathcal{V}\rightarrow\Omega^{(1,1)}/\partial_{x}\Omega^{(1,1)}:v\mapsto\iota_{v}\omega=\int\sum_{k,\ell}a_{k,\ell}[u]\left(\partial_{x}^{k}v[u]\,\delta u_{x^{\ell}}-\partial_{x}^{\ell}v[u]\,\delta u_{x^{k}}\right)\wedge\mbox{\rm d}x.$ ###### Definition 4.1. A closed $(2,1)$-form $\omega$ on $\mathcal{V}$ is called _pre-symplectic_. Equivalently we can require the form to be vertically closed, i.e. closed with respect to $\delta$. Since the horizontal space is 1-dimensional ($x$ is the only independent variable) every $(a,1)$-form is closed with respect to the horizontal exterior derivative d, so only vertical closedness is a nontrivial property. We choose to work with pre-symplectic forms instead of symplectic forms, because the non-degeneracy required of a symplectic form is a subtle issue in the present context. Consider for example the pre-symplectic form $\omega=\int\delta u\wedge\delta u_{x}\wedge\mbox{\rm d}x$. It is degenerate because $\int\iota_{v}\omega=\int(v\,\delta u_{x}-v_{x}\,\delta u)\wedge\mbox{\rm d}x=\int-2v_{x}\,\delta u\wedge\mbox{\rm d}x,$ which is zero whenever $v[u]$ is constant. However, if we restrict our attention to compactly supported fields, then a constant must be zero, so the restriction of $\omega$ to the space of compactly supported fields is non- degenerate. ###### Definition 4.2. A _Hamiltonian vector field_ with Hamilton functional ${\textstyle\int}H\,\mbox{\rm d}x$ is an element $v\in\mathcal{V}$ satisfying the relation $\int\iota_{v}\omega=\int\delta H\wedge\mbox{\rm d}x.$ Note that if $\omega$ is degenerate, we cannot guarantee existence or uniqueness of a Hamiltonian vector field in general. ### 4.3 From pluri-Lagrangian to Hamiltonian systems We will consider two different types of Lagrangian 2-forms. The first type are those where for every $j$ the coefficient $\mathcal{L}_{1j}$ is linear in $u_{t_{j}}$. This is the case for the 2-form for the potential KdV hierarchy from Example 2.5 and for the Lagrangian 2-forms of many other hierarchies like the AKNS hierarchy [21] and the modified KdV, Schwarzian KdV and Krichever- Novikov hierarchies [30]. The second type satisfy the same property for $j>2$, but have a coefficient $\mathcal{L}_{12}$ that is quadratic in $u_{t_{2}}$, as is the case for the Boussinesq hierarchy from Example 2.6. #### 4.3.1 When all $\mathcal{L}_{1j}$ are linear in $u_{t_{j}}$ Consider a Lagrangian 2-form $\mathcal{L}\llbracket u\rrbracket=\sum_{i<j}\mathcal{L}_{ij}\llbracket u\rrbracket\,\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}$, where for all $j$ the variational derivative $\frac{\delta_{1}{\mathcal{L}_{1j}}}{\delta{u_{t_{j}}}}$ does not depend on any $t_{j}$-derivatives, hence we can write $\frac{\delta_{1}{\mathcal{L}_{1j}}}{\delta{u_{t_{j}}}}=p[u]$ for some function $p[u]$ depending on on an arbitrary number of space derivatives, but not on any time-derivatives. We use single square brackets $[\cdot]$ to indicate dependence on space derivatives only. Note that $p$ does not depend on the index $j$. This is imposed on us by the multi-time Euler- Lagrange equation stating that $\frac{\delta_{1}{\mathcal{L}_{1j}}}{\delta{u_{t_{j}}}}$ is independent of $j$. Starting from these assumptions and possibly adding a full $x$-derivative (recall that $x=t_{1}$) we find that the coefficients $\mathcal{L}_{1j}$ are of the form $\mathcal{L}_{1j}\llbracket u\rrbracket=p[u]u_{j}-h_{j}[u],$ (19) where $u_{j}$ is shorthand notation for the derivative $u_{t_{j}}$. Coefficients of this form appear in many prominent examples, like the potential KdV hierarchy and several hierarchies related to it [28, 29, 30] as well as the AKNS hierarchy [21]. Their Euler-Lagrange equations are $\mathcal{E}_{p}u_{j}-\frac{\delta_{1}{h_{j}[u]}}{\delta{u}}=0,$ (20) where $\mathcal{E}_{p}$ is the differential operator $\mathcal{E}_{p}=\sum_{k=0}^{\infty}\left((-1)^{k}\partial_{x}^{k}\frac{\partial{p}}{\partial{u_{x^{k}}}}-\frac{\partial{p}}{\partial{u_{x^{k}}}}\partial_{x}^{k}\right).$ We can also write $\mathcal{E}_{p}=\mathsf{D}_{p}^{*}-\mathsf{D}_{p}$, where $\mathsf{D}_{p}$ is the Fréchet derivative of $p$ and $\mathsf{D}_{p}^{*}$ its adjoint [17, Eqs (5.32) resp. (5.79)]. Consider the pre-symplectic form $\begin{split}\omega&=-\delta p[u]\wedge\delta u\wedge\mbox{\rm d}x\\\ &=-\sum_{k=1}^{\infty}\frac{\partial{p}}{\partial{u_{x^{k}}}}\delta u_{x^{k}}\wedge\delta u\wedge\mbox{\rm d}x.\end{split}$ (21) Inserting the vector field $X=\chi\frac{\partial{}}{\partial{u}}$ we find $\displaystyle\int\iota_{X}\omega$ $\displaystyle=\int\sum_{k=0}^{\infty}\left(\frac{\partial{p}}{\partial{u_{x^{k}}}}\chi\,\delta u_{x^{k}}\wedge\mbox{\rm d}x-\frac{\partial{p}}{\partial{u_{x^{k}}}}\chi_{x^{k}}\,\delta u\wedge\mbox{\rm d}x\right)$ $\displaystyle=\int\sum_{k=0}^{\infty}\left((-1)^{k}\partial_{x}^{k}\\!\left(\frac{\partial{p}}{\partial{u_{x^{k}}}}\chi\right)-\frac{\partial{p}}{\partial{u_{x^{k}}}}\chi_{x^{k}}\right)\delta u\wedge\mbox{\rm d}x$ $\displaystyle=\int\mathcal{E}_{p}\chi\,\delta u\wedge\mbox{\rm d}x.$ From the Hamiltonian equation of motion $\int\iota_{X}\omega=\int\delta h_{j}[u]\wedge\mbox{\rm d}x$ we now obtain that the Hamiltonian vector field $X=\chi\frac{\partial{}}{\partial{u}}$ associated to $h_{j}$ satisfies $\mathcal{E}_{p}\chi=\frac{\delta_{1}{h_{j}}}{\delta{u}},$ which corresponds the Euler-Lagrange equation (20) by identifying $\chi=u_{t_{j}}$. This observation was made previously in the context of loop spaces in [16, Section 1.3]. The Poisson bracket associated to the symplectic operator $\mathcal{E}_{p}$ is formally given by $\left\\{{\textstyle\int}f\,\mbox{\rm d}x,{\textstyle\int}g\,\mbox{\rm d}x\right\\}=-\int\frac{\delta{f}}{\delta{u}}\,\mathcal{E}_{p}^{-1}\,\frac{\delta{g}}{\delta{u}}\,\mbox{\rm d}x.$ (22) If the pre-symplectic form is degenerate, then $\mathcal{E}_{p}$ will not be invertible. In this case $\mathcal{E}_{p}^{-1}$ can be considered as a pseudo- differential operator and the Poisson bracket is called _non-local_ [16, 7]. Note that $\\{\cdot,\cdot\\}$ does not satisfy the Leibniz rule because there is no multiplication on the space $\mathcal{F}$ of formal integrals. However, we can recover the Leibniz rule in one entry by introducing $[f,g]=-\sum_{k=0}^{\infty}\frac{\partial{f}}{\partial{u_{x^{k}}}}\partial_{x}^{k}\,\mathcal{E}_{p}^{-1}\,\frac{\delta{g}}{\delta{u}}.$ Then we have $\left\\{{\textstyle\int}f\,\mbox{\rm d}x,{\textstyle\int}g\,\mbox{\rm d}x\right\\}=\int[f,g]\,\mbox{\rm d}x$ and $[fg,h]=f[g,h]+[f,h]g.$ In summary, we have the following result: ###### Theorem 4.3. Assume that $\frac{\delta_{1}{h_{j}[u]}}{\delta{u}}$ is in the image of $\mathcal{E}_{p}$ and has a unique inverse (possibly in some equivalence class) for each $j$. Then the evolutionary PDEs $u_{j}=\mathcal{E}_{p}^{-1}\frac{\delta_{1}{h_{j}[u]}}{\delta{u}},$ which imply the Euler-Lagrange equations (20) of the Lagrangians (19), are Hamiltonian with respect to the symplectic form (21) and the Poisson bracket (22), with Hamilton functions $h_{j}$. This theorem applies without assuming any kind of consistency of the system of multi-time Euler-Lagrange equations. Of course we are mostly interested in the case where the multi-time Euler-Lagrange equations are equivalent to an integrable hierarchy. In almost all known examples (see e.g. [28, 21, 30]) the multi-time Euler-Lagrange equations consist of an integrable hierarchy in its evolutionary form and differential consequences thereof. Hence the general picture suggested by these examples is that the multi-time Euler-Lagrange equations are equivalent to the equations $u_{j}=\mathcal{E}_{p}^{-1}\frac{\delta_{1}{h_{j}[u]}}{\delta{u}}$ form Theorem 4.3. In light of these observations, we emphasize the following consequence of Theorem 4.3. ###### Corollary 4.4. If the multi-time Euler-Lagrange equations are evolutionary, then they are Hamiltonian. ###### Example 4.5. The pluri-Lagrangian structure for the potential KdV hierarchy, given in Example 2.5, has $p=\frac{1}{2}u_{x}$. Hence $\mathcal{E}_{p}=-\partial_{x}\frac{\partial{p}}{\partial{u_{x}}}-\frac{\partial{p}}{\partial{u_{x}}}\partial_{x}=-\partial_{x}$ and $\left\\{{\textstyle\int}f\,\mbox{\rm d}x,{\textstyle\int}g\,\mbox{\rm d}x\right\\}=\int\frac{\delta{f}}{\delta{u}}\partial_{x}^{-1}\frac{\delta{g}}{\delta{u}}\,\mbox{\rm d}x.$ Here we assume that $\frac{\delta{g}}{\delta{u}}$ is in the image of $\partial_{x}$. Then $\partial_{x}^{-1}\frac{\delta{g}}{\delta{u}}$ is uniquely defined by the convention that it does not contain a constant term. If $f$ and $g$ depend only on derivatives of $u$, not on $u$ itself, this becomes the Gardner bracket [10] $\left\\{{\textstyle\int}f\,\mbox{\rm d}x,{\textstyle\int}g\,\mbox{\rm d}x\right\\}=\int\left(\partial_{x}\frac{\delta{f}}{\delta{u_{x}}}\right)\frac{\delta{g}}{\delta{u_{x}}}\,\mbox{\rm d}x.$ The Hamilton functions are $\displaystyle h_{2}[u]$ $\displaystyle=\frac{1}{2}u_{x}u_{t_{2}}-\mathcal{L}_{12}=u_{x}^{3}+\frac{1}{2}u_{x}u_{xxx},$ $\displaystyle h_{3}[u]$ $\displaystyle=\frac{1}{2}u_{x}u_{t_{3}}-\mathcal{L}_{13}=\frac{5}{2}u_{x}^{4}-5u_{x}u_{xx}^{2}+\frac{1}{2}u_{xxx}^{2},$ $\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$ A related derivation of the Gardner bracket from the multi-symplectic perspective was given in [11]. It can also be obtained from the Lagrangian structure by Dirac reduction [15]. ###### Example 4.6. The Schwarzian KdV hierarchy, $\displaystyle u_{2}$ $\displaystyle=-\frac{3u_{11}^{2}}{2u_{1}}+u_{111},$ $\displaystyle u_{3}$ $\displaystyle=-\frac{45u_{11}^{4}}{8u_{1}^{3}}+\frac{25u_{11}^{2}u_{111}}{2u_{1}^{2}}-\frac{5u_{111}^{2}}{2u_{1}}-\frac{5u_{11}u_{1111}}{u_{1}}+u_{11111},$ $\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$ has a pluri-Lagrangian structure with coefficients [29] $\displaystyle\mathcal{L}_{12}$ $\displaystyle=\frac{u_{3}}{2u_{1}}-\frac{u_{11}^{2}}{2u_{1}^{2}},$ $\displaystyle\mathcal{L}_{13}$ $\displaystyle=\frac{u_{5}}{2u_{1}}-\frac{3u_{11}^{4}}{8u_{1}^{4}}+\frac{u_{111}^{2}}{2u_{1}^{2}},$ $\displaystyle\mathcal{L}_{23}$ $\displaystyle=-\frac{45u_{11}^{6}}{32u_{1}^{6}}+\frac{57u_{11}^{4}u_{111}}{16u_{1}^{5}}-\frac{19u_{11}^{2}u_{111}^{2}}{8u_{1}^{4}}+\frac{7u_{111}^{3}}{4u_{1}^{3}}-\frac{3u_{11}^{3}u_{1111}}{4u_{1}^{4}}-\frac{3u_{11}u_{111}u_{1111}}{2u_{1}^{3}}$ $\displaystyle\quad+\frac{u_{1111}^{2}}{2u_{1}^{2}}+\frac{3u_{11}^{2}u_{11111}}{4u_{1}^{3}}-\frac{u_{111}u_{11111}}{2u_{1}^{2}}+\frac{u_{111}u_{113}}{u_{1}^{2}}-\frac{3u_{11}^{3}u_{13}}{2u_{1}^{4}}+\frac{2u_{11}u_{111}u_{13}}{u_{1}^{3}}$ $\displaystyle\quad-\frac{u_{1111}u_{13}}{u_{1}^{2}}+\frac{u_{11}u_{15}}{u_{1}^{2}}-\frac{27u_{11}^{4}u_{3}}{16u_{1}^{5}}+\frac{17u_{11}^{2}u_{111}u_{3}}{4u_{1}^{4}}-\frac{7u_{111}^{2}u_{3}}{4u_{1}^{3}}-\frac{3u_{11}u_{1111}u_{3}}{2u_{1}^{3}}$ $\displaystyle\quad+\frac{u_{11111}u_{3}}{2u_{1}^{2}}+\frac{u_{11}^{2}u_{5}}{4u_{1}^{3}}-\frac{u_{111}u_{5}}{2u_{1}^{2}},$ $\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$ In this example we have $p=\frac{1}{2u_{x}}$, hence $\mathcal{E}_{p}=-\partial_{x}\frac{\partial{p}}{\partial{u_{x}}}-\frac{\partial{p}}{\partial{u_{x}}}\partial_{x}=\frac{1}{u_{x}^{2}}\partial_{x}-\frac{u_{xx}}{u_{x}^{3}}=\frac{1}{u_{x}}\partial_{x}\frac{1}{u_{x}}$ and $\mathcal{E}_{p}^{-1}=u_{x}\partial_{x}^{-1}u_{x}.$ This nonlocal operator seems to be the simplest Hamiltonian operator for the SKdV equation, see for example [9, 31]. The Hamilton functions for the first two equations of the hierarchy are $h_{2}=\frac{u_{11}^{2}}{2u_{1}^{2}}\qquad\text{and}\qquad h_{3}=\frac{3u_{11}^{4}}{8u_{1}^{4}}-\frac{u_{111}^{2}}{2u_{1}^{2}}.$ #### 4.3.2 When $\mathcal{L}_{12}$ is quadratic in $u_{t_{2}}$ Consider a Lagrangian 2-form $\mathcal{L}\llbracket u\rrbracket=\sum_{i<j}\mathcal{L}_{ij}\llbracket u\rrbracket\,\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}$ with $\mathcal{L}_{12}=\frac{1}{2}\alpha[u]u_{2}^{2}-V[u],$ (23) and, for all $j\geq 3$, $\mathcal{L}_{1j}$ of the form $\mathcal{L}_{1j}\llbracket u\rrbracket=\alpha[u]u_{2}u_{j}-h_{j}[u,u_{2}],$ (24) where $[u,u_{2}]=(u,u_{2},u_{1},u_{12},u_{11},u_{112},\ldots)$ since the single bracket $[\cdot]$ denotes dependence on the fields and their $x$-derivatives only (recall that $x=t_{1}$). To write down the full set of multi-time Euler-Lagrange equations we need to specify all $\mathcal{L}_{ij}$, but for the present discussion it is sufficient to consider the equations $\frac{\delta_{12}{\mathcal{L}_{12}}}{\delta{u}}=0\quad\Leftrightarrow\quad\alpha[u]u_{22}=-\frac{\mbox{\rm d}\alpha[u]}{\mbox{\rm d}t_{2}}u_{2}+\frac{1}{2}\sum_{k=0}^{\infty}(-1)^{k}\partial_{x}^{k}\left(\frac{\partial{\alpha[u]}}{\partial{u_{x^{k}}}}u_{2}^{2}\right)-\frac{\delta_{1}{V[u]}}{\delta{u}}$ and $\frac{\delta_{1j}{\mathcal{L}_{1j}}}{\delta{u_{2}}}=0\quad\Leftrightarrow\quad\alpha[u]u_{j}=\frac{\delta_{1}{h_{j}[u,u_{2}]}}{\delta{u_{2}}}.$ We assume that all other multi-time Euler-Lagrange equations are consequences of these. Since $\mathcal{L}_{12}$ is non-degenerate, the Legendre transform is invertible and allows us to identify $\pi=\alpha[u]u_{2}$. Consider the canonical symplectic form on formal integrals, where now the momentum $\pi$ enters as a second field, $\omega=\delta\pi\wedge\delta u\wedge\mbox{\rm d}x.$ This defines the Poisson bracket $\left\\{{\textstyle\int}f\,\mbox{\rm d}x,{\textstyle\int}g\,\mbox{\rm d}x\right\\}=-\int\left(\frac{\delta{f}}{\delta{\pi}}\frac{\delta{g}}{\delta{u}}-\frac{\delta{f}}{\delta{u}}\frac{\delta{g}}{\delta{\pi}}\right)\mbox{\rm d}x.$ (25) The coefficients $\mathcal{L}_{1j}\llbracket u\rrbracket=\alpha[u]u_{2}u_{j}-h_{j}[u,u_{2}]$ are linear in their velocities $u_{j}$, hence they are Hamiltonian with respect to the pre-symplectic form $\delta(\alpha[u]u_{2})\wedge\delta u\wedge\mbox{\rm d}x,$ which equals $\omega$ if we identify $\pi=\alpha[u]u_{2}$. Hence we find the following result. ###### Theorem 4.7. A hierarchy described by a Lagrangian 2-form with coefficients of the form (23)–(24) is Hamiltonian with respect to the canonical Poisson bracket (25), with Hamilton functions $H_{2}[u,\pi]=\frac{1}{2}\frac{\pi^{2}}{\alpha[u]}+V[u]$ and $H_{j}[u,\pi]=h_{j}\\!\left[u,\frac{\pi}{\alpha[u]}\right]$ for $j\geq 3$. ###### Example 4.8. The Lagrangian 2-form for the Boussinesq hierarchy from Example 2.6 leads to $\displaystyle H_{2}$ $\displaystyle=\frac{1}{2}\pi^{2}+2u_{1}^{3}+\frac{3}{2}u_{11}^{2},$ $\displaystyle H_{3}$ $\displaystyle=-6u_{1}^{4}-27u_{1}u_{11}^{2}+6u\pi_{1}\pi-\frac{9}{2}u_{111}^{2}-\frac{3}{2}\pi_{1}^{2},$ where the Legendre transform identifies $\pi=u_{2}$. Indeed we have $\displaystyle\left\\{{\textstyle\int}H_{2}\,\mbox{\rm d}x,{\textstyle\int}u\,\mbox{\rm d}x\right\\}$ $\displaystyle={\textstyle\int}\pi\,\mbox{\rm d}x={\textstyle\int}u_{2}\,\mbox{\rm d}x,$ $\displaystyle\left\\{{\textstyle\int}H_{2}\,\mbox{\rm d}x,{\textstyle\int}\pi\,\mbox{\rm d}x\right\\}$ $\displaystyle={\textstyle\int}(12u_{1}u_{11}-3u_{111})\,\mbox{\rm d}x={\textstyle\int}\pi_{2}\,\mbox{\rm d}x,$ and $\displaystyle\left\\{{\textstyle\int}H_{3}\,\mbox{\rm d}x,{\textstyle\int}u\,\mbox{\rm d}x\right\\}$ $\displaystyle={\textstyle\int}(-6u_{1}\pi+3\pi_{11})\,\mbox{\rm d}x={\textstyle\int}u_{3}\,\mbox{\rm d}x,$ $\displaystyle\left\\{{\textstyle\int}H_{3}\,\mbox{\rm d}x,{\textstyle\int}\pi\,\mbox{\rm d}x\right\\}$ $\displaystyle={\textstyle\int}\left(-72u_{1}^{2}u_{11}+108u_{11}u_{111}+54u_{1}u_{1111}-6\pi\pi_{1}-9u_{111111}\right)\mbox{\rm d}x$ $\displaystyle={\textstyle\int}\pi_{3}\,\mbox{\rm d}x.$ ### 4.4 Closedness and involutivity Let us now have a look at the relation between the closedness of the Lagrangian 2-form and the involutivity of the corresponding Hamiltonians. ###### Proposition 4.9. On solutions of the multi-time Euler-Lagrange equations of a Lagrangian 2-form with coefficients $\mathcal{L}_{1j}$ given by Equation (19), there holds $\\{h_{i},h_{j}\\}=\int\left(p_{i}u_{j}-p_{j}u_{i}\right)\mbox{\rm d}x=\int\left(\frac{\mbox{\rm d}\mathcal{L}_{1i}}{\mbox{\rm d}t_{j}}-\frac{\mbox{\rm d}\mathcal{L}_{1j}}{\mbox{\rm d}t_{i}}\right)\mbox{\rm d}x,$ (26) where the Poisson bracket is given by Equation (22). * Proof. On solutions of the Euler-Lagrange equations we have $\displaystyle\int\frac{\mbox{\rm d}\mathcal{L}_{1i}}{\mbox{\rm d}t_{j}}\,\mbox{\rm d}x$ $\displaystyle=\int\left(\frac{\delta_{1}{\mathcal{L}_{1i}}}{\delta{u}}u_{j}+\frac{\partial{\mathcal{L}_{1i}}}{\partial{u_{i}}}u_{ij}\right)\mbox{\rm d}x$ $\displaystyle=\int\left(\left(\frac{\mbox{\rm d}}{\mbox{\rm d}t_{i}}\frac{\delta_{1i}{\mathcal{L}_{1i}}}{\delta{u_{i}}}\right)u_{j}+\frac{\partial{\mathcal{L}_{1i}}}{\partial{u_{i}}}u_{ij}\right)\mbox{\rm d}x$ $\displaystyle=\int\left(p_{i}u_{j}+pu_{ij}\right)\mbox{\rm d}x.$ It follows that $\int\left(\frac{\mbox{\rm d}\mathcal{L}_{1i}}{\mbox{\rm d}t_{j}}-\frac{\mbox{\rm d}\mathcal{L}_{1j}}{\mbox{\rm d}t_{i}}\right)\mbox{\rm d}x=\int\left(p_{i}u_{j}-p_{j}u_{i}\right)\mbox{\rm d}x.$ (27) On the other hand we have that $\displaystyle\int\left(\frac{\mbox{\rm d}\mathcal{L}_{1i}}{\mbox{\rm d}t_{j}}-\frac{\mbox{\rm d}\mathcal{L}_{1j}}{\mbox{\rm d}t_{i}}\right)\mbox{\rm d}x$ $\displaystyle=\int\left(\frac{\mbox{\rm d}}{\mbox{\rm d}t_{j}}(pu_{i}-h_{i})-\frac{\mbox{\rm d}}{\mbox{\rm d}t_{i}}(pu_{j}-h_{j})\right)\mbox{\rm d}x$ $\displaystyle=-\int\left(p_{i}u_{j}-p_{j}u_{i}\right)\mbox{\rm d}x+2\\{h_{i},h_{j}\\}.$ Combined with Equation (27), this implies both identities in Equation (26). ∎ ###### Proposition 4.10. On solutions of the multi-time Euler-Lagrange equations of a Lagrangian 2-form with coefficients $\mathcal{L}_{1j}$ given by Equations (23)–(24), there holds $\\{H_{i},H_{j}\\}=\int\left(\pi_{i}u_{j}-\pi_{j}u_{i}\right)\mbox{\rm d}x=\int\left(\frac{\mbox{\rm d}\mathcal{L}_{1i}}{\mbox{\rm d}t_{j}}-\frac{\mbox{\rm d}\mathcal{L}_{1j}}{\mbox{\rm d}t_{i}}\right)\mbox{\rm d}x,$ where the Poisson bracket is given by Equation (25) and the Hamilton functions $H_{j}$ are given in Theorem 4.7. * Proof. Analogous to the proof of Proposition 4.9, with $p[u]$ replaced by the field $\pi$. ∎ ###### Theorem 4.11. Let $\mathcal{L}$ be a Lagrangian 2-form with coefficients $\mathcal{L}_{1j}$ given by Equation (19) or by Equations (23)–(24). Consider the corresponding Hamiltonian structures, given by $H_{1j}=h_{j}$ or $H_{1j}=H_{j}$, as in Theorems 4.3 and 4.7 respectively. There holds $\\{H_{1i},H_{1j}\\}=0$ if and only if $\int\left(\frac{\mbox{\rm d}\mathcal{L}_{ij}}{\mbox{\rm d}t_{1}}-\frac{\mbox{\rm d}\mathcal{L}_{1j}}{\mbox{\rm d}t_{i}}+\frac{\mbox{\rm d}\mathcal{L}_{1i}}{\mbox{\rm d}t_{j}}\right)\mbox{\rm d}x=0$ on solutions of the multi-time Euler-Lagrange equations. * Proof. Recall that $t_{1}=x$, hence $\frac{\mbox{\rm d}}{\mbox{\rm d}t_{1}}=\partial_{x}$. By definition of the formal integral as an equivalence class, we have $\int\partial_{x}\mathcal{L}_{ij}\,\mbox{\rm d}x=0$. Hence the claim follows from Proposition 4.9 or Proposition 4.10. ∎ It is known that $\mbox{\rm d}\mathcal{L}\llbracket u\rrbracket$ is constant in the set of solutions $u$ to the multi-time Euler-Lagrange equations (see Proposition A.2). In most examples, one can verify using a trivial solution that this constant is zero. ###### Corollary 4.12. If a Lagrangian 2-form, with coefficients $\mathcal{L}_{1j}\llbracket u\rrbracket$ given by Equation (19) or by Equations (23)–(24), is closed for a solution $u$ to the pluri-Lagrangian problem, then $\\{H_{1i},H_{1j}\\}=0$ for all $i,j$. All examples of Lagrangian 2-forms discussed so far satisfy $\mbox{\rm d}\mathcal{L}=0$ on solutions. We now present a system where this is not the case. ###### Example 4.13. Consider a perturbation of the Boussinesq Lagrangian, obtained by adding $cu$ for some constant $c\in\mathbb{R}$, $\mathcal{L}_{12}=\frac{1}{2}u_{2}^{2}-2u_{1}^{3}-\frac{3}{2}u_{11}^{2}+cu,$ combined with the Lagrangian coefficients $\displaystyle\mathcal{L}_{13}$ $\displaystyle=u_{2}(u_{3}-1)$ $\displaystyle\mathcal{L}_{23}$ $\displaystyle=(6u_{1}^{2}-3u_{111})(u_{3}-1).$ The corresponding multi-time Euler-Lagrange equations consist of a perturbed Boussinesq equation, $u_{22}=12u_{1}u_{11}-3u_{1111}+c$ and $u_{3}=1.$ We have $\frac{\mbox{\rm d}\mathcal{L}_{12}}{\mbox{\rm d}t_{3}}-\frac{\mbox{\rm d}\mathcal{L}_{13}}{\mbox{\rm d}t_{2}}+\frac{\mbox{\rm d}\mathcal{L}_{23}}{\mbox{\rm d}t_{1}}=c$ on solutions, hence $\mbox{\rm d}\mathcal{L}$ is nonzero. The multi-time Euler-Lagrange equations are equivalent to the canonical Hamiltonian systems with $\displaystyle H_{2}$ $\displaystyle=\frac{1}{2}\pi^{2}+2u_{1}^{3}+\frac{3}{2}u_{11}^{2}-cu$ $\displaystyle H_{3}$ $\displaystyle=\pi.$ They are not in involution, but rather $\left\\{{\textstyle\int}H_{2}\,\mbox{\rm d}x,{\textstyle\int}H_{3}\,\mbox{\rm d}x\right\\}=\int\left(12u_{11}u_{1}-3u_{1111}+c\right)\mbox{\rm d}x=\int c\,\mbox{\rm d}x.$ Note that if we would allow the fields in $\mathcal{V}$ to depend explicitly on $x$, then we would find ${\textstyle\int}c\,\mbox{\rm d}x={\textstyle\int}\partial_{x}(cx)\,\mbox{\rm d}x=0$. Note that this is not a property of the Lagrangian form, but of the function space we work in. Allowing fields that depend on $x$ affects the definition of the formal integral ${\textstyle\int}(\cdot)\,\mbox{\rm d}x$ as an equivalence class modulo $x$-derivatives. If dependence on $x$ is allowed, then there is no such thing as a nonzero constant functional in this equivalence class. However, in our definition of $\mathcal{V}$, fields can only depend on $x$ through $u$, hence $c$ is not an $x$-derivative and ${\textstyle\int}c\,\mbox{\rm d}x$ is not the zero element of $\mathcal{F}$. ### 4.5 Additional (nonlocal) Poisson brackets Even though the closedness property in Section 4.4 involves all coefficients of a Lagrangian 2-form $\mathcal{L}$, so far we have only used the first row of coefficients $\mathcal{L}_{1j}$ to construct Hamiltonian structures. A similar procedure can be carried out for other $\mathcal{L}_{ij}$, but the results are not entirely satisfactory. In particular, it will not lead to true bi-Hamiltonian structures. Because of this slightly disappointing outcome, we will make no effort to present the most general statement possible. Instead we make some convenient assumptions on the form of the coefficients $\mathcal{L}_{ij}$. Consider a Lagrangian 2-form $\mathcal{L}$ such that for all $i<j$ the coefficient $\mathcal{L}_{ij}$ only contains derivatives with respect to $t_{1}$, $t_{i}$ and $t_{j}$ (no “alien derivatives” in the terminology of [29]). In addition, assume that $\mathcal{L}_{ij}$ can be written as the sum of terms that each contain at most one derivative with respect to $t_{i}$ (if $i>1$) or $t_{j}$. In particular, $\mathcal{L}_{ij}$ does not contain higher derivatives with respect to $t_{i}$ (if $i>1$) or $t_{j}$, but mixed derivatives with respect to $t_{1}$ and $t_{i}$ or $t_{1}$ and $t_{j}$ are allowed. There is no restriction on the amount of $t_{1}$-derivatives. To get a Hamiltonian description of the evolution along the time direction $t_{j}$ from the Lagrangian $\mathcal{L}_{ij}$, we should consider both $t_{1}$ and $t_{i}$ as space coordinates. Hence we will work on the space $\mathcal{V}\big{/}\left(\partial_{1}\\!\mathcal{V}+\partial_{i}\\!\mathcal{V}\right).$ For $i>1$, consider the momenta $p^{[i]}[u]=\frac{\delta_{1i}{\mathcal{L}_{ij}}}{\delta{u_{j}}}.$ From the assumption that each term of $\mathcal{L}_{ij}$ contains at most one time-derivative it follows that $p^{[i]}$ only depends on $u$ and its $x$-derivatives. Note that $p^{[i]}$ is independent of $j$ because of the multi-time Euler-Lagrange equation (6). The variational derivative in the definition of $p^{[i]}$ is in the directions $1$ and $i$, corresponding to the formal integral, whereas the Lagrangian coefficient has indices $i$ and $j$. However, we can also write $p^{[i]}[u]=\frac{\delta_{1}{\mathcal{L}_{ij}}}{\delta{u_{j}}}$ because of the assumption on the derivatives that occur in $\mathcal{L}_{ij}$, which excludes mixed derivatives with respect to $t_{i}$ and $t_{j}$. As Hamilton function we can take $H_{ij}=p^{[i]}u_{j}-\mathcal{L}_{ij}.$ Its formal integral $\int H_{ij}\,\mbox{\rm d}x\wedge\mbox{\rm d}t_{i}$ does not depend on any $t_{j}$-derivatives. Since we are working with 2-dimensional integrals, we should take a $(2,2)$-form as symplectic form. In analogy to Equation (13) we take $\omega_{i}=-\delta p^{[i]}\wedge\delta u\wedge\mbox{\rm d}x\wedge\mbox{\rm d}t_{i}.$ A Hamiltonian vector field $X=\chi\frac{\partial{}}{\partial{u}}$ satisfies $\int\iota_{X}\omega_{i}=\int\delta H_{ij}\wedge\mbox{\rm d}x\wedge\mbox{\rm d}t_{i}$ hence $\mathcal{E}_{p^{[i]}}\chi=\frac{\delta_{1i}{H_{ij}}}{\delta{u}},$ where $\mathcal{E}_{p^{[i]}}$ is the differential operator $\mathcal{E}_{p^{[i]}}=\sum_{k=0}^{\infty}\left((-1)^{k}\partial_{x}^{k}\frac{\partial{p^{[i]}}}{\partial{u_{x^{k}}}}-\frac{\partial{p^{[i]}}}{\partial{u_{x^{k}}}}\partial_{x}^{k}\right)$ The corresponding (nonlocal) Poisson bracket is $\left\\{{\textstyle\int}f\,\mbox{\rm d}x\wedge\mbox{\rm d}t_{i},{\textstyle\int}g\,\mbox{\rm d}x\wedge\mbox{\rm d}t_{i}\right\\}_{i}=-\int\frac{\delta_{1i}{f}}{\delta{u}}\,\mathcal{E}_{p^{[i]}}^{-1}\,\frac{\delta_{1i}{g}}{\delta{u}}\,\mbox{\rm d}x\wedge\mbox{\rm d}t_{i}.$ Note that $H$ is not skew-symmetric, $H_{ij}\neq H_{ji}$. The space of functionals $\mathcal{V}\big{/}\left(\partial_{1}\\!\mathcal{V}+\partial_{i}\\!\mathcal{V}\right)$, on which the Poisson bracket $\\{\cdot,\cdot\\}_{i}$ is defined, depends on $i$ and is different from the space of functionals for the bracket $\\{\cdot,\cdot\\}$ from Equation (25). Hence no pair of these brackets are compatible with each other in the sense of a bi-Hamiltonian system. As before, we can relate Poisson brackets between the Hamilton functionals to coefficients of $\mbox{\rm d}\mathcal{L}$. ###### Proposition 4.14. Assume that for all $i,j>1$, $\mathcal{L}_{ij}$ does not depend on any second or higher derivatives with respect to $t_{i}$ and $t_{j}$. On solutions of the Euler-Lagrange equations there holds that, for $i,j,k>1$, $\begin{split}\int\left(\frac{\mbox{\rm d}\mathcal{L}_{ij}}{\mbox{\rm d}t_{k}}-\frac{\mbox{\rm d}\mathcal{L}_{ik}}{\mbox{\rm d}t_{j}}\right)\mbox{\rm d}x\wedge\mbox{\rm d}t_{i}&=\int\left(p^{[i]}_{j}u_{k}-p^{[i]}_{k}u_{j}\right)\mbox{\rm d}x\wedge\mbox{\rm d}t_{i}\\\ &=\left\\{{\textstyle\int}H_{ij}\,\mbox{\rm d}x\wedge\mbox{\rm d}t_{i},{\textstyle\int}H_{ik}\,\mbox{\rm d}x\wedge\mbox{\rm d}t_{i}\right\\}_{i}.\end{split}$ (28) * Proof. Analogous to the proof of Proposition 4.9. ∎ ###### Example 4.15. For the potential KdV equation (see Example 2.5) we have $p^{[2]}=\frac{\delta_{1}{\mathcal{L}_{23}}}{\delta{u_{3}}}=\frac{3}{2}u_{111}+\frac{3}{2}u_{1}^{2},$ hence $\displaystyle\mathcal{E}_{p^{[2]}}$ $\displaystyle=-\partial_{1}\frac{\partial{p^{[i]}}}{\partial{u_{1}}}-\partial_{1}^{3}\frac{\partial{p^{[i]}}}{\partial{u_{111}}}-\frac{\partial{p^{[i]}}}{\partial{u_{1}}}\partial_{1}-\frac{\partial{p^{[i]}}}{\partial{u_{111}}}\partial_{1}^{3}$ $\displaystyle=-3\partial_{1}u_{1}-\frac{3}{2}\partial_{1}^{3}-3u_{1}\partial_{1}-\frac{3}{2}\partial_{1}^{3}$ $\displaystyle=-3\partial_{1}^{3}-6u_{1}\partial_{1}-3u_{11}.$ We have $\displaystyle H_{23}$ $\displaystyle=p^{[2]}u_{3}-\mathcal{L}_{23}$ $\displaystyle=-3u_{1}^{5}+\frac{15}{2}u_{1}^{2}u_{11}^{2}-10u_{1}^{3}u_{111}+5u_{1}^{3}u_{3}-\frac{7}{2}u_{11}^{2}u_{111}-3u_{1}u_{111}^{2}+6u_{1}u_{11}u_{1111}$ $\displaystyle\quad-\frac{3}{2}u_{1}^{2}u_{11111}-10u_{1}u_{11}u_{12}+\frac{5}{2}u_{11}^{2}u_{2}+5u_{1}u_{111}u_{2}+\frac{1}{2}u_{1111}^{2}-\frac{1}{2}u_{111}u_{11111}$ $\displaystyle\quad+\frac{1}{2}u_{111}u_{112}-\frac{1}{2}u_{1}u_{113}-u_{1111}u_{12}+\frac{1}{2}u_{11}u_{13}+\frac{1}{2}u_{11111}u_{2}+u_{111}u_{3},$ where the terms involving $t_{3}$-derivatives cancel out when the Hamiltonian is integrated. Its variational derivative is $\displaystyle\frac{\delta_{12}{H_{23}}}{\delta{u}}$ $\displaystyle=60u_{1}^{3}u_{11}+75u_{11}^{3}+300u_{1}u_{11}u_{111}+75u_{1}^{2}u_{1111}-30u_{1}^{2}u_{12}-30u_{1}u_{11}u_{2}$ $\displaystyle\quad+120u_{111}u_{1111}+72u_{11}u_{11111}+24u_{1}u_{111111}-30u_{1}u_{1112}-45u_{11}u_{112}$ $\displaystyle\quad-25u_{111}u_{12}-5u_{1111}u_{2}+2u_{11111111}-5u_{111112}.$ On solutions this simplifies to $\displaystyle\frac{\delta_{12}{H_{23}}}{\delta{u}}$ $\displaystyle=-210u_{1}^{3}u_{11}-195u_{11}^{3}-690u_{1}u_{11}u_{111}-150u_{1}^{2}u_{1111}-210u_{111}u_{1111}$ $\displaystyle\quad-123u_{11}u_{11111}-36u_{1}u_{111111}-3u_{11111111}$ $\displaystyle=\mathcal{E}_{p^{[2]}}\left(10u_{1}^{3}+5u_{11}^{2}+10u_{1}u_{111}+u_{11111}\right)$ $\displaystyle=\mathcal{E}_{p^{[2]}}u_{3}.$ Hence $\frac{\mbox{\rm d}}{\mbox{\rm d}t_{3}}\int u\,\mbox{\rm d}x\wedge\mbox{\rm d}t_{2}=\int\mathcal{E}_{p^{[2]}}^{-1}\frac{\delta_{12}{H_{23}}}{\delta{u}}\,\mbox{\rm d}x\wedge\mbox{\rm d}t_{2}=\left\\{{\textstyle\int}H_{23}\,\mbox{\rm d}x\wedge\mbox{\rm d}t_{2},{\textstyle\int}u\,\mbox{\rm d}x\wedge\mbox{\rm d}t_{2}\right\\}_{2}.$ ### 4.6 Comparison with the covariant approach In Section 4.5 we derived Poisson brackets $\\{\cdot,\cdot\\}_{i}$, associated to each time variable $t_{i}$. This was somewhat cumbersome because we had a priori assigned $x=t_{1}$ as a distinguished variable. The recent work [6] explores the relation of pluri-Lagrangian structures to covariant Hamiltonian structures. The meaning of “covariant” here is that all variables are on the same footing; there is no distinguished $x$ variable. More details on covariant field theory, and its connection to the distinguished-variable (or “instantaneous”) perspective, can be found in [13]. The main objects in the covariant Hamiltonian formulation of [6] are: * • A “symplectic multiform” $\Omega$, which can be expanded as $\Omega=\sum_{j}\omega_{j}\wedge\mbox{\rm d}t_{j},$ where each $\omega_{j}$ is a vertical 2-form in the variational bicomplex. * • A “Hamiltonian multiform” $\mathcal{H}=\sum_{i<j}\mathtt{H}_{ij}\,\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}$ which gives the equations of motion through $\delta\mathcal{H}=\sum_{j}\mbox{\rm d}t_{j}\wedge\xi_{j}\lrcorner\,\Omega,$ (29) where $\delta$ is the vertical exterior derivative in the variational bicomplex, $\xi_{j}$ denotes the vector field corresponding to the $t_{j}$-flow, and $\lrcorner$ denotes the interior product. This equation should be understood as a covariant version of the instantaneous Hamiltonian equation $\delta H=\xi\lrcorner\,\omega$. On the equations of motion there holds $\mbox{\rm d}\mathcal{H}=0$ if and only if $\mbox{\rm d}\mathcal{L}=0$. Since the covariant Hamiltonian equation (29) is of a different form than the instantaneous Hamiltonian equation we use, the coefficients $\mathtt{H}_{ij}$ of the Hamiltonian multiform $\mathcal{H}$ are also different from the $H_{ij}$ we found in Sections 4.3–4.5. Our $H_{ij}$ are instantaneous Hamiltonians where $t_{1}$ and $t_{i}$ are considered as space variables and the Legendre transformation has been applied with respect to $t_{j}$. * • A “multi-time Poisson bracket” $\\{|\cdot,\cdot|\\}$ which defines a pairing between functions or (a certain type of) horizontal one-forms, defined by $\\{|F,G|\\}=(-1)^{r}\xi_{F}\delta G,$ where $\xi_{F}$ is the Hamiltonian (multi-)vector field associated to $F$, and $r$ is the horizontal degree of $F$ (which is either 0 or 1). The equations of motion can be written as $\mbox{\rm d}F=\sum_{i<j}\\{|\mathtt{H}_{ij},F|\\}\,\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}.$ Single-time Poisson brackets are obtained in [6] by expanding the multi-time Poisson bracket as $\left\\{\left|{\textstyle\sum_{j}}F_{j}\,\mbox{\rm d}t_{j},{\textstyle\sum_{j}}G_{j}\,\mbox{\rm d}t_{j}\right|\right\\}=\sum_{j}\\{F_{j},G_{j}\\}_{j}\,\mbox{\rm d}t_{j}$ where $\\{f,g\\}_{j}=-\xi_{f}^{j}\lrcorner\,\delta g\qquad\text{and}\qquad\xi_{f}^{j}\lrcorner\,\omega_{j}=\delta f.$ (30) These are fundamentally different from the Poisson brackets of Sections 4.3–4.5 because they act on different function spaces. Equation (30) assumes that $\delta f$ lies in the image of $\omega_{j}$ (considered as a map from vertical vector fields to vertical one-forms). For example, for the potential KdV hiararchy one has $\omega_{1}=\delta v\wedge\delta v_{1}$, hence the Poisson bracket $\\{\cdot,\cdot\\}_{1}$ can only be applied to functions of $v$ and $v_{1}$, not to functions depending on any higher derivatives. Similar conditions on the function space apply to the higher Poisson brackets corresponding to $\omega_{j}$, $j\geq 2$. On the other hand, the Poisson brackets of Sections 4.3–4.5 are defined on an equivalence class of functions modulo certain derivatives, without further restrictions on the functions in this class. In summary, the single-time Poisson brackets of [6] are constructed with a certain elegance in a covariant way, but they are defined only in a restricted function space. They are different from our Poisson brackets of Section 4.3–4.5, which have no such restrictions, but break covariance already in the definition of the function space as an equivalence class. It is not clear how to pass from one picture to the other, or if their respective benefits can be combined into a single approach. ## 5 Conclusions We have established a connection between pluri-Lagrangian systems and integrable Hamiltonian hierarchies. In the case of ODEs, where the pluri- Lagrangian structure is a 1-form, this connection was already obtained in [26]. Our main contribution is its generalization to the case of 2-dimensional PDEs, described by Lagrangian 2-forms. Presumably, this approach extends to Lagrangian $d$-forms of any dimension $d$, but the details of this are postponed to future work. A central property in the theory of pluri-Lagrangian systems is that the Lagrangian form is (almost) closed on solutions. We showed that closedness is equivalent to the corresponding Hamilton functions being in involution. Although one can obtain several Poisson brackets (and corresponding Hamilton functions) from one Lagrangian 2-form, these do not form a bi-Hamiltonian structure and it is not clear if a recursion operator can be obtained from them. Hence it remains an open question to find a fully variational description of bi-Hamiltonian hierarchies. ### Acknowledgements The author would like to thank Frank Nijhoff for his inspiring questions and comments, Matteo Stoppato for helpful discussions about the covariant Hamiltonian approach, Yuri Suris for his constructive criticism on early drafts of this paper, and the anonymous referees for their thoughtful comments. The author is funded by DFG Research Fellowship VE 1211/1-1. Part of the work presented here was done at TU Berlin, supported by the SFB Transregio 109 “Discretization in Geometry and Dynamics”. ## Appendix A Pluri-Lagrangian systems and the variational bicomplex In this appendix we study the pluri-Lagrangian principle using the variational bicomplex, described in Section 4.1. We provide proofs that the multi-time Euler-Lagrange equations from Section 2 are sufficient conditions for criticality. Alternative proofs of this fact can be found in [28] and [23, Appendix A]. ###### Proposition A.1. The field $u$ is a solution to the pluri-Lagrangian problem of a $d$-form $\mathcal{L}\llbracket u\rrbracket$ if locally there exists a $(1,d-1)$-form $\Theta$ such that $\delta\mathcal{L}\llbracket u\rrbracket=\mbox{\rm d}\Theta$. * Proof. Consider a field $u$ such that such a $(1,d-1)$-form $\Theta$ exists. Consider any $d$-manifold $\Gamma$ and any variation $v$ that vanishes (along with all its derivatives) on the boundary $\partial\Gamma$. Note that the horizontal exterior derivative d anti-commutes with the interior product operator $\iota_{V}$, where $V$ is the prolonged vertical vector field $V=\operatorname{pr}(v\partial/\partial_{u})$ defined by the variation $v$. It follows that $\int_{\Gamma}\iota_{V}\delta\mathcal{L}=-\int_{\Gamma}\mbox{\rm d}\left(\iota_{V}\Theta\right)=-\int_{\partial\Gamma}\iota_{V}\Theta=0,$ hence $u$ solves the pluri-Lagrangian problem. ∎ If we are dealing with a classical Lagrangian problem from mechanics, $\mathcal{L}=L(u,u_{t})\,\mbox{\rm d}t$, we have $\Theta=-\frac{\partial{L}}{\partial{u_{t}}}\delta u$, which is the pull back to the tangent bundle of the canonical 1-form $\sum_{i}p_{i}\,\mbox{\rm d}q_{i}$ on the cotangent bundle. Often we want the Lagrangian form to be closed when evaluated on solutions. As we saw in Theorems 3.7 and 4.11, this implies that the corresponding Hamiltonians are in involution. We did not include this in the definition of a pluri-Lagrangian system, because our definition already implies a slightly weaker property. ###### Proposition A.2. The horizontal exterior derivative $\mbox{\rm d}\mathcal{L}$ of a pluri- Lagrangian form is constant on connected components of the set of critical fields for $\mathcal{L}$. * Proof. Critical points satisfy locally $\delta\mathcal{L}=\mbox{\rm d}\Theta\qquad\Rightarrow\qquad\mbox{\rm d}\delta\mathcal{L}=0\qquad\Rightarrow\qquad\delta\mbox{\rm d}\mathcal{L}=0.$ Hence for any variation $v$ the Lie derivative of $\mbox{\rm d}\mathcal{L}$ along its prolongation $V=\operatorname{pr}(v\partial/\partial_{u})$ is $\iota_{V}\delta(\mbox{\rm d}\mathcal{L})=0$. Therefore, if a solution $u$ can be continuously deformed into another solution $\bar{u}$, then $\mbox{\rm d}\mathcal{L}\llbracket u\rrbracket=\mbox{\rm d}\mathcal{L}\llbracket\bar{u}\rrbracket$. ∎ Now let us prove the sufficiency of the multi-time Euler-Lagrange equations for 1-forms and 2-forms, as given in Theorems 2.2 and 2.4. For different approaches to the multi-time Euler-Lagrange equations, including proofs of necessity, see [28] and [23]. * Proof of sufficiency in Theorem 2.2.. We calculate the vertical exterior derivative $\delta\mathcal{L}$ of the Lagrangian 1-form, modulo the multi-time Euler-Lagrange Equations (1) and (2). We have $\displaystyle\delta\mathcal{L}$ $\displaystyle=\sum_{j=1}^{N}\sum_{I}\frac{\partial{\mathcal{L}_{j}}}{\partial{u_{I}}}\,\delta u_{I}\wedge\mbox{\rm d}t_{j}$ $\displaystyle=\sum_{j=1}^{N}\sum_{I}\left(\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{I}}}+\partial_{j}\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{It_{j}}}}\right)\delta u_{I}\wedge\mbox{\rm d}t_{j}.$ Rearranging this sum, we find $\delta\mathcal{L}=\sum_{j=1}^{N}\left[\sum_{I\not\ni t_{j}}\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{I}}}\delta u_{I}\wedge\mbox{\rm d}t_{j}+\sum_{I}\left(\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{It_{j}}}}\delta u_{It_{j}}\wedge\mbox{\rm d}t_{j}+\left(\partial_{j}\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{It_{j}}}}\right)\delta u_{I}\wedge\mbox{\rm d}t_{j}\right)\right].$ On solutions of Equation (2), we can define the generalized momenta $p^{I}=\frac{\delta_{j}{\mathcal{L}_{j}}}{\delta{u_{It_{j}}}}.$ Using Equations (1) and (2) it follows that $\displaystyle\delta\mathcal{L}$ $\displaystyle=\sum_{j=1}^{N}\sum_{I}\left(p^{I}\delta u_{It_{j}}\wedge\mbox{\rm d}t_{j}+\left(\partial_{j}p^{I}\right)\delta u_{I}\wedge\mbox{\rm d}t_{j}\right)=-d\left(\sum_{I}p^{I}\delta u_{I}\right).$ This implies by Proposition A.1 that $u$ solves the pluri-Lagrangian problem. ∎ * Proof of sufficiency in Theorem 2.4.. We calculate the vertical exterior derivative $\delta\mathcal{L}$, $\displaystyle\delta\mathcal{L}$ $\displaystyle=\sum_{i<j}\sum_{I}\frac{\partial{\mathcal{L}_{ij}}}{\partial{u_{I}}}\,\delta u_{I}\wedge\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}$ $\displaystyle=\sum_{i<j}\sum_{I}\left(\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{I}}}+\partial_{i}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}}}}+\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{j}}}}+\partial_{i}\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}t_{j}}}}\right)\delta u_{I}\wedge\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}$ (31) We will rearrange this sum according to the times occurring in the multi-index $I$. We have $\displaystyle\sum_{I}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{I}}}\,\delta u_{I}=\sum_{I\not\ni t_{i},t_{j}}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{I}}}\,\delta u_{I}+\sum_{I\not\ni t_{j}}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}}}}\,\delta u_{It_{i}}$ $\displaystyle\hskip 85.35826pt+\sum_{I\not\ni t_{i}}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{j}}}}\,\delta u_{It_{j}}+\sum_{I}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}t_{j}}}}\,\delta u_{It_{i}t_{j}},$ $\displaystyle\sum_{I}\partial_{i}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}}}}\,\delta u_{I}=\sum_{I\not\ni t_{j}}\partial_{i}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}}}}\,\delta u_{I}+\sum_{I}\partial_{i}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}t_{j}}}}\,\delta u_{It_{j}},$ $\displaystyle\sum_{I}\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{j}}}}\,\delta u_{I}=\sum_{I\not\ni t_{i}}\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{j}}}}\,\delta u_{I}+\sum_{I}\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}t_{j}}}}\,\delta u_{It_{i}}.$ Modulo the multi-time Euler-Lagrange equations (5)–(7), we can write these expressions as $\displaystyle\sum_{I}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{I}}}\,\delta u_{I}=\sum_{I\not\ni t_{j}}p_{j}^{I}\,\delta u_{It_{i}}-\sum_{I\not\ni t_{i}}p_{i}^{I}\,\delta u_{It_{j}}+\sum_{I}(n_{j}^{I}-n_{i}^{I})\,\delta u_{It_{i}t_{j}},$ $\displaystyle\sum_{I}\partial_{i}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}}}}\,\delta u_{I}=\sum_{I\not\ni t_{j}}\partial_{i}p_{j}^{I}\,\delta u_{I}+\sum_{I}\partial_{i}(n_{j}^{I}-n_{i}^{I})\,\delta u_{It_{j}},$ $\displaystyle\sum_{I}\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{j}}}}\,\delta u_{I}=\sum_{I\not\ni t_{i}}-\partial_{j}p_{i}^{I}\,\delta u_{I}+\sum_{I}\partial_{j}(n_{j}^{I}-n_{i}^{I})\,\delta u_{It_{i}},$ $\displaystyle\sum_{I}\partial_{i}\partial_{j}\frac{\delta_{ij}{\mathcal{L}_{ij}}}{\delta{u_{It_{i}t_{j}}}}\,\delta u_{I}=\sum_{I}\partial_{i}\partial_{j}(n_{j}^{I}-n_{i}^{I})\delta u_{I}.$ where $\displaystyle p_{j}^{I}=\frac{\delta_{1j}{\mathcal{L}_{1j}}}{\delta{u_{It_{1}}}}\qquad\text{for }I\not\ni t_{j},$ $\displaystyle n_{j}^{I}=\frac{\delta_{1j}{\mathcal{L}_{1j}}}{\delta{u_{It_{1}t_{j}}}}.$ Note that here the indices of $p$ and $n$ are labels, not derivatives. Hence on solutions to equations (5)–(7), Equation (31) is equivalent to $\displaystyle\delta\mathcal{L}$ $\displaystyle=\sum_{i<j}\Bigg{[}\sum_{I\not\ni t_{j}}\left(p_{j}^{I}\,\delta u_{It_{i}}+\partial_{i}p_{j}^{I}\,\delta u_{I}\right)-\sum_{I\not\ni t_{i}}\left(p_{i}^{I}\,\delta u_{It_{j}}+\partial_{j}p_{i}^{I}\,\delta u_{I}\right)$ $\displaystyle\hskip 42.67912pt+\sum_{I}\Big{(}(n_{j}^{I}-n_{i}^{I})\,\delta u_{It_{i}t_{j}}+\partial_{j}(n_{j}^{I}-n_{i}^{I})\,\delta u_{It_{i}}$ $\displaystyle\hskip 85.35826pt+\partial_{i}(n_{j}^{I}-n_{i}^{I})\,\delta u_{It_{j}}+\partial_{i}\partial_{j}(n_{j}^{I}-n_{i}^{I})\,\delta u_{I}\Big{)}\Bigg{]}\wedge\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}.$ Using the anti-symmetry of the wedge product, we can write this as $\displaystyle\delta\mathcal{L}$ $\displaystyle=\sum_{i,j=1}^{N}\Bigg{[}\sum_{I\not\ni t_{j}}\left(p_{j}^{I}\,\delta u_{It_{i}}+\partial_{i}p_{j}^{I}\,\delta u_{I}\right)$ $\displaystyle\hskip 42.67912pt+\sum_{I}\Big{(}n_{j}^{I}\,\delta u_{It_{i}t_{j}}+\partial_{j}n_{j}^{I}\,\delta u_{It_{i}}+\partial_{i}n_{j}^{I}\,\delta u_{It_{j}}+\partial_{i}\partial_{j}n_{j}^{I}\,\delta u_{I}\Big{)}\Bigg{]}\wedge\mbox{\rm d}t_{i}\wedge\mbox{\rm d}t_{j}$ $\displaystyle=\sum_{j=1}^{N}\Bigg{[}\sum_{I\not\ni t_{j}}-\mbox{\rm d}\left(p_{j}^{I}\,\delta u_{I}\wedge\mbox{\rm d}t_{j}\right)+\sum_{I}-d\left(n_{j}^{I}\,\delta u_{It_{j}}\wedge\mbox{\rm d}t_{j}+\partial_{j}n_{j}^{I}\,\delta u_{I}\wedge\mbox{\rm d}t_{j}\right)\Bigg{]}.$ It now follows by Proposition A.1 that $u$ is a critical field. ∎ ## References * Anderson [1992] Anderson I. 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2024-09-04T02:54:59.265619

2003.05402

arxiv-papers

# FuDGE: A Method to Estimate a Functional Differential Graph in a High- Dimensional Setting Boxin Zhao<EMAIL_ADDRESS> Booth School of Business The University of Chicago Chicago, IL 60637, USA Y. Samuel Wang<EMAIL_ADDRESS> Department of Statistics and Data Science Cornell University Ithaca, NY 14853, USA Mladen Kolar<EMAIL_ADDRESS> Booth School of Business The University of Chicago Chicago, IL 60637, USA ###### Abstract We consider the problem of estimating the difference between two functional undirected graphical models with shared structures. In many applications, data are naturally regarded as a vector of random functions rather than a vector of scalars. For example, electroencephalography (EEG) data are more appropriately treated as functions of time. In such a problem, not only can the number of functions measured per sample be large, but each function is itself an infinite dimensional object, making estimation of model parameters challenging. This is further complicated by the fact that the curves are usually only observed at discrete time points. We first define a functional differential graph that captures the differences between two functional graphical models and formally characterize when the functional differential graph is well defined. We then propose a method, FuDGE, that directly estimates the functional differential graph without first estimating each individual graph. This is particularly beneficial in settings where the individual graphs are dense, but the differential graph is sparse. We show that FuDGE consistently estimates the functional differential graph even in a high-dimensional setting for both fully observed and discretely observed function paths. We illustrate the finite sample properties of our method through simulation studies. We also propose a competing method, the Joint Functional Graphical Lasso, which generalizes the Joint Graphical Lasso to the functional setting. Finally, we apply our method to EEG data to uncover differences in functional brain connectivity between a group of individuals with alcohol use disorder and a control group. Keywords: differential graph estimation, functional data analysis, multivariate functional data, probabilistic graphical models, structure learning ## 1 Introduction We consider a setting where we observe two samples of multivariate functional data, $X_{i}(t)$ for $i=1,\ldots,n_{X}$ and $Y_{i}(t)$ for $i=1,\ldots,n_{Y}$. The primary goal is to determine if and how the underlying populations—specifically their conditional dependency structures—differ. As a motivating example, consider electroencephalography (EEG) data where the electrical activity of multiple regions of the brain can be measured simultaneously across a period of time. Given samples from the general population, fitting a graphical model to the observed measurements would allow a researcher to determine which regions of the brain are dependent after conditioning on all other regions. The EEG data analyzed in Section 6.2 consists of two samples: one from a control group and the other from a group of individuals with alcohol use disorder (AUD). Using this data, researchers may be interested in explicitly comparing the two groups and investigating the complex question of how brain functional connectivity patterns in the AUD group differ from those in the control group. The conditional independence structure within multivariate data is commonly represented by a graphical model (Lauritzen, 1996). Let $G=\\{V,E\\}$ denote an undirected graph where $V$ is the set of vertices with $|V|=p$ and $E\subset V^{2}$ is the set of edges. At times, we also denote $V$ as $[p]=\\{1,2,\dots,p\\}$. When the data consist of random vectors $X=(X_{1},\dots,X_{p})^{\top}$, we say that $X$ satisfies the pairwise Markov property with respect to $G$ if $X_{v}\centernot\perp\\!\\!\\!\perp X_{w}\mid\\{X_{u}\\}_{u\in V\setminus\\{v,w\\}}$ holds if and only if $\\{v,w\\}\in E$. When $X$ follows a multivariate Gaussian distribution with covariance $\Sigma=\Theta^{-1}$, then $\Theta_{vw}\neq 0$ if and only if $\\{v,w\\}\in E$. Thus, recovering the structure of an undirected graph from multivariate Gaussian data is equivalent to estimating the support of the precision matrix, $\Theta$. When the primary interest is in characterizing the difference between the conditional independence structure of two populations, the object of interest may be the _differential graph_ , $G_{\Delta}=\\{V,E_{\Delta}\\}$. When $X$ and $Y$ follow multivariate normal distributions with covariance matrices $\Sigma^{X}$ and $\Sigma^{Y}$, let $\Delta=\Theta^{X}-\Theta^{Y}$, where $\Theta^{X}=(\Sigma^{X})^{-1}$ and $\Theta^{Y}=(\Sigma^{Y})^{-1}$ are the precision matrices of $X$ and $Y$ respectively. The differential graph is then defined by letting $E_{\Delta}=\left\\{\\{v,w\\}\,:\,\Delta_{v,w}\neq 0\right\\}$. This type of differential model for vector-valued data has been adopted in Zhao et al. (2014), Xu and Gu (2016), and Cai (2017). In the motivating example of EEG data, the electrical activity is observed over a period of time. When measurements smoothly vary over time, it may be more natural to consider the observations as arising from an underlying function. This is particularly true when data from different subjects are observed at different time points. Furthermore, when characterizing conditional independence, it is likely that the activity of each region depends not only on what is occurring simultaneously in the other regions, but also on what has previously occurred in other regions; this suggests that a functional graphical model might be appropriate. In this paper, we define a differential graph for functional data that we refer to as a functional differential graphical model. Similar to differential graphs for vector-valued data, functional differential graphical models characterize the differences in the conditional dependence structures of two distributions of multivariate curves. We build on the functional graphical model developed in Qiao et al. (2019). However, while Qiao et al. (2019) required that the observed functions lie in a finite-dimensional space in order for the functional graphical model to be well defined, the functional differential graphical models may be well defined even in certain cases where the observed functions live in an infinite-dimensional space. We propose an algorithm called FuDGE to estimate the differential graph and show that this procedure enjoys many benefits, similar to differential graph estimation in the vector-valued setting. Most notably, we show that under suitable conditions, the proposed method can consistently recover the differential graph even in the high-dimensional setting where $p$, the number of observed variables, may be larger than $n$, the number of observed samples. A conference version of this paper was presented at the Conference on Neural Information Processing Systems (Zhao et al., 2019). Compared to the conference version, this paper includes the following new results: * • We give a new definition for a differential graph for functional data, which allows us to circumvent the unnatural assumption made in the previous version and take a truly functional approach. Specifically, instead of defining the differential graph based on the difference between conditional covariance functions, we use the limit of the norm of the difference between finite- dimensional precision matrices. * • We include new theoretical guarantees for discretely observed curves. In practice, we can only observe the functions at discrete time points, so this extends the theoretical guarantees to a practical estimation procedure. Discrete observations bring an additional source of error when the estimated curves are used in functional PCA. In Theorem 4, we give an error bound for estimating the covariance matrix of the PCA score vectors under mild conditions. * • We introduce the Joint Functional Graphical Lasso, which is a generalization of the Joint Graphical Lasso (Danaher et al., 2014) to the functional data setting. Empirically, we show that the procedure performs competitively in some settings, but is generally outperformed by the FuDGE procedure. The software implementation can be found at https://github.com/boxinz17/FuDGE. The repository also contains the code to reproduce the simulation results. ### 1.1 Related Work The work we develop lies at the intersection of two different lines of literature: graphical models for functional data and direct estimation of differential graphs. There are many previous works studying the structure estimation of a static undirected graphical model (Chow and Liu, 1968; Yuan and Lin, 2007; Cai et al., 2011; Meinshausen and Bühlmann, 2006; Yu et al., 2016, 2019; Vogel and Fried, 2011). Previous methods have also been proposed for characterizing conditional independence for multivariate observations recorded over time. For example, Talih and Hengartner (2005), Xuan and Murphy (2007), Ahmed and Xing (2009), Song et al. (2009a), Song et al. (2009b), Kolar et al. (2010b), Kolar et al. (2009), Kolar and Xing (2009), Zhou et al. (2010), Yin et al. (2010), Kolar et al. (2010a), Kolar and Xing (2011), Kolar and Xing (2012), Wang and Kolar (2014), Lu et al. (2018) studied methods for dynamic graphical models that assume the data are independently sampled at different time points, but generated by related distributions. In these works, the authors proposed procedures to estimate a series of graphs which represent the conditional independence structure at each time point; however, they assume the observed data does not encode “longitudinal” dependence. In contrast, Qiao et al. (2019); Zhu et al. (2016); Li and Solea (2018); Zhang et al. (2018) considered the setting where the data data are multivariate random functions. Most similar to the setting we consider, Qiao et al. (2019) assumed that the data are distributed as a multivariate Gaussian process (MGP) and use a graphical lasso type procedure on the estimated functional principal component scores. Zhu et al. (2016) also assumed an MGP, but proposed a Bayesian procedure. Crucially, however, both required that the covariance kernel can essentially be represented by a finite dimensional object. Zapata et al. (2019) showed that under various notions of separability—roughly when the covariance kernel can be decomposed into covariance across time and covariance across nodes—the conditional independence of the MGP is well defined even when the functional data are truly infinite dimensional and that the conditional independence graph can be recovered by the union of a (potentially infinitely) countable number of graphs over finite dimensional objects. In a different approach, Li and Solea (2018) did not assume that the random functions are Gaussian, and instead used the notion of additive conditional independence to define a graphical model for the random functions. Finally, Qiao et al. (2020) also assumed that the data are random functions, but also allowed for the dependency structure to change smoothly across time—similar to a dynamic graphical model. We also draw on recent literature which has shown that when the object of interest is the difference between two distributions, directly estimating the difference can provide improvements over first estimating each distribution and then taking the difference. Most notably, when estimating the difference in graphs in the high-dimensional setting, even if each individual graph does not satisfy the appropriate sparsity conditions, the differential graph may still be recovered consistently. Zhao et al. (2014) considered data drawn from two Gaussian graphical models, and they showed that even if both underlying graphs are dense, if the difference between the precision matrices of each distribution is sparse, the differential graph can still be recovered in the high-dimensional setting. Liu et al. (2014) proposed procedure based on KLIEP (Sugiyama et al., 2008) that estimates the differential graph by directly modeling the ratio of two densities. They did not assume Gaussianity, but required that both distributions lie in some exponential family. Fazayeli and Banerjee (2016) extended this idea to estimate the differences in Ising models. Wang et al. (2018) and Ghoshal and Honorio (2019) also proposed direct difference estimators for directed graphs when the data are generated by linear structural equation models that share a common topological ordering. ### 1.2 Notation Let $|\cdot|_{p}$ denote the vector $p$-norm and $\|\cdot\|_{p}$ denote the matrix/operator $p$-norm. For example, for a $p\times 1$ vector $a=(a_{1},a_{2},\dots,a_{p})^{\top}$, we have $|a|_{1}=\sum_{j}|a_{j}|$, $|a|_{2}=(\sum_{j}|a^{2}_{j}|)^{1/2}$ and $|a|_{\infty}=\max_{j}|a_{j}|$. For a $p\times{q}$ matrix $A$ with entries $a_{jk}$, $|A|_{1}=\sum_{j,k}|a_{jk}|$, $\|A\|_{1}=\max_{k}\sum_{j}|a_{jk}|$, $|A|_{\infty}=\max_{j,k}|a_{jk}|$, and $\|A\|_{\infty}=\max_{j}\sum_{k}|a_{jk}|$. Let $\left\lVert A\right\rVert_{\text{F}}=(\sum_{j,k}a^{2}_{jk})^{1/2}$ be the Frobenius norm of $A$. When $A$ is symmetric, let $\mathrm{tr}(A)=\sum_{j}a_{jj}$ denote the trace of A. Let $\lambda_{\min}(A)$ and $\lambda_{\max}(A)$ denote the minimum and maximum eigenvalues, respectively. Let $a_{n}\asymp{b_{n}}$ denote that $0<C_{1}\leq{\inf_{n}|a_{n}/b_{n}|}\leq{\sup_{n}|a_{n}/b_{n}|}\leq C_{2}<\infty$ for some positive constants $C_{1}$ and $C_{2}$. We assume that all random functions belong to a separable Hilbert space $\mathbb{H}$. For any two functions $f_{1},f_{2}\in\mathbb{H}$, we define their inner product as $\langle f_{1},f_{2}\rangle=\int f_{1}(t)f_{2}(t)dt$. The induced norm is $\|f_{1}\|=\|f_{1}\|_{\mathcal{L}^{2}}=\\{\int f_{1}^{2}(t)dt\\}^{1/2}$. For a function vector $f(t)=(f_{1}(t),f_{2}(t),\dots,f_{p}(t))^{\top}$, we let $\|f\|_{\mathcal{L}^{2},2}=(\sum^{p}_{j=1}\|f_{j}\|^{2})^{1/2}$ denote its $\mathcal{L}^{2},2$-norm. For a bivariate function $g(s,t)$, we define the Hilbert-Schmidt norm of $g(s,t)$ as $\|g\|_{\text{HS}}=\int\int\\{g(s,t)\\}^{2}dsdt$. Typically, we will use $f(\cdot)$ (and similarly $g(\cdot,*)$) to denote the entire function $f$, while we use $f(t)$ (and similarly $g(s,t)$) to mean the value of $f$ evaluated at $t$. For a vector space $\mathbb{V}$, we use $\mathbb{V}^{\bot}$ to denote its orthogonal complement. For $v_{1},\ldots,v_{K}\in\mathbb{V}$, and $v=(v_{1},\ldots,v_{K})^{\top}$, we use ${\rm Span}\left\\{v_{1},v_{2},\dots,v_{K}\right\\}={\rm Span}\left(v\right)$ to denote the vector subspace spanned by $v_{1},\ldots,v_{K}$. ## 2 Functional Differential Graphical Models In this section, we give a review of functional graphical models and introduce the notion of a functional differential graphical model. ### 2.1 Functional Graphical Model Suppose $X_{i}(\cdot)=\left(X_{i1}(\cdot),X_{i2}(\cdot),\dots,X_{ip}(\cdot)\right)^{\top}$ is a p-dimensional _multivariate Gaussian processes (MGP)_ with mean zero and common domain $\mathcal{T}$, where $\mathcal{T}$ is a closed interval of the real line with length $\lvert\mathcal{T}\rvert$.111We assume mean zero and a common domain $\mathcal{T}$ to simplify the notation, but the methodology and theory generalize to non-zero means and different time domains. Each observation, for $i=1,2,\ldots,n$, is i.i.d. In addition, assume that for $j\in V$, $X_{ij}(\cdot)$ is a random element of a separable Hilbert space $\mathbb{H}$. Qiao et al. (2019), define the conditional cross-covariance function for $X_{i}(\cdot)$ as ${}C^{X}_{jl}(s,t)\;=\;\mathrm{Cov}\left(X_{ij}(s),X_{il}(t)\,\mid\,\\{X_{ik}(\cdot)\\}_{k\neq j,l}\right).$ (1) If $C^{X}_{jl}(s,t)=0$ for all $s,t\in\mathcal{T}$, then the random functions $X_{j}(\cdot)$ and $X_{l}(\cdot)$ are conditionally independent given the other random functions, and the graph $G_{X}=\\{V,E_{X}\\}$ represents the pairwise Markov properties of $X_{i}(\cdot)$ if $E_{X}=\left\\{(j,l)\,:\,j<l\text{ and }\|C^{X}_{jl}\|_{\text{HS}}\neq 0\right\\}.$ (2) In general, we cannot directly estimate (2), since $X_{i}(\cdot)$ may be an infinite dimensional object. Thus, before applying a statistical estimation procedure, dimension reduction is typically required. Qiao et al. (2019) used _functional principal component analysis_ (FPCA) to project each observed function onto an orthonormal function basis defined by a finite number of eigenfunctions. Their procedure then estimates the conditional independence structure from the “projection scores” of this basis. We outline their approach below. However, in contrast to Qiao et al. (2019), we do not restrict ourselves to dimension reduction by projecting onto the FPCA basis, and in our discussion we instead consider a general function subspace. Let $\mathbb{V}^{M_{j}}_{j}\subseteq\mathbb{H}$ be a subspace of a separable Hilbert space $\mathbb{H}$ with dimension $M_{j}\in\mathbb{N}^{+}$ for all $j=1,2,\dots,p$. Our theory easily generalizes to the setting where $M_{j}$ may differ, but to simplify notation, we assume $M_{j}=M$ for all $j$ and simply write $\mathbb{V}^{M}_{j}$ instead of $\mathbb{V}^{M_{j}}_{j}$. Let $\mathbb{V}^{M}_{[p]}\coloneqq\mathbb{V}^{M}_{1}\otimes\mathbb{V}^{M}_{2}\otimes\dots\otimes\mathbb{V}^{M}_{p}$. For any function $g(\cdot)\in\mathbb{H}$ and a subspace $\mathbb{F}\subseteq\mathbb{H}$, let $\pi(g(\cdot);\mathbb{F})\in\mathbb{F}$ denote the projection of the function $g(\cdot)$ onto the subspace $\mathbb{F}$, and let $\pi(X_{i}(\cdot);\mathbb{V}^{M}_{[p]})=\left(\pi(X_{i1}(\cdot);\mathbb{V}^{M}_{1}),\pi(X_{i2}(\cdot);\mathbb{V}^{M}_{2}),\dots,\pi(X_{ip}(\cdot);\mathbb{V}^{M}_{p})\right)^{\top}.$ When the choice of the subspace is clear from the context, we will use the following shorthand notation: $X^{\pi}_{ij}(\cdot)=\pi(X_{ij}(\cdot);\mathbb{V}^{M}_{j})$, $j=1,2,\dots,p$, and $X^{\pi}_{i}(\cdot)=\pi(X_{i}(\cdot);\mathbb{V}^{M}_{[p]})$. Similar to the definitions in (1) and (2), we define the conditional independence graph of $X^{\pi}(\cdot)$ as $E^{\pi}_{X}=\left\\{\\{j,l\\}\,:\,j<l\text{ and }\|C^{X,\pi}_{jl}\|_{\text{HS}}\neq 0\right\\},$ (3) where ${}C^{X,\pi}_{jl}(s,t)\;=\;\mathrm{Cov}\left(X^{\pi}_{ij}(s),X^{\pi}_{il}(t)\,\mid\,\\{X^{\pi}_{ik}(\cdot)\\}_{k\neq j,l}\right).$ (4) Note that $E^{\pi}_{X}$ depends on the choice of $\mathbb{V}^{M}_{[p]}$ through the projection operator $\pi$, and as we discuss below, $E_{X}^{\pi}$ may be recovered from the observed samples. When the data arise from an MGP, we can estimate the projected graphical structure by studying the precision matrix of projection score vectors (defined below) with _any_ orthonormal function basis of the subspace $\mathbb{V}^{M}_{[p]}$. Let $e^{M}_{j}=(e_{j1}(\cdot),e_{j2}(\cdot),\dots,e_{jM}(\cdot))^{\top}$ be any orthonormal function basis of $\mathbb{V}^{M}_{j}$ and let $e^{M}(\cdot)=\\{e^{M}_{j}\\}^{p}_{j=1}$ be orthonormal function basis of $\mathbb{V}^{M}_{[p]}$. Let $a^{X}_{ijk}=\int_{\mathcal{T}}X_{ij}(t)e_{jk}(t)dt$ denote the projection score of $X_{ij}(\cdot)$ onto $e_{jk}(\cdot)$ and let $\displaystyle a^{X,M}_{ij}=(a^{X}_{ij1},a^{X}_{ij2},\dots,a^{X}_{ijM})^{\top}\;\text{ and }\;a^{X,M}_{i}=((a^{X,M}_{i1})^{\top},\ldots,(a^{X,M}_{ip})^{\top})^{\top}\in{\mathbb{R}^{pM}}.$ Since $X_{i}(\cdot)$ is a $p$-dimensional MGP, $a^{X,M}_{i}$ follows a multivariate Gaussian distribution and we denote the covariance matrix of that distribution as $\Sigma^{X,M}=(\Theta^{X,M})^{-1}\in\mathbb{R}^{pM\times pM}$. Each function $X_{ij}(\cdot)$ is associated with $M$ rows and columns of $\Sigma^{X,M}$ corresponding to $a_{ij}^{X,M}$. We use $\Theta_{jl}^{X,M}$ to refer to the $M\times M$ sub-matrix of $\Theta^{X,M}$ corresponding to functions $X_{ij}(\cdot)$ and $X_{il}(\cdot)$. Lemma 1, from Qiao et al. (2019), shows that the conditional independence structure of the projected functional data can be obtained from the block sparsity of $\Theta^{X,M}$. ###### Lemma 1 [Qiao et al. (2019)] Let $\Theta^{X,M}$ denote the inverse covariance of the projection scores. Then, $X^{\pi}_{ij}(s)\perp\\!\\!\\!\perp X^{\pi}_{il}(t)\mid\\{X^{\pi}_{ik}(\cdot)\\}_{k\neq j,l}$ for all222More precisely, we only need the conditional independence to hold for all $s,t\in{\cal T}$ except for a subset of $\mathcal{T}^{2}$ with zero measure. $s,t\in{\cal T}$ if and only if $\Theta_{jl}^{X,M}\equiv 0$. This implies that $E^{\pi}_{X}$—as defined in (3)—can be equivalently defined as $E^{\pi}_{X}\;=\;\left\\{\\{j,l\\}\,:\,j<l\text{ and }\|\Theta^{X,M}_{jl}\|_{F}\neq 0\right\\}.$ (5) While Qiao et al. (2019) only considered projections onto the span of the FPCA basis (that is, the eigenfunctions of $X_{ij}(\cdot)$ corresponding to $M$ largest eigenvalues), the result trivially extends to the more general case of _any subspace_ and _any orthonormal function basis_ of that subspace. Although $\Theta^{X,M}$ depends on the specific basis onto which $X_{i}(\cdot)$ is projected, the edge set $E^{\pi}_{X}$ only depends on the subspace $\mathbb{V}^{M}_{[p]}$, that is, the span of the basis onto which $X_{i}(\cdot)$ is projected. Thus, Lemma 1 implies that although the entries of $\Theta^{X,M}$ may change when using different orthonormal function bases to represent $\mathbb{V}^{M}_{[p]}$, the block sparsity pattern of $\Theta^{X,M}$ only depends on the span of the selected basis. When $X_{i}(\cdot)\neq X^{\pi}_{i}(\cdot)$, $E^{\pi}_{X}$ may not be the same as $E_{X}$; furthermore, it may not be the case that $E^{\pi}_{X}\subseteq E_{X}$ or $E_{X}\subseteq E^{\pi}_{X}$. Thus, Condition 2 of Qiao et al. (2019) requires a finite $M^{\star}<\infty$ such that $X_{ij}$ lies in $\mathbb{V}^{M^{\star}}_{[p]}$ almost surely. When $M=M^{\star}$, then $X_{i}(\cdot)=X_{i}^{\pi}(\cdot)$ and $E^{\pi}_{X}=E_{X}$. Under this assumption, to estimate $E^{\pi}_{X}=E_{X}$, Qiao et al. (2019) proposed the functional graphical lasso estimator (fglasso), which solves the following objective: $\hat{\Theta}^{X,M}=\operatorname*{arg\,max}_{\Theta^{X,M}}{\left\\{\log{\text{det}\left(\Theta^{X,M}\right)}-\mathrm{tr}\left(S^{X,M}\Theta^{X,M}\right)-\gamma_{n}\sum_{j\neq l}{\left\lVert\Theta^{X,M}_{jl}\right\rVert_{\text{F}}}\right\\}}.$ (6) In (6), $\Theta^{X,M}$ is a symmetric positive definite matrix, $\Theta^{X,M}_{jl}\in\mathbb{R}^{M\times M}$ corresponds to the $(j,l)$ sub- matrix of $\Theta^{X,M}$, $\gamma_{n}$ is a non-negative tuning parameter, and $S^{X,M}$ is an estimator of $\Sigma^{X,M}$. The matrix $S^{X,M}$ is obtained by using FPCA on the empirical covariance functions (see Section 2.3 for details). The resulting estimated edge set for the functional graph is $\hat{E}_{X}^{\pi}=\left\\{\\{j,l\\}\,:\,j<l\text{ and }\left\lVert\hat{\Theta}^{X,M}_{jl}\right\rVert_{\text{F}}>0\right\\}.$ (7) We also note that the objective in (6) was earlier used in Kolar et al. (2013) and Kolar et al. (2014) for estimation of graphical models from multi- attribute data. However, the requirement that $X_{i}(\cdot)$ lies in a subspace with finite dimension may be violated in many practical applications and negates one of the primary benefits of considering the observations as functions. Unfortunately, the extension to infinite-dimensional data is nontrivial, and indeed Condition 2 in Qiao et al. (2019) requires that the observed functional data lies within a finite-dimensional span. To see why, we first note that $\Sigma^{X,M^{\star}}$ is always a compact operator on $\mathbb{R}^{pM^{\star}}$. Thus, as $M^{\star}\to\infty$, the smallest eigenvalue of $\Sigma^{X,M^{\star}}$ will go to zero. As a consequence, $\Sigma^{X,M^{\star}}$ becomes increasingly ill-conditioned, and $\Theta^{X,M^{\star}}$, the inverse of $\Sigma^{X,M^{\star}}$ will become ill- defined when $M^{\star}=\infty$. This behaviour makes the estimation of a functional graphical model —at least through the basis expansion approach proposed by Qiao et al. (2019)—generally infeasible for truly infinite- dimensional functional data. When the data is truly infinite-dimensional, the best we can do is to estimate a finite-dimensional approximation and hope that it captures the relevant information. ### 2.2 Functional Differential Graphical Models: Finite Dimensional Setting In this paper, instead of estimating the conditional independence structure of a single MGP, we are interested in characterizing the difference between two MGPs, $X$ and $Y$. For brevity, we will typically only explicitly define the notation for $X$; however, the reader should infer that all notation for $Y$ is defined analogously. As described in the introduction, Li et al. (2007) and Zhao et al. (2014) consider the setting where $X$ and $Y$ are multivariate Gaussian vectors, and define the differential graph $G_{\Delta}=\\{V,E_{\Delta}\\}$ by letting $E_{\Delta}=\left\\{(v,w)\,:\,v<w\text{ and }\Delta_{vw}\neq 0\right\\}$ (8) where $\Delta=(\Sigma^{X})^{-1}-(\Sigma^{Y})^{-1}$ and $\Sigma^{X},\Sigma^{Y}$ are the covariance matrices of $X$ and $Y$. We extend this definition to the functional data setting and define functional differential graphical models. To develop the intuition, we first start by defining the differential graph with respect to their finite-dimensional projections, that is, with respect to $X^{\pi}_{i}(t)$ and $Y^{\pi}_{i}(t)$ for some choice of $\mathbb{V}^{M}_{[p]}$. As implied by Lemma 1, in the functional graphical model setting, the $M\times M$ blocks of the precision matrix of the projection scores play a similar role to the individual entries of a precision matrix in the vector-valued Gaussian graphical model setting. Thus, we also define a functional differential graphical model by the difference of the precision matrices of the projection scores. Note that for each $j\in V$, we require that both $a^{X}_{ij}$ and $a^{Y}_{ij}$ are computed by the same function basis of $\mathbb{V}^{M}_{j}$. Let $\Theta^{X,M}=\left(\Sigma^{X,M}\right)^{-1}$ and $\Theta^{Y,M}=\left(\Sigma^{Y,M}\right)^{-1}$ be the precision matrices for the projection scores for $X$ and $Y$, respectively, where the inverse should be understood as the pseudo-inverse when $\Sigma^{X,M}$ or $\Sigma^{Y,M}$ are not invertible. The functional differential graphical model is defined as $\Delta^{M}=\Theta^{X,M}-\Theta^{Y,M}.$ (9) Let $\Delta^{M}_{jl}$ be the $(j,l)$-th $M\times M$ block of $\Delta^{M}$ and define the edges of the functional differential graph of the projected data as: $E^{\pi}_{\Delta}\,=\,\left\\{(j,l)\,:\,j<l\text{ and }\,\|\Delta^{M}_{jl}\|_{F}>0\right\\}.$ (10) While the entries of $\Delta^{M}$ depend on the choice of orthonormal function basis, the definition of $E^{\pi}_{\Delta}$ is invariant to the particular basis and only depends on the span. The following lemma formally states this result. ###### Lemma 2 Suppose that ${\rm span}(e^{M}(\cdot))={\rm span}(\tilde{e}^{M}(\cdot))$ for two orthonormal bases $e^{M}(\cdot)$ and $\tilde{e}^{M}(\cdot)$. Let $E_{\Delta}^{\pi}$ and $E_{\Delta}^{\tilde{\pi}}$ be defined by (10) when projecting $X$ and $Y$ onto $e^{M}(\cdot)$ and $\tilde{e}^{M}(\cdot)$, respectively. Then, $E_{\Delta}^{\pi}=E_{\Delta}^{\tilde{\pi}}$. Proof See Appendix B.1. We have several comments regarding $E^{\pi}_{\Delta}$ defined in (10). ##### Projecting $X$ and $Y$ onto different subspaces: While we project both $X$ and $Y$ onto the same subspace $\mathbb{V}^{M}_{[p]}$, our definition can be easily generalized to a setting where we project $X$ onto $\mathbb{V}^{X,M}_{[p]}$ and $Y$ onto $\mathbb{V}^{Y,M}_{[p]}$, with $\mathbb{V}^{X,M}_{[p]}\neq\mathbb{V}^{Y,M}_{[p]}$. For instance, naively following the procedure of Qiao et al. (2019), we could perform FPCA on $X$ and $Y$ separately, and subsequently we could use the difference between the precision matrices of projection scores to define the functional differential graph. Although defining the functional differential graph using this alternative approach may be suitable for some applications, it may result in the undesirable case where $(j,l)\in E_{\Delta}^{\pi}$ even though $C_{jl}^{X,\pi}(\cdot,*)=C_{jl}^{Y,\pi}(\cdot,*)$, $C_{jj}^{X,\pi}(\cdot,*)=C_{ll}^{Y,\pi}(\cdot,*)$, and $C_{ll}^{\setminus j,X,\pi}(\cdot,*)=C_{ll}^{\setminus j,Y,\pi}(\cdot,*)$. Therefore, we restrict our discussion to the setting where both $X$ and $Y$ are projected onto the same subspace. ##### Connection to Multi-Attribute Graphical Models: The selection of a specific functional subspace is connected to multi- attribute graphical models (Kolar et al., 2014). If we treat the random function $X_{ij}(\cdot)$ as representing an infinite number of attributes, then $X^{\pi}_{ij}(\cdot)$ will be an approximation using $M$ attributes. The chosen attributes are given by the subspace $\mathbb{V}^{M}_{j}$. While we allow different nodes to choose different attributes by allowing $\mathbb{V}^{M}_{j}$ to vary across $j$, we require that the same attributes are used to represent both $X$ and $Y$ by restricting $\mathbb{V}^{M}_{[p]}$ to be the same for $X$ and $Y$. The specific choice of $\mathbb{V}^{M}_{[p]}$, can extract different attributes from the data. For instance, using the subspace spanned by the Fourier basis can be viewed as extracting frequency information, while using the subspace spanned by the eigenfunctions—as introduced in the next section—can be viewed as extracting the dominant modes of variation. Given definition (10) and Lemma 2, there are two main questions to be answered: First, how do we choose $\mathbb{V}^{M}_{[p]}$? Second, what happens when $X$ and $Y$ are infinite-dimensional? We answer the first question in Section 2.3 and the second question in Section 2.4. ### 2.3 Choosing Functional Subspace via FPCA As discussed in Section 2.2, the choice of $\mathbb{V}^{M}_{[p]}$ in Definition 10 decides—roughly speaking—the attributes or dimensions in which we compare the conditional independence structures of $X$ and $Y$. In some applications, we may have a very good prior knowledge about this choice. However, in many cases we may not have strong prior knowledge. In this section, we describe our recommended “default choice” that uses FPCA on the combined $X$ and $Y$ observations. In particular, suppose there exist subspaces $\\{\mathbb{V}^{M^{\star}}_{j}\\}_{j\in V}$ such that $\mathbb{V}^{M^{\star}}_{j}$ has dimension $M^{\star}<\infty$ and $X_{ij}(t),Y_{ij}(t)\in\mathbb{V}^{M^{\star}}_{j}$ for all $j\in V$. Then, FPCA—when given population values—recovers this subspace. Similar to the way principal component analysis provides the $L_{2}$ optimal lower dimensional representation of vector-valued data, FPCA provides the $L_{2}$ optimal finite dimensional representation of functional data. Let $K^{X}_{jj}(t,s)=\mathrm{Cov}(X_{ij}(t),X_{ij}(s))$ denote the covariance function for $X_{ij}$ where $j\in V$. Then, there exist orthonormal eigenfunctions and eigenvalues $\\{\phi^{X}_{jk}(t),\lambda^{X}_{jk}\\}_{k\in\mathbb{N}}$ such that $\int_{\mathcal{T}}K^{X}_{jj}(s,t)\phi_{jk}^{X}(t)dt=\lambda_{jk}^{X}\phi_{jk}^{X}(s)$ for all $k\in\mathbb{N}$ (Hsing and Eubank, 2015). Since $K^{X}_{jj}(s,t)$ is symmetric and non-negative definite, we assume, without loss of generality, that $\\{\lambda^{X}_{js}\\}_{s\in\mathbb{N}^{+}}$ is non-negative and non- increasing. By the Karhunen-Loève expansion (Hsing and Eubank, 2015, Theorem7.3.5), $X_{ij}(t)$ can be expressed as $X_{ij}(t)=\sum_{k=1}^{\infty}a^{X}_{ijk}\phi^{X}_{jk}(t)$, where the principal component scores satisfy $a^{X}_{ijk}=\int_{\mathcal{T}}X_{ij}(t)\phi^{X}_{jk}(t)dt$ and $a^{X}_{ijk}\sim N(0,\lambda_{jk}^{X})$ with $E(a^{X}_{ijk}a^{X}_{ijl})=0$ if $k\neq l$. Because the eigenfunctions are orthonormal, the $L_{2}$ projection of $X_{ij}$ onto the span of the first $M$ eigenfunctions is $X^{M}_{ij}(t)=\sum_{k=1}^{M}a^{X}_{ijk}\phi^{X}_{jk}(t)$. Similarly, we can define $K^{Y}_{jj}(t,s)$, $\\{\phi^{Y}_{jk}(t),\lambda^{Y}_{jk}\\}_{k\in\mathbb{N}}$ and $Y^{M}_{ij}(t)$. Let $K_{jj}(s,t)=K^{X}_{jj}(s,t)+K^{Y}_{jj}(s,t)$ and let $\\{\phi_{jk}(t),\lambda_{jk}\\}_{k\in\mathbb{N}}$ be the eigenfunction- eigenvalue pairs of $K_{jj}(s,t)$. Lemma 3 implies that $X_{ij}(\cdot)$ and $Y_{ij}(\cdot)$ lie within the span of the eigenfunctions corresponding to the non-zero eigenvalues of $K_{jj}$. Furthermore, this subspace is minimal in the sense that no subspace with smaller dimension contains $X_{ij}(\cdot)$ and $Y_{ij}(\cdot)$ almost surely. Thus, the FPCA basis of $K_{jj}$ provides a good default choice for dimension reduction. ###### Lemma 3 Let $|\mathbb{V}|$ denote the dimension of a subspace $\mathbb{V}$ and suppose $M^{\prime}_{j}=\inf\\{|\mathbb{V}|:\mathbb{V}\subseteq\mathbb{H},X_{ij}(\cdot),Y_{ij}(\cdot)\in\mathbb{V}\,\text{almost surely}\\}.$ Let $\\{\phi_{jk}(t),\lambda_{jk}\\}_{k\in\mathbb{N}}$ be the eigenfunction- eigenvalue pairs of $K_{jj}(s,t)$ and $M^{\star}_{j}=\sup\\{M\in\mathbb{N}^{+}:\lambda_{jM}>0\\}.$ Then $M^{\prime}_{j}=M^{\star}_{j}$ and $X_{ij},Y_{ij}\in{\rm Span}\\{\phi_{j1}(\cdot),\phi_{j2}(\cdot),\dots,\phi_{j,M^{\star}_{j}}(\cdot)\\}$ almost surely. Proof See Appendix B.2. ### 2.4 Infinite Dimensional Functional Data In Section 2.2, we defined a functional differential graph for functional data that have finite-dimensional representation. In this section, we present a more general definition that also allows for infinite-dimensional functional data. As discussed in Section 2.1, when the data are infinite-dimensional, estimating a functional graphical model is not straightforward because the precision matrix of the scores does not have a well-defined limit as $M$, the dimension of the projected data, increases to $\infty$. When estimating the differential graph, however, although $\|\Theta^{X,M}\|_{\text{F}}\to\infty$ and $\|\Theta^{Y,M}\|_{\text{F}}\to\infty$ as $M\to\infty$, it is still possible for $\|\Theta^{X,M}-\Theta^{Y,M}\|_{\text{F}}$ to be bounded as $M\to\infty$. For instance, $x_{n},y_{n}\in\mathbb{R}$ may both tend to infinity, but $\lim_{n}x_{n}-y_{n}$ may still exist and be bounded. Furthermore, even when $\|\Theta^{X,M}-\Theta^{Y,M}\|_{\text{F}}\rightarrow\infty$, it is still possible for the difference $\Theta^{X,M}-\Theta^{Y,M}$ to be informative. This observation leads to Definition 1 below. To simplify notation, in the rest of the paper, we assume that $X_{ij}(\cdot)$ and $Y_{ij}(\cdot)$ live in an $M^{\star}$ dimensional space where $M^{\star}\leq\infty$. Recall that $\\{\phi^{X}_{jk}(\cdot),\lambda^{X}_{jk}\\}_{k\in\mathbb{N}}$ and $\\{\phi^{Y}_{jk}(\cdot),\lambda^{Y}_{jk}\\}_{k\in\mathbb{N}}$ denote the eigenpairs of $K_{jj}^{X}$ and $K_{jj}^{Y}$ respectively. ###### Definition 1 (Differential Graph Matrix and Comparability) The MGPs $X$ and $Y$ are comparable if, for all $j\in[p]$, $K_{jj}^{X}$ and $K_{jj}^{Y}$ have $M^{\star}$ non-zero eigenvalues and $\mathrm{span}\left(\\{\phi_{jk}^{X}\\}_{k=1}^{M^{\star}}\right)=\mathrm{span}\left(\\{\phi_{jk}^{Y}\\}_{k=1}^{M^{\star}}\right)$. Furthermore, for every $(j,l)\in V^{2}$ and a projection subspace sequence $\left\\{\mathbb{V}^{M}_{[p]}\right\\}_{M\geq 1}$ satisfying that $\lim_{M\to M^{\star}}\mathbb{V}^{M}_{j}=\mathrm{span}\left(\\{\phi_{jk}^{X}\\}_{k=1}^{M^{\star}}\right)$, we have either: $\lim_{M\to M^{\star}}\|\Delta^{M}_{jl}\|_{\text{F}}=0\qquad\text{or}\qquad\lim\inf_{M\to M^{\star}}\|\Delta^{M}_{jl}\|_{\text{F}}>0.$ In this case, we define the differential graph matrix (DGM) $D=(D_{jl})_{(j,l)\in V^{2}}\in\mathbb{R}^{p\times p}$, where $D_{jl}=\lim\inf_{M\to M^{\star}}\|\Delta^{M}_{jl}\|_{\text{F}}.$ (11) We say that $X$ and $Y$ are incomparable, if for some $j$, $K_{jj}^{X}$ and $K_{jj}^{Y}$ have a different number of non-zero eigenvalues, or if $\mathrm{span}\left(\\{\phi_{jk}^{X}\\}_{k=1}^{M^{\star}}\right)\neq\mathrm{span}\left(\\{\phi_{jk}^{Y}\\}_{k=1}^{M^{\star}}\right)$, or if there exists some $(j,l)$ such that given $\left\\{\mathbb{V}^{M}_{[p]}\right\\}_{M\geq 1}$ satisfying that $\lim_{M\to M^{\star}}\mathbb{V}^{M}_{j}=\mathrm{span}\left(\\{\phi_{jk}^{X}\\}_{k=1}^{M^{\star}}\right)$, we have $\lim\inf_{M\to M^{\star}}\|\Delta^{M}_{jl}\|_{\text{F}}=0,\qquad\text{but}\qquad\lim\sup_{M\to M^{\star}}\|\Delta^{M}_{jl}\|_{\text{F}}>0.$ In Definition 1 we say $\lim_{M\to M^{\star}}\mathbb{V}^{M}_{j}=\mathrm{span}\left(\\{\phi_{jk}^{X}\\}_{k=1}^{M^{\star}}\right)$, to mean the following: For any $\epsilon>0$ and all $g\in\mathrm{span}\left(\\{\phi_{jk}^{X}\\}_{k=1}^{M^{\star}}\right)$, there exists $M^{\prime}=M^{\prime}(\epsilon)<\infty$ such that $\|g-g^{M}_{P}\|<\epsilon$ for all $M\geq M^{\prime}$, where $g^{M}_{P}$ denotes the projection of $g$ onto the subspace of $\mathbb{V}^{M}_{j}$. When $M^{\star}<\infty$, the conditional independence structure in $X_{i}$ and $Y_{i}$ can be completely captured by a finite dimensional representation. When $M^{\star}=\infty$, as $M\to\infty$, $\Delta^{M}_{jl}$ approaches the difference of two matrices with unbounded eigenvalues. Nonetheless, when $X$ and $Y$ are comparable, the limits are still informative. This would suggest that by using a sufficiently large subspace, we can capture such a difference arbitrarily well. On the other hand, if the MGPs are not comparable, then using a larger subspace may not improve the approximation regardless of the sample size. For this reason, in the rest of the paper, we only focus on the setting where $X$ and $Y$ are comparable. To our knowledge, there is no existing procedure to estimate a graphical model for functional data when the functions are infinite-dimensional. Thus, it is not straightforward to determine whether the comparability condition is stronger or weaker than what might be required for estimating the graphs separately and then comparing post hoc. Nonetheless, we hope to provide some intuition for the reader. Suppose $X$ and $Y$ are of the same dimension, $M^{\star}$. If $M^{\star}<\infty$ and the functional graphical model for each sample could be estimated separately (that is, $\|\Theta^{X,M}\|_{F}<\infty$ and $\|\Theta^{Y,M}\|_{F}<\infty$), then $X$ and $Y$ are comparable when the minimal basis which spans $X$ and $Y$ is the same. Thus, the functional differential graph is also well defined. On the other hand, the conditions required by Qiao et al. (2019, Condition 2) for consistent estimation are not satisfied when $M^{\star}=\infty$, since $\lim_{M\rightarrow\infty}\|\Theta^{X,M}\|_{F}=\infty$ due to the compactness of the covariance operator. However, $X$ and $Y$ may still be comparable depending on the limiting behavior of $\Theta^{X,M}$ and $\Theta^{Y,M}$. Thus, there are settings where the differential graph may exist and be consistently recovered even when each individual graph cannot be recovered (even when $p$ is fixed). However, when one MGP is finite-dimensional and the other is infinite- dimensional, then the MGPs are incomparable. To see this, without loss of generality, we assume that MGP $X$ has infinite dimension $M^{X}_{j}=M^{\star}_{X}=\infty$ for all $j\in V$ and MGP $Y$ has finite dimension $M^{Y}_{j}=M^{\star}_{Y}<\infty$ for all $j\in V$. Then $\Theta^{Y,M}$ is ill-defined when $M>M^{\star}_{Y}$ and recovering the differential graph is not straightforward. We now define the notion of a functional differential graph. ###### Definition 2 When two MGPs $X$ and $Y$ are comparable, we define their functional differential graph as an undirected graph $G_{\Delta}=\\{V,E_{\Delta}\\}$, where $E_{\Delta}$ is defined as $E_{\Delta}=\left\\{\\{j,l\\}\,:\,j<l\text{ and }D_{jl}>0\right\\}.$ (12) ###### Remark 1 The functional graphical model defined by Qiao et al. (2019) uses the conditional covariance function $C_{jl}^{X}(\cdot,*)$ given in (1). Thus, it would be quite natural to use the conditional covariance functions directly to define a differential graph where $E_{\Delta}=\left\\{\\{j,l\\}\;:\;j<l\text{ and }C_{jl}^{X}(\cdot,*)\neq C_{jl}^{Y}(\cdot,*)\right\\}.$ (13) Unfortunately, this definition does not always coincide with the one we propose in Definition 2. Nevertheless, the functional differential graph given in Definition 2 has many nice statistical properties and retains important features of the graph defined in (13). The primary statistical benefit of the graph defined in Definition 2 is that it can be directly estimated without estimating each conditional independence function: $C^{X}_{jl}(\cdot,\cdot)$ and $C^{Y}_{jl}(\cdot,\cdot)$. Similar to the vector-valued case considered by (Zhao et al., 2014), this allows for a much lower sample complexity when each individual graph is dense but the difference is sparse. In some settings, there may not be enough samples to estimate each individual graph accurately, but the difference may still be recovered. This result is demonstrated in Theorem 1. The statistical advantages of our estimand unfortunately come at the cost of a slightly less precise characterization of the difference in the conditional covariance functions. However, many of the key characteristics are still preserved. Suppose $X_{i}$ and $Y_{i}$ are both $M^{\star}$-dimensional with $M^{\star}<\infty$ and further suppose that $\\{\phi_{jm}(\cdot)\phi_{lm^{\prime}}(*)\\}_{m,m^{\prime}\in[M^{\star}]\times[M^{\star}]}$ is a linearly independent set of functions. Suppose the conditional covariance functions for $j,l\in V$ are unchanged so that $C_{jj}^{X}(\cdot,*)=C_{jj}^{Y}(\cdot,*)$ and $C_{ll}^{\backslash j,X}(\cdot,*)=C_{ll}^{\backslash j,Y}(\cdot,*)$, where $C_{ll}^{\backslash j,X}(\cdot,*)\coloneqq{\rm Cov}(X_{l}(\cdot),X_{l}(*)\,|\,X_{k}(\cdot),k\neq j,l)$ and $C_{ll}^{\backslash j,Y}(\cdot,*)$ is defined similarly; then, $\Delta_{jl}=0$ if and only if $C_{jl}^{X}(\cdot,*)=C_{jl}^{Y,\pi}(\cdot,*)$. When this holds for all pairs $j,l\in V$, then the definitions of a differential graph in Definition 2 and (13) are equivalent. When the conditional covariance functions may change so that $C_{jj}^{X}(\cdot,*)\neq C_{jj}^{Y}(\cdot,*)$, then we still have that $\Delta_{jl}\neq 0$ if $C_{jl}^{X,\pi}(\cdot,*)=0$ and $C_{jl}^{Y,\pi}(\cdot,*)\neq 0$ (or vice versa). Thus, even in this more general setting, the functional differential graph given in Definition 2 captures all qualitative differences between the conditional covariance functions $C_{jl}^{X}(\cdot,*)$ and $C_{jl}^{Y}(\cdot,*)$. Our objective is to directly estimate $E_{\Delta}$ without first estimating $E_{X}$ or $E_{Y}$. Since the functions we consider may be infinite- dimensional objects, in practice, what we can directly estimate is actually $E^{\pi}_{\Delta}$ defined in (10). We will use a sieve estimator to estimate $\Delta^{M}$, where $M$ grows with the sample size $n$. When $M^{\star}=M$, then $E^{\pi}_{\Delta}=E_{\Delta}$. When $M<M^{\star}\leq\infty$, then this is generally not true; however, we would expect the graphs to be similar when $M$ is large enough compared with $M^{\star}$. Thus, by constructing a suitable estimator of $\Delta^{M}$, we can still recover $E_{\Delta}$. ### 2.5 Illustration of comparability We provide few examples that illustrate the notion of comparability. In the first two examples, the graphs are comparable, while in the third example, the graphs are incomparable. First, we state a lemma that will be helpful in the following discussions. The lemma follows directly from the properties of the multivariate normal and the inverse of block matrices. ###### Lemma 4 Let $H^{X,M}_{jl}=\mathrm{Cov}(a^{X,M}_{ij},a^{X,M}_{il}\mid a^{X,M}_{ik},k\neq j,l)$ and $H^{\backslash l,X,M}_{jj}=\mathrm{Var}(a^{X,M}_{ij}\mid a^{X,M}_{ik},k\neq j,l)$. For any $j\in V$, we have $\Theta^{X,M}_{jj}=(H^{X,M}_{jj})^{-1}$. For any $(j,l)\in V^{2}$ and $j\neq l$, we have $\Theta^{X,M}_{jl}=-(H^{X,M}_{jj})^{-1}H^{X,M}_{jl}(H^{\backslash j,X,M}_{ll})^{-1}$. Proof See Appendix B.3. The following proposition follows directly from Lemma 4. ###### Proposition 1 Assume that for any $(j,l)\in V^{2}$ and $j\neq l$, we have $a^{X}_{ijm}\perp\\!\\!\\!\perp a^{X}_{ijm^{\prime}}\mid a^{X,M}_{ik},k\neq j\qquad\text{and}\qquad a^{X}_{ijm}\perp\\!\\!\\!\perp a^{X}_{ijm^{\prime}}\mid a^{X,M}_{ik},k\neq j,l,$ for any $M$ and $1\leq m\neq m^{\prime}\leq M$. We then have $\Theta^{X,M}_{jj}={\rm diag}\left(\frac{1}{{\rm Var}\left(a^{X}_{ij1}\mid a^{X,M}_{ik},k\neq j\right)},\dots,\frac{1}{{\rm Var}\left(a^{X}_{ijM}\mid a^{X,M}_{ik},k\neq j\right)}\right)$ and $\Theta^{X,M}_{jl,mm^{\prime}}=\frac{{\rm Cov}\left(a^{X}_{ijm},a^{X}_{ilm^{\prime}}\mid a^{X,M}_{ik},k\neq j,l\right)}{{\rm Var}\left(a^{X}_{ijm}\mid a^{X,M}_{ik},k\neq j\right){\rm Var}\left(a^{X}_{ilm^{\prime}}\mid a^{X,M}_{ik},k\neq j\right)}\overset{\Delta}{=}\bar{v}^{X,jl,M}_{mm^{\prime}},$ for any $M$ and $1\leq m\neq m^{\prime}\leq M$. In addition, if $a^{Y}_{ijm}\perp\\!\\!\\!\perp a^{Y}_{ijm^{\prime}}\mid a^{Y,M}_{ik},k\neq j\quad\text{and}\quad a^{Y}_{ijm}\perp\\!\\!\\!\perp a^{Y}_{ijm^{\prime}}\mid a^{Y,M}_{ik},k\neq j,l,$ for any $M$ and $1\leq m\neq m^{\prime}\leq M$, then $\displaystyle\Theta^{X,M}_{jj}-\Theta^{Y,M}_{jj}$ $\displaystyle={\rm diag}\left(\left\\{\frac{{\rm Var}\left(a^{Y}_{ijm}\mid a^{Y,M}_{ik},k\neq j\right)-{\rm Var}\left(a^{X}_{ijm}\mid a^{X,M}_{ik},k\neq j\right)}{{\rm Var}\left(a^{X}_{ijm}\mid a^{X,M}_{ik},k\neq j\right){\rm Var}\left(a^{Y}_{ijm}\mid a^{Y,M}_{ik},k\neq j\right)}\right\\}^{M}_{m=1}\right)$ $\displaystyle\overset{\Delta}{=}{\rm diag}\left(\bar{w}^{j,M}_{1},\bar{w}^{j,M}_{2},\dots,\bar{w}^{j,M}_{M}\right)$ and $\displaystyle\Theta^{X,M}_{jl,mm^{\prime}}-\Theta^{Y,M}_{jl,mm^{\prime}}$ $\displaystyle=\frac{{\rm Cov}\left(a^{X}_{ijm},a^{X}_{ilm^{\prime}}\mid a^{X,M}_{ik},k\neq j,l\right)}{{\rm Var}\left(a^{X}_{ijm}\mid a^{X,M}_{ik},k\neq j\right){\rm Var}\left(a^{X}_{ilm^{\prime}}\mid a^{X,M}_{ik},k\neq j\right)}$ $\displaystyle\qquad\qquad\qquad-\frac{{\rm Cov}\left(a^{Y}_{ijm},a^{Y}_{ilm^{\prime}}\mid a^{Y,M}_{ik},k\neq j,l\right)}{{\rm Var}\left(a^{Y}_{ijm}\mid a^{Y,M}_{ik},k\neq j\right){\rm Var}\left(a^{Y}_{ilm^{\prime}}\mid a^{Y,M}_{ik},k\neq j\right)}$ $\displaystyle=\bar{v}^{Y,jl,M}_{mm^{\prime}}-\bar{v}^{X,jl,M}_{mm^{\prime}}\overset{\Delta}{=}\bar{z}^{jl,M}_{mm^{\prime}},$ for any $M$ and $1\leq m\neq m^{\prime}\leq M$. With the notation defined in Proposition 1, we have that $\|\Delta^{M}_{jj}\|^{2}_{\text{HS}}=\sum^{M}_{m=1}\left(\bar{w}^{j,M}_{m}\right)^{2}\qquad\text{and}\qquad\|\Delta^{M}_{jl}\|^{2}_{\text{HS}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{jl,M}_{mm^{\prime}}\right)^{2}.$ (14) As a result, we have the following condition for comparability. ###### Proposition 2 Under the assumptions in Proposition 1, assume that MGPs $X$ and $Y$ are $M^{\star}$-dimensional, with $1\leq M^{\star}\leq\infty$, and lie in the same space. Then they are comparable if and only if for every $(j,l)\in V\times V$, we have either $\lim\inf_{M\to M^{\star}}\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{jl,M}_{mm^{\prime}}\right)^{2}>0\qquad\text{or}\qquad\lim_{M\to M^{\star}}\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{jl,M}_{mm^{\prime}}\right)^{2}=0,$ (15) where $\bar{z}^{jl,M}_{mm^{\prime}}$ are defined in Proposition 1. We now give an infinite-dimensional comparable example. ###### Example 1 Assume that $\\{\epsilon^{X}_{i1k}\\}_{k\geq 1}$, $\\{\epsilon^{X}_{i2k}\\}_{k\geq 1}$, and $\\{\epsilon^{X}_{i3k}\\}_{k\geq 1}$ are all independent mean zero Gaussian variables with ${\rm Var}(\epsilon^{X}_{ijk})=\sigma^{2}_{X,jk}$, $j=1,2,3$, $k\geq 1$ for all $i$. For any $k\geq 1$, let $a^{X}_{i1k}=a^{X}_{i2k}+\epsilon^{X}_{i1k},\quad a^{X}_{i2k}=\epsilon^{X}_{i2k},\quad a^{X}_{i3k}=a^{X}_{i2k}+\epsilon^{X}_{i3k}.$ Let $a^{X,M}_{ij}=(a^{X}_{ij1},\cdots,a^{X}_{ijM})^{\top}$, $j=1,2,3$. We then define $X_{ij}(t)=\sum^{\infty}_{k=1}a^{X}_{ijk}b_{k}(t)$, $j=1,2,3$, where $\\{b_{k}(t)\\}^{\infty}_{k=1}$ is some orthonormal function basis of $\mathbb{H}$. We define $\\{\epsilon^{Y}_{ijk}\\}_{k\geq 1}$, $\\{a^{Y}_{ijk}\\}_{k\geq 1}$, $a^{Y,M}_{ij}$, and $Y_{ij}(t)$, $j=1,2,3$, similarly. The graph structure of $X$ and $Y$ is shown in Figure 1. Since $a^{X,M}_{ij}$ follows a multivariate Gaussian distribution, for any $M\geq 2$, $1\leq m,m^{\prime}\leq M$ and $m\neq m^{\prime}$: $\displaystyle{\rm Var}\left(a^{X}_{i1m}\mid a^{X,M}_{i2},a^{X,M}_{i3}\right)=\sigma^{2}_{X,1m},$ $\displaystyle{\rm Var}\left(a^{X}_{i3m}\mid a^{X,M}_{i1},a^{X,M}_{i2}\right)=\sigma^{2}_{X,3m},$ $\displaystyle{\rm Var}\left(a^{X}_{i2m}\mid a^{X,M}_{i1},a^{X,M}_{i3}\right)=\frac{\sigma^{2}_{X,1m}\sigma^{2}_{X,2m}\sigma^{2}_{X,3m}}{\sigma^{2}_{X,1m}\sigma^{2}_{X,2m}+\sigma^{2}_{X,1m}\sigma^{2}_{X,3m}+\sigma^{2}_{X,2m}\sigma^{2}_{X,3m}},$ and $\displaystyle{\rm Var}\left(a^{X}_{i1m}\mid a^{X,M}_{i2}\right)=\sigma^{2}_{X,1m},$ $\displaystyle{\rm Var}\left(a^{X}_{i1m}\mid a^{X,M}_{i3}\right)=\frac{\sigma^{2}_{X,1m}\sigma^{2}_{X,2m}+\sigma^{2}_{X,1m}\sigma^{2}_{X,3m}+\sigma^{2}_{X,2m}\sigma^{2}_{X,3m}}{\sigma^{2}_{2m}+\sigma^{2}_{3m}},$ $\displaystyle{\rm Var}\left(a^{X}_{i3m}\mid a^{X,M}_{i2}\right)=\sigma^{2}_{X,3m},$ $\displaystyle{\rm Var}\left(a^{X}_{i3m}\mid a^{X,M}_{i1}\right)=\frac{\sigma^{2}_{X,1m}\sigma^{2}_{X,2m}+\sigma^{2}_{X,1m}\sigma^{2}_{X,3m}+\sigma^{2}_{X,2m}\sigma^{2}_{X,3m}}{\sigma^{2}_{2m}+\sigma^{2}_{1m}},$ $\displaystyle{\rm Var}\left(a^{X}_{i2m}\mid a^{X,M}_{i1}\right)=\frac{\sigma^{2}_{X,1m}\sigma^{2}_{X,2m}}{\sigma^{2}_{X,1m}+\sigma^{2}_{X,2m}},$ $\displaystyle{\rm Var}\left(a^{X}_{i2m}\mid a^{X,M}_{i3}\right)=\frac{\sigma^{2}_{X,3m}\sigma^{2}_{X,2m}}{\sigma^{2}_{X,3m}+\sigma^{2}_{X,2m}}.$ In addition, we also have $\displaystyle{\rm Cov}(a^{X}_{i1m},a^{X}_{3m^{\prime}}\mid a^{X,M}_{i2})=0,$ $\displaystyle{\rm Cov}(a^{X}_{i1m},a^{X}_{i2m^{\prime}}\mid a^{X,M}_{i3})=\mathbbm{1}(m=m^{\prime})\cdot\frac{\sigma^{2}_{X,3m}\sigma^{2}_{X,2m}}{\sigma^{2}_{X,3m}+\sigma^{2}_{X,2m}},$ $\displaystyle{\rm Cov}(a^{X}_{i2m},a^{X}_{i3m^{\prime}}\mid a^{X,M}_{i3})=\mathbbm{1}(m=m^{\prime})\cdot\frac{\sigma^{2}_{X,1m}\sigma^{2}_{X,2m}}{\sigma^{2}_{X,1m}+\sigma^{2}_{X,2m}}.$ 123 Figure 1: The conditional independence graph for both $X$ and $Y$ in Example 1. The differential graph between $X$ and $Y$ has the same structure. Similar results hold for $Y$. Suppose that $\sigma^{2}_{X,jk},\sigma^{2}_{Y,jk}\asymp k^{-\alpha}\quad\text{and}\quad|\sigma^{2}_{X,jk}-\sigma^{2}_{Y,jk}|\asymp k^{-\beta},\quad j=1,2,3,$ where $\alpha,\beta>0$ and $\beta>\alpha$. Then $\displaystyle\bar{z}^{13,M}_{mm^{\prime}}=0,$ $\displaystyle\bar{z}^{12,M}_{mm^{\prime}}=\mathbbm{1}(m=m^{\prime})\frac{\sigma^{2}_{X,1m}-\sigma^{2}_{Y,1m}}{\sigma^{2}_{X,1m}\cdot\sigma^{2}_{Y,1m}}\asymp\mathbbm{1}(m=m^{\prime})\cdot m^{-(\beta-\alpha)},$ $\displaystyle\bar{z}^{23,M}_{mm^{\prime}}=\mathbbm{1}(m=m^{\prime})\frac{\sigma^{2}_{X,3m}-\sigma^{2}_{Y,3m}}{\sigma^{2}_{X,3m}\cdot\sigma^{2}_{Y,3m}}\asymp\mathbbm{1}(m=m^{\prime})\cdot m^{-(\beta-\alpha)}.$ This implies that $\displaystyle\|\Delta^{M}_{13}\|^{2}_{\text{F}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{13,M}_{mm^{\prime}}\right)^{2}=0,$ (16) $\displaystyle\|\Delta^{M}_{12}\|^{2}_{\text{F}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{12,M}_{mm^{\prime}}\right)^{2}\asymp\sum^{M}_{m=1}\frac{1}{m^{\beta-\alpha}},$ $\displaystyle\|\Delta^{M}_{23}\|^{2}_{\text{F}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{23,M}_{mm^{\prime}}\right)^{2}\asymp\sum^{M}_{m=1}\frac{1}{m^{\beta-\alpha}}.$ When $\beta>\alpha+1$, we have $0<\lim_{M\to\infty}\|\Delta^{M}_{12}\|_{\text{F}}=\lim_{M\to\infty}\|\Delta^{M}_{23}\|_{\text{F}}<\infty$. When $\beta\leq\alpha+1$, we have $\lim_{M\to\infty}\|\Delta^{M}_{12}\|_{\text{F}}=\lim_{M\to\infty}\|\Delta^{M}_{23}\|_{\text{F}}=\infty$. In both cases the two graphs are comparable. The following example describes a sequence of MGPs that are comparable; however, the differential graph is intrinsically hard to estimate. ###### Example 2 We define $\\{\epsilon^{X}_{ijk}\\}_{k\geq 1}$, $\\{a^{X}_{ijk}\\}_{k\geq 1}$, $\\{\epsilon^{Y}_{ijk}\\}_{k\geq 1}$, and $\\{a^{Y}_{ijk}\\}_{k\geq 1}$ as in Example 1. Let $X_{ij}(t)=\sum^{M^{\star}}_{k=1}a^{X}_{ijk}b_{k}(t)$ and $Y_{ij}(t)=\sum^{M^{\star}}_{k=1}a^{Y}_{ijk}b_{k}(t)$, $j=1,2,3$, where $M^{\star}$ is a positive integer. Suppose that $\sigma^{2}_{X,jk},\sigma^{2}_{Y,jk}\asymp k^{-\alpha}\quad\text{and}\quad|\sigma^{2}_{X,jk}-\sigma^{2}_{Y,jk}|\asymp\mathbbm{1}(k=M^{\star})k^{-\beta},\quad j=1,2,3,$ where $\alpha,\beta>0$ and $\beta>\alpha$. Following the argument in Example 1, for any $1\leq M\leq M^{\star}$, we have $\displaystyle\bar{z}^{13,M}_{mm^{\prime}}=0,$ $\displaystyle\bar{z}^{12,M}_{mm^{\prime}}=\mathbbm{1}(m=m^{\prime})\mathbbm{1}(m=M^{\star})\cdot\frac{\sigma^{2}_{X,1m}-\sigma^{2}_{Y,1m}}{\sigma^{2}_{X,1m}\cdot\sigma^{2}_{Y,1m}}\asymp\mathbbm{1}(m=m^{\prime})\mathbbm{1}(m=M^{\star})\cdot m^{-(\beta_{1}-2\alpha_{1})},$ $\displaystyle\bar{z}^{23,M}_{mm^{\prime}}=\mathbbm{1}(m=m^{\prime})\mathbbm{1}(m=M^{\star})\cdot\frac{\sigma^{2}_{X,3m}-\sigma^{2}_{Y,3m}}{\sigma^{2}_{X,3m}\cdot\sigma^{2}_{Y,3m}}\asymp\mathbbm{1}(m=m^{\prime})\mathbbm{1}(m=M^{\star})\cdot m^{-(\beta_{3}-2\alpha_{3})}.$ This implies that $\displaystyle\|\Delta^{M}_{13}\|^{2}_{\text{F}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{13,M}_{mm^{\prime}}\right)^{2}=0,$ (17) $\displaystyle\|\Delta^{M}_{12}\|^{2}_{\text{F}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{12,M}_{mm^{\prime}}\right)^{2}\asymp M^{-2(\beta-2\alpha)}\mathbbm{1}(M=M^{\star}),$ $\displaystyle\|\Delta^{M}_{23}\|^{2}_{\text{F}}=\sum^{M}_{m^{\prime}=1}\sum^{M}_{m=1}\left(\bar{z}^{23,M}_{mm^{\prime}}\right)^{2}\asymp M^{-2(\beta-2\alpha)}\mathbbm{1}(M=M^{\star}).$ Based on the calculation above, we observe that estimation of the differential graph here is intrinsically hard. For any $M<M^{\star}$, we have $\|\Delta^{M}_{12}\|_{\text{F}}=\|\Delta^{M}_{23}\|_{\text{F}}=0$. Thus, when $M<M^{\star}$ is used for estimation, the resulting target graph $E^{\pi}_{\Delta}$ would be empty. However, by Definition 1 and Definition 2, we have $D_{12}=D_{23}\asymp(M^{\star})^{-2(\beta-2\alpha)}>0$ and $E_{\Delta}=\\{(1,2),(2,3)\\}$. In practice, if $M^{\star}$ is very large and we do not have enough samples to accurately estimate $\Delta^{M}$ for a large $M$, then it is hopeless for us to estimate the differential graph correctly. Moreover, the situation is worse if $\beta>2\alpha$, since $D_{12}$ and $D_{23}$—the signal strength—vanish as $M^{\star}$ increases. Figure 2 shows how the signal strength (defined as $D_{12}$) changes as $M^{\star}$ increases for three cases: $\beta<2\alpha$, $\beta=2\alpha$, and $\beta>2\alpha$. This problem is intrinsically hard because the difference between two graphs only occurs between components with the smallest positive eigenvalue. To capture this difference, we have to use a large number of basis $M$ to approximate the functional data, which is statistically expensive. As we increase $M$, no useful information is captured until $M=M^{\star}$. Furthermore, if the difference between eigenvalues decreases fast relative to the decrease of eigenvalues, the signal strength will be very weak when the intrinsic dimension is large. This example shows that the estimation of functional differential graphical models is harder compared to the scalar case. Figure 2: Signal Strength $D_{12}\asymp(M^{\star})^{-2(\beta-2\alpha)}$ in Example 2. In Example 1, we characterized a pair of infinite-dimensional MGPs which are comparable, and in Example 2 we discussed a sequence of models which are all comparable, but increasingly difficult to recover. The following example demonstrates that there are infinite-dimensional MGPs that may share the same eigenspace, but are still not comparable. ###### Example 3 We construct two MGPs that are both infinite-dimensional and have the same eigenspace, but are incomparable. As with the previous two examples, let $V=\\{1,2,3\\}$. We assume that $X$ and $Y$ share a common set of eigenfunctions: $\\{\phi_{m}\\}_{m=1}^{\infty}$ for $j=1,2,3$. We construct the distribution of the scores of $X$ and $Y$ as follows. For for any $m\in\mathbb{N}^{+}$, let $a^{X}_{i\,\cdot\,m}$ denote the vector of scores $(a^{X}_{i1m},a^{X}_{i2m},a^{X}_{i3m})$ and define $a^{Y}_{i\,\cdot\,m}$ analogously. For any natural number $z$, we first assume that $a^{X}_{i\,\cdot\,(3z-2)},a^{X}_{i\,\cdot\,(3z-1)},a^{X}_{i\,\cdot\,(3z)}\perp\\!\\!\\!\perp\\{a^{X}_{i\,\cdot\,k}\\}_{k\neq 3z,3z-1,3z-2}.$ (18) Thus, the conditional independence graph for the individual scores is a set of disconnected subgraphs corresponding to $\\{a^{X}_{i\,\cdot\,(3z-2)},a^{X}_{i\,\cdot,(3z-1)},a^{X}_{i\,\cdot\,(3z)}\\}$ for $z\in\mathbb{N}^{+}$. We make the analogous assumption for the scores of $Y$. Within the sets $\\{a^{X}_{i\,\cdot\,(3z-2)},a^{X}_{i\,\cdot\,(3z-1)},a^{X}_{i\,\cdot\,(3z)}\\}$ and $\\{a^{Y}_{i\,\cdot\,(3z-2)},a^{Y}_{i\,\cdot\,(3z-1)},a^{Y}_{i\,\cdot\,(3z)}\\}$, we assume that the conditional independence graph has the structure shown in Figure 3. By construction, when projecting onto the span of the first $M$ functions, the edge set of individual functional graphical models for $X^{\pi}$ and $Y^{\pi}$ is not stable as $M\rightarrow\infty$. In particular, for both $X$ and $Y$, the edges $(1,2)$ and $(2,3)$ will persist; however, the edge $(1,3)$ will either appear or be absent depending on $M$. $a^{X}_{i1(3z-2)}$$a^{X}_{i2(3z-2)}$$a^{X}_{i3(3z-2)}$$a^{X}_{i1(3z-1)}$$a^{X}_{i2(3z-1)}$$a^{X}_{i3(3z-1)}$$a^{X}_{i1(3z)}$$a^{X}_{i2(3z)}$$a^{X}_{i3(3z)}$ (a) CI graph for $X$ scores $a^{Y}_{i1(3z-2)}$$a^{Y}_{i2(3z-2)}$$a^{Y}_{i3(3z-2)}$$a^{Y}_{i1(3z-1)}$$a^{Y}_{i2(3z-1)}$$a^{Y}_{i3(3z-1)}$$a^{Y}_{i1(3z)}$$a^{Y}_{i2(3z)}$$a^{Y}_{i3(3z)}$ (b) CI graph for $Y$ scores Figure 3: CI graph for the individual scores for two incomparable MGPs. If $M\mod 3=1$, which corresponds to the first row in Figure 3 where $M=3z-2$ for some $z\in\mathbb{N}^{+}$, then $\\{a^{X}_{i1k}\\}_{k<M}\perp\\!\\!\\!\perp\\{a^{X}_{i3k}\\}_{k<M}\mid\\{a^{X}_{i2k}\\}_{k\leq M}\quad\text{ and }\quad\\{a^{Y}_{i1k}\\}_{k<M}\perp\\!\\!\\!\perp\\{a^{Y}_{i3k}\\}_{k<M}\mid\\{a^{Y}_{i2k}\\}_{k\leq M}.$ (19) However, $a^{X}_{i1M}\not\perp\\!\\!\\!\perp a^{X}_{i3M}\mid\\{a^{X}_{i2k}\\}_{k\leq M}$ since we do not condition on $a^{X}_{i2(M+1)}$. Similarly, $a^{Y}_{i1M}\not\perp\\!\\!\\!\perp a^{Y}_{i3M}\mid\\{a^{Y}_{i2k}\\}_{k\leq M}$ since we do not condition on $a^{Y}_{i2(M+2)}$. Thus, the edge $(1,3)$ is in the functional graphical model for both $X^{\pi}$ and $Y^{\pi}$; however, the specific values of $\Theta^{X,M}$ and $\Theta^{Y,M}$ may differ. In contrast to the previous case, when $M\mod 3=2$, which corresponds to the second row in Figure 3 where $M=3z-1$ for some $z\in\mathbb{N}^{+}$, the functional graphical models for $X^{\pi}$ and $Y^{\pi}$ now differ. Note that, $\\{a^{X}_{i1k}\\}_{k\leq M}\perp\\!\\!\\!\perp\\{a^{X}_{i3k}\\}_{k\leq M}\mid\\{a^{X}_{i2k}\\}_{k\leq M}$. Thus, the edge $(1,3)$ is absent in the functional graphical model for $X^{\pi}$ and $\Theta^{X,M}_{1,3}=0$. Considering $Y^{\pi}$, we have that $\\{a^{Y}_{i1k}\\}_{k<M-1}\perp\\!\\!\\!\perp\\{a^{Y}_{i3k}\\}_{k<M-1}\mid\\{a^{Y}_{i2k}\\}_{k\leq M}$. However, because we do not condition on $a^{Y}_{i2(M+1)}$ (the node in the third row of Figure 3), the $(1,3)$ edge exists in the functional graphical model for $Y^{\pi}$ since $a^{Y}_{i1(M-1)}\not\perp\\!\\!\\!\perp a^{Y}_{i3(M-1)}\mid\\{a^{Y}_{i2k}\\}_{k\leq M}$. In this setting where $M\mod 3=2$, for all $z\in\mathbb{N}^{+}$, we set the covariance of the scores to be $z^{-\beta}\times\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern 2.5pt\kern-5.0pt\left[\kern 0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{ \halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&& \kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr 5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle a^{Y}_{i1(3z-2)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle a^{Y}_{i1(3z-1)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle a^{Y}_{i1(3z)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle a^{Y}_{i2(3z-2)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle a^{Y}_{i2(3z-1)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle a^{Y}_{i2(3z)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle a^{Y}_{i3(3z-2)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle a^{Y}_{i3(3z-1)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle a^{Y}_{i3(3z)}\\\a^{Y}_{i1(3z-2)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 3/2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1/2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i1(3z-1)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i1(3z)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i2(3z-2)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 8$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i2(3z-1)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 4$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i2(3z)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i3(3z-2)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1/2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle-1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 3/2$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i3(3z-1)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0\\\a^{Y}_{i3(3z)}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 1\\\$\hfil\kern 5.0pt\crcr}}}}\right]$}},$ (20) where $\beta>0$ is a parameter which determines the decay rate of the eigenvalues (see Assumption 3). We then set all other elements of the covariance to be $0$. The support of the inverse of this matrix corresponds to the edges of the graph in Figure 3. However, when we consider the marginal distribution of the first $M$ scores and invert the corresponding covariance, $\Theta^{Y,M}_{1,3}$ is $0$ everywhere except for the element corresponding to $a^{Y}_{i,1,M-1}$ and $a^{Y}_{i,3,M-1}$, that is, nodes in the top row of Figure 3, which is equal to $-1/4\times((M+1)/3)^{\beta}$. Thus, $\|\Delta^{M}_{1,3}\|_{F}=1/4\times((M+1)/3)^{\beta}$ and $\lim\sup_{M\rightarrow\infty}\|\Delta^{M}_{1,3}\|_{F}=\infty$. Finally, when $M\mod 3=0$, that is, $M=3z$ for some $z\in\mathbb{N}^{+}$, the $(1,3)$ edge is absent in both functional graphical models for $X^{\pi}$ and $Y^{\pi}$ because $\\{a^{X}_{i1k}\\}_{k\leq M}\perp\\!\\!\\!\perp\\{a^{X}_{i3k}\\}_{k\leq M}\mid\\{a^{X}_{i2k}\\}_{k\leq M}\ \text{ and }\ \\{a^{Y}_{i1k}\\}_{k\leq M}\perp\\!\\!\\!\perp\\{a^{Y}_{i3k}\\}_{k\leq M}\mid\\{a^{Y}_{i2k}\\}_{k\leq M}.$ Thus, $\Theta^{X,M}_{1,3}=\Theta^{Y,M}_{1,3}=\Delta^{M}_{1,3}=0$. This implies that $\lim\inf_{M\rightarrow\infty}\|\Delta^{M}_{1,3}\|_{F}=0$. Because $\lim\inf_{M\rightarrow\infty}\|\Delta^{M}_{1,3}\|_{F}=0$, but $\lim\sup_{M\rightarrow\infty}\|\Delta^{M}_{1,3}\|_{F}=\infty$, $X$ and $Y$ are incomparable. The notion of comparability illustrates the intrinsic difficulty of dealing with functional data. However, it also illustrates when we can still hope to estimate the differential network consistently. We have formally stated when two infinite-dimensional functional graphical models will be comparable and have given conditions and examples of comparability. Unfortunately, these conditions cannot be checked using observational data. For this reason, we mainly discuss the methodology and theoretical properties for estimation of $E^{\pi}_{\Delta}$. Prior knowledge about the problem at hand should be used to decide whether two infinite-dimensional functional graphs are comparable. This is similar to other assumptions common in the graphical modeling literature, such as “faithfulness” (Spirtes et al., 2000), that are critical to graph recovery, but can not be verified. ## 3 Functional Differential Graph Estimation: FuDGE In this section, we detail our methodology for estimating a functional differential graph. Unfortunately, in most situations, there may not be prior knowledge on which subspace to use to define the functional differential graph. In such situations, we suggest using the principle component scores of $K_{jj}(s,t)=K^{X}_{jj}(s,t)+K^{Y}_{jj}(s,t)$, $j\in V$ as a default choice. In addition, each observed function is only recorded (potentially with measurement error) at discrete time points. In Section 3.1 we consider this practical setting. Of course, if an appropriate basis for dimension reduction is known in advance or if the functions are fully observed at all time points, then the estimated objects can always be replaced with their known/observed counterparts. ### 3.1 Estimating the covariance of the scores For each $X_{ij}$, suppose we have measurements at time points $t_{ijk}$, $k=1,\ldots,T$,333For simplicity, we assume that all functions have the same number of observations, however, our method and theory can be trivially extended to allow a different number of observations for each function. and the recorded data, $h_{ijk}$, are the function values with random noise. That is, $h_{ijk}=g_{ij}(t_{ijk})+\epsilon_{ijk},$ (21) where $g_{ij}$ can denote either $X_{ij}$ or $Y_{ij}$ and the unobserved noise $\epsilon_{ijk}$ is i.i.d. Gaussian with mean $0$ and variance $\sigma^{2}_{0}$. Without loss of generality, we assume that $t_{ij1}<\ldots<t_{ijT}$ for any $1\leq i\leq n$ and $1\leq j\leq p$. We do not assume that $t_{ijk}=t_{i^{\prime}jk}$ for $i\neq i^{\prime}$, so that each observation may be observed on a different grid. We first use a basis expansion to estimate a least squares approximation of the whole curve $X_{ij}(t)$ (see Section 4.2 in Ramsay and Silverman (2005)). Specifically, given an initial basis function vector $b(t)=(b_{1}(t),\dots,b_{L}(t))^{\top}$—for example, the B-spline or Fourier basis—our estimated approximation for $X_{ij}(t)$ is given by: $\displaystyle\hat{X}_{ij}(t)$ $\displaystyle=\hat{\beta}_{ij}^{\top}b(t),$ (22) $\displaystyle\hat{\beta}_{ij}$ $\displaystyle=\left(B^{\top}_{ij}B_{ij}\right)^{-1}B^{\top}_{ij}h_{ij},$ where $h_{ij}=(h_{ij1},h_{ij2},\dots,h_{ijT})^{\top}$ and $B_{ij}$ is the design matrix for $g_{ij}$: $B_{ij}=\left[\begin{matrix}b_{1}(t_{ij1})&\cdots&b_{L}(t_{ij1})\\\ \vdots&\ddots&\vdots\\\ b_{1}(t_{ijT})&\cdots&b_{L}(t_{ijT})\end{matrix}\right]\in\mathbb{R}^{T\times L}.$ (23) The computational complexity of the basis expansion procedure is $O(npT^{3}L^{3})$, and in practice, there are many efficient package implementations of this step; for example, fda (Ramsay et al., 2020). We repeat this process for the observed $Y$ functions. After obtaining $\\{\hat{X}_{ij}(t)\\}_{j\in V,i=1,2,\dots,n_{X}}$ and $\\{\hat{Y}_{ij}(t)\\}_{j\in V,i=1,2,\dots,n_{Y}}$, we use them as inputs to the FPCA procedure. Specifically, we first estimate the sum of the covariance functions by $\hat{K}_{jj}(s,t)=\hat{K}^{X}_{jj}(s,t)+\hat{K}^{Y}_{jj}(s,t)=\frac{1}{n_{X}}\sum^{n_{X}}_{i=1}\hat{X}_{ij}(s)\hat{X}_{ij}(t)+\frac{1}{n_{Y}}\sum^{n_{Y}}_{i=1}\hat{Y}_{ij}(s)\hat{Y}_{ij}(t).$ (24) Using $\hat{K}_{jj}(s,t)$ as the input to FPCA, we can estimate the corresponding eigenfunctions $\hat{\phi}_{jk}(t)$, $k=1,\ldots,M$, $j=1,\ldots,p$. Given the estimated eigenfunctions, we compute the estimated projection scores $\displaystyle\hat{a}^{X}_{ijk}$ $\displaystyle=\int_{\mathcal{T}}\hat{X}_{ij}(t)\hat{\phi}_{jk}(t)dt\qquad\text{and}\qquad\hat{a}^{Y}_{ijk}=\int_{\mathcal{T}}Y_{ij}(t)\hat{\phi}_{jk}(t)dt,$ and collect them into vectors $\displaystyle a^{X,M}_{ij}$ $\displaystyle=(a^{X}_{ij1},a^{X}_{ij2},\dots,a^{X}_{ijM})^{\top}\in{\mathbb{R}^{M}}\qquad\text{and}\qquad a^{X,M}_{i}$ $\displaystyle=((a^{X,M}_{i1})^{\top},\ldots,(a^{X,M}_{ip})^{\top})^{\top}\in{\mathbb{R}^{pM}},$ $\displaystyle a^{Y,M}_{ij}$ $\displaystyle=(a^{Y}_{ij1},a^{Y}_{ij2},\dots,a^{Y}_{ijM})^{\top}\in{\mathbb{R}^{M}}\qquad\text{and}\qquad a^{Y,M}_{i}$ $\displaystyle=((a^{Y,M}_{i1})^{\top},\ldots,(a^{Y,M}_{ip})^{\top})^{\top}\in{\mathbb{R}^{pM}}.$ Finally, we estimate the covariance matrices of the score vectors, $\Sigma^{X,M}$ and $\Sigma^{Y,M}$, as $\displaystyle S^{X,M}=\frac{1}{n_{X}}\sum^{n_{X}}_{i=1}\hat{a}^{X,M}_{i}(\hat{a}^{X,M}_{i})^{\top}\qquad\text{and}\qquad S^{Y,M}=\frac{1}{n_{Y}}\sum^{n_{Y}}_{i=1}\hat{a}^{Y,M}_{i}(\hat{a}^{Y,M}_{i})^{\top}.$ ### 3.2 FuGDE: Functional Differential Graph Estimation We now describe the FuDGE algorithm for Functional Differential Graph Estimation. To estimate $\Delta^{M}$, we solve the following optimization program: $\hat{\Delta}^{M}\in\operatorname*{arg\,min}_{\Delta\in{\mathbb{R}^{pM\times pM}}}L(\Delta)+\lambda_{n}\sum_{\\{i,j\\}\in V^{2}}\|\Delta_{ij}\|_{F},$ (25) where $L(\Delta)=\mathrm{tr}\left[\frac{1}{2}S^{Y,M}\Delta^{\top}{S^{X,M}}\Delta-\Delta^{\top}\left(S^{Y,M}-S^{X,M}\right)\right]$ and $S^{X,M}$ and $S^{Y,M}$ are obtained as described in Section 3.1. We construct the loss function, $L(\Delta)$, so that the true parameter value, $\Delta^{M}=\left(\Sigma^{X,M}\right)^{-1}-\left(\Sigma^{Y,M}\right)^{-1}$, minimizes the population loss $\mathbb{E}\left[L(\Delta)\right]$, which for a differentiable and convex loss function, is equivalent to selecting $L$ such that $\mathbb{E}\left[\nabla L(\Delta^{M})\right]=0$. Since $\Delta^{M}$ satisfies $\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M}-(\Sigma^{Y,M}-\Sigma^{X,M})=0,$ a choice for $\nabla L(\Delta)$ is $\nabla{L(\Delta^{M})}=S^{X,M}\Delta^{M}{S^{Y,M}}-\left(S^{Y,M}-S^{X,M}\right)$ (26) so that $\mathbb{E}\left[\nabla L(\Delta^{M})\right]=\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M}-(\Sigma^{Y,M}-\Sigma^{X,M})=0.$ Given this choice of $\nabla L(\Delta)$, $L(\Delta)$ in (25) directly follows from properties of the differential of the trace function. The chosen loss is quadratic (see (B.9) in appendix) and leads to an efficient algorithm. Similar loss functions are used in Xu and Gu (2016), Yuan et al. (2017), Na et al. (2019), and Zhao et al. (2014). We also include the additional group lasso penalty (Yuan and Lin, 2006) to promote blockwise sparsity in $\hat{\Delta}^{M}$. The objective in (25) can be solved by a proximal gradient method detailed in Algorithm 1. Finally, we form $\hat{E}_{\Delta}$ by thresholding $\hat{\Delta}^{M}$ so that: ${}\hat{E}_{\Delta}=\left\\{\\{j,l\\}\,:\,\|\hat{\Delta}^{M}_{jl}\|_{F}>\epsilon_{n}\;\;\text{or}\;\;\|\hat{\Delta}^{M}_{lj}\|_{F}>\epsilon_{n}\right\\}.$ (27) The thresholding step in (27) is used for theoretical purposes. Specifically, it helps correct for the bias induced by the finite-dimensional truncation and relaxes commonly used assumptions for the graph structure recovery, such as the irrepresentability or incoherence condition (van de Geer and Bühlmann, 2009). In practice, one can simply set $\epsilon_{n}=0$, as we do in the simulations. ### 3.3 Optimization Algorithm for FuDGE Algorithm 1 Functional differential graph estimation 0: $S^{X,M},S^{Y,M},\lambda_{n},\eta$. 0: $\hat{\Delta}^{M}$. Initialize $\Delta^{(0)}=0_{pM}$. repeat $A=\Delta-\eta\nabla L(\Delta)=\Delta-\eta\left(S^{X,M}\Delta S^{Y,M}-\left(S^{Y,M}-S^{X,M}\right)\right)$ for $1\leq{i,j}\leq{p}$ do $\Delta_{jl}\leftarrow\left(\frac{\|A_{jl}\|_{F}-\lambda_{n}\eta}{\|A_{jl}\|_{F}}\right)_{+}\cdot A_{jl}$ end for until Converge The optimization program (25) can be solved by a proximal gradient method (Parikh and Boyd, 2014) summarized in Algorithm 1. Specifically, at each iteration, we update the current value of $\Delta$, denoted as $\Delta^{\text{old}}$, by solving the following problem: ${}\Delta^{\text{new}}=\operatorname*{arg\,min}_{\Delta}\left(\frac{1}{2}\left\|\Delta-\left(\Delta^{\text{old}}-\eta\nabla L\left(\Delta^{\text{old}}\right)\right)\right\|_{F}^{2}+\eta\cdot\lambda_{n}\sum^{p}_{j,l=1}\|\Delta_{jl}\|_{F}\right),$ (28) where $\nabla L(\Delta)$ is defined in (26) and $\eta$ is a user specified step size. Note that $\nabla L(\Delta)$ is Lipschitz continuous with Lipschitz constant $\lambda^{S}_{\max}=\|S^{Y,M}\otimes S^{X,M}\|_{2}=\lambda_{\max}(S^{Y,M})\lambda_{\max}(S^{X,M})$. Thus, for any step size $\eta$ such that $0<\eta\leq 1/\lambda^{S}_{\max}$, the proximal gradient method is guaranteed to converge (Beck and Teboulle, 2009). The update in (28) has a closed-form solution: ${}\Delta^{\text{new}}_{jl}=\left[\left(\|A^{\text{old}}_{jl}\|_{F}-\lambda_{n}\eta\right)/\|A^{\text{old}}_{jl}\|_{F}\right]_{+}\cdot A^{\text{old}}_{jl},\qquad 1\leq{j,l}\leq{p},$ (29) where $A^{\text{old}}=\Delta^{\text{old}}-\eta\nabla L(\Delta^{\text{old}})$ and $x_{+}=\max\\{0,x\\},x\in{\mathbb{R}}$, represents the positive part of $x$. Detailed derivations are given in the appendix. Note that although the true $\Delta^{M}$ is symmetric, we do not explicitly enforce symmetry in $\hat{\Delta}^{M}$ in Algorithm 1. After performing FPCA, the proximal gradient descent method converges in $O\left(\lambda^{S}_{\max}/\text{tol}\right)$ iterations, where tol is a user specified optimization error tolerance, and each iteration takes $O((pM)^{3})$ operations; see Tibshirani (2010) for a convergence analysis of the general proximal gradient descent algorithm. ### 3.4 Selection of Tuning Parameters There are four tuning parameters that need to be chosen for implementing FuDGE: $L$ (dimension of the basis used to estimate the curves from the discretely observed data), $M$ (dimension of subspace to estimate the projection scores), $\lambda_{n}$ (regularization parameter to tune the block sparsity of $\Delta^{M}$), and $\epsilon_{n}$ (thresholding parameter for $\hat{E}_{\Delta}$). While we need the thresholding parameter $\epsilon_{n}$ in (27) to establish theoretical results, in practice, we simply set $\epsilon_{n}=0$. To select $M$, we follow the procedure in Qiao et al. (2019). More specifically, for each discretely-observed curve, we first estimate the underlying functions by fitting an $L$-dimensional B-spline basis. Both $M$ and $L$ are then chosen by 5-fold cross-validation as discussed in Qiao et al. (2019). Finally, to choose $\lambda_{n}$, we recommend using selective cross- validation (SCV) (She, 2012). Given a value of $\lambda_{n}$, we use the entire data set to estimate a sparsity pattern. Then, fixing the sparsity pattern, we use a typical cross-validation procedure to calculate the CV error. Ultimately, we choose the value of $\lambda_{n}$ that results in the sparsity pattern that minimizes the CV error. In addition to SCV, if we have some prior knowledge about the number of edges in the differential graph, we can also choose $\lambda_{n}$ that results in a desired level of sparsity of the differential graph. ## 4 Theoretical Properties In this section, we provide theoretical guarantees for FuDGE. We first give a deterministic result for $\hat{E}_{\Delta}$ defined in (27) when the max-norm of the difference between the estimates $S^{X,M},S^{Y,M}$ and their corresponding parameters, $\Sigma^{X,M},\Sigma^{Y,M}$, is bounded by $\delta_{n}$. We then show that when projecting the data onto either a fixed basis or an estimated basis—under some mild conditions—$\delta_{n}$ can be controlled and the bias of the finite-dimensional projection decreases fast enough that $E_{\Delta}$ can be consistently recovered. ### 4.1 Deterministic Guarantees for $\hat{E}_{\Delta}$ In this section, we assume that $S^{X,M},S^{Y,M}$ are good estimates of $\Sigma^{X,M},\Sigma^{Y,M}$ and give a deterministic result in Theorem 1. Let $n=\min\\{n_{X},n_{Y}\\}$. We assume that the following holds. ###### Assumption 1 The matrices $S^{X,M},S^{Y,M}$ are estimates of $\Sigma^{X,M},\Sigma^{Y,M}$ that satisfy $\max\left\\{|S^{X,M}-\Sigma^{X,M}|_{\infty},|S^{Y,M}-\Sigma^{Y,M}|_{\infty}\right\\}\leq\delta_{n}.$ (30) We also require that $E_{\Delta}$ is sparse. This does not preclude the case where $E_{X}$ and $E_{Y}$ are dense, as long as there are not too many differences in the precision matrices. This assumption is also required when estimating a differential graph from vector-valued data; for example, see Condition 1 in Zhao et al. (2014). ###### Assumption 2 There are $s$ edges in the differential graph; that is, $|E_{\Delta}|=s$ and $s\ll p$. We introduce the following three quantities that characterize the problem instance and will be used in Theorem 1 below: $\nu_{1}=\nu_{1}(M)=\min_{(j,l)\in E_{\Delta}}\|\Delta^{M}_{jl}\|_{F},\quad\nu_{2}=\nu_{2}(M)=\max_{(j,l)\in E^{C}_{\Delta}}\|\Delta^{M}_{jl}\|_{F},$ and $\tau=\tau(M)=\nu_{1}(M)-\nu_{2}(M).$ (31) Roughly speaking, $\nu_{1}(M)$ indicates the “signal strength” present when using the $M$-dimensional representation and $\nu_{2}(M)$ measures the bias. By Definition 1, when $X$ and $Y$ are comparable, we have $\lim\inf_{M\to M^{\star}}\nu_{1}(M)>0$ and $\lim_{M\to M^{\star}}\nu_{2}(M)=0$. Therefore, for a large enough $M$, we have $\tau>0$. However, a smaller $\tau$ implies that the differential graph is harder to recover. Before we give the deterministic result in Theorem 1, we first define additional quantities that will be used in subsequent results. Let $\displaystyle\sigma_{\max}$ $\displaystyle=\max\\{|\Sigma^{X,M}|_{\infty},|\Sigma^{Y,M}|_{\infty}\\},$ (32) $\displaystyle\lambda^{*}_{\min}$ $\displaystyle=\lambda_{\min}\left(\Sigma^{X,M}\right)\times\lambda_{\min}\left(\Sigma^{Y,M}\right),\text{ and }$ $\displaystyle\Gamma^{2}_{n}$ $\displaystyle=\frac{9\lambda^{2}_{n}s}{\kappa^{2}_{\mathcal{L}}}+\frac{2\lambda_{n}}{\kappa_{\mathcal{L}}}(\omega^{2}_{\mathcal{L}}+2p^{2}\nu_{2}),$ where $\displaystyle\lambda_{n}$ $\displaystyle=\;2M\left[\left(\delta_{n}^{2}+2\delta_{n}\sigma_{\max}\right)\left|\Delta^{M}\right|_{1}+2\delta_{n}\right],$ (33) $\displaystyle\kappa_{\mathcal{L}}$ $\displaystyle=\;(1/2)\lambda^{*}_{\min}-8M^{2}s\left(\delta_{n}^{2}+2\delta_{n}\sigma_{\max}\right),$ $\displaystyle\omega_{\mathcal{L}}$ $\displaystyle=\;4Mp^{2}\nu_{2}\sqrt{\delta_{n}^{2}+2\delta_{n}\sigma_{\max}},$ and $\delta_{n}$ is defined in Assumption 1. Note that $\Gamma_{n}$—which measures the estimation error of $\|\hat{\Delta}^{M}-\Delta^{M}\|_{\text{F}}$—implicitly depends on $\delta_{n}$ through $\lambda_{n}$, $\kappa_{\mathcal{L}}$, and $\omega_{\mathcal{L}}$. We observe that $\Gamma_{n}$ decreases to zero as $\delta_{n}$ goes to zero. The quantity $\kappa_{\mathcal{L}}$ is the maximum restricted eigenvalue from the analysis framework of Negahban et al. (2012). Finally, $\omega_{\mathcal{L}}$ is the tolerance parameter that comes from the fact that $\nu_{2}$ might be larger than zero, and it will decrease to zero as $\nu_{2}$ goes to zero. ###### Theorem 1 Given Assumptions 1 and 2, when $\nu_{1}(M),\nu_{2}(M),\delta_{n},\lambda_{n},\sigma_{\max},M$ and $s$ satisfy $\displaystyle 0<\Gamma_{n}<\tau/2\qquad\text{and}\qquad\delta_{n}<(1/4)\sqrt{\left(\lambda^{*}_{\min}+16M^{2}s(\sigma_{\max})^{2}\right)/\left(M^{2}s\right)}-\sigma_{\max},$ (34) then setting $\epsilon_{n}\in\left[\nu_{2}+\Gamma_{n},\nu_{1}-\Gamma_{n}\right)$ ensures that $\hat{E}_{\Delta}=E_{\Delta}$. As shown in Section 4.2, under a few additional conditions, Assumption 1 holds for a sequence of $\delta_{n}$ that decreases to $0$ as $n$ goes to infinity. Thus, as $M$ and $n$ both increase to infinity, we have $\nu_{2}+\Gamma_{n}\approx 0$ and $\nu_{1}-\Gamma_{n}\approx\min_{(j,l)\in E_{\Delta}}D_{jl}$, and we only require $0\leq\epsilon_{n}<\min_{(j,l)\in E_{\Delta}}D_{jl}$. ### 4.2 Theoretical Guarantees for $S^{X,M}$ and $S^{Y,M}$ In this section, we prove that under some mild conditions, (30) will hold with high probability for specific values of $\delta_{n}$. We discuss the results in two cases: the case where the curves are fully observed and the case where the curves are only observed at discrete time points. #### 4.2.1 Fully Observed Curves In this section, we discuss the case where each curve is fully observed. We first consider the case where the basis defining the differential graph are known in advance; that is, the exact form of $\\{e_{jk}\\}_{k\geq 1}$ for all $j\in V$ is known. In this case, the projection score vectors $a^{X,M}_{i}$ and $a^{Y,M}_{i}$ can be exactly recovered for all $i=1,2,\dots,n$. By the assumption that $X_{i}(t)$ and $Y_{i}(t)$ are $p$-dimensional multivariate Gaussian processes with mean zero, we then have $a^{X,M}_{i}\sim N(0,\Sigma^{X,M})$ and $a^{Y,M}_{i}\sim N(0,\Sigma^{Y,M})$. The following result follows directly from standard results on the sample covariance of multivariate Gaussian variables. ###### Theorem 2 Assume that $S^{X,M}$ and $S^{Y,M}$ are computed as in Section 3.1, except the basis functions $\\{e_{jk}\\}_{k\geq 1}$, $j\in V$, are fixed and known in advance. Recall that $n=\min\\{n_{X},n_{Y}\\}\quad\text{and}\quad\sigma_{\max}=\max\\{|\Sigma^{X,M}|_{\infty},|\Sigma^{Y,M}|_{\infty}\\}.$ Fix $\iota\in(0,1]$. Suppose that $n$ is large enough so that $\delta_{n}=\sigma_{\max}\sqrt{\frac{C_{1}}{n}\log\left(\frac{8p^{2}M^{2}}{\iota}\right)}\leq C_{2},$ for some universal constants $C_{1},C_{2}>0$. Then (30) holds with probability at least $1-\iota$. Proof The proof follows directly from Lemma 1 of Ravikumar et al. (2011) and a union bound. [2mm] With fully observed curves and known basis functions, it follows from Theorem 2 that $\delta_{n}\asymp\sqrt{\log(p^{2}M^{2})/n}$ with high probability. As assumed in Section 2.2 (and also in Qiao et al. (2019)), when $\lambda^{X}_{jm^{\prime}}=\lambda^{Y}_{jm^{\prime}}=0$ for all $j$ and $m^{\prime}>M$ (where $M$ is allowed to grow with $n$), then $\nu_{2}(M)=0$, $\tau(M)=\nu_{1}(M)=\min_{(j,l)\in E_{\Delta}}D_{jl}>0$, and $E_{\Delta}=E^{\pi}_{\Delta}$. We can recover $E_{\Delta}$ with high probability even in the high-dimensional setting, as long as $\max\left\\{\frac{sM^{2}\log(p^{2}M^{2})|\Delta^{M}|_{1}^{2}/((\lambda^{\star}_{\min})^{2}\tau^{2})}{n},\frac{sM^{2}\log(p^{2}M^{2})/\lambda^{\star}_{\min}}{n}\right\\}\rightarrow 0.$ Even with an infinite number of positive eigenvalues, high-dimensional consistency is still possible for quickly increasing $\nu_{1}$ and quickly decaying $\nu_{2}$. We then consider the case where the curves are fully observed, but we do not have any prior knowledge on which orthonormal function basis should be used. In this case, as discussed in Section 2.3, we recommend using the eigenfunctions of $K_{jj}(\cdot,*)=K^{X}_{jj}(\cdot,*)+K^{Y}_{jj}(\cdot,*)$ as basis functions. We use FPCA to estimate the eigenfuctions of $K_{jj}(\cdot,*)$ and make the following assumption. ###### Assumption 3 Let $\\{\lambda_{jk},\phi_{jk}(\cdot)\\}$ be the eigenpairs of $K_{jj}(\cdot,*)=K^{X}_{jj}(\cdot,*)+K^{Y}_{jj}(\cdot,*)$, $j\in V$, and suppose that $\lambda_{jk}$ are non-increasing in $k$. 1. (i) Suppose $\max_{j\in{V}}\sum_{k=1}^{\infty}\lambda_{jk}<\infty$ and assume that there exists a constant $\beta>1$ such that, for each $k\in\mathbb{N}$, $\lambda_{jk}\asymp{k^{-\beta}}$ and $d_{jk}\lambda_{jk}=O(k)$ uniformly in $j\in{V}$, where $d_{jk}=2\sqrt{2}\max\\{(\lambda_{j(k-1)}-\lambda_{jk})^{-1},(\lambda_{jk}-\lambda_{j(k+1)})^{-1}\\}$, $k\geq 2$, and $d_{j1}=2\sqrt{2}(\lambda_{j1}-\lambda_{j2})^{-1}$. 2. (ii) For all $k$, $\phi_{jk}(\cdot)$’s are continuous on the compact set $\mathcal{T}$ and satisfy $\max_{j\in{V}}\sup_{s\in{\mathcal{T}}}\sup_{k\geq{1}}|\phi_{jk}(s)|_{\infty}=O(1).$ This assumption was used in Qiao et al. (2019, Condition 1). We have the following result. ###### Theorem 3 Suppose Assumption 3 holds and the basis functions are estimated using FPCA of $K_{jj}(\cdot,*)$ with fully observed curves. Fix $\iota\in(0,1]$. Suppose $n$ is large enough so that $\delta_{n}=M^{1+\beta}\sqrt{\frac{\log\left(2C_{2}p^{2}M^{2}/\iota\right)}{n}}\leq C_{1},$ for some universal constants $C_{1},C_{2}>0$. Then (30) holds with probability at least $1-\iota$. Proof The proof follows directly from Theorem 1 of Qiao et al. (2019) and the fact that $\|\hat{K}_{jj}(\cdot,*)-K_{jj}(\cdot,*)\|_{\text{HS}}\leq\|\hat{K}^{X}_{jj}(\cdot,*)-K^{X}_{jj}(\cdot,*)\|_{\text{HS}}+\|\hat{K}^{Y}_{jj}(\cdot,*)-K^{Y}_{jj}(\cdot,*)\|_{\text{HS}}$. [2mm] It follows from Theorem 3 that $\delta_{n}\asymp M^{1+\beta}\sqrt{\log(p^{2}M^{2}/)/n}$ with high probability. Compared with Theorem 2, there is an additional $M^{1+\beta}$ term that arises from FPCA estimation error. Similarly, when $\lambda^{X}_{jm^{\prime}}=\lambda^{Y}_{jm^{\prime}}=0$ for all $j$ and $m^{\prime}>M$, we can recover $E_{\Delta}$ with high probability as long as $\max\left\\{\frac{sM^{(4+2\beta)}\log(p^{2}M^{2})|\Delta^{M}|_{1}^{2}/((\lambda^{\star}_{\min})^{2}\tau^{2})}{n},\frac{sM^{(4+2\beta)}\log(p^{2}M^{2})/\lambda^{\star}_{\min}}{n}\right\\}\rightarrow 0.$ #### 4.2.2 Discretely-Observed Curves Finally, we discuss the case when the curves are only observed at discrete time points—possibly with measurement error. Following Chapter 1 of Kokoszka and Reimherr (2017), we first estimate each curve from the available observations by basis expansion; then we use the estimated curves to form empirical covariance functions from which we estimate the eigenfunctions using FPCA. The estimated eigenfunctions are then used to calculate the scores. Recall the model for discretely observed functions given in (21): $h_{ijk}=g_{ij}(t_{ijk})+\epsilon_{ijk},$ where $g_{ij}$ denotes either $X_{ij}$ or $Y_{ij}$, $\epsilon_{ijk}$ are i.i.d. Gaussian noise with mean $0$ and variance $\sigma^{2}_{0}$. Assume that $t_{ij1}<\dots<t_{ijT}$ for any $1\leq i\leq n$ and $1\leq j\leq p$. Note that we do not need $X$ and $Y$ to be observed at the same time points and we use $t_{ijk}$ to represent either $t^{X}_{ijk}$ or $t^{Y}_{ijk}$. Furthermore, recall that we first compute a least squares estimator of $X_{ij}(\cdot)$ and $Y_{ij}(\cdot)$ by projecting it onto the basis $b(\cdot)=\left(b_{1}(\cdot),\ldots,b_{L}(\cdot)\right)$. First, we assume that as we increase the number of basis functions, we can approximate any function in $\mathbb{H}$ arbitrarily well. ###### Assumption 4 We assume that $\\{b_{l}\\}^{\infty}_{l=1}$ is a complete orthonormal system (CONS) (See Definition 2.4.11 of Hsing and Eubank, 2015) of $\mathbb{H}$, that is, $\overline{{\rm Span}\left(\\{b_{l}\\}^{\infty}_{l=1}\right)}=\mathbb{H}$. Assumption 4 requires that the basis functions are orthonormal. When this assumption is violated—for example, when using the B-splines basis—we can always first use an orthonormalization process, such as Gram-Schmidt, to convert the basis to an orthonormal one. For B-splines, there are many algorithms that can efficiently provide orthonormalization (Liu et al., 2019). To establish theoretical guarantees for the least squares estimator, we require smoothness in both the curves we are trying to estimate as well as the basis functions we use. ###### Assumption 5 We assume that the basis functions $\\{b_{l}(\cdot)\\}^{\infty}_{l=1}$ satisfy the following conditions. $D_{0,b}\coloneqq\sup_{l\geq 1}\sup_{t\in\mathcal{T}}\lvert b_{l}(t)\rvert<\infty,\qquad D_{1,b}(l)\coloneqq\sup_{t\in\mathcal{T}}\lvert b^{\prime}_{l}(t)\rvert<\infty,\qquad D_{1,b,L}\coloneqq\max_{1\leq l\leq L}D_{1,b}(l).$ (35) We also require that the curves $g_{ij}$ satisfy the following smoothness condition: $\max_{1\leq j\leq p}\sum^{\infty}_{m=1}\mathbb{E}\left[\left(\langle g_{ij},b_{m}\rangle\right)^{2}\right]D^{2}_{1,b}(m)<\infty.$ (36) To better understand Assumption 5, we use the Fourier basis as an example. Let $\mathcal{T}=[0,1]$ and $b_{m}(t)=\sqrt{2}\cos(2\pi mt)$, $0\leq t\leq 1$ and $m\in\mathbb{N}$. Thus, $\\{b_{m}(t)\\}^{\infty}_{m=0}$ then constitutes an orthonormal basis of $\mathbb{H}=\mathcal{L}^{2}[0,1]$. We then have $b^{\prime}(t)=-2\sqrt{2}\pi m\sin(2\pi mt)$, $D_{0,b}=\sqrt{2}$, $D_{1,b}(m)=2\sqrt{2}\pi m$ and $D_{1,b,L}=2\sqrt{2}\pi L$. In this case, (36) is equivalent to $\max_{1\leq j\leq p}\sum^{\infty}_{m=1}\mathbb{E}\left[\left(\langle g_{ij},b_{m}\rangle\right)^{2}\right]m^{2}<\infty.$ On the other hand, $g_{ij}(t)=\sum^{\infty}_{m=1}\langle g_{ij},b_{m}\rangle b_{m}(t)$ and $g^{\prime}_{ij}(t)=\sum^{\infty}_{m=1}\langle g_{ij},b_{m}\rangle b^{\prime}_{m}(t)$. Suppose that, $\mathbb{E}\left[\|g^{\prime}_{ij}\|^{2}\right]<\infty$. Then $\mathbb{E}\left[\|g^{\prime}_{ij}\|^{2}\right]=\sum^{\infty}_{m=1}\mathbb{E}\left[\left(\langle g_{ij},b_{m}\rangle\right)^{2}\right]\|b^{\prime}_{m}\|^{2}\asymp\sum^{\infty}_{m=1}\mathbb{E}\left[\left(\langle g_{ij},b_{m}\rangle\right)^{2}\right]m^{2}.$ (37) Therefore, $\max_{1\leq j\leq p}\mathbb{E}\left[\|g^{\prime}_{ij}\|^{2}\right]<\infty$, which is a commonly used assumption in nonparameteric statistics (e.g., Section 7.2 of Wasserman (2006)), implies (36). Finally, we require each function to be observed at time points that are “evenly spaced.” Formally, we require the following assumption. ###### Assumption 6 The observation time points $\\{t_{ijk}:1\leq i\leq n,1\leq j\leq p,1\leq k\leq T\\}$ satisfy $\max_{1\leq i\leq n}\max_{1\leq j\leq p}\max_{1\leq k\leq T+1}\left|\frac{t_{ijk}-t_{ij(k-1)}}{\lvert\mathcal{T}\rvert}-\frac{1}{T}\right|\leq\frac{\zeta_{0}}{T^{2}},$ (38) where $t_{ij0}$ and $t_{ij(T+1)}$ are endpoints of $\mathcal{T}$ for any $1\leq i\leq n$, $1\leq j\leq p$, and $\zeta_{0}$ is a positive constant that does not depend on $i$ or $j$. Any $g_{ij}$ can be decomposed into $g_{ij}=g^{\shortparallel}_{ij}+g^{\bot}_{ij}$, where $g_{ij}^{\shortparallel}\in{\rm Span}(b)$ and $g_{ij}^{\bot}\in{\rm Span}(b)^{\bot}$. We denote the eigenvalues of the covariance operator of $g_{ij}$ as $\\{\lambda_{jk}\\}_{k\geq 1}$ and $\lambda_{j0}=\sum^{\infty}_{k=1}\lambda_{jk}$; and denote the eigenvalues of the covariance operator of $g^{\bot}_{ij}$ as $\\{\lambda^{\bot}_{jk}\\}_{k\geq 1}$ and $\lambda^{\bot}_{j0}=\sum^{\infty}_{k=1}\lambda^{\bot}_{jk}$. Note that under Assumption 3, we have $\max_{1\leq j\leq p}\lambda_{j0}<\infty$. Let $1<\lambda_{0,\max}<\infty$ be a constant such that $\max_{1\leq j\leq p}\lambda_{j0}\leq\lambda_{0,\max}$. Let $B_{ij}$ be the design matrix of $g_{ij}$ as defined in (23) and let $\lambda^{B}_{\min}=\min_{1\leq i\leq n,1\leq j\leq p}\left\\{\lambda_{\min}(B^{\top}_{ij}B_{ij})\right\\}$. We define $\displaystyle\tilde{\psi}_{1}(T,L)=\frac{\sigma_{0}L}{\sqrt{\lambda^{B}_{\min}}},\quad\tilde{\psi}_{2}(T,L)=\frac{L^{2}}{(\lambda^{B}_{\min})^{2}}\left(\lambda_{0}\left(\tilde{c}_{1}D^{2}_{1,b,L}+\tilde{c_{2}}\right)\tilde{\psi}_{3}(L)+\tilde{c}_{1}\tilde{\psi}_{4}(L)\right),$ (39) $\displaystyle\tilde{\psi}_{3}(L)\;=\;\max_{1\leq j\leq p}\left(\lambda^{\bot}_{j0}/\lambda_{j0}\right),\quad\tilde{\psi}_{4}(L)=\max_{1\leq j\leq p}\sum_{m>L}\mathbb{E}\left[\left(\langle g_{ij},b_{m}\rangle\right)^{2}\right]D^{2}_{1,b}(m),$ (40) $\displaystyle\Phi(T,L)=\min\left\\{1/\tilde{\psi}_{1}(T,L),1/\sqrt{\tilde{\psi}_{3}(L)}\right\\},$ (41) where $\tilde{c}_{1}=18D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}$ and $\tilde{c_{2}}=36D^{4}_{0,b}(2\zeta_{0}+1)^{2}$. We now use superscripts or subscripts to indicate the specific quantities for $X$ and $Y$. In this way, we define $L_{X}$, $L_{Y}$, $T_{X}$, $T_{Y}$, $\tilde{\psi}^{X}_{1}$-$\tilde{\psi}^{X}_{4}$, $\tilde{\psi}^{Y}_{1}$-$\tilde{\psi}^{Y}_{4}$, and $\Phi^{X},\Phi^{Y}$. In addition, let $T=\min\\{T_{X},T_{Y}\\}$, $L=\min\\{L_{X},L_{Y}\\}$, $\bar{\psi}_{k}=\max\\{\tilde{\psi}^{X}_{k},\tilde{\psi}^{Y}_{k}\\}$, $k=1,\cdots,4$, $\bar{\Phi}=\min\\{\Phi^{X},\Phi^{Y}\\}$, and let $n$, $\beta$ be defined as in Section 4.1. ###### Theorem 4 Assume the observation model given in (21). Suppose Assumption 3 holds, and Assumption 4-6 hold for both $X$ and $Y$. Suppose $T$ and $L$ are large enough so that $\displaystyle\bar{\psi}_{1}(T,L)\leq\gamma_{1}\frac{\delta_{n}}{M^{1+\beta}},\quad\bar{\psi}_{3}(L)\leq\gamma_{3}\frac{\delta_{n}^{2}}{M^{2+2\beta}}$ (42) where $\delta_{n}=\max\left\\{\frac{M^{1+\beta}\log\left(4\bar{C}_{1}np/\iota\right)}{\bar{C}_{2}\bar{\Phi}(T,L)},M^{1+\beta}\sqrt{\frac{1}{C_{6}}\bar{\psi}_{2}(T,L)\log\left(\frac{C_{5}npL}{\iota}\right)},\right.\\\ \left.M^{1+\beta}\sqrt{\frac{\log\left(4\bar{C}_{3}p^{2}M^{2}/\iota\right)}{\bar{C}_{4}n}}\right\\},$ (43) $\bar{C}_{1}=\max\\{C^{X}_{1},C^{Y}_{1}\\}$, $\bar{C}_{2}=\min\\{C^{X}_{2},C^{Y}_{2}\\}$, $\bar{C}_{3}=\max\\{C^{X}_{3},C^{Y}_{3}\\}$, $\bar{C}_{4}=\min\\{C^{X}_{4},C^{Y}_{4}\\}$, $\bar{C}_{5}=\max\\{C^{X}_{5},C^{Y}_{6}\\}$, $\bar{C}_{6}=\min\\{C^{X}_{6},C^{Y}_{6}\\}$. $\gamma^{X}_{k}$, $\gamma^{Y}_{k}$, $k=1,2,3$, and $C^{X}_{k}$, $C^{Y}_{k}$, $k=1,\cdots,6$ are constants that do not depend on $n$, $p$, and $M$. Then $\max\left\\{|S^{X,M}-\Sigma^{X,M}|_{\infty},|S^{Y,M}-\Sigma^{Y,M}|_{\infty}\right\\}\leq\delta_{n}$ (44) holds with probability at least $1-\iota$. Proof See Appendix B.5. [2mm] The rate $\delta_{n}$ in Theorem 4 is comprised of three terms. The first two terms correspond to the error incurred by measuring the curves at discrete locations and are approximation errors. The third term, which also appears in Theorem 3, is the sampling error. We provide some intuition on how $\tilde{\psi}_{1}$, $\tilde{\psi}_{2}$, $\tilde{\psi}_{3}$, and $\tilde{\psi}_{4}$ depend on $T$ and $L$. Note that we choose an orthonormal basis. Then as $T\to\infty$, we have $\displaystyle\frac{1}{T}B^{\top}_{ij}B_{ij}$ $\displaystyle=\frac{1}{T}\sum^{T}_{k=1}\left[\begin{matrix}b^{2}_{1}(t_{ijk})&b_{1}(t_{ijk})b_{2}(t_{ijk})&\cdots&b_{1}(t_{ijk})b_{L}(t_{ijk})\\\ \vdots&\vdots&\ddots&\vdots\\\ b_{L}(t_{ijk})b_{1}(t_{ijk})&b_{L}(t_{ijk})b_{2}(t_{ijk})&\cdots&b^{2}_{L}(t_{ijk})\end{matrix}\right]$ $\displaystyle\approx\left[\begin{matrix}\|b_{1}\|^{2}&\langle b_{1},b_{2}\rangle&\cdots&\langle b_{1},b_{L}\rangle\\\ \vdots&\vdots&&\vdots\\\ \langle b_{L},b_{1}\rangle&\langle b_{L},b_{2}\rangle&\cdots&\|b_{L}\|^{2}\end{matrix}\right]$ $\displaystyle=\left[\begin{matrix}1&0&\cdots&0\\\ \vdots&\vdots&&\vdots\\\ 0&0&\cdots&1\end{matrix}\right].$ Thus, as $T$ grows, we would expect $\lambda_{\min}(B^{\top}_{ij}B_{ij})\approx T$ for any $1\leq j\leq p$ and $1\leq i\leq n$. This implies that $\tilde{\psi}_{1}(T,L)\approx L/\sqrt{T}$ and $\tilde{\psi}_{2}(T,L)\approx\left(D^{2}_{1,b,L}\tilde{\psi}_{3}(L)+\tilde{\psi}_{4}(L)\right)L^{2}/T^{2}$. Furthermore, $D^{2}_{1,b,L}\asymp L^{2}$ when we use Fourier basis. To understand $\tilde{\psi}_{3}(L)$ and $\tilde{\psi}_{4}(L)$, note that $\lambda^{\bot}_{j0}=\mathbb{E}[\|g^{\bot}_{ij}\|^{2}]=\mathbb{E}_{g_{ij}}[\mathbb{E}_{\epsilon}[\|g^{\bot}_{ij}\|^{2}\mid g_{ij}]]$. Under Assumption 4, $\lambda^{\bot}_{j0}\to 0$ as $L\to\infty$; however, the speed at which $\lambda^{\bot}_{j0}$ goes to zero will depend on $\mathbb{H}$ and the choice of the basis functions. For example, for a fixed $g_{ij}$, by well known approximation results (see, for example, Barron and Sheu (1991)), if $g_{ij}$ has $r$-th continuous and square integrable derivatives, $\|g^{\bot}_{ij}\|^{2}\approx 1/L^{r}$ for frequently used bases such as the Legendre polynomials, B-splines, and Fourier basis. Thus, roughly speaking, we should have $\tilde{\psi}_{3}(L)\approx 1/L^{r}$ when $\mathbb{H}$ is a Sobolev space of order $r$. When $g_{ij}$ is an infinitely differentiable function and all derivatives can be uniformly bounded, then $\|g^{\bot}_{ij}\|^{2}\approx\exp(-L)$ and thus $\tilde{\psi}_{3}(L)\approx\exp(-L)$. Similarly, we have $\tilde{\psi}_{4}(L)\approx 1/L^{r-1}$ if $g_{ij}$ has $r$-th continuous and square integrable derivatives; and $\tilde{\psi}_{4}(L)\approx\exp(-L)$ if $g_{ij}$ is an infinitely differentiable function and all derivatives can be uniformly bounded. To roughly show how $M$, $T$, $L$ and $n$ may co-vary, we assume that $p$ and $s$ are fixed, and all elements of $\mathbb{H}$ have $r$-th continuous and square integrable derivatives. Then FuDGE will recover the differential graph with high probability, if $M\ll n^{1/(2+2\beta)}$, $\sqrt{T}/L\gg M^{1+\beta}$, $T\gg L^{2-r/2}$, and $L\gg M^{(1+\beta)/r}$. As pointed out by a reviewer, the noise term in (21) will create a nugget effect in the covariance, meaning that $\text{Var}(h_{ijk})=\text{Var}(g_{ij}(t_{ijk}))+\sigma^{2}_{0}$. This nugget effect leads to bias in the estimated eigenvalues (variances of the scores). In our theorem, the nugget effect is reflected by $\sigma_{0}$ in $\tilde{\psi}_{1}$. When $\sigma_{0}$ is large, adding a regularization term when estimating the eigenvalues can improve the estimation of FPCA scores and their covariance matrices (see Chapter 6 of Hsing and Eubank (2015)). However, adding a regularization term increases the number of tuning parameters that need to be chosen. An alternative approach to estimating the covariance matrix is through local polynomial regression (Zhang and Wang, 2016). Since the focus of the paper is on the estimation of differential functional graphical models, we do not explore ways to improve the estimation of FPCA scores. However, we recognize that there are alternative approaches that can perform better in some cases. ## 5 Joint Functional Graphical Lasso In this section, we introduce two variants of a Joint Functional Graphical Lasso (JFGL) estimator which we compare empirically to our proposed FuDGE procedure in Section 6.1. Danaher et al. (2014) proposed the Joint Graphical Lasso (JGL) to estimate multiple related Gaussian graphical models from different classes simultaneously. Given $Q\geq 2$ data sets, where the $q$-th data set consists of $n_{q}$ independent random vectors drawn from $N(\mu_{q},\Sigma_{q})$, JGL simultaneously estimates $\\{\Theta\\}=\\{\Theta^{(1)},\Theta^{(2)},\dots,\Theta^{(Q)}\\}$, where $\Theta^{(q)}=\Sigma^{-1}_{q}$ is the precision matrix of the $q$-th data set. Specifically, JGL constructs an estimator $\\{\hat{\Theta}\\}=\\{\hat{\Theta}^{(1)},\hat{\Theta}^{(2)},\dots,\hat{\Theta}^{(Q)}\\}$ by solving the penalized log-likelihood: $\\{\hat{\Theta}\\}=\operatorname*{arg\,min}_{\\{\Theta\\}}\left\\{-\sum^{Q}_{q=1}n_{q}\left(\log\text{det}\Theta^{(q)}-\text{trace}\left(S^{(q)}\Theta^{(q)}\right)\right)+P(\\{\Theta\\})\right\\},$ (45) where $S^{(q)}$ is the sample covariance of the $q$-th data set and $P(\\{\Theta\\})$ is a penalty function. The fused graphical lasso (FGL) is obtained by setting $P(\\{\Theta\\})=\lambda_{1}\sum^{Q}_{q=1}\sum_{i\neq j}|\Theta^{(q)}_{ij}|+\lambda_{2}\sum_{q<q^{\prime}}\sum_{i\neq j}|\Theta^{(q)}_{ij}-\Theta^{(q^{\prime})}_{ij}|,$ (46) while the group graphical lasso (GGL) is obtained by setting $P(\\{\Theta\\})=\lambda_{1}\sum^{Q}_{q=1}\sum_{i\neq j}|\Theta^{(q)}_{ij}|+\lambda_{2}\sum_{i\neq j}\sqrt{\sum^{Q}_{q=1}\left(\Theta^{(q)}_{ij}\right)^{2}}.$ (47) The terms $\lambda_{1}$ and $\lambda_{2}$ are non-negative tuning parameters, while $\Theta^{(q)}_{ij}$ denotes the $(i,j)$-th entry of $\Theta^{(q)}$. For both penalties, the first term is the lasso penalty, which encourages sparsity for the off-diagonal entries of all precision matrices; however, FGL and GGL differ in the second term. For FGL, the second term encourages the off- diagonal entries of precision matrices among all classes to be similar, which means that it encourages not only similar network structure, but also similar edge values. For GGL, the second term is a group lasso penalty, which encourages the support of the precision matrices to be similar, but allows the specific values to differ. A similar approach can be used for estimating the precision matrix of the score vectors. In contrast to the direct estimation procedure proposed in Section 3, we could first estimate $\hat{\Theta}^{X,M}$ and $\hat{\Theta}^{Y,M}$ using a joint graphical lasso objective, and then take the difference to estimate $\Delta$. In the functional graphical model setting, we are interested in the block sparsity, so we modify the entry-wise penalties to a block-wise penalty. Specifically, we propose solving the objective function in (45), where $S^{(q)}$ and $\Theta^{(q)}$ denote the sample covariance and estimated precision of the projection scores for the $q$-th group. Note that now $S^{(q)}$, $\Theta^{(q)}$ and $\hat{\Theta}^{(q)}$, $q=1,\ldots,Q$ are all $pM\times pM$ matrices. Similar to the GGL and FGL procedures, we define the Grouped Functional Graphical Lasso (GFGL) and Fused Functional Graphical Lasso (FFGL) penalties for functional graphs. Specifically, let $\Theta^{(q)}_{jl}$ denote the $(j,l)$-th $M\times M$ block matrix, the GFGL penalty is $P(\\{\Theta\\})=\lambda_{1}\sum^{Q}_{q=1}\sum_{j\neq l}\|\Theta^{(q)}_{jl}\|_{\text{F}}+\lambda_{2}\sum_{j\neq l}\sqrt{\sum^{Q}_{q=1}\|\Theta^{(q)}_{jl}\|^{2}_{\text{F}}},$ (48) where $\lambda_{1}$ and $\lambda_{2}$ are non-negative tuning parameters. The FFGL penalty can be defined in two ways. The first way is to use the Frobenius norm for the second term: $P(\\{\Theta\\})=\lambda_{1}\sum^{Q}_{q=1}\sum_{j\neq l}\|\Theta^{(q)}_{jl}\|_{\text{F}}+\lambda_{2}\sum_{q<q^{\prime}}\sum_{j,l}\|\Theta^{(q)}_{jl}-\Theta^{(q^{\prime})}_{jl}\|_{\text{F}}.$ (49) The second way is to keep the element-wise $L_{1}$ norm as in FGL: $P(\\{\Theta\\})=\lambda_{1}\sum^{Q}_{q=1}\sum_{j\neq l}\|\Theta^{(q)}_{jl}\|_{\text{F}}+\lambda_{2}\sum_{q<q^{\prime}}\sum_{j,l}|\Theta^{(q)}_{jl}-\Theta^{(q^{\prime})}_{jl}|_{1},$ (50) where $\lambda_{1}$ and $\lambda_{2}$ are non-negative tuning parameters. The Joint Functional Graphical Lasso accommodates an arbitrary $Q$. However, when estimating the functional differential graph, we set $Q=2$. We will refer to (49) as FFGL and to (50) as FFGL2. The algorithms for solving GFGL, FFGL, and FFGL2 are given in Appendix A. ## 6 Experiments We examine the performance of FuDGE using both simulations and a real data set.444Code to replicate the simulations is available at https://github.com/boxinz17/FuDGE. ### 6.1 Simulations Given a graph $G_{X}$, we generate samples of $X$ such that $X_{ij}(t)=b^{\prime}(t)^{\top}\delta^{X}_{ij}$. The coefficients $\delta^{X}_{i}=((\delta^{X}_{i1})^{\top},\ldots,(\delta^{X}_{ip})^{\top})^{\top}\in{\mathbb{R}^{mp}}$ are drawn from $N\left(0,(\Omega^{X})^{-1}\right)$ where $\Omega_{X}$ is described below. In all cases, $b^{\prime}(t)$ is an $m$-dimensional basis with disjoint support over $[0,1]$ such that for $k=1,\ldots m$: $b^{\prime}_{k}(t)=\begin{cases}\cos\left(10\pi\left(x-(2k-1)/10\right)\right)+1&\text{if }(k-1)/m\leq{x}<k/m;\\\ 0&\text{otherwise}.\end{cases}$ (51) To generate noisy observations at discrete time points, we sample data $h^{X}_{ijk}=X_{ij}(t_{k})+e_{ijk},\quad e_{ijk}\sim N(0,0.5^{2}),$ for $200$ evenly spaced time points $0=t_{1}\leq\ldots\leq t_{200}=1$. $Y_{ij}(t)$ and $h^{Y}_{ijk}$ are sampled in an analogous procedure. We use $m=5$ for the experiments below, except for the simulation where we explore the effect of $m$ on empirical performance. We consider three different simulation settings for constructing $G_{X}$ and $G_{Y}$. In each setting, we let $n_{X}=n_{Y}=100$ and $p=30,60,90,120$, and we replicate the procedure 30 times for each $p$ and model setting. Model 1: This model is similar to the setting considered in Zhao et al. (2014), but modified to the functional case. We generate the support of $\Omega^{X}$ according to a graph with $p(p-1)/10$ edges and a power-law degree distribution with an expected power parameter of 2. Although the graph is sparse with only 20% of all possible edges present, the power-law structure mimics certain real-world graphs by creating hub nodes with large degree (Newman, 2003). For each nonzero block, we set $\Omega^{X}_{jl}=\delta^{\prime}I_{5}$, where $\delta^{\prime}$ is sampled uniformly from $\pm[0.2,0.5]$. To ensure positive definiteness, we further scale each off-diagonal block by $1/2,1/3,1/4,1/5$ for $p=30,60,90,120$ respectively. Each diagonal element of $\Omega^{X}$ is set to $1$ and the matrix is symmetrized by averaging it with its transpose. To get $\Omega^{Y}$, we first select the top 2 hub nodes in $G_{X}$ (i.e., the nodes with top 2 largest degree), and for each hub node we select the top (by magnitude) 20% of edges. For each selected edge, we set $\Omega^{Y}_{jl}=\Omega^{X}_{jl}+W$ where $W_{kk^{\prime}}=0$ for $|k-k^{\prime}|\leq{2}$, and $W_{kk^{\prime}}=c$ otherwise, where $c$ is generated in the same way as $\delta^{\prime}$. For all other blocks, $\Omega^{Y}_{jl}=\Omega^{X}_{jl}$. Model 2: We first generate a tridiagonal block matrix $\Omega^{*}_{X}$ with $\Omega^{*}_{X,jj}=I_{5}$, $\Omega^{*}_{X,j,j+1}=\Omega^{*}_{X,j+1,j}=0.6I_{5}$, and $\Omega^{*}_{X,j,j+2}=\Omega^{*}_{X,j+2,j}=0.4I_{5}$ for $j=1,\ldots,p$. All other blocks are set to 0. We form $G_{Y}$ by adding four edges to $G_{X}$. Specifically, we first let $\Omega^{*}_{Y,jl}=\Omega^{*}_{X,jl}$ for all blocks, then for $j=1,2,3,4$, we set $\Omega^{*}_{Y,j,j+3}=\Omega^{*}_{Y,j+3,j}=W$, where $W_{kk^{\prime}}=0.1$ for all $1\leq k,k^{\prime}\leq M$. Finally, we set $\Omega^{X}=\Omega^{*}_{X}+\delta I$, $\Omega^{Y}=\Omega^{*}_{Y}+\delta I$, where $\delta=\max\left\\{|\min(\lambda_{\min}(\Omega^{*}_{X}),0)|,|\min(\lambda_{\min}(\Omega^{*}_{Y}),0)|\right\\}+0.05$. Model 3: We generate $\Omega^{*}_{X}$ according to an Erdös-Rényi graph. We first set $\Omega^{*}_{X,jj}=I_{5}$. With probability $.8$, we set $\Omega^{*}_{X,jl}=\Omega^{*}_{X,lj}=0.1I_{5}$, and set it to $0$ otherwise. Thus, we expect 80% of all possible edges to be present. Then, we form $G_{Y}$ by randomly adding $s$ new edges to $G_{X}$, where $s=3$ for $p=30$, $s=4$ for $p=60$, $s=5$ for $p=90$, and $s=6$ for $p=120$. We set each corresponding block $\Omega^{*}_{Y,jl}=W$, where $W_{kk^{\prime}}=0$ when $|k-k^{\prime}|\leq{1}$ and $W_{kk^{\prime}}=c$ otherwise. We let $c=2/5$ for $p=30$, $c=4/15$ for $p=60$, $c=1/5$ for $p=90$, and $c=4/25$ for $p=120$. Finally, we set $\Omega^{X}=\Omega^{*}_{X}+\delta I$, $\Omega^{Y}=\Omega^{*}_{Y}+\delta I$, where $\delta=\max\left\\{|\min(\lambda_{\min}(\Omega^{*}_{X}),0)|,|\min(\lambda_{\min}(\Omega^{*}_{Y}),0)|\right\\}+0.05$. Figure 4: Average ROC curves across 30 simulations. Different columns correspond to different models, different rows correspond to different dimensions. We compare FuDGE with four competing methods. The first competing method (denoted by _multiple_ in Figure 4) ignores the functional nature of the data. We select 15 equally spaced time points, and at each time point, we implement a direct difference estimation procedure (Zhao et al., 2014) to estimate the graph at that time point. Specifically, for each $t$, $X_{i}(t)$ and $Y_{i}(t)$ are simply $p$-dimensional random vectors, and we use their sample covariances in (25) to obtain a $p\times p$ matrix $\hat{\Delta}$. This produces 15 differential graphs, and we use a majority vote to form a single differential graph. The ROC curve is obtained by changing the $L_{1}$ penalty, $\lambda_{n}$, used for all time points. The other three competing methods all estimate two functional graphical models using either the Joint Graphical Lasso or Functional Joint Graphical Lasso introduced in Section 5. For each method, we first estimate the sample covariances of the FPCA scores for $X$ and $Y$. The second competing method (denoted as _FGL_) ignores the block structure in precision matrices and applies the fused graphical lasso method directly. The third and fourth competing methods do account for the block structure and apply FFGL and FFGL2 defined in Section 5. To draw an ROC curve, we follow the same approach as in Zhao et al. (2014). We first fix $\lambda_{1}=0.1$, which controls the overall sparsity in each graph; we then form an ROC curve by varying across $\lambda_{2}$, which controls the similarity between two graphs. For each setting and method, the ROC curve averaged across the $30$ replications is shown in Figure 4. We see that FuDGE clearly has the best overall performance in recovering the support of the differential graph for all cases. We also note that the explicit consideration of block structure in the joint graphical methods does not seem to make a substantial difference as the performance of FGL is comparable to FFGL and FFGL2. ##### The effect of the number of basis functions: To examine how the estimation accuracy is associated with the dimension of the functional data, we repeat the experiment under Model 1 with $p=30$ and vary the number of basis functions used to generate the data in (51). In each case, the number of principal components selected by the cross-validation is $M=4$. In Figure 5, we see that as the gap between the true dimension $m$ and the number of dimensions used $M$ increases, the performance of FuDGE degrades slightly, but is still relatively robust. This is because the FPCA procedure is data adaptive and produces an eigenfunction basis that approximates the true functions well with a relatively small number of basis functions. Figure 5: ROC curves for Model 1 with $p=30$ and changing number of basis functions $m$. Each curve is drawn by averaging across 30 simulations. The number of eigenfunctions, $M$, selected by the cross-validation is 4 in each replication. ### 6.2 Neuroscience Application We apply our method to electroencephalogram (EEG) data obtained from a study (Zhang et al., 1995; Ingber, 1997), which included 122 total subjects; 77 individuals with alcohol use disorder (AUD) and 45 in the control group. Specifically, the EEG data was measured by placing $p=64$ electrodes on various locations on the subject’s scalp and measuring voltage values across time. We follow the preprocessing procedure in Knyazev (2007) and Zhu et al. (2016), which filters the EEG signals at $\alpha$ frequency bands between 8 and 12.5 Hz. Qiao et al. (2019) estimate separate functional graphs for each group, but we directly estimate the differential graph using FuDGE. We choose $\lambda_{n}$ so that the estimated differential graph has approximately 1% of possible edges. The estimated edges of the differential graph are shown in Figure 6. In this setting, an edge in the differential graph suggests that the communication pattern between two different regions of the brain may be affected by alcohol use disorder. However, the differential graph does not indicate exactly how the communication pattern has changed. For instance, the edge between P4 and P6 suggests that AUD affects the communication pattern between those two regions; however, it could be that those two regions are associated (conditionally) in the control group, but not the AUD group or vice versa. It could also be that the two regions are associated (conditionally) in both groups, but the conditional covariance is different. Nonetheless, many interesting observations can be gleaned from the results and may generate interesting hypotheses that could be investigated more thoroughly in an experimental setting. Figure 6: Estimated differential graph for EEG data. The anterior region is the top of the figure and the posterior region is the bottom of the figure. We give two specific observations. First, edges are generally between nodes located in the same region—either the anterior region or the posterior region—and there is no edge that crosses between regions. This observation is consistent with the result in Qiao et al. (2019) where there are no connections between the anterior and posterior regions for both groups. We also note that electrode X, lying in the middle left region has a high degree in the estimated differential graph. While there is no direct connection between the anterior and posterior regions, this region may play a role in helping the two parts communicate and may be heavily affected by AUD. Similarly, P08 in the anterior region also has a high degree and is connected to other nodes in the anterior region, which may indicate that this region can be an information exchange center for anterior regions, which, at the same time, may be heavily affected by AUD. ## 7 Discussion We proposed a method to directly estimate the differential graph for functional graphical models. In certain settings, direct estimation allows for the differential graph to be recovered consistently, even if each underlying graph cannot be consistently recovered. Experiments on simulated data also show that preserving the functional nature of the data rather than treating the data as multivariate scalars can also result in better estimation of the differential graph. A key step in the procedure is first representing the functions with an $M$-dimensional basis using FPCA. Definition 1 ensures that there exists some $M$ large enough so that the signal, $\nu_{1}(M)$, is larger than the bias, $\nu_{2}(M)$, due to using a finite dimensional representation. Intuitively, $\tau=\nu_{1}(M)-\nu_{2}(M)$ is tied to the eigenvalue decay rate; however, we defer derivation of the explicit connection for future work. In addition, we have provided a method for direct estimation of the differential graph, but the development of methods that allow for inference and hypothesis testing in functional differential graphs would be fruitful avenues for future work. For example, Kim et al. (2019) has developed inferential tools for high- dimensional Markov networks, and future work may extend their results to the functional graph setting. ## Acknowledgements We thank the associate editor and reviewers for their helpful feedback which has greatly improved the manuscript. This work is partially supported by the William S. Fishman Faculty Research Fund at the University of Chicago Booth School of Business. This work was completed in part with resources provided by the University of Chicago Research Computing Center. ## Appendix A Derivation of Optimization Algorithm In this section, we derive the key steps for the optimization algorithms. ### A.1 Optimization Algorithm for FuDGE We derive the closed-form updates for the proximal method stated in (29). In particular, recall that for all $1\leq{j,l}\leq{p}$, we have $\Delta^{\text{new}}_{jl}\;=\;\left[\left(\|A^{\text{old}}_{jl}\|_{F}-\lambda_{n}\eta\right)/\|A^{\text{old}}_{jl}\|_{F}\right]_{+}\times A^{\text{old}}_{jl},$ where $A^{\text{old}}=\Delta^{\text{old}}-\eta\nabla L(\Delta^{\text{old}})$ and $x_{+}=\max\\{0,x\\}$, $x\in{\mathbb{R}}$ represents the positive part of $x$. Proof [Proof of (29)] Let $A^{\text{old}}=\Delta^{\text{old}}-\eta\nabla L(\Delta^{\text{old}})$ and let $f_{jl}$ denote the loss decomposed over each $j,l$ block so that ${}f_{jl}(\Delta_{jl})\;=\;\frac{1}{2\lambda_{n}\eta}\|\Delta_{jl}-A^{\text{old}}_{jl}\|^{2}_{F}+\|\Delta_{jl}\|_{F}$ (A.1) and $\Delta^{\text{new}}_{jl}\;=\;\operatorname*{arg\,min}_{\Delta_{jl}\in{\mathbb{R}^{M\times{M}}}}f_{jl}(\Delta_{jl}).$ (A.2) The loss $f_{jl}(\Delta_{jl})$ is convex, so the first-order optimality condition implies that: ${}0\in\partial f_{jl}\left(\Delta^{\text{new}}_{jl}\right),$ (A.3) where $\partial f_{jl}\left(\Delta_{jl}\right)$ is the subdifferential of $f_{jl}$ at $\Delta_{jl}$: ${}\partial f_{jl}(\Delta_{jl})\;=\;\frac{1}{\lambda_{n}\eta}\left(\Delta_{jl}-A^{\text{old}}_{jl}\right)+Z_{jl},$ (A.4) where ${}Z_{jl}\;=\;\begin{cases}\frac{\Delta_{jl}}{\|\Delta_{jl}\|_{F}}\qquad&\text{ if }\Delta_{jl}\neq{0}\\\\[10.0pt] \left\\{Z_{jl}\in{\mathbb{R}^{M\times{M}}}\colon\|Z_{jl}\|_{F}\leq{1}\right\\}\qquad&\text{ if }\Delta_{jl}=0.\end{cases}$ (A.5) Claim 1 If $\|A^{\text{old}}_{jl}\|_{F}>\lambda_{n}\eta>0$, then $\Delta^{\text{new}}_{jl}\neq{0}$. We verify this claim by proving the contrapositive. Suppose $\Delta^{\text{new}}_{jl}={0}$. Then by (A.3) and (A.5), there exists a $Z_{jl}\in{\mathbb{R}^{M\times{M}}}$ such that $\|Z_{jl}\|_{F}\leq{1}$ and $0=-\frac{1}{\lambda_{n}\eta}A^{\text{old}}_{jl}+Z_{jl}.$ Thus, $\|A^{\text{old}}_{jl}\|_{F}=\|\lambda_{n}\eta\cdot Z_{jl}\|_{F}\leq{\lambda_{n}\eta}$, so that Claim 1 holds. Combining Claim 1 with (A.3) and (A.5), for any $j,l$ such that $\|A^{\text{old}}_{jl}\|_{F}>\lambda_{n}\eta$, we have $0=\frac{1}{\lambda_{n}\eta}\left(\Delta^{\text{new}}_{jl}-A^{\text{old}}_{jl}\right)+\frac{\Delta^{\text{new}}_{jl}}{\|\Delta^{\text{new}}_{jl}\|_{F}},$ which is solved by ${}\Delta^{\text{new}}_{jl}=\frac{\|A^{\text{old}}_{jl}\|_{F}-\lambda_{n}\eta}{\|A^{\text{old}}_{jl}\|_{F}}A^{\text{old}}_{jl}.$ (A.6) Claim 2 If $\|A^{\text{old}}_{jl}\|_{F}\leq\lambda_{n}\eta$, then $\Delta^{\text{new}}_{jl}=0$. Again, we verify the claim by proving the contrapositive. Suppose $\Delta^{\text{new}}_{jl}\neq 0$. Then the first-order optimality implies the updates in (A.6). However, taking the Frobenius norm on both sides of the equation gives $\|\Delta^{\text{new}}_{jl}\|_{F}=\|A^{\text{old}}_{jl}\|_{F}-\lambda_{n}\eta$, which implies that $\|A^{\text{old}}_{jl}\|_{F}-\lambda_{n}\eta\geq{0}$. The updates in (29) immediately follow by combining Claim 2 and (A.6). [2mm] ### A.2 Solving the Joint Functional Graphical Lasso As in Danaher et al. (2014), we use the alternating directions method of multipliers (ADMM) algorithm to solve (45); see Boyd et al. (2011) for a detailed exposition of ADMM. To solve (45), we first rewrite the problem as: $\max_{\\{\Theta\\},\\{Z\\}}\left\\{-\sum^{Q}_{q=1}n_{q}\left(\log\text{det}\Theta^{(q)}-\text{trace}\left(S^{(q)}\Theta^{(q)}\right)\right)+P(\\{Z\\})\right\\},$ subject to $\Theta^{(q)}\succ 0$ and $Z^{(q)}=\Theta^{(q)}$, where $\\{Z\\}=\\{Z^{(1)},Z^{(2)},\dots,Z^{(Q)}\\}$. The scaled augmented Lagrangian (Boyd et al., 2011) is given by $L_{\rho}\left(\\{\Theta\\},\\{Z\\},\\{U\\}\right)=-\sum^{Q}_{q=1}n_{q}\left(\log\text{det}\Theta^{(q)}-\text{trace}\left(S^{(q)}\Theta^{(q)}\right)\right)+P(\\{Z\\})\\\ +\frac{\rho}{2}\sum^{Q}_{q=1}\|\Theta^{(q)}-Z^{(q)}+U^{(q)}\|^{2}_{\text{F}},$ (A.7) where $\rho>0$ is a tuning parameter and $\\{U\\}=\\{U^{(1)},U^{(2)},\dots,U^{(Q)}\\}$ are dual variables. The ADMM algorithm will then solve (A.7) by iterating the following three steps. At the $i$-th iteration, they are as follows: 1. 1. $\\{\Theta_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{\Theta\\}}L_{\rho}\left(\\{\Theta\\},\\{Z_{(i-1)}\\},\\{U_{(i-1)}\\}\right)$. 2. 2. $\\{Z_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{Z\\}}L_{\rho}\left(\\{\Theta_{(i)}\\},\\{Z\\},\\{U_{(i-1)}\\}\right)$. 3. 3. $\\{U_{(i)}\\}\leftarrow\\{U_{(i-1)}\\}+(\\{\Theta_{(i)}\\}-\\{Z_{(i)}\\})$. We now give more details for the above three steps. ADMM algorithm for solving the joint functional graphical lasso problem (a) Initialize the variables: $\Theta^{(q)}_{(0)}=I_{pM}$, $U^{(q)}_{(0)}=0_{pM}$, and $Z^{(q)}_{(0)}=0_{pM}$ for $q=1,\ldots,Q$. (b) Select a scalar $\rho>0$. (c) For $i=1,2,3,\dots$ until convergence (i) For $q=1,\ldots,Q$, update $\Theta^{(q)}_{(i)}$ as the minimizer (with respect to $\Theta^{(q)}$) of $-n_{q}\left(\log\text{det}\Theta^{(q)}-\text{trace}\left(S^{(q)}\Theta^{(q)}\right)\right)+\frac{\rho}{2}\|\Theta^{(q)}-Z^{(q)}_{(i-1)}+U^{(q)}_{(i-1)}\|^{2}_{\text{F}}$ Letting $VDV^{\top}$ denote the eigendecomposition of $S^{(q)}-\rho Z^{(q)}_{(i-1)}/n_{q}+\rho U^{(q)}_{(i-1)}/n_{q}$, then the solution is given by $V\tilde{D}V^{\top}$ (Witten and Tibshirani, 2009), where $\tilde{D}$ is the diagonal matrix with $j$-th diagonal element being $\frac{n_{q}}{2\rho}\left(-D_{jj}+\sqrt{D^{2}_{jj}+4\rho/n_{q}}\right),$ where $D_{jj}$ is the $(j,j)$-th entry of $D$. (ii) Update $\\{Z_{(i)}\\}$ as the minimizer (with respect to $\\{Z\\}$) of $\min_{\\{Z\\}}\frac{\rho}{2}\sum^{Q}_{q=1}\|Z^{(q)}-A^{(q)}\|^{2}_{\text{F}}+P(\\{Z\\}),$ (A.8) where $A^{(q)}=\Theta^{(q)}_{(i)}+U^{(q)}_{(i-1)}$, $q=1,\ldots,Q$. (iii) $U^{(q)}_{(i)}\leftarrow U^{(q)}_{(i-1)}+(\Theta^{(q)}_{(i)}-Z^{(q)}_{(i)})$, $q=1,\ldots,Q$. There are three things worth noticing. 1. The key step is to solve (A.8), which depends on the form of penalty term $P(\cdot)$; 2. This algorithm is guaranteed to converge to the global optimum when $P(\cdot)$ is convex (Boyd et al., 2011); 3. The positive-definiteness constraint on $\\{\hat{\Theta}\\}$ is naturally enforced by step (c) (i). ### A.3 Solutions to (A.8) for Joint Functional Graphical Lasso We provide solutions to (A.8) for three problems (GFGL, FFGL, FFGL2) defined by (48), (49) and (50). #### A.3.1 Solution to (A.8) for GFGL Let the solution for $\min_{\\{Z\\}}\frac{\rho}{2}\sum^{Q}_{q=1}\|Z^{(q)}-A^{(q)}\|^{2}_{\text{F}}+\lambda_{1}\sum^{Q}_{q=1}\sum_{j\neq l}\|Z^{(q)}_{jl}\|_{\text{F}}+\lambda_{2}\sum_{j\neq l}\left(\sum^{Q}_{q=1}\|Z^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2}$ (A.9) be denoted as $\\{\hat{Z}\\}=\\{\hat{Z}^{(1)},\hat{Z}^{(2)},\dots,\hat{Z}^{(Q)}\\}$. Let $Z^{(q)}_{jl}$, $\hat{Z}^{(q)}_{jl}$ be $(j,l)$-th $M\times M$ block of $Z^{(q)}$ and $\hat{Z}^{(q)}$, $q=1,\ldots,Q$. Then, for $j=1,\ldots,p$, we have $\hat{Z}^{(q)}_{jj}=A^{(q)}_{jj},\qquad q=1,\ldots,Q,$ (A.10) and, for $j\neq l$, we have $\hat{Z}^{(q)}_{jl}=\left(\frac{\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho}{\|A^{(q)}_{jl}\|_{\text{F}}}\right)_{+}\left(1-\frac{\lambda_{2}}{\rho\sqrt{\sum^{Q}_{q=1}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}}\right)_{+}A^{(q)}_{jl},$ (A.11) where $q=1,\ldots,Q$. Details of the update are given in Appendix A.4. #### A.3.2 Solution to (A.8) for FFGL For FFGL, there is no simple closed form solution. When $Q=2$, (A.8) becomes $\min_{\\{Z\\}}\;\frac{\rho}{2}\sum^{2}_{q=1}\|Z^{(q)}-A^{(q)}\|^{2}_{\text{F}}+\lambda_{1}\left(\sum^{2}_{q=1}\sum_{j\neq l}\|Z^{(q)}_{jl}\|_{\text{F}}\right)+\lambda_{2}\sum_{j,l}\|Z^{(1)}_{jl}-Z^{(2)}_{jl}\|_{\text{F}}.$ For each $1\leq j,l\leq p$, we compute $\hat{Z}^{(1)}_{jl}$, $\hat{Z}^{(2)}_{jl}$ by solving $\min_{\\{Z^{(1)}_{jl},Z^{(2)}_{jl}\\}}\;\frac{1}{2}\sum^{2}_{q=1}\|Z^{(q)}_{jl}-A^{(q)}_{jl}\|^{2}_{\text{F}}+\frac{\lambda_{1}}{\rho}\mathbbm{1}_{j\neq l}\sum^{2}_{q=1}\|Z^{(q)}_{jl}\|_{\text{F}}+\frac{\lambda_{2}}{\rho}\|Z^{(1)}_{jl}-Z^{(2)}_{jl}\|_{\text{F}},$ (A.12) where $\mathbbm{1}_{j\neq l}=1$ when $j\neq l$ and $0$ otherwise. When $j=l$, by Lemma 6, we have the following closed form updates for $\\{\hat{Z}^{(1)}_{jj},\hat{Z}^{(2)}_{jj}\\}$, $j=1,\ldots,p$. If $\|A^{(1)}_{jj}-A^{(2)}_{jj}\|_{\text{F}}\leq 2\lambda_{2}/\rho$, then $\hat{Z}^{(1)}_{jj}=\hat{Z}^{(2)}_{jj}=\frac{1}{2}\left(A^{(1)}_{jj}+A^{(2)}_{jj}\right).$ If $\|A^{(1)}_{jj}-A^{(2)}_{jj}\|_{\text{F}}>2\lambda_{2}/\rho$, then $\displaystyle\hat{Z}^{(1)}_{jj}$ $\displaystyle=A^{(1)}_{jj}-\frac{\lambda_{2}/\rho}{\|A^{(1)}_{jj}-A^{(2)}_{jj}\|_{\text{F}}}\left(A^{(1)}_{jj}-A^{(2)}_{jj}\right),$ $\displaystyle\hat{Z}^{(2)}_{jj}$ $\displaystyle=A^{(2)}_{jj}+\frac{\lambda_{2}/\rho}{\|A^{(1)}_{jj}-A^{(2)}_{jj}\|_{\text{F}}}\left(A^{(1)}_{jj}-A^{(2)}_{jj}\right).$ For $j\neq l$, we get $\\{\hat{Z}^{(1)}_{jl},\hat{Z}^{(2)}_{jl}\\}$ using the ADMM algorithm again. We construct the scaled augmented Lagrangian as: ${L^{\prime}}_{\rho^{\prime}}\left(\\{W\\},\\{R\\},\\{V\\}\right)=\frac{1}{2}\sum^{2}_{q=1}\|W^{(q)}-B^{(q)}\|_{\text{F}}+\frac{\lambda_{1}}{\rho}\sum^{2}_{q=1}\|W^{(q)}\|_{\text{F}}\\\ +\frac{\lambda_{2}}{\rho}\|R^{(1)}-R^{(2)}\|_{\text{F}}+\frac{\rho^{\prime}}{2}\sum^{2}_{q=1}\|W^{(q)}-R^{(q)}+V^{(q)}\|^{2}_{\text{F}},$ where $\rho^{\prime}>0$ is a tuning parameter, $B^{(q)}=A^{(q)}_{jl}$, $q=1,2$, and $W^{q},R^{(q)},V^{(q)}\in\mathbb{R}^{M\times M}$, $q=1,2$. $\\{W\\}=\\{W^{(1)},W^{(2)}\\}$, $\\{R\\}=\\{R^{(1)},R^{(2)}\\}$, and $\\{V\\}=\\{V^{(1)},V^{(2)}\\}$. The detailed ADMM algorithm is described as below: ADMM algorithm for solving (A.12) for $j\neq l$ (a) Initialize the variables: $W^{(q)}_{(0)}=I_{M}$, $R^{(q)}_{(0)}=0_{M}$, and $V^{(q)}_{(0)}=0_{M}$ for $q=1,2$. Let $B^{(q)}=A^{(q)}_{jl}$, $q=1,2$. (b) Select a scalar $\rho^{\prime}>0$. (c) For $i=1,2,3,\dots$ until convergence (i) $\\{W_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{W\\}}{L^{\prime}}_{\rho^{\prime}}\left(\\{W\\},\\{R_{(i-1)}\\},\\{V_{(i-1)}\\}\right)$. This is equivalent to $\\{W_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{W\\}}\frac{1}{2}\sum^{2}_{q=1}\|W^{(q)}-C^{(q)}\|^{2}_{\text{F}}+\frac{\lambda_{1}}{\rho(1+\rho^{\prime})}\sum^{2}_{q=1}\|W^{(q)}\|_{\text{F}},$ where $C^{(q)}=\frac{1}{1+\rho^{\prime}}\left[B^{(q)}+\rho^{\prime}\left(R^{(q)}_{(i-1)}-V^{(q)}_{(i-1)}\right)\right].$ Similar to (28), we have $W^{(q)}_{(i)}\leftarrow\left(\frac{\|C^{(q)}\|_{\text{F}}-\lambda_{1}/(\rho(1+\rho^{\prime}))}{\|C^{(q)}\|_{\text{F}}}\right)_{+}\cdot C^{(q)},\qquad q=1,2.$ (ii) $\\{R_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{R\\}}{L^{\prime}}_{\rho^{\prime}}\left(\\{W_{(i)}\\},\\{R\\},\\{V_{(i-1)}\\}\right)$. This is equivalent to $\\{R_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{R\\}}\frac{1}{2}\sum^{2}_{q=1}\|R^{(q)}-D^{(q)}\|^{2}_{\text{F}}+\frac{\lambda_{2}}{\rho\rho^{\prime}}\|R^{(1)}-R^{(2)}\|_{\text{F}},$ where $D^{(q)}=W^{(q)}_{(i)}+V^{(q)}_{(i-1)}$. By Lemma 6, if $\|D^{(1)}-D^{(2)}\|_{\text{F}}\leq 2\lambda_{2}/(\rho\rho^{\prime})$, then $R^{(1)}_{(i)}=R^{(2)}_{(i)}\leftarrow\frac{1}{2}\left(D^{(1)}+D^{(2)}\right),$ and if $\|D^{(1)}-D^{(2)}\|_{\text{F}}>2\lambda_{2}/(\rho\rho^{\prime})$, then $\displaystyle R^{(1)}\leftarrow D^{(1)}-\frac{\lambda_{2}/(\rho\rho^{\prime})}{\|D^{(1)}-D^{(2)}\|_{\text{F}}}\left(D^{(1)}-D^{(2)}\right),$ $\displaystyle R^{(2)}\leftarrow D^{(2)}+\frac{\lambda_{2}/(\rho\rho^{\prime})}{\|D^{(1)}-D^{(2)}\|_{\text{F}}}\left(D^{(1)}-D^{(2)}\right).$ (iii) $V^{(q)}_{(i)}\leftarrow V^{(q)}_{(i-1)}+W^{(q)}_{(i)}-R^{(q)}_{(i)}$, $q=1,2$. #### A.3.3 Solution to (A.8) for FFGL2 For FFGL2, there is also no closed form solution. Similar to Section A.3.2, we compute a closed form solution for $\\{\hat{Z}^{(1)}_{jj},\hat{Z}^{(2)}_{jj}\\}$, $j=1,\ldots,p$, and use an ADMM algorithm to compute $\\{\hat{Z}^{(1)}_{jl},\hat{Z}^{(2)}_{jl}\\}$, $1\leq j\neq l\leq p$. For any $1\leq j,l\leq p$, we solve: $\min_{\\{Z^{(1)}_{jl},Z^{(2)}_{jl}\\}}\;\frac{1}{2}\sum^{2}_{q=1}\|Z^{(q)}_{jl}-A^{(q)}_{jl}\|^{2}_{\text{F}}+\frac{\lambda_{1}}{\rho}\mathbbm{1}_{j\neq l}\sum^{2}_{q=1}\|Z^{(q)}_{jl}\|_{\text{F}}+\frac{\lambda_{2}}{\rho}\sum_{1\leq a,b\leq M}|Z^{(1)}_{jl,ab}-Z^{(2)}_{jl,ab}|,$ (A.13) where $\mathbbm{1}_{j\neq l}=1$ when $j\neq l$ and $0$ otherwise. By Lemma 6, when $j=l$ we have $\left(\hat{Z}^{(1)}_{jj,ab},\hat{Z}^{(2)}_{jj,ab}\right)=\left\\{\begin{aligned} &\left(A^{(1)}_{jl,ab}-\lambda_{2}/\rho,A^{(2)}_{jl,ab}+\lambda_{2}/\rho\right)\quad\text{if}\;A^{(1)}_{jl,ab}>A^{(2)}_{jl,ab}+2\lambda_{2}/\rho\\\ &\left(A^{(1)}_{jl,ab}+\lambda_{2}/\rho,A^{(2)}_{jl,ab}-\lambda_{2}/\rho\right)\quad\text{if}\;A^{(1)}_{jl,ab}<A^{(2)}_{jl,ab}-2\lambda_{2}/\rho\\\ &\left(\left(A^{(1)}_{jl,ab}+A^{(2)}_{jl,ab}\right)/2,\left(A^{(1)}_{jl,ab}+A^{(2)}_{jl,ab}\right)/2\right)\quad\text{if}\;\left|A^{(1)}_{jl,ab}-A^{(2)}_{jl,ab}\right|\leq 2\lambda_{2}/\rho,\end{aligned}\right.$ where subscripts $(a,b)$ denote the $(a,b)$-th entry, $1\leq a,b\leq M$ and $j=1,\ldots,p$. For $j\neq l$, we get $\\{\hat{Z}^{(1)}_{jl},\hat{Z}^{(2)}_{jl}\\}$, $1\leq j\neq l\leq p$ by using an ADMM algorithm. Let $B^{(q)}=A^{(q)}_{jl}$, $q=1,2$. We first construct the scaled augmented Lagrangian: ${L^{\prime}}_{\rho^{\prime}}\left(\\{W\\},\\{R\\},\\{V\\}\right)=\frac{1}{2}\sum^{2}_{q=1}\|W^{(q)}-B^{(q)}\|_{\text{F}}+\frac{\lambda_{1}}{\rho}\sum^{2}_{q=1}\|W^{(q)}\|_{\text{F}}\\\ +\frac{\lambda_{2}}{\rho}\sum_{a,b}|R^{(1)}_{a,b}-R^{(2)}_{a,b}|+\frac{\rho^{\prime}}{2}\sum^{2}_{q=1}\|W^{(q)}-R^{(q)}+V^{(q)}\|^{2}_{\text{F}},$ where $\rho^{\prime}>0$ is a tuning parameter, $W^{q},R^{(q)},V^{(q)}\in\mathbb{R}^{M\times M}$, $q=1,2$, $\\{W\\}=\\{W^{(1)},W^{(2)}\\}$, $\\{R\\}=\\{R^{(1)},R^{(2)}\\}$, and $\\{V\\}=\\{V^{(1)},V^{(2)}\\}$. The detailed ADMM algorithm is described as below: ADMM algorithm for solving (A.13) for $j\neq l$ (a) Initialize the variables: $W^{(q)}_{(0)}=I_{M}$, $R^{(q)}_{(0)}=0_{M}$, and $V^{(q)}_{(0)}=0_{M}$ for $q=1,2$. Let $B^{(q)}=A^{(q)}_{jl}$, $q=1,2$. (b) Select a scalar $\rho^{\prime}>0$. (c) For $i=1,2,3,\dots$ until convergence (i) $\\{W_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{W\\}}.{L^{\prime}}_{\rho^{\prime}}\left(\\{W\\},\\{R_{(i-1)}\\},\\{V_{(i-1)}\\}\right)$ This is equivalent to $\\{W_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{W\\}}\frac{1}{2}\sum^{2}_{q=1}\|W^{(q)}-C^{(q)}\|^{2}_{\text{F}}+\frac{\lambda_{1}}{\rho(1+\rho^{\prime})}\sum^{2}_{q=1}\|W^{(q)}\|_{\text{F}},$ where $C^{(q)}=\frac{1}{1+\rho^{\prime}}\left[B^{(q)}+\rho^{\prime}\left(R^{(q)}_{(i-1)}-V^{(q)}_{(i-1)}\right)\right].$ Similar to (28), we have $W^{(q)}_{(i)}\leftarrow\left(\frac{\|C^{(q)}\|_{\text{F}}-\lambda_{1}/(\rho(1+\rho^{\prime}))}{\|C^{(q)}\|_{\text{F}}}\right)_{+}\cdot C^{(q)},\qquad q=1,2.$ (ii) $\\{R_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{R\\}}{L^{\prime}}_{\rho^{\prime}}\left(\\{W_{(i)}\\},\\{R\\},\\{V_{(i-1)}\\}\right)$ This is equivalent to $\\{R_{(i)}\\}\leftarrow\operatorname*{arg\,min}_{\\{R\\}}\frac{1}{2}\sum^{2}_{q=1}\|R^{(q)}-D^{(q)}\|^{2}_{\text{F}}+\frac{\lambda_{2}}{\rho\rho^{\prime}}\sum_{a,b}\left|R^{(1)}_{ab}-R^{(2)}_{ab}\right|,$ where $D^{(q)}=W^{(q)}_{(i)}+V^{(q)}_{(i-1)}$. Then by Lemma 6, we have $\left(R^{(1)}_{(i),ab},R^{(2)}_{(i),ab}\right)=\left\\{\begin{aligned} &\left(D^{(1)}_{ab}-\lambda_{2}/(\rho\rho^{\prime}),D^{(2)}_{ab}+\lambda_{2}/(\rho\rho^{\prime})\right)\quad\text{if}\;D^{(1)}_{ab}>D^{(2)}_{ab}+2\lambda_{2}/(\rho\rho^{\prime})\\\ &\left(D^{(1)}_{ab}+\lambda_{2}/(\rho\rho^{\prime}),D^{(2)}_{ab}-\lambda_{2}/(\rho\rho^{\prime})\right)\quad\text{if}\;D^{(1)}_{ab}<D^{(2)}_{ab}-2\lambda_{2}/(\rho\rho^{\prime})\\\ &\left(\left(D^{(1)}_{ab}+D^{(2)}_{ab}\right)/2,\left(D^{(1)}_{ab}+D^{(2)}_{ab}\right)/2\right)\quad\text{if}\;\left|D^{(1)}_{ab}-D^{(1)}_{ab}\right|\leq 2\lambda_{2}/(\rho\rho^{\prime}),\end{aligned}\right.$ where subscripts $(a,b)$ denote the $(a,b)$-th entry, $1\leq a,b\leq M$ and $1\leq j,l\leq p$. (iii) $V^{(q)}_{(i)}\leftarrow V^{(q)}_{(i-1)}+W^{(q)}_{(i)}-R^{(q)}_{(i)}$, $q=1,2$. ### A.4 Derivation of (A.10) and (A.11) We provide proof of (A.10) and (A.11). Note that for any $1\leq j,l\leq p$, we can obtain $\hat{Z}^{(1)}_{jl},\hat{Z}^{(2)}_{jl},\dots,\hat{Z}^{(Q)}_{jl}$ by solving $\operatorname*{arg\,min}_{Z^{(1)}_{jl},Z^{(2)}_{jl},\dots,Z^{(Q)}_{jl}}\frac{\rho}{2}\sum^{Q}_{q=1}\|Z^{(q)}_{jl}-A^{(q)}_{jl}\|^{2}_{\text{F}}+\lambda_{1}\mathbbm{1}_{j\neq l}\sum^{Q}_{q=1}\|Z^{(q)}_{jl}\|_{\text{F}}+\lambda_{2}\mathbbm{1}_{j\neq l}\left(\sum^{Q}_{q=1}\|Z^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2},$ (A.14) where $\mathbbm{1}_{j\neq l}=1$ when $j\neq l$ and $0$ otherwise. By (A.14), we have that $\hat{Z}^{(q)}_{jj}=A^{(q)}_{jj}$ for any $j=1,\ldots,p$ and $q=1,\ldots,Q$, which is (A.10). We then prove (A.11). Denote the objective function in (A.14) as $\tilde{L}_{jl}$. Then, for $j\neq l$, the subdifferential of $\tilde{L}_{jl}$ with respect to $Z^{(q)}_{jl}$ is $\partial_{Z^{(q)}_{jl}}\tilde{L}_{jl}=\rho(Z^{(q)}_{jl}-A^{(q)}_{jl})+\lambda_{1}G^{(q)}_{jl}+\lambda_{2}D^{(q)}_{jl},$ where $G^{(q)}_{jl}=\left\\{\begin{aligned} &\frac{Z^{(q)}_{jl}}{\|Z^{(q)}_{jl}\|_{\text{F}}}\quad\text{when}\;Z^{(q)}_{jl}\neq 0\\\ &\\{G^{(q)}_{jl}\in\mathbb{R}^{M\times M}:\|G^{(q)}_{jl}\|_{\text{F}}\leq 1\\}\quad\text{otherwise}\end{aligned}\right.,$ and $D^{(q)}_{jl}=\left\\{\begin{aligned} &\frac{Z^{(q)}_{jl}}{\left(\sum^{Q}_{q=1}\|Z^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2}}\quad\text{when}\;\sum^{Q}_{q=1}\|Z^{(q)}_{jl}\|^{2}_{\text{F}}>0\\\ &\\{D^{(q)}_{jl}\in\mathbb{R}^{M\times M}:\sum^{Q}_{q=1}\|D^{(q)}_{jl}\|^{2}_{\text{F}}\leq 1\\}\quad\text{otherwise}\end{aligned}\right..$ To obtain the optimum, we need $0\in\partial_{Z^{(q)}_{jl}}\tilde{L}_{jl}(\hat{Z}^{(q)}_{jl})$ for all $q=1,\ldots,Q$. We now split our discussion into two cases. (a) When $\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}=0$, or equivalently, $\hat{Z}^{(q)}_{jl}=0$ for all $q=1,\ldots,Q$. In this case, there exists $G^{(q)}_{jl}$, where $\|G^{(q)}_{jl}\|_{\text{F}}\leq 1$, for all $q=1,\ldots,Q$; and also $D^{(q)}_{jl}$, where $\sum^{Q}_{q=1}\|D^{(q)}_{jl}\|^{2}_{\text{F}}\leq 1$, such that $0=-\rho\cdot A^{(q)}_{jl}+\lambda_{1}G^{(q)}_{jl}+\lambda_{2}D^{(q)}_{jl},$ which implies that $D^{(q)}_{jl}=\frac{\rho}{\lambda_{2}}\left(A^{(q)}_{jl}-\frac{\lambda_{1}}{\rho}G^{(q)}_{jl}\right).$ Thus, we have $\displaystyle\|D^{(q)}_{jl}\|_{\text{F}}$ $\displaystyle=\frac{\rho}{\lambda_{2}}\left\|A^{(q)}_{jl}-\frac{\lambda_{1}}{\rho}G^{(q)}_{jl}\right\|_{\text{F}}\geq\frac{\rho}{\lambda_{2}}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\frac{\lambda_{1}}{\rho}\|G^{(q)}_{jl}\|_{\text{F}}\right)_{+}$ $\displaystyle\geq\frac{\rho}{\lambda_{2}}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\frac{\lambda_{1}}{\rho}\right)_{+},$ which implies that $\frac{\rho^{2}}{\lambda^{2}_{2}}\sum^{Q}_{q=1}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\frac{\lambda_{1}}{\rho}\right)^{2}_{+}\leq\sum^{Q}_{q=1}\|D^{(q)}_{jl}\|^{2}_{\text{F}}\leq 1,$ and then we have $\sqrt{\sum^{Q}_{q=1}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}\leq\lambda_{2}/\rho.$ (A.15) (b) When $\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}>0$. For those $q$’s such that $\hat{Z}^{(q)}_{jl}=0$, there exists $G^{(q)}_{jl}$, where $\|G^{(q)}_{jl}\|_{\text{F}}=1$, such that $0=-\rho A^{(q)}_{jl}+\lambda_{1}G^{(q)}_{jl}.$ Thus, we have $\|A^{(q)}_{jl}\|_{\text{F}}=\frac{\lambda_{1}}{\rho}\|G^{(q)}_{jl}\|_{\text{F}}\leq\frac{\lambda_{1}}{\rho},$ which implies that $\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)_{+}=0.$ (A.16) On the other hand, for those $q$’s such that $\hat{Z}^{(q)}_{jl}\neq 0$, we have $0=\rho\left(\hat{Z}^{(q)}_{jl}-A^{(q)}_{jl}\right)+\lambda_{1}\frac{\hat{Z}^{(q)}_{jl}}{\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}}+\lambda_{2}\frac{\hat{Z}^{(q)}_{jl}}{\left(\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2}},$ which implies that $A^{(q)}_{jl}=\hat{Z}^{(q)}_{jl}\left(1+\frac{\lambda_{1}}{\rho\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}}+\frac{\lambda_{2}}{\rho\left(\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2}}\right),$ (A.17) and $\|A^{(q)}_{jl}\|_{\text{F}}=\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}+\lambda_{1}/\rho+(\lambda_{2}/\rho)\cdot\frac{\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}}{\left(\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2}}.$ (A.18) By (A.18), we have $\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)_{+}>\frac{\lambda_{2}}{\rho}\cdot\frac{\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}}{\sqrt{\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}}}>0.$ (A.19) By (A.16) and (A.19), we have $\displaystyle\sum^{Q}_{q=}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}$ $\displaystyle=\sum_{q:\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}\neq 0}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}$ (A.20) $\displaystyle>\frac{\lambda^{2}_{2}}{\rho^{2}}\sum_{q:\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}\neq 0}\frac{\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}}{\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}}$ $\displaystyle>\lambda^{2}_{2}/\rho^{2}.$ We now make the following claims. Claim 1. $\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}=0\Leftrightarrow\sqrt{\sum^{Q}_{q=}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}\leq\lambda_{2}/\rho$. This claim is easily shown by (A.15) and (A.20). Claim 2. When $\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}>0$, we have $\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}=0\Leftrightarrow\|A^{(q)}_{jl}\|_{\text{F}}\leq\lambda_{1}/\rho$. This claim is easily shown by (A.16) and (A.19). Claim 3. When $\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}\neq 0$, then we have $\hat{Z}^{(q)}_{jl}=\left(\frac{\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho}{\|A^{(q)}_{jl}\|_{\text{F}}}\right)\left(1-\frac{\lambda_{2}}{\rho\sqrt{\sum^{Q}_{q=}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}}\right)A^{(q)}_{jl}.$ To prove this claim, note that by Claim 2 and (A.18), we have $\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)_{+}=\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}\left(1+\frac{\lambda_{2}}{\rho\left(\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}\right)^{1/2}}\right)$ for $q=1,\ldots,Q$. Thus, $\sqrt{\sum^{Q}_{q=1}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}=\sqrt{\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}}+\lambda_{2}/\rho,$ which implies that $\sqrt{\sum^{Q}_{q=1}\|\hat{Z}^{(q)}_{jl}\|^{2}_{\text{F}}}=\sqrt{\sum^{Q}_{q=1}\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}-\lambda_{2}/\rho.$ Thus, by (A.18), we have $\displaystyle\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}$ $\displaystyle=\frac{\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho}{1+\frac{\lambda_{2}/\rho}{\sqrt{\sum^{Q}_{q^{\prime}=1}\left(\|A^{(q^{\prime})}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}-\lambda_{2}/\rho}}$ $\displaystyle=\left(1-\frac{\lambda_{2}}{\rho\sqrt{\sum^{Q}_{q^{\prime}=1}\left(\|A^{(q^{\prime})}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}}\right)\left(\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right).$ This way, combined with (A.17), we then have $\hat{Z}^{(q)}_{jl}=\frac{\|\hat{Z}^{(q)}_{jl}\|_{\text{F}}}{\|A^{(q)}_{jl}\|_{\text{F}}}A^{(q)}_{jl}=\left(\frac{\|A^{(q)}_{jl}\|_{\text{F}}-\lambda_{1}/\rho}{\|A^{(q)}_{jl}\|_{\text{F}}}\right)\left(1-\frac{\lambda_{2}}{\rho\sqrt{\sum^{Q}_{q^{\prime}=1}\left(\|A^{(q^{\prime})}_{jl}\|_{\text{F}}-\lambda_{1}/\rho\right)^{2}_{+}}}\right)A^{(q)}_{jl}.$ Finally, combining Claims 1-3, we obtain (A.11). ## Appendix B Main Technical Proofs We give proofs of the results given in the main text. ### B.1 Proof of Lemma 2 We only need to prove that when we use two sets of orthonormal function basis $e^{M}(t)=\\{e^{M}_{j}(t)\\}^{p}_{j=1}$ and $\tilde{e}^{M}(t)=\\{\tilde{e}^{M}_{j}(t)\\}^{p}_{j=1}$ to expand the same subspace $\mathbb{V}^{M}_{[p]}$, the definition of $E^{\pi}_{\Delta}$ will not be changed. Since both $e^{M}_{j}(t)=(e^{M}_{j1}(t),e^{M}_{j2}(t),\dots,e^{M}_{jM}(t))^{\top}$ and $\tilde{e}^{M}_{j}(t)=(\tilde{e}^{M}_{j1}(t),\tilde{e}^{M}_{j2}(t),\dots,\tilde{e}^{M}_{jM}(t))^{\top}$ are orthonormal function basis of $\mathbb{V}^{M}_{j}$, there must exist an orthonormal matrix $U_{j}\in\mathbb{R}^{M\times M}$ satisfying $U^{\top}_{j}U_{j}=U_{j}U^{\top}_{j}=I_{M}$, such that $\tilde{e}^{M}_{j}(t)=U_{j}e^{M}_{j}(t)$. Let $a^{X,M}_{ij}$ be the projection score vectors of $X_{ij}(t)$ onto $e^{M}_{j}(t)$ and $\tilde{a}^{X,M}_{ij}$ be the projection score vectors of $X_{ij}(t)$ onto $\tilde{e}^{M}_{j}(t)$. Then $\tilde{a}^{X,M}_{ij}=U_{j}a^{X,M}_{ij}$. Denote $U={\rm diag}\\{U_{1},U_{2},\dots,U_{p}\\}\in\mathbb{R}^{pM\times pM}.$ We then have $\displaystyle\tilde{a}^{X,M}_{i}$ $\displaystyle=((\tilde{a}^{X,M}_{i1})^{\top},(\tilde{a}^{X,M}_{i2})^{\top},\dots,(\tilde{a}^{X,M}_{ip})^{\top})^{\top}$ $\displaystyle=((a^{X,M}_{i1})^{\top}U^{\top}_{1},(a^{X,M}_{i2})^{\top}U^{\top}_{2},\dots,(a^{X,M}_{ip})^{\top}U^{\top}_{p})^{\top}=Ua^{X,M}_{i}$ and $\tilde{\Sigma}^{X,M}={\rm Cov}\left(\tilde{a}^{X,M}\right)=U{\rm Cov}\left(\tilde{a}^{X,M}\right)U^{\top}=U\Sigma^{X,M}U^{\top}.$ Thus $\tilde{\Theta}^{X,M}=\left(\tilde{\Sigma}^{X,M}\right)^{-1}=U\left(\Sigma^{X,M}\right)^{-1}U^{\top}=U\Theta^{X,M}U^{\top}.$ Therefore, $\tilde{\Theta}^{X,M}_{jl}=U_{j}\Theta^{X,M}_{jl}U^{\top}_{l}$ for all $j,l\in V^{2}$ and, thus, $\|\tilde{\Theta}^{X,M}_{jl}\|_{\text{F}}=\|\Theta^{X,M}_{jl}\|_{\text{F}}$ for all $j,l\in V^{2}$. This implies the final result. ### B.2 Proof of Lemma 3 We first show that $X_{ij},Y_{ij}\in{\rm Span}\left\\{\phi_{j1},\dots,\phi_{jM^{\star}_{j}}\right\\}$ almost surely. Let $M^{X}_{j}=\sup\\{M\in\mathbb{N}^{+}:\lambda^{X}_{jM}>0\\}.$ By Karhunen–Loève theorem, we have $X_{ij}=\sum^{M^{X}_{j}}_{k=1}\langle X_{ij},\phi^{X}_{jk}\rangle\phi^{X}_{jk}$ almost surely. Thus, we have $X_{ij}\in{\rm Span}\left\\{\phi^{X}_{j1},\dots,\phi^{X}_{j,M^{X}_{j}}\right\\}$ almost surely. For any $1\leq k\leq M^{X}_{j}$, we have that $\int_{\mathcal{T}}K_{jj}(s,t)\phi^{X}_{k}(s)\phi^{X}_{k}(t)dsdt\geq\int_{\mathcal{T}}K^{X}_{jj}(s,t)\phi^{X}_{k}(s)\phi^{X}_{k}(t)dsdt=\lambda^{X}_{jk}>0,$ which implies that $\phi^{X}_{k}\in{\rm Span}\left\\{\phi_{j1},\dots,\phi_{jM^{\star}_{j}}\right\\}$. Thus, we have ${\rm Span}\left\\{\phi^{X}_{j1},\dots,\phi^{X}_{j,M^{X}_{j}}\right\\}\subseteq{\rm Span}\left\\{\phi_{j1},\dots,\phi_{jM^{\star}_{j}}\right\\}$ and $X_{ij}\in{\rm Span}\left\\{\phi_{j1},\dots,\phi_{jM^{\star}_{j}}\right\\}$ almost surely. Similarly, we have that $Y_{ij}\in{\rm Span}\left\\{\phi_{j1},\dots,\phi_{jM^{\star}_{j}}\right\\}$ almost surely. Next, we show that $M^{\prime}_{j}=M^{\star}_{j}$ by contradiction. By the definition of $M^{\prime}_{j}$, we have that $M^{\prime}_{j}\leq M^{\star}_{j}$. If $M^{\prime}_{j}\neq M^{\star}_{j}$, then we have $\mathbb{V}^{M^{\prime}_{j}}_{j}\subseteq\mathbb{H}$ such that $M^{\prime}_{j}<M^{\star}_{j}$ and $X_{ij},Y_{ij}\in\mathbb{V}^{M^{\prime}_{j}}_{j}$ almost surely. This implies that there exists $\phi\in{\rm Span}\left\\{\phi_{j1},\dots,\phi_{jM^{\star}_{j}}\right\\}\setminus\mathbb{V}^{M^{\prime}_{j}}_{j}$ such that $\displaystyle\mathbb{E}\left[\left(\langle\phi_{jk}(t),X_{ij}(t)\rangle\right)^{2}\right]=0\quad\text{and}\quad\mathbb{E}\left[\left(\langle\phi_{jk}(t),Y_{ij}(t)\rangle\right)^{2}\right]=0$ $\displaystyle\Rightarrow$ $\displaystyle\int_{\mathcal{T}}K^{X}_{jj}(s,t)\phi_{jk}(s)\phi_{jk}(t)dsdt=0\quad\text{and}\quad\int_{\mathcal{T}}K^{Y}_{jj}(s,t)\phi_{jk}(s)\phi_{jk}(t)dsdt=0$ $\displaystyle\Rightarrow$ $\displaystyle\int_{\mathcal{T}}K_{jj}(s,t)\phi_{jk}(s)\phi_{jk}(t)dsdt=0,$ $\displaystyle\Rightarrow$ $\displaystyle\lambda_{jk}=0,$ which contradicts the definition of $M^{\star}_{j}$. Thus, we must have $M^{\prime}_{j}=M^{\star}_{j}$. ### B.3 Proof of Lemma 4 Let $U=V\backslash\\{j,l\\}$, and $a^{X,M}_{U}=\left((a^{X,M}_{j})^{\top},j\in U\right)^{\top}$. Without loss of generality, assume that $\Sigma^{X,M}$ and $\Theta^{X,M}$ take the following block structure: $\displaystyle\Sigma^{X,M}=\left[\begin{matrix}\Sigma^{X,M}_{jj}&\Sigma^{X,M}_{jl}&\Sigma^{X,M}_{jU}\\\ \Sigma^{X,M}_{lj}&\Sigma^{X,M}_{ll}&\Sigma^{X,M}_{lU}\\\ \Sigma^{X,M}_{Uj}&\Sigma^{X,M}_{Ul}&\Sigma^{X,M}_{UU}\\\ \end{matrix}\right],\quad\Theta^{X,M}=\left[\begin{matrix}\Theta^{X,M}_{jj}&\Theta^{X,M}_{jl}&\Theta^{X,M}_{jU}\\\ \Theta^{X,M}_{lj}&\Theta^{X,M}_{ll}&\Theta^{X,M}_{lU}\\\ \Theta^{X,M}_{Uj}&\Theta^{X,M}_{Ul}&\Theta^{X,M}_{UU}\\\ \end{matrix}\right].$ Let $P$ denote the submatrix: $P=\left[\begin{matrix}\Theta^{X,M}_{jj}&\Theta^{X,M}_{jl}\\\ \Theta^{X,M}_{lj}&\Theta^{X,M}_{ll}\end{matrix}\right].$ By standard results for the multivariate Gaussian (Heckler, 2005), we have $\displaystyle\mathrm{Var}\left(a^{X,M}_{j}\mid a^{X,M}_{k},k\neq j\right)=H^{X,M}_{jj}=(\Theta^{X,M}_{jj})^{-1},$ $\displaystyle\mathrm{Var}\left(\left[\begin{matrix}a^{X,M}_{j}\\\ a^{X,M}_{l}\end{matrix}\right]\mid a^{X,M}_{U}\right)=P^{-1}=\left[\begin{matrix}(P^{-1})_{11}&(P^{-1})_{12}\\\ (P^{-1})_{21}&(P^{-1})_{22}\end{matrix}\right].$ Thus, the first statement directly follows from the first equation. To prove the second statement, we only need to note that $\displaystyle H^{X,M}_{jl}$ $\displaystyle=\mathrm{Cov}\left(a^{X,M}_{j},a^{X,M}_{l}\mid a^{X,M}_{U}\right)$ $\displaystyle=(P^{-1})_{12}$ $\displaystyle=-(\Theta^{X,M}_{jj})^{-1}\Theta^{X,M}_{jl}(P^{-1})_{22}$ $\displaystyle=-H^{X,M}_{jj}\Theta_{jl}^{X,M}H^{\backslash j,X,M}_{ll},$ where the second to last equation follows from the $2\times 2$ block matrix inverse and the last equation follows from the property of multivariate Gaussian. This completes the proof. ### B.4 Proof of Theorem 1 We provide the proof of Theorem 1, following the framework introduced in Negahban et al. (2012). We start by introducing some notation. We use $\otimes$ to denote the Kronecker product. For $\Delta\in\mathbb{R}^{pM\times pM}$, let $\theta=\operatorname{vec}(\Delta)\in{\mathbb{R}^{p^{2}M^{2}}}$ and $\theta^{*}=\operatorname{vec}({\Delta^{M}})$, where $\Delta^{M}$ is defined in Section 2.2. Let $\mathcal{G}=\\{G_{t}\\}_{t=1,\ldots,N_{\mathcal{G}}}$ be a set of indices, where $N_{\mathcal{G}}=p^{2}$ and $G_{t}\subset\\{1,2,\cdots,p^{2}M^{2}\\}$ is the set of indices for $\theta$ that correspond to the $t$-th $M\times M$ submatrix of $\Delta^{M}$. Thus, if $t=(j-1)p+l$, then $\theta_{G_{t}}=\operatorname{vec}{(\Delta_{jl})}\in{\mathbb{R}^{M^{2}}}$, where $\Delta_{jl}$ is the $(j,l)$-th $M\times{M}$ submatrix of $\Delta$. Denote the group indices of $\theta^{*}$ that belong to blocks corresponding to $E_{\Delta}$ as $S_{\mathcal{G}}\subseteq{\\{1,2,\cdots,N_{\mathcal{G}}\\}}$. Note that we define $S_{\mathcal{G}}$ using $E_{\Delta}$ and not $E_{\Delta^{M}}$. Therefore, as stated in Assumption 2, $|S_{\mathcal{G}}|=s$. We further define the subspace $\mathcal{M}$ as ${}\mathcal{M}\coloneqq{\\{\theta\in{\mathbb{R}^{p^{2}M^{2}}}\mid\theta_{G_{t}}=0\text{ for all }t\notin{S_{\mathcal{G}}}\\}}.$ (B.1) Its orthogonal complement with respect to the Euclidean inner product is $\mathcal{M}^{\bot}\coloneqq{\\{\theta\in{\mathbb{R}^{p^{2}M^{2}}}\mid\theta_{G_{t}}=0\text{ for all }t\in{S_{\mathcal{G}}}\\}}.$ (B.2) For a vector $\theta$, let $\theta_{\mathcal{M}}$ and $\theta_{\mathcal{M}^{\bot}}$ be the projection of $\theta$ on the subspaces $\mathcal{M}$ and $\mathcal{M}^{\bot}$, respectively. Let $\langle\cdot,\cdot\rangle$ represent the Euclidean inner product. Let ${}\mathcal{R}(\theta)\coloneqq{\sum_{t=1}^{N_{\mathcal{G}}}|\theta_{G_{t}}|_{2}}\triangleq{|\theta|_{1,2}}.$ (B.3) For any $v\in{\mathbb{R}^{p^{2}M^{2}}}$, the dual norm of $\mathcal{R}$ is given by ${}\mathcal{R}^{*}(v)\coloneqq\sup_{u\in{\mathbb{R}^{p^{2}M^{2}}\backslash{\\{0\\}}}}\frac{\langle{u},{v}\rangle}{\mathcal{R}(u)}=\sup_{\mathcal{R}(u)\leq{1}}\langle{u},{v}\rangle.$ (B.4) The subspace compatibility constant of $\mathcal{M}$ with respect to $\mathcal{R}$ is defined as ${}\Psi(\mathcal{M})\coloneqq{\sup_{u\in{\mathcal{M}\backslash\\{0\\}}}}\frac{\mathcal{R}(u)}{|u|_{2}}.$ (B.5) Proof By Lemma 5 and Assumption 1, we have $|(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{Y,M}\otimes{\Sigma^{X,M}})|_{\infty}\leq\delta_{n}^{2}+2\delta_{n}\sigma_{\max}$ (B.6) and $|\operatorname{vec}{(S^{Y,M}-S^{X,M})}-\operatorname{vec}{(\Sigma^{Y,M}-\Sigma^{X,M})}|_{\infty}\leq 2\delta_{n}.$ (B.7) Problem (25) can be written in the following form: $\hat{\theta}_{\lambda_{n}}\in\operatorname*{arg\,min}_{\theta\in{\mathbb{R}^{p^{2}M^{2}}}}\mathcal{L}(\theta)+\lambda_{n}\mathcal{R}(\theta),$ (B.8) where ${}\mathcal{L}(\theta)=\frac{1}{2}\theta^{\top}(S^{Y,M}\otimes{S^{X,M}})\theta-\theta^{\top}\operatorname{vec}({S^{Y,M}-S^{X,M}}).$ (B.9) The loss $\mathcal{L}(\theta)$ is convex and differentiable with respect to $\theta$, and it can be easily verified that $\mathcal{R}(\cdot)$ defines a vector norm. For $h\in\mathbb{R}^{p^{2}M^{2}}$, the error of the first-order Taylor series expansion of $\mathcal{L}$ is: $\displaystyle\delta{\mathcal{L}}(h,\theta^{*})\coloneqq\mathcal{L}(\theta^{*}+h)-\mathcal{L}(\theta^{*})-\langle\nabla\mathcal{L}(\theta^{*}),h\rangle=\frac{1}{2}h^{\top}(S^{Y,M}\otimes{S^{X,M}})h.$ (B.10) From (B.9), we see that $\nabla{\mathcal{L}}(\theta)=(S^{Y,M}\otimes{S^{X,M}})\theta-\operatorname{vec}({S^{Y,M}-S^{X,M}})$. By Lemma 9, we have ${}\mathcal{R}^{*}(\nabla{\mathcal{L}}(\theta^{*}))=\max_{t=1,2,\cdots,N_{\mathcal{G}}}\left|\left[(S^{Y,M}\otimes{S^{X,M}})\theta^{*}-\operatorname{vec}({S^{Y,M}-S^{X,M}})\right]_{G_{t}}\right|_{2}.$ (B.11) We now establish an upper bound for $\mathcal{R}^{*}(\nabla{\mathcal{L}}(\theta^{*}))$. First, note that $(\Sigma^{Y,M}\otimes{\Sigma^{X,M}})\theta^{*}-\operatorname{vec}({\Sigma^{Y,M}-\Sigma^{X,M}})=\operatorname{vec}({\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M}-(\Sigma^{Y,M}-\Sigma^{X,M})})=0.$ Letting $(\cdot)_{jl}$ denote the $(j,l)$-th submatrix, we have $\displaystyle\left|\left[(S^{Y,M}\otimes{S^{X,M}})\theta^{*}-\operatorname{vec}({S^{Y,M}-S^{X,M}})\right]_{G_{t}}\right|_{2}$ (B.12) $\displaystyle=\left|\left[(S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}})\theta^{*}-\operatorname{vec}{((S^{Y,M}-\Sigma^{Y,M})-(S^{X,M}-\Sigma^{X,M}))}\right]_{G_{t}}\right|_{2}$ $\displaystyle={\|(S^{X,M}\Delta^{M}S^{Y,M}-\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M})_{jl}-(S^{Y,M}-\Sigma^{Y,M})_{jl}-(S^{X,M}-\Sigma^{X,M})_{jl}\|_{F}}$ $\displaystyle\leq{\|(S^{X,M}\Delta^{M}S^{Y,M}-\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M})_{jl}\|_{F}+\|(S^{Y,M}-\Sigma^{Y,M})_{jl}\|_{F}+\|(S^{X,M}-\Sigma^{X,M})_{jl}\|_{F}}.$ For any $M\times{M}$ matrix $A$, $\|A\|_{F}\leq{M|A|_{\infty}}$, so $\displaystyle\left|\left[(S^{Y,M}\otimes{S^{X,M}})\theta^{*}-\operatorname{vec}({S^{Y,M}-S^{X,M}})\right]_{G_{t}}\right|_{2}$ $\displaystyle\leq M\left[\left|(S^{X,M}\Delta^{M}S^{Y,M}-\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M})_{jl}\right|_{\infty}+\left|(S^{Y,M}-\Sigma^{Y,M})_{jl}\right|_{\infty}+\left|(S^{X,M}-\Sigma^{X,M})_{jl}\right|_{\infty}\right]$ $\displaystyle\leq M\left[\left|S^{X,M}\Delta^{M}S^{Y,M}-\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M}\right|_{\infty}+|S^{Y,M}-\Sigma^{Y,M}|_{\infty}+|S^{X,M}-\Sigma^{X,M}|_{\infty}\right].$ For any $A\in{\mathbb{R}^{k\times{k}}}$ and $v\in{\mathbb{R}^{k}}$, we have $|Av|_{\infty}\leq{|A|_{\infty}|v|_{1}}$. Thus, we further have $\displaystyle|S^{X,M}\Delta^{M}S^{Y,M}-\Sigma^{X,M}\Delta^{M}\Sigma^{Y,M}|_{\infty}$ $\displaystyle=|[(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{X,M}\otimes{\Sigma^{Y,M}})]\operatorname{vec}{(\Delta^{M})}|_{\infty}$ $\displaystyle\leq{|(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{X,M}\otimes{\Sigma^{Y,M}})|_{\infty}}|\operatorname{vec}{(\Delta^{M})}|_{1}$ $\displaystyle=|(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{X,M}\otimes{\Sigma^{Y,M}})|_{\infty}|\Delta^{M}|_{1}.$ Combining the inequalities gives an upper bound uniform over $\mathcal{G}$ (i.e., for all $G_{t}$): $\displaystyle\left|\left[(S^{Y,M}\otimes{S^{X,M}})\theta^{*}-\operatorname{vec}({S^{Y,M}-S^{X,M}})\right]_{G_{t}}\right|_{2}$ $\displaystyle\leq M\left[|(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{X,M}\otimes{\Sigma^{Y,M}})|_{\infty}|\Delta^{M}|_{1}+|S^{Y,M}-\Sigma^{Y,M}|_{\infty}+|S^{X,M}-\Sigma^{X,M}|_{\infty}\right],$ which implies $\displaystyle\mathcal{R}^{*}\left(\nabla{\mathcal{L}}(\theta^{*})\right)\leq$ $\displaystyle M[|(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{X,M}\otimes{\Sigma^{Y,M}})|_{\infty}|\Delta^{M}|_{1}+$ (B.13) $\displaystyle|S^{Y,M}-\Sigma^{Y,M}|_{\infty}+|S^{X,M}-\Sigma^{X,M}|_{\infty}].$ Assuming $|S^{X,M}-\Sigma^{X,M}|_{\infty}\leq{\delta_{n}}$ and $|S^{Y,M}-\Sigma^{Y,M}|_{\infty}\leq\delta_{n}$ implies ${}\mathcal{R}^{*}\left(\nabla{\mathcal{L}}(\theta^{*})\right)\leq{M[(\delta_{n}^{2}+2\delta_{n}\sigma_{\max})|\Delta^{M}|_{1}+2\delta_{n}]}.$ (B.14) Setting ${}\lambda_{n}=2M\left[\left(\delta_{n}^{2}+2\delta_{n}\sigma_{\max}\right)\left|\Delta^{M}\right|_{1}+2\delta_{n}\right],$ (B.15) then implies that $\lambda_{n}\geq{2\mathcal{R}^{*}\left(\nabla{\mathcal{L}}(\theta^{*})\right)}$. Thus, invoking Lemma 1 in Negahban et al. (2012), $h=\hat{\theta}_{\lambda_{n}}-\theta^{*}$ must satisfy ${}\mathcal{R}(h_{\mathcal{M}^{\bot}})\leq{3\mathcal{R}(h_{\mathcal{M}})}+4\mathcal{R}(\theta^{*}_{\mathcal{M}^{\bot}}),$ (B.16) where $\mathcal{M}$ is defined in (B.1). Equivalently, ${}|h_{\mathcal{M}^{\bot}}|_{1,2}\leq{3|h_{\mathcal{M}}|_{1,2}}+4|\theta^{*}_{\mathcal{M}^{\bot}}|_{1,2}.$ (B.17) By the definition of $\nu_{2}$, we have ${}|\theta^{*}_{\mathcal{M}^{\bot}}|_{1,2}=\sum_{t\notin{\mathcal{S}_{\mathcal{G}}}}|\theta^{*}_{G_{t}}|_{2}\leq\left(p(p+1)/2-s\right)\nu_{2}\leq p^{2}\nu_{2}.$ (B.18) Next, we show that $\delta\mathcal{L}(h,\theta^{*})$, as defined in (B.10), satisfies the Restricted Strong Convexity property defined in definition 2 in Negahban et al. (2012). That is, we show an inequality of the form: $\delta\mathcal{L}(h,\theta^{*})\geq{\kappa_{\mathcal{L}}|h|^{2}_{2}}-\omega^{2}_{\mathcal{L}}\left(\theta^{*}\right)$ whenever $h$ satisfies (B.17). By using Lemma 7, we have $\displaystyle\theta^{\top}(S^{Y,M}\otimes{S^{X,M}})\theta$ $\displaystyle=\theta^{\top}(\Sigma^{Y,M}\otimes{\Sigma^{X,M}})\theta+\theta^{\top}(S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}})\theta$ $\displaystyle\geq{\theta^{\top}(\Sigma^{Y,M}\otimes{\Sigma^{X,M}})\theta-|\theta^{\top}(S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}})\theta|}$ $\displaystyle\geq{\lambda^{*}_{\min}}|\theta|^{2}_{2}-M^{2}|S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}}|_{\infty}|\theta|^{2}_{1,2},$ where the last inequality holds because Lemma 7 and $\lambda^{*}_{\min}=\lambda_{\min}(\Sigma^{X,M})\times{\lambda_{\min}(\Sigma^{Y,M})}=\lambda_{\min}(\Sigma^{Y,M}\otimes{\Sigma^{X,M}})>0$. Thus, $\displaystyle\delta\mathcal{L}(h,\theta^{*})$ $\displaystyle=\frac{1}{2}h^{\top}(S^{Y,M}\otimes{S^{X,M}})h$ $\displaystyle\geq{\frac{1}{2}\lambda^{*}_{\min}}|h|^{2}_{2}-\frac{1}{2}M^{2}|S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}}|_{\infty}|h|^{2}_{1,2}.$ By Lemma 8 and (B.17), we have $\displaystyle|h|^{2}_{1,2}$ $\displaystyle=(|h_{\mathcal{M}}|_{1,2}+|h_{\mathcal{M}^{\bot}}|_{1,2})^{2}\leq 16({|h_{\mathcal{M}}|_{1,2}}+|\theta^{*}_{\mathcal{M}^{\bot}}|_{1,2})^{2}$ $\displaystyle\leq 16(\sqrt{s}|h|_{2}+p^{2}\nu_{2})^{2}\leq 32s|h|^{2}_{2}+32p^{4}\nu_{2}^{2}.$ Combining with the equation above, we get $\displaystyle\delta\mathcal{L}(h,\theta^{*})$ $\displaystyle\geq{\left[\frac{1}{2}\lambda^{*}_{\min}-16M^{2}s|S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}}|_{\infty}\right]}|h|^{2}_{2}$ (B.19) $\displaystyle\qquad\qquad-16M^{2}p^{4}\nu_{2}^{2}|S^{Y,M}\otimes{S^{X,M}}-\Sigma^{Y,M}\otimes{\Sigma^{X,M}}|_{\infty}$ $\displaystyle\geq\left[\frac{1}{2}\lambda^{*}_{\min}-8M^{2}s\left(\delta^{2}_{n}+2\delta^{2}_{n}\sigma_{\max}\right)\right]|h|^{2}_{2}$ $\displaystyle\qquad\qquad-16M^{2}p^{4}\nu_{2}^{2}\left(\delta^{2}_{n}+2\delta_{n}\sigma_{\max}\right).$ Thus, appealing to (B.6), the Restricted Strong Convexity property holds with $\displaystyle\kappa_{\mathcal{L}}$ $\displaystyle\;=\;\frac{1}{2}\lambda^{*}_{\min}-8M^{2}s\left(\delta^{2}+2\delta_{n}\sigma_{\max}\right),$ (B.20) $\displaystyle\omega_{\mathcal{L}}$ $\displaystyle\;=\;4Mp^{2}\nu_{2}\sqrt{\delta_{n}^{2}+2\delta_{n}\sigma_{\max}}.$ When $\delta_{n}<\frac{1}{4}\sqrt{\frac{\lambda^{*}_{\min}+16M^{2}s(\sigma_{\max})^{2}}{M^{2}s}}-\sigma_{\max}$ as we assumed in the theorem, then $\kappa_{\mathcal{L}}>0$. By Theorem 1 of Negahban et al. (2012) and Lemma 8, letting $\lambda_{n}=2M\left[\left(\delta_{n}^{2}+2\delta_{n}\sigma_{\max}\right)|\Delta^{M}|_{1}+2\delta_{n}\right]$, as in (B.15), ensures $\displaystyle\|\hat{\Delta}^{M}-\Delta^{M}\|^{2}_{F}$ $\displaystyle=|\hat{\theta}_{\lambda_{n}}-\theta^{*}|^{2}_{2}$ (B.21) $\displaystyle\leq{9\frac{\lambda^{2}_{n}}{\kappa^{2}_{\mathcal{L}}}}\Psi^{2}(\mathcal{M})+\frac{\lambda_{n}}{\kappa_{\mathcal{L}}}\left(2\omega^{2}_{\mathcal{L}}+4\mathcal{R}(\theta^{*}_{\mathcal{M}^{\bot}})\right)$ $\displaystyle=\frac{9\lambda^{2}_{n}s}{\kappa^{2}_{\mathcal{L}}}+\frac{2\lambda_{n}}{\kappa_{\mathcal{L}}}(\omega^{2}_{\mathcal{L}}+2p^{2}\nu_{2})$ $\displaystyle=\Gamma^{2}_{n}.$ We then prove that $\hat{E}_{\Delta}=E_{\Delta}$. Recall that we have assumed that $0<\Gamma_{n}<\tau/2=(\nu_{1}-\nu_{2})/2$ and $\nu_{2}+\Gamma_{n}\leq\epsilon_{n}<\nu_{1}-\Gamma_{n}$. Note that we have $\|\hat{\Delta}^{M}_{jl}-\Delta^{M}_{jl}\|_{F}\leq{\|\hat{\Delta}^{M}-\Delta^{M}\|_{F}}\leq\Gamma_{n}$ for any $(j,l)\in{V^{2}}$. Recall that ${}E_{\Delta}\;=\;\\{(j,l)\in{V^{2}}:\;j\neq{l},D_{jl}>0\\}.$ (B.22) We first prove that $E_{\Delta}\subseteq{\hat{E}_{{\Delta}}}$. For any $(j,l)\in{E_{\Delta}}$, by the definition of $\nu_{1}$ in Section 4.1, we have $\displaystyle\|\hat{\Delta}^{M}_{jl}\|_{F}$ $\displaystyle\geq{\|\Delta^{M}_{jl}\|_{F}-\|\hat{\Delta}_{jl}^{M}-\Delta^{M}_{jl}\|_{F}}$ $\displaystyle\geq\nu_{1}-\Gamma_{n}$ $\displaystyle>\epsilon_{n}.$ The last inequality holds because we have assumed that $\epsilon_{n}<\nu_{1}-\Gamma_{n}$. Thus, by definition of $\hat{E}_{{\Delta}}$ in (27), we have $(j,l)\in{\hat{E}_{{\Delta}}}$, which further implies that $E_{\Delta}\subseteq{\hat{E}_{{\Delta}}}$. We then show $\hat{E}_{{\Delta}}\subseteq{E_{\Delta}}$. Let $\hat{E}^{c}_{{\Delta}}$ and $E^{c}_{\Delta}$ denote the complement set of $\hat{E}_{{\Delta}}$ and $E_{\Delta}$. For any $(j,l)\in{E^{c}_{\Delta}}$, which also means that $(l,j)\in{E^{c}_{\Delta}}$, by definition of $\nu_{2}$, we have that $\displaystyle\|\hat{\Delta}^{M}_{jl}\|_{F}$ $\displaystyle\leq{\|\Delta^{M}_{jl}\|_{F}+\|\hat{\Delta}_{jl}^{M}-\Delta^{M}_{jl}\|_{F}}$ $\displaystyle\leq\nu_{2}+\Gamma_{n}$ $\displaystyle\leq\epsilon_{n}.$ Again, the last inequality holds because because we have assumed that $\epsilon_{n}\geq\nu_{2}+\Gamma_{n}$. Thus, by definition of $\hat{E}_{{\Delta}}$, we have $(j,l)\notin{\hat{E}_{{\Delta}}}$ or $(j,l)\in{\hat{E}^{c}_{{\Delta}}}$. This implies that $E^{c}_{\Delta}\subseteq{\hat{E}^{c}_{{\Delta}}}$, or $\hat{E}_{{\Delta}}\subseteq{E_{\Delta}}$. Combing with previous conclusion that $E_{\Delta}\subseteq{\hat{E}_{{\Delta}}}$, the proof is complete. [2mm] ### B.5 Proof of Theorem 4 We only need to prove that $\displaystyle P\left(\lvert S^{M}-\Sigma^{M}\rvert_{\infty}>\delta\right)$ $\displaystyle\leq C_{1}np\exp\\{-C_{2}\Phi(T,L)M^{-(1+\beta)}\delta\\}$ (B.23) $\displaystyle+C_{3}(pM)^{2}\exp\\{-C_{4}nM^{-2(1+\beta)}\delta^{2}\\}$ $\displaystyle+C_{5}npL\exp\left\\{-\frac{C_{6}M^{-2(1+\beta)}\delta^{2}}{\tilde{\psi}_{2}(T,L)}\right\\},$ where $S^{M}$ can be understood as either $S^{X,M}$ or $S^{Y,M}$ and $\Sigma^{M}$ can be understood as either $\Sigma^{X,M}$ or $\Sigma^{Y,M}$, with $C_{k}=C^{X}_{k}$ or $C_{k}=C^{Y}_{k}$ for $k=1,2,3,4$ accordingly. To see that (B.23) implies (43), we first note that (B.23) implies that $\displaystyle P\left(\lvert S^{X,M}-\Sigma^{X,M}\rvert_{\infty}\leq\delta\,\text{and}\,\lvert S^{Y,M}-\Sigma^{Y,M}\rvert_{\infty}\leq\delta\right)$ $\displaystyle\geq$ $\displaystyle 1-P\left(\lvert S^{X,M}-\Sigma^{X,M}\rvert_{\infty}>\delta\right)-P\left(\lvert S^{Y,M}-\Sigma^{Y,M}\rvert_{\infty}>\delta\right)$ $\displaystyle\geq$ $\displaystyle 1-C^{X}_{1}pM\exp\\{-C^{X}_{2}\Phi(T,L)M^{-(1+\beta)}\delta\\}-C^{X}_{3}(pM)^{2}\exp\\{-C^{X}_{4}nM^{-2(1+\beta)}\delta^{2}\\}-$ $\displaystyle C^{Y}_{1}pM\exp\\{-C^{Y}_{2}\Phi(T,L)M^{-(1+\beta)}\delta\\}-C^{Y}_{3}(pM)^{2}\exp\\{-C^{Y}_{4}nM^{-2(1+\beta)}\delta^{2}\\}$ $\displaystyle\geq$ $\displaystyle 1-2\bar{C}_{1}pM\exp\\{-\bar{C}_{2}\Phi(T,L)M^{-(1+\beta)}\delta\\}-2\bar{C}_{3}(pM)^{2}\exp\\{-\bar{C}_{4}nM^{-2(1+\beta)}\delta^{2}\\},$ where $\bar{C}_{k}$ for $k=1,2,3,4$ are defined in Theorem 4. Thus, by letting the last two terms in the last line of the above equation all to be $\iota/2$, we then have (43). This way, the rest of the proof will focus on proving (B.23). Denote $(j,l)$-th submatrix of $S^{M}$ as $S^{M}_{jl}$, and $(k,m)$-th entry of $S^{M}_{jl}$ as $\hat{\sigma}_{jl,km}$, thus we have $S^{M}=(\hat{\sigma}_{jl,km})_{1\leq j,l\leq p,\leq k,m\leq M}$; similarly, let $\Sigma^{M}=(\sigma_{jl,km})_{1\leq j,l\leq p,\leq k,m\leq M}$. Then, by the definition of $S^{M}$ and $\Sigma^{M}$, we have $\displaystyle\hat{\sigma}_{jl,km}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\hat{a}_{ijk}\hat{a}_{ilm}$ $\displaystyle\sigma_{jl,km}$ $\displaystyle=\mathbb{E}\left[a_{ijk}a_{ilm}\right].$ Note that $\displaystyle\hat{a}_{ijk}$ $\displaystyle=\langle\hat{g}_{ij},\hat{\phi}_{jk}\rangle$ $\displaystyle=\langle g_{ij}+\hat{g}_{ij}-g_{ij},\phi_{jk}+\hat{\phi}_{jk}-\phi_{jk}\rangle$ $\displaystyle=\langle g_{ij},\phi_{jk}\rangle+\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle$ $\displaystyle=a_{ijk}+\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle.$ Thus, we have $\displaystyle\hat{\sigma}_{jl,km}-\sigma_{jl,km}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\left(\hat{a}_{ijk}\hat{a}_{ilm}-\sigma_{jl,km}\right)$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\left[a_{ijk}+\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\right]\times$ $\displaystyle\left[a_{ijk}+\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle+\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\right]-\sigma_{jl,km}$ $\displaystyle=\sum^{16}_{u=1}I_{u},$ where $\displaystyle I_{1}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\left(a_{ijk}a_{ilm}-\mathbb{E}(a_{ijk}a_{ilm})\right),$ $\displaystyle I_{2}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}a_{ijk}\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle,$ $\displaystyle I_{3}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}a_{ijk}\langle g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$ $\displaystyle I_{4}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}a_{ijk}\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$ $\displaystyle I_{5}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}a_{ilm}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle,$ $\displaystyle I_{6}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle,$ $\displaystyle I_{7}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$ $\displaystyle I_{8}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$ $\displaystyle I_{9}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle a_{ilm},$ $\displaystyle I_{10}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle,$ $\displaystyle I_{11}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$ $\displaystyle I_{12}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$ $\displaystyle I_{13}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle a_{ilm},$ $\displaystyle I_{14}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle,$ $\displaystyle I_{15}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle,$ $\displaystyle I_{16}$ $\displaystyle=\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle.$ Note that $I_{u}$, $u=1,\ldots,16$ depend on $j,l,k,m$. To simplify the notation, we do not denote this fact explicitly. Thus, for any $0<\delta\leq 1$, when for any $1\leq j,l\leq p$ and $1\leq k,m\leq M$, if $\lvert I_{u}\rvert\leq\delta/16$, $u=1,\ldots,16$, we will have $\lvert S^{M}-\Sigma^{M}\rvert_{\infty}\leq\delta$. This way, for the rest of the paper, we only need to calculate the probability of $\lvert I_{u}\rvert\leq\delta/16$, $u=1,\ldots,16$, $1\leq j,l\leq p$ and $1\leq k,m\leq M$. Before we proceed to calculate the probability, we need a bit more notation. By Assumption 3 (i), we have constants $d_{1},d_{2}>0$, such that $\lambda_{jk}\leq d_{1}k^{-\beta}$, $d_{jk}\leq d_{2}k^{1+\beta}$ for any $j=1,\ldots,p$ and $k\geq 1$. Let $d_{0}=\max\\{1,\sqrt{d_{1}},d_{2}\\}$, let $\xi_{ijk}=\lambda^{-1/2}_{jk}a_{ijk}$ so that $\xi_{ijk}\sim N(0,1)$ i.i.d. for $i=1,\ldots,n$, and denote $\displaystyle\delta_{1}$ $\displaystyle=\frac{\delta}{144d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}},$ (B.24) $\displaystyle\delta_{2}$ $\displaystyle=9\lambda_{0,\max}\delta_{1}=\frac{\delta}{16d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}},$ where $\lambda_{0,\max}=\max_{j\in V}\sum^{\infty}_{k=1}\lambda_{jk}$. Recall that $\hat{K}_{jj}$, $j=1,\ldots,p$ are defined as in (24). We define five events $A_{1}$-$A_{5}$ as below: $\displaystyle A_{1}$ $\displaystyle:\;\lVert\hat{g}_{ij}-g_{ij}\rVert\leq\delta_{1},\quad\forall i=1,\ldots,n\ \forall j=1,\ldots,p,$ (B.25) $\displaystyle A_{2}$ $\displaystyle:\;\lVert\hat{K}_{jj}-K_{jj}\rVert_{\text{HS}}\leq\delta_{2}\quad\forall j=1,\ldots,p,$ $\displaystyle A_{3}$ $\displaystyle:\;\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}\leq\frac{3}{2}\quad\forall j=1,\ldots,p\ \forall k=1,\ldots,M,$ $\displaystyle A_{4}$ $\displaystyle:\;\frac{1}{n}\sum^{n}_{i=1}\lVert g_{ij}\rVert^{2}\leq 2\lambda_{0,\max}\quad\forall j=1,\ldots,p,$ $\displaystyle A_{5}$ $\displaystyle:\;\lvert\frac{1}{n}\sum^{n}_{i=1}a_{ijk}a_{ilm}-\sigma_{jl,km}\rvert\leq\frac{\delta}{16}\quad\forall 1\leq j,l\leq\ 1\leq k,m\leq M.$ Without loss of generality, we assume that $\langle\hat{\phi}_{jl},\phi_{jl}\rangle\geq 0$ for any $1\leq j\leq p$ and $1\leq k\leq M$ (If this is not true, we only need to use $-\phi_{jl}$ to substitute $\phi_{jl}$). Then, by Lemma 10-Lemma 25, when $A_{1}$-$A_{5}$ hold simultaneously, we have $\lvert I_{u}\rvert\leq\delta/16$ for all $u=1,\ldots,16$, $1\leq j,l\leq p$ and $\ 1\leq k,m\leq M$. This way, we have $\displaystyle P\left(\lvert S^{M}-\Sigma^{M}\rvert_{\infty}\leq\delta\right)$ $\displaystyle\geq P\left(\lvert I_{u}\rvert\leq\delta/16,\;\text{for all}\;1\leq u\leq 16,1\leq j,l\leq\ 1\leq k,m\leq M\right)$ $\displaystyle\geq P\left(\bigcap^{5}_{w=1}A_{w}\right).$ Or equivalently, $P\left(\lvert S^{M}-\Sigma^{M}\rvert_{\infty}>\delta\right)\leq P\left(\bigcup^{5}_{w=1}\bar{A}_{w}\right)\leq\sum^{5}_{w=1}P\left(\bar{A}_{w}\right),$ (B.26) where the last inequality follows Boole’s inequality, and $\bar{A}$ means the complement of $A$. This way, we then only need to give an upper bound for $P(\bar{A}_{w})$, $w=1,\ldots,5$. The $P(\bar{A}_{1})$ follows directly from Theorem 5. Note that by Theorem 5 and definition of $\tilde{\psi}_{1}$-$\tilde{\psi}_{4}$, we have $\displaystyle P(\bar{A}_{1})=$ $\displaystyle P\left(\lVert\hat{g}_{ij}-g_{ij}\rVert>\delta_{1}\;\exists 1\leq i\leq n,1\leq j\leq p\right)$ $\displaystyle\leq$ $\displaystyle 2(np)\left\\{\exp\left(-\frac{\delta_{1}^{2}}{72\tilde{\psi}^{2}_{1}(T,L)+6\sqrt{2}\tilde{\psi}_{1}(T,L)\delta_{1}}\right)\right.$ $\displaystyle+$ $\displaystyle L\exp\left(-\frac{\delta_{1}^{2}}{\tilde{\psi}_{2}(T,L)}\right)$ $\displaystyle+$ $\displaystyle\left.\exp\left(-\frac{\delta_{1}^{2}}{72\lambda_{0,\max}\tilde{\psi}_{3}(L)+6\sqrt{2\lambda_{0,\max}\tilde{\psi}_{3}(L)}\delta_{1}}\right)\right\\}.$ Let $\gamma_{1}=\sqrt{2}/(12\times 144d^{2}_{0}3\sqrt{3\lambda_{0,\max}})$, and $\gamma_{3}=1/(72\lambda_{0,\max}\times(144d^{2}_{0}\sqrt{3\lambda_{0,\max}})^{2})$, then when $\tilde{\psi}_{1}<\gamma_{1}\cdot\delta/M^{1+\beta}$, and $\tilde{\psi}_{3}<\gamma_{3}\cdot\delta^{2}/M^{2+2\beta}$, we have $72\tilde{\psi}^{2}_{1}<6\sqrt{2}\tilde{\psi}_{1}\delta_{1}$ and $72\lambda_{0,\max}\tilde{\psi}_{3}<6\sqrt{2\lambda_{0,\max}\tilde{\psi}_{3}}\delta_{1}$, which implies that $\displaystyle P(\bar{A}_{1})$ (B.27) $\displaystyle\leq 2np\left\\{\exp\left(-\frac{\delta_{1}}{12\sqrt{2}\tilde{\psi}_{1}(T,L)}\right)+\exp\left(-\frac{\delta_{1}}{12\sqrt{2\lambda_{0,\max}}\sqrt{\tilde{\psi}_{3}(L)}}\right)+L\exp\left(-\frac{\delta_{1}^{2}}{\tilde{\psi}_{2}(T,L)}\right)\right\\}$ $\displaystyle\overset{(i)}{\leq}2np\left\\{\exp\left(-\frac{\delta_{1}}{12\sqrt{2}}\Phi(T,L)\right)+\exp\left(-\frac{\delta_{1}}{12\sqrt{2\lambda_{0,\max}}}\Phi(T,L)\right)+L\exp\left(-\frac{\delta_{1}^{2}}{\tilde{\psi}_{2}(T,L)}\right)\right\\}$ $\displaystyle\overset{(ii)}{\leq}4np\exp\left(-\frac{\delta_{1}}{12\sqrt{2\lambda_{0,\max}}}\Phi(T,L)\right)+2npL\exp\left(-\frac{\delta_{1}^{2}}{\tilde{\psi}_{2}(T,L)}\right)$ $\displaystyle=4np\exp\left(-\frac{1}{1728\sqrt{6}\lambda_{0,\max}d^{2}_{0}}\cdot\frac{\delta}{M^{1+\beta}}\cdot\Phi(T,L)\right)$ $\displaystyle+2npL\exp\left(-\frac{\delta^{2}}{6228d^{4}_{0}\lambda_{0,\max}M^{2+2\beta}\tilde{\psi}_{2}(T,L)}\right),$ where $(i)$ follows the definition of $\Phi(T,L)$ and $(ii)$ follows the fact that $\lambda_{0,\max}>1$. Before we calculate $P(\bar{A}_{2})$, we first compute $P(\bar{A}_{4})$. Note that by Jensen’s inequality, for any two real values $z_{1},z_{2}$ and any positive integer $k$, we have $(z_{1}+z_{2})^{k}\leq\left(|z_{1}|+|z_{2}|\right)^{k}=2^{k}\left(\frac{1}{2}|z_{1}|+\frac{1}{2}|z_{2}|\right)^{k}\leq 2^{k-1}\left(|z_{1}|+|z_{2}|\right),$ where the last line is because Jensen’s inequality with convex function $\varphi(t)=t^{k}$, $k$ is a positive integer. Since for any $i=1,\ldots,n$ and $j=1,2\dots,p$, we have $\mathbb{E}[\|g_{ij}\|^{2}]=\lambda_{j0}$. Then, by Jensen’s inequality and Lemma 31, for any $k\geq 2$, we have $\displaystyle\mathbb{E}\left[\left(\|g_{ij}\|^{2}-\lambda_{j0}\right)^{k}\right]$ $\displaystyle\leq 2^{k-1}\left(\mathbb{E}\left[\|g_{ij}\|^{2k}+\lambda^{k}_{j0}\right]\right)$ $\displaystyle\leq 2^{k-1}\left((2\lambda_{j0})^{k}k!+\lambda^{k}_{j0}\right)$ $\displaystyle\leq(4\lambda_{j0})^{k}k!,$ where the second inequality is because Lemma 31. Thus, $\sum^{n}_{i=1}\mathbb{E}\left[\left(\|g_{ij}\|^{2}-\lambda_{j0}\right)^{k}\right]\leq\frac{k!}{2}n\times(32\lambda^{2}_{j0})\times(4\lambda_{j0})^{k-2}.$ Then by Lemma 29, for any $\epsilon>0$, we have $P\left(\left|\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|^{2}-\lambda_{j0}\right|>\epsilon\right)\leq 2\exp\left(-\frac{n\epsilon^{2}}{64\lambda^{2}_{j0}+8\lambda_{j0}\epsilon}\right).$ This way, we further get $\displaystyle P\left(\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|^{2}>2\lambda_{0,\max}\right)$ $\displaystyle\leq P\left(\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|^{2}>2\lambda_{j0}\right)$ $\displaystyle\leq P\left(\left|\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|^{2}-\lambda_{j0}\right|>\lambda_{j0}\right)$ $\displaystyle\leq 2\exp\left(-\frac{n}{72}\right).$ Since the above inequality holds for any $j=1,\ldots,p$, we then have $P(\bar{A}_{4})=P\left(\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|^{2}>2\lambda_{0,\max},\;\exists j=1,\ldots,p\right)\leq 2p\exp\left(-\frac{n}{72}\right).$ (B.28) For $P(\bar{A}_{2})$, we first let $\hat{K}^{g}_{jj}(s,t)=\frac{1}{n}\sum^{n}_{i=1}g_{ij}(s)g_{ij}(t),$ for all $j\in V$ and $K_{jj}(s,t)=\mathbb{E}[g_{ij}(s)g_{ij}(t)]$, and also let $A^{\prime}_{2}:\;\lVert\hat{K}^{g}_{jj}-K^{g}_{jj}\rVert_{\text{HS}}\leq\delta_{2}\quad\forall j=1,\ldots,p.$ Note that $\displaystyle\|\hat{K}^{g}_{jj}(s,t)-K^{g}_{jj}(s,t)\|_{\text{HS}}$ $\displaystyle=\left\|\frac{1}{n}\sum^{n}_{i=1}\left[\hat{g}_{ij}(s)-g_{ij}(s)+g_{ij}(s)\right]\left[\hat{g}_{ij}(t)-g_{ij}(t)+g_{ij}(t)\right]-K^{g}_{jj}(s,t)\right\|_{\text{HS}}$ $\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\|\hat{g}_{ij}-g_{ij}\|^{2}+\frac{2}{n}\sum^{n}_{i=1}\|\hat{g}_{ij}-g_{ij}\|\cdot\|g_{ij}\|+\left\|\frac{1}{n}\sum^{n}_{i=1}\left[g_{ij}(s)g_{ij}(t)-K^{g}_{jj}(s,t)\right]\right\|_{\text{HS}}.$ Let $\displaystyle A_{6}$ $\displaystyle:\;\left\|\frac{1}{n}\sum^{n}_{i=1}\left[g_{ij}(s)g_{ij}(t)-K^{g}_{jj}(s,t)\right]\right\|_{\text{HS}}\leq 4\lambda_{0,\max}\delta_{1},\;\forall j=1,\ldots,p.$ We claim that when $A_{1}\cap A_{4}\cap A_{6}\Rightarrow A^{\prime}_{2}$. To prove it, note that by Jensen’s inequality, we have $\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\left\|g_{ij}\right\|^{2}},$ thus, when $A_{4}$ holds, we have $(1/n)\sum^{n}_{i=1}\left\|g_{ij}\right\|\leq\sqrt{2\lambda_{0,\max}}$ for any $j=1,\ldots,p$. This way, when $A_{1}$, $A_{4}$ and $A_{6}$ hold simultaneously, we have $\|\hat{K}^{g}_{jj}(s,t)-K^{g}_{jj}(s,t)\|_{\text{HS}}\leq\delta^{2}_{1}+2\sqrt{2\lambda_{0,\max}}\delta_{1}+4\lambda_{0,\max}\delta_{1}\leq 9\lambda_{0,\max}\delta_{1},$ which is $A_{2}$. This way, we have proved $A_{1}\cap A_{4}\cap A_{6}\Rightarrow A^{\prime}_{2}$, which implies that $\bar{A^{\prime}}_{2}\Rightarrow\bar{A}_{1}\cup\bar{A}_{4}\cup\bar{A}_{6}$, and thus $P(\bar{A^{\prime}}_{2})\leq P(\bar{A}_{1})+P(\bar{A}_{4})+P(\bar{A}_{6})$. $P(\bar{A}_{1})$ has been given by (B.27) and $P(\bar{A}_{4})$ has been given by (B.28), thus we only need to compute $P(\bar{A}_{6})$. By Lemma 32, for any $j=1,\ldots,p$, we have $P\left(\left\|\frac{1}{n}\sum^{n}_{i=1}\left[g_{ij}(s)g_{ij}(t)-K^{g}(s,t)\right]\right\|_{\text{HS}}>4\lambda_{0,\max}\delta_{1}\right)\leq 2\exp\left(-\frac{n\delta^{2}_{1}}{6}\right),$ thus $P(\bar{A}_{6})\leq 2p\exp\left(-\frac{n\delta^{2}_{1}}{6}\right)=2p\exp\left(-\frac{1}{373248d^{4}_{0}\lambda^{2}_{0,\max}}\times n\frac{\delta^{2}}{M^{2+2\beta}}\right).$ (B.29) This way, by combining (B.27), (B.28) and (B.29), we have $\displaystyle P(\bar{A^{\prime}}_{2})\leq$ $\displaystyle 4pM\exp\left(-\frac{1}{1728\sqrt{6}\lambda_{0,\max}d^{2}_{0}}\cdot\frac{\delta}{M^{1+\beta}}\cdot\Phi(T,L)\right)+2p\exp\left(-\frac{n}{72}\right)$ $\displaystyle+$ $\displaystyle 2p\exp\left(-\frac{1}{373248d^{4}_{0}\lambda^{2}_{0,\max}}\times n\frac{\delta^{2}}{M^{2+2\beta}}\right).$ Finally, since $\|\hat{K}_{j}j(s,t)-K_{jj}(s,t)\|_{\text{HS}}\leq\|\hat{K}^{X}_{j}j(s,t)-K^{X}_{jj}(s,t)\|_{\text{HS}}+\|\hat{K}^{Y}_{j}j(s,t)-K^{Y}_{jj}(s,t)\|_{\text{HS}}$, we have $P(\bar{A}_{2})\leq P(\bar{A^{\prime}}_{X,2})+P(\bar{A^{\prime}}_{Y,2})$, where $A^{\prime}_{X,2}$ and $A^{\prime}_{Y,2}$ are defined similarly as $A^{\prime}_{2}$ with $g$ to be $X$ and $Y$. Thus, we have $\displaystyle P(\bar{A}_{2})\leq$ $\displaystyle 8pM\exp\left(-\frac{1}{1728\sqrt{6}\lambda_{0,\max}d^{2}_{0}}\cdot\frac{\delta}{M^{1+\beta}}\cdot\Phi(T,L)\right)+4p\exp\left(-\frac{n}{72}\right)$ $\displaystyle+$ $\displaystyle 4p\exp\left(-\frac{1}{373248d^{4}_{0}\lambda^{2}_{0,\max}}\times n\frac{\delta^{2}}{M^{2+2\beta}}\right).$ For $P(\bar{A}_{3})$, by Page 28-29 of Boucheron et al. (2013), and note that $\sum^{n}_{i=1}\xi^{2}_{ijk}\sim\chi^{2}_{n}$ for any $j=1,\ldots,p$ and $k=1,\ldots,M$, we have that for any $\epsilon>0$, we have $P\left(\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}-1>\epsilon\right)\leq\exp\left(-\frac{n\epsilon^{2}}{4+4\epsilon}\right).$ Thus, by letting $\epsilon=1/2$, we have $P\left(\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}>\frac{3}{2}\right)\leq\exp\left(-\frac{n}{24}\right),$ which implies that $P(\bar{A}_{3})\leq pM\exp\left(-\frac{n}{24}\right).$ (B.30) Finally, for $P(\bar{A}_{5})$, we first claim that for any $\epsilon>0$ and $1\leq j,l\leq p$, $1\leq k,m\leq M$, we have $P\left(\left|\frac{1}{n}\sum^{n}_{i=1}a_{ijk}a_{ilm}-\sigma_{jl,km}\right|>\epsilon\right)\leq 2\exp\left(-\frac{n\epsilon^{2}}{64d^{2}_{0}+8d_{0}\epsilon}\right).$ We now prove this claim. Note that $\displaystyle\mathbb{E}\left[\left(a_{ijk}a_{ilm}-\mathbb{E}(a_{ijk}a_{ilm})\right)^{k}\right]$ $\displaystyle=\lambda^{k/2}_{jk}\lambda^{k/2}_{lm}\mathbb{E}\left[\left(\xi_{ijk}\xi_{ilm}-\mathbb{E}(\xi_{ijk}\xi_{ilm})\right)^{k}\right]$ $\displaystyle\leq d^{k}_{0}\mathbb{E}\left[\left(\xi_{ijk}\xi_{ilm}-\mathbb{E}(\xi_{ijk}\xi_{ilm})\right)^{k}\right],$ and $\displaystyle\mathbb{E}\left[\left(\xi_{ijk}\xi_{ilm}-\mathbb{E}(\xi_{ijk}\xi_{ilm})\right)^{k}\right]$ $\displaystyle\leq 2^{k-1}\left(\mathbb{E}\left[|\xi_{ijk}\xi_{ilm}|^{k}\right]+|\mathbb{E}(\xi_{ijk}\xi_{ilm})|^{k}\right)$ $\displaystyle\leq 2^{k-1}\left(\mathbb{E}[\xi^{2k}_{ij1}]+1\right)$ $\displaystyle\leq 2^{k-1}(2^{k}k!+1)$ $\displaystyle\leq 4^{k}k!,$ thus $\mathbb{E}\left[\left(a_{ijk}a_{ilm}-\mathbb{E}(a_{ijk}a_{ilm})\right)^{k}\right]\leq(4d_{0})^{k}k!.$ The claim then follows directly from Lemma 29. By letting $\epsilon=\delta/16$, $P\left(\left|\frac{1}{n}\sum^{n}_{i=1}a_{ijk}a_{ilm}-\sigma_{jl,km}\right|>\frac{\delta}{16}\right)\leq 2\exp\left(-\frac{n\delta^{2}}{16^{2}\times 64\times d^{2}_{0}+128d_{0}\delta}\right)\leq 2\exp\left(-\frac{n\delta^{2}}{16512d^{2}_{0}}\right)$ holds for any $1\leq j,l\leq p$ and $1\leq k,m\leq M$, which further implies that $P\left(\bar{A}_{5}\right)\leq 2(pM)^{2}\exp\left(-\frac{n\delta^{2}}{16512d^{2}_{0}}\right).$ (B.31) Let $C_{1}=12$, $C_{2}=1/(1728\sqrt{6}\lambda_{0,\max})$, $C_{3}=9$, $C_{4}=1/(373248d^{4}_{0}\lambda^{2}_{0,\max})$, $C_{5}=2$, and $C_{6}=1/(6228d^{4}_{0}\lambda_{0,\max})$, then the final result follows by combining (B.27)-(B.31). ## Appendix C More Theorems In this section, we introduce more theorems along with their proofs. ### C.1 Theorem 5 and Its Proof In this section, we give a non-asymptotic error bound for our basis expansion estimated function. This theorem is used in proving Theorem 4. For a random function $g(t)$, where $t\in\mathcal{T}$, a closed interval of real line, and lying in a separable Hilbert space $\mathbb{H}$, we have noisy discrete observations at time points $t_{1},t_{2},\dots,t_{T}$ generated from the model below: $h_{k}=g(t_{k})+\epsilon_{k},$ (C.1) where $\epsilon_{k}\overset{\text{i.i.d.}}{\sim}N(0,\sigma^{2}_{0})$ for $k=1,\ldots,T$. Let $b(t)=(b_{1}(t),b_{2}(t),\dots,b_{L}(t))^{\top}$ be basis function vector. We use basis expansion to get $\hat{g}(t)=\hat{\beta}^{\top}b(t)$, the estimator of $g(t)$, where $\hat{\beta}\in\mathbb{R}^{L}$ is obtained by minimizing the least square loss: $\hat{\beta}=\operatorname*{arg\,min}_{\beta\in\mathbb{R}^{L}}\sum^{T}_{k=1}\left(\beta^{\top}b(t_{k})-h_{k}\right)^{2}.$ (C.2) We define the design matrix $B$ as $B=\left[\begin{matrix}b_{1}(t_{1})&\cdots&b_{L}(t_{1})\\\ \vdots&\ddots&\vdots\\\ b_{1}(t_{T})&\cdots&b_{L}(t_{T})\end{matrix}\right]\in\mathbb{R}^{T\times L},$ (C.3) so that $\hat{\beta}=\left(B^{\top}B\right)^{-1}B^{\top}h,$ (C.4) where $h=(h_{1},h_{2},\dots,h_{T})^{\top}\in\mathbb{R}^{T}$. We assume that $g(t)=\sum^{\infty}_{m=1}\beta^{*}_{m}b_{m}(t)$, and we can decompose $g(t)$ as $g=g^{\shortparallel}+g^{\bot}$, where $g^{\shortparallel}\in{\rm Span}(b)$ and $g^{\bot}\in{\rm Span}(b)^{\bot}$. Let $\lambda_{0}\coloneqq\mathbb{E}[\|g\|^{2}]$ and $\lambda^{\bot}_{0}\coloneqq\mathbb{E}[\|g^{\bot}\|^{2}]$. Then it is easy to check that $\lambda_{0}=\sum^{\infty}_{m=1}\mathbb{E}[(\beta^{*}_{m})^{2}]$ and $\lambda^{\bot}_{0}=\sum^{\infty}_{m>L}\mathbb{E}[(\beta^{*}_{m})^{2}]$. We assume that the basis functions $\\{b_{l}(t)\\}^{\infty}_{l=1}$ compose a complete orthonormal system (CONS) of $\mathbb{H}$, that is, $\overline{{\rm Span}\left(\\{b_{l}\\}^{\infty}_{l=1}\right)}=\mathbb{H}$ (see Definition 2.4.11 of Hsing and Eubank (2015)), and have continuous derivative functions with $D_{0,b}\coloneqq\sup_{l\geq 1}\sup_{t\in\mathcal{T}}\lvert b_{l}(t)\rvert<\infty,\qquad D_{1,b}(l)\coloneqq\sup_{t\in\mathcal{T}}\lvert b^{\prime}_{l}(t)\rvert<\infty,\qquad D_{1,b,L}\coloneqq\max_{1\leq l\leq L}D_{1,b}(l).$ (C.5) We further assume that the observation time points $\\{t_{k}:1\leq k\leq T\\}$ satisfy $\max_{1\leq k\leq T+1}\left|\frac{t_{k}-t_{(k-1)}}{\lvert\mathcal{T}\rvert}-\frac{1}{T}\right|\leq\frac{\zeta_{0}}{T^{2}},$ (C.6) where $t_{0}$ and $t_{(T+1)}$ are endpoints of $\mathcal{T}$ and $\zeta_{0}$ is a positive constant. Besides, we assume that $\sum^{\infty}_{m=1}\mathbb{E}\left[(\beta^{*}_{m})^{2}\right]D^{2}_{1,b}(m)<\infty$, we then define $\psi_{4}(L)=\sum_{m>L}\mathbb{E}\left[(\beta^{*}_{m})^{2}\right]D^{2}_{1,b}(m).$ Let $\displaystyle\psi_{1}(T,L)$ $\displaystyle=\frac{\sigma_{0}L}{\sqrt{\lambda_{\min}\left(B^{\top}B\right)}},\qquad\psi_{3}(L)=\lambda^{\bot}_{0}/\lambda_{0},$ and $\displaystyle\psi_{2}(T,L)$ $\displaystyle=\frac{1}{(\lambda^{B}_{\min})^{2}}\left(18\lambda_{0}\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b,L}+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]L^{2}\psi_{3}(L)\right.$ $\displaystyle\left.\qquad\qquad+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}L^{2}\psi_{4}(L)\right),$ We then have the following theorem. ###### Theorem 5 For any $\delta>0$, we have $P\left(\lVert g-\hat{g}\rVert>\delta\right)\leq 2\exp\left(-\frac{\delta^{2}}{72\psi^{2}_{1}(T,L)+6\sqrt{2}\psi_{1}(T,L)\delta}\right)+L\exp\left(-\frac{\delta^{2}}{\psi_{2}(T,L)}\right)\\\ +2\exp\left(-\frac{\delta^{2}}{72\lambda_{0}\psi_{3}(L)+6\sqrt{2\lambda_{0}}\sqrt{\psi_{3}(L)}\delta}\right).$ (C.7) Proof Throughout the proof, we often use the technique to first treat $g$ as a fixed function, that is, we consider probability conditioned on $g$, so the only randomness comes from $\epsilon_{k}$, $k=1,\ldots,T$. We will then include the randomness from $g$. Note that since $\epsilon_{k}$ is independent of $g$, thus the conditional distribution of $\epsilon_{k}$ is the same with unconditional distribution. For a fixed $g$, since $\overline{{\rm Span}\left(\\{b_{l}\\}^{\infty}_{l=1}\right)}=\mathbb{H}$, we can assume that $g(t)=\sum^{\infty}_{l=1}\beta^{*}_{l}b_{l}(t)$ where $\beta^{*}_{l}=\langle g,b_{l}\rangle=\int_{\mathcal{T}}g(t)b_{l}(t)dt$. Let $\beta^{*}=(\beta^{*}_{1},\cdots,\beta^{*}_{L})^{\top}\in\mathbb{R}^{L}$, we then have $g^{\shortparallel}(t)=(\beta^{*})^{\top}b(t)=\sum^{L}_{l=1}\beta^{*}_{l}b_{l}(t)$ and $g^{\bot}(t)=\sum_{l>L}\beta^{*}_{l}b_{l}(t)$. Thus, we have $h_{k}=g(t_{k})+\epsilon_{k}=(\beta^{*})^{\top}b(t_{k})+g^{\bot}(t_{k})+\epsilon_{k}.$ Let $h^{\bot}=\left(g^{\bot}(t_{1}),g^{\bot}(t_{2}),\dots,g^{\bot}(t_{T})\right)^{\top}$, $\epsilon=\left(\epsilon_{1},\epsilon_{2},\dots,\epsilon_{T}\right)^{\top}$, we then have $h=B\beta^{*}+h^{\bot}+\epsilon.$ Thus, $\mathbb{E}(\mathbb{\hat{\beta}})=\beta^{*}+\left(B^{\top}B\right)^{-1}B^{\top}h^{\bot},$ and $\displaystyle\hat{g}(t)-g(t)$ $\displaystyle=\hat{g}(t)-g^{\shortparallel}(t)-g^{\bot}(t)$ $\displaystyle=\hat{g}(t)-(\beta^{*})^{\top}b(t)-g^{\bot}(t)$ $\displaystyle=\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)^{\top}b(t)+\left(\left(B^{\top}B\right)^{-1}B^{\top}h^{\bot}\right)^{\top}b(t)-g^{\bot}(t).$ By Lemma 26, we then have $\displaystyle\lVert\hat{g}-g\rVert$ $\displaystyle\leq\lVert\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)^{\top}b(t)\rVert+\lVert\left(\left(B^{\top}B\right)^{-1}B^{\top}h^{\bot}\right)^{\top}b(t)\rVert+\lVert g^{\bot}\rVert$ $\displaystyle\leq\lvert\hat{\beta}-\mathbb{E}(\hat{\beta})\rvert_{2}\times\lVert b\rVert_{\mathcal{L}^{2},2}+\lvert\left(B^{\top}B\right)^{-1}B^{\top}h^{\bot}\rvert_{2}\times\lVert b\rVert_{\mathcal{L}^{2},2}+\lVert g^{\bot}\rVert$ $\displaystyle\leq\lvert\hat{\beta}-\mathbb{E}(\hat{\beta})\rvert_{2}\times\lVert b\rVert_{\mathcal{L}^{2},2}+\frac{1}{\lambda_{\min}(B^{\top}B)}\times\left\lvert B^{\top}h^{\bot}\right\rvert_{2}\times\lVert b\rVert_{\mathcal{L}^{2},2}+\lVert g^{\bot}\rVert.$ Let $\displaystyle J_{1}$ $\displaystyle=\lvert\hat{\beta}-\mathbb{E}(\hat{\beta})\rvert_{2}\times\lVert_{2}b\rVert_{\mathcal{L}^{2},2}$ (C.8) $\displaystyle J_{2}$ $\displaystyle=\frac{1}{\lambda_{\min}(B^{\top}B)}\times\lvert B^{\top}h^{\bot}\rvert_{2}\times\lVert b\rVert_{\mathcal{L}^{2},2}$ $\displaystyle J_{3}$ $\displaystyle=\lVert g^{\bot}\rVert,$ where $\lvert\mathcal{T}\rvert$ denotes the length of the interval, then $\lVert\hat{g}-g\rVert\leq J_{1}+J_{2}+J_{3}.$ (C.9) Since this equation holds for any $g\in\mathbb{H}$, thus when we include the randomness from $g$, the above equation holds with probability one. We then bound $J_{1}$, $J_{2}$ and $J_{3}$ individually. First, for $J_{1}$, recall that $\lVert b\rVert_{\mathcal{L}^{2},2}=\sqrt{L}$ and $\psi_{1}(T,L)=\sigma_{0}\lVert b\rVert_{\mathcal{L}^{2},2}\sqrt{L}/\sqrt{\lambda_{\min}\left(B^{\top}B\right)}$, then for any $\delta>0$, we claim that $P\left(J_{1}>\delta\right)\leq 2\exp\left(-\frac{\delta^{2}}{8\psi^{2}_{1}(T,L)+2\sqrt{2}\psi_{1}(T,L)\delta}\right).$ (C.10) To prove this result, we first treat $g$ as fixed, then note that by standard linear regression theory, we have $\hat{\beta}\sim N_{L}\left(\mathbb{E}(\hat{\beta}),\sigma^{2}_{0}\left(B^{\top}B\right)^{-1}\right).$ Thus, $\frac{1}{\sigma_{0}}\left(B^{\top}B\right)^{1/2}\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)\sim N_{L}\left(0,I_{L}\right)$ Since $\displaystyle J_{1}$ $\displaystyle=\lvert\hat{\beta}-\mathbb{E}(\hat{\beta})\rvert_{2}\times\lVert b\rVert_{\mathcal{L}^{2},2}$ $\displaystyle=\lvert\left(B^{\top}B\right)^{-1/2}\left(B^{\top}B\right)^{1/2}\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)\rvert_{2}\times\lVert b\rVert_{\mathcal{L}^{2},2}$ $\displaystyle\leq\frac{1}{\sqrt{\lambda_{\min}\left(B^{\top}B\right)}}\lvert\left(B^{\top}B\right)^{1/2}\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)\rvert_{2}\times\lVert b\rVert_{\mathcal{L}^{2},2}$ $\displaystyle=\frac{\sigma_{0}\lVert b\rVert_{\mathcal{L}^{2},2}}{\sqrt{\lambda_{\min}\left(B^{\top}B\right)}}\lvert\frac{1}{\sigma_{0}}\left(B^{\top}B\right)^{1/2}\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)\rvert_{2},$ we have $\displaystyle P(J_{1}>\delta)$ $\displaystyle\leq P\left(\frac{\sigma_{0}\lVert b\rVert_{\mathcal{L}^{2},2}}{\sqrt{\lambda_{\min}\left(B^{\top}B\right)}}\lvert\frac{1}{\sigma_{0}}\left(B^{\top}B\right)^{1/2}\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)\rvert_{2}>\delta\right)$ $\displaystyle=P\left(\lvert\frac{1}{\sigma_{0}}\left(B^{\top}B\right)^{1/2}\left(\hat{\beta}-\mathbb{E}(\hat{\beta})\right)\rvert_{2}>\frac{\delta}{\sigma_{0}\lVert b\rVert_{\mathcal{L}^{2},2}/\sqrt{\lambda_{\min}\left(B^{\top}B\right)}}\right)$ $\displaystyle\overset{(i)}{\leq}2\exp\left(-\frac{\left(\delta/\left(\sigma_{0}\lVert b\rVert_{\mathcal{L}^{2},2}/\sqrt{\lambda_{\min}\left(B^{\top}B\right)}\right)\right)^{2}}{8L+2\sqrt{2}\left(\left(\delta/\left(\sigma_{0}\lVert b\rVert_{\mathcal{L}^{2},2}/\sqrt{\lambda_{\min}\left(B^{\top}B\right)}\right)\right)\right)}\right)$ $\displaystyle=2\exp\left(-\frac{\delta^{2}}{8\psi^{2}_{1}(T,L)+2\sqrt{2}\psi_{1}(T,L)\delta}\right),$ where $(i)$ follows Lemma 28. Now if we treat $g$ as random, we only need to note that $\displaystyle P\left(J_{1}>\delta\right)$ $\displaystyle=\mathbb{E}_{g}\left[P\left(J_{1}>\delta_{2}|g\right)\right]$ $\displaystyle=\mathbb{E}_{g}\left[2\exp\left(-\frac{\delta^{2}}{8\psi^{2}_{1}(T,L)+2\sqrt{2}\psi_{1}(T,L)\delta}\right)\right]$ $\displaystyle=2\exp\left(-\frac{\delta^{2}}{8\psi^{2}_{1}(T,L)+2\sqrt{2}\psi_{1}(T,L)\delta}\right).$ Next, for $J_{2}$, we claim that for any $\delta>0$, we have $\mathbb{P}\left(J_{2}>\delta\right)\leq L\exp\left(-\frac{9\delta^{2}}{\psi_{2}(T,L)}\right).$ (C.11) We use $(B^{\top}h^{\bot})_{l}$ to denote the $l$-th element of vector $B^{\top}h^{\bot}$, then we have $(B^{\top}h^{\bot})_{l}=\sum^{T}_{k=1}b_{l}(t_{k})g^{\bot}(t_{k})=\sum_{m>L}\beta^{*}_{m}\sum^{T}_{k=1}b_{l}(t_{k})b_{m}(t_{k}).$ Since $g$ is a Gaussian random function with mean zero, we then have $(B^{\top}h^{\bot})_{l}$ to be a Gaussian random variable. Besides, we have $\mathbb{E}\left[(B^{\top}h^{\bot})_{l}\right]=0$ and $\mathbb{E}\left[(B^{\top}h^{\bot})^{2}_{l}\right]=\sum_{m>L}\mathbb{E}\left[\beta^{*2}_{m}\right]\left(\sum^{T}_{k=1}b_{l}(t_{k})b_{m}(t_{k})\right)^{2}$ (C.12) By definition of $D_{0,b}$, $D_{1,b}(\cdot)$, for any $l<m$, we have that $\sup_{t\in\mathcal{T}}(b_{l}(t)b_{m}(t))\leq D^{2}_{0,b}$, and $\sup_{t\in\mathcal{T}}(b_{l}(t)b_{m}(t))^{\prime}=\sup_{t\in\mathcal{T}}\\{b^{\prime}_{l}(t)b_{m}(t)+b_{l}(t)b^{\prime}_{m}(t)\\}\leq D_{0,b}(D_{1,b}(l)+D_{1,b}(m))$. Note that $\int_{\mathcal{T}}b_{l}(t)b_{m}(t)dt=0$ for any $l<m$, then by Lemma 30, we have $\displaystyle\left|\frac{1}{T}\sum^{T}_{k=1}b_{l}(t_{k})b_{m}(t_{k})\right|$ $\displaystyle=$ $\displaystyle\left|\frac{1}{T}\sum^{T}_{k=1}b_{l}(t_{k})b_{m}(t_{k})-\frac{1}{|\mathcal{T}|}\int_{\mathcal{T}}b_{l}(t)b_{m}(t)dt\right|$ $\displaystyle\leq$ $\displaystyle\frac{D_{0,b}(D_{1,b}(l)+D_{1,b}(m))(\zeta_{0}+1)^{2}|\mathcal{T}|/2+D^{2}_{0,b}(2\zeta_{0}+1)}{T}$ for all $1\leq l<m<\infty$, which implies that $\left|\sum^{T}_{k=1}b_{l}(t_{k})b_{m}(t_{k})\right|\leq\frac{1}{2}D_{0,b}(\zeta_{0}+1)^{2}|\mathcal{T}|(D_{1,b}(l)+D_{1,b}(m))+D^{2}_{0,b}(2\zeta_{0}+1).$ Then we have $\displaystyle\left(\sum^{T}_{k=1}b_{l}(t_{k})b_{m}(t_{k})\right)^{2}$ $\displaystyle\leq\left(\frac{1}{2}D_{0,b}(\zeta_{0}+1)^{2}|\mathcal{T}|(D_{1,b}(l)+D_{1,b}(m))+D^{2}_{0,b}(2\zeta_{0}+1)\right)^{2}$ $\displaystyle\leq\frac{1}{2}D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}(D_{1,b}(l)+D_{1,b}(m))^{2}+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}$ $\displaystyle\leq D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}(D^{2}_{1,b}(l)+D^{2}_{1,b}(m))+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}.$ By (C.12), we then have $\displaystyle\mathbb{E}\left[(B^{\top}h^{\bot})^{2}_{l}\right]$ $\displaystyle\leq\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b}(l)+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]\sum_{m>L}\mathbb{E}\left[\beta^{*2}_{m}\right]$ $\displaystyle+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}\sum_{m>L}\mathbb{E}\left[\beta^{*2}_{m}\right]D^{2}_{1,b}(m)$ $\displaystyle\leq\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b}(l)+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]\lambda^{\bot}_{0}+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}\psi_{4}(L)$ $\displaystyle\leq\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b,L}+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]\lambda^{\bot}_{0}+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}\psi_{4}(L)$ $\displaystyle=\lambda_{0}\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b,L}+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]\psi_{3}(L)+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}\psi_{4}(L)$ Thus, by tail bound of Gaussian random variable (Section 2.1.2 of Wainwright (2019)), we have $\displaystyle\mathbb{P}\left((B^{\top}h^{\bot})_{l}>\delta\right)\leq$ $\displaystyle\exp\left(-\frac{\delta^{2}}{2\lambda_{0}\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b,L}+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]\psi_{3}(L)+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}\psi_{4}(L)}\right).$ Recall that $\displaystyle\psi_{2}(T,L)$ $\displaystyle=\frac{1}{(\lambda^{B}_{\min})^{2}}\left(18\lambda_{0}\left[D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}D^{2}_{1,b,L}+2D^{4}_{0,b}(2\zeta_{0}+1)^{2}\right]L^{2}\psi_{3}(L)\right.$ $\displaystyle\left.\qquad\qquad+D^{2}_{0,b}(\zeta_{0}+1)^{4}|\mathcal{T}|^{2}L^{2}\psi_{4}(L)\right),$ and note that $\|b\|_{\mathcal{L}^{2},2}=\sqrt{L}$, then we have $\displaystyle\mathbb{P}\left(J_{2}>\delta\right)$ $\displaystyle\leq\mathbb{P}\left(\lvert B^{\top}h^{\bot}\rvert_{2}>\frac{\lambda^{B}_{\min}\delta}{\sqrt{L}}\right)\leq\mathbb{P}\left(\max_{1\leq l\leq L}(B^{\top}h^{\bot})_{l}>\frac{\lambda^{B}_{\min}\delta}{L}\right)$ (C.13) $\displaystyle\leq L\exp\left(-\frac{9\delta^{2}}{\psi_{2}(T,L)}\right).$ Finally, for $J_{3}$, by Lemma 31 and definition of $\psi_{3}(L)$, we have $\mathbb{E}\left[\|g^{\bot}\|^{2k}\right]\leq(2\lambda_{0}\psi_{3}(L))^{k}k!.$ This way, by Jensesn’s inequality, we have $\mathbb{E}\left[\|g^{\bot}\|^{k}\right]=\mathbb{E}\left[\sqrt{\|g^{\bot}\|^{2k}}\right]\leq\sqrt{\mathbb{E}\left[\|g^{\bot}\|^{2k}\right]}\leq\left(\sqrt{2\lambda_{0}\psi_{3}(L)}\right)^{k}k!.$ Thus, by Lemma 29, we have $P\left(J_{3}>\delta\right)=P\left(\|g^{\bot}\|>\delta\right)\leq 2\exp\left(-\frac{\delta^{2}}{8\lambda_{0}\psi_{3}(L)+2\sqrt{2\lambda_{0}}\sqrt{\psi_{3}(L)}\delta}\right).$ (C.14) The final result then follows (C.10), (C.13) and (C.14), and the fact that $\mathbb{P}\left(J_{1}+J_{2}+J_{3}>\delta\right)\leq\mathbb{P}\left(J_{1}>\delta/3\right)+\mathbb{P}\left(J_{2}>\delta/3\right)+\mathbb{P}\left(J_{3}>\delta/3\right).$ [2mm] ## Appendix D Lemmas and their proofs In this section, we introduce some useful lemmas along with their proofs. ###### Lemma 5 Let $\sigma_{\max}=\max\\{|\Sigma^{X,M}|_{\infty},\ |\Sigma^{Y,M}|_{\infty}\\}$. Suppose that $|S^{X,M}-\Sigma^{X,M}|_{\infty}\leq\delta,\qquad|S^{Y,M}-\Sigma^{Y,M}|_{\infty}\leq\delta,$ (D.1) for some $\delta\geq 0$. Then $\displaystyle|(S^{Y,M}\otimes{S^{X,M}})-(\Sigma^{Y,M}\otimes{\Sigma^{X,M}})|_{\infty}\leq\delta^{2}+2\delta\sigma_{\max},$ (D.2) and $\displaystyle|\operatorname{vec}{(S^{Y,M}-S^{X,M})}-\operatorname{vec}{(\Sigma^{Y,M}-\Sigma^{X,M})}|_{\infty}\leq 2\delta.$ (D.3) Proof Note that for any $(j,l),(j^{\prime},l^{\prime})\in V^{2}$ and $1\leq k,k^{\prime},m,m^{\prime}\leq M$, by (D.1), we have $\displaystyle\left|S^{X,M}_{jl,km}S^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}-\Sigma^{X,M}_{jl,km}\Sigma^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}\right|$ $\displaystyle\leq\left|S^{X,M}_{jl,km}-\Sigma^{X,M}_{jl,km}\right|\cdot\left|S^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}-\Sigma^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}\right|+\left|\Sigma^{X,M}_{jl,km}\right|\cdot\left|S^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}-\Sigma^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}\right|$ $\displaystyle\quad+\left|\Sigma^{Y,M}_{j^{\prime}l^{\prime},k^{\prime}m^{\prime}}\right|\cdot\left|S^{X,M}_{jl,km}-\Sigma^{X,M}_{jl,km}\right|$ $\displaystyle\leq\left|S^{X,M}-\Sigma^{X,M}\right|_{\infty}\left|S^{Y,M}-\Sigma^{Y,M}\right|_{\infty}+\sigma_{\max}\left|S^{Y,M}-\Sigma^{Y,M}\right|_{\infty}+\sigma_{\max}\left|S^{X,M}-\Sigma^{X,M}\right|_{\infty}$ $\displaystyle\leq\delta^{2}+2\delta\sigma_{\max}.$ For (D.3), note that $\displaystyle\left|\operatorname{vec}{(S^{Y,M}-S^{X,M})}-\operatorname{vec}{(\Sigma^{Y,M}-\Sigma^{X,M})}\right|_{\infty}$ $\displaystyle=\left|(S^{X,M}-\Sigma^{X,M})-(S^{Y,M}-\Sigma^{Y,M})\right|_{\infty}$ $\displaystyle\leq|S^{X,M}-\Sigma^{X,M}|_{\infty}+|S^{Y,M}-\Sigma^{Y,M}|_{\infty}$ $\displaystyle\leq 2\delta.$ [2mm] ###### Lemma 6 For $Z^{(1)},Z^{(2)},A^{(1)},A^{(2)}\in\mathbb{R}^{M\times M}$. Denote the solution of $\operatorname*{arg\,min}_{\\{Z^{(1)},Z^{(2)}\\}}\;\frac{1}{2}\sum^{2}_{q=1}\|Z^{(q)}-A^{(q)}\|^{2}_{\text{F}}+\lambda\|Z^{(1)}-Z^{(2)}\|_{\text{F}}$ (D.4) as $\\{\hat{Z}^{(1)},\hat{Z}^{(2)}\\}$, where $\lambda>0$ is a constant. Then when $\|A^{(1)}-A^{(2)}\|_{\text{F}}\leq 2\lambda$, we have $\hat{Z}^{(1)}=\hat{Z}^{(2)}=\frac{1}{2}\left(A^{(1)}+A^{(2)}\right),$ (D.5) and when $\|A^{(1)}-A^{(2)}\|_{\text{F}}>2\lambda$, we have $\displaystyle\hat{Z}^{(1)}=A^{(1)}-\frac{\lambda}{\|A^{(1)}-A^{(2)}\|_{\text{F}}}\left(A^{(1)}-A^{(2)}\right)$ (D.6) $\displaystyle\hat{Z}^{(2)}=A^{(2)}+\frac{\lambda}{\|A^{(1)}-A^{(2)}\|_{\text{F}}}\left(A^{(1)}-A^{(2)}\right).$ Proof The subdifferential of the objective function in (D.4) is $G^{(1)}(Z^{(1)},Z^{(2)})\coloneq\partial_{Z^{(1)}}=Z^{(1)}-A^{(1)}+\lambda T(Z^{(1)},Z^{(2)}),$ (D.7) $G^{(2)}(Z^{(1)},Z^{(2)})\coloneq\partial_{Z^{(2)}}=Z^{(2)}-A^{(2)}-\lambda T(Z^{(1)},Z^{(2)}),$ (D.8) where $T(Z^{(1)},Z^{(2)})=\left\\{\begin{aligned} &\frac{Z^{(1)}-Z^{(2)}}{\|Z^{(1)}-Z^{(2)}\|_{\text{F}}}\quad\text{if}\;Z^{(1)}\neq Z^{(2)}\\\ &\left\\{T\in\mathbb{R}^{M\times M}:\|T\|_{\text{F}}\leq 1\right\\}\quad\text{if}\;Z^{(1)}=Z^{(2)}\end{aligned}\right..$ (D.9) The optimal condition is: $0\in G^{(q)}(Z^{(1)},Z^{(2)})\quad q=1,2.$ (D.10) Claim $\hat{Z}^{(1)}\neq\hat{Z}^{(2)}$ if and only if $\|A^{(1)}-A^{(2)}\|_{\text{F}}>2\lambda$. We first prove the necessaity, that is, when $\hat{Z}^{(1)}\neq\hat{Z}^{(2)}$, we prove that $\|A^{(1)}-A^{(2)}\|_{\text{F}}>2\lambda$. By (D.7)-(D.10), we have $\hat{Z}^{(1)}-\hat{Z}^{(2)}-\left(A^{(1)}-A^{(2)}\right)-2\lambda\frac{\hat{Z}^{(1)}-\hat{Z}^{(2)}}{\|\hat{Z}^{(1)}-\hat{Z}^{(2)}\|_{\text{F}}}=0,$ which implies that $\|A^{(1)}-A^{(2)}\|_{\text{F}}=2\lambda+\|\hat{Z}^{(1)}-\hat{Z}^{(2)}\|_{\text{F}}>2\lambda.$ We then prove the sufficiency, that is, when $\|A^{(1)}-A^{(2)}\|_{\text{F}}>2\lambda$, we prove $\hat{Z}^{(1)}\neq\hat{Z}^{(2)}$. Note that by (D.7)-(D.10), we have $\hat{Z}^{(1)}+\hat{Z}^{(2)}=A^{(1)}+A^{(2)}.$ If $\hat{Z}^{(1)}=\hat{Z}^{(2)}$, we then have $\hat{Z}^{(1)}=\hat{Z}^{(2)}=\frac{A^{(1)}+A^{(2)}}{2}.$ By (D.7) and (D.10), we have $\|\hat{Z}^{(1)}-A^{(1)}\|_{\text{F}}=\frac{1}{2}\|A^{(1)}-A^{(2)}\|_{\text{F}}=\lambda\|T(\hat{Z}^{(1)},\hat{Z}^{(2)})\|_{\text{F}}\leq\lambda,$ which implies that $\|A^{(1)}-A^{(2)}\|_{\text{F}}\leq 2\lambda,$ and this contradicts the assumption that $\|A^{(1)}-A^{(2)}\|_{\text{F}}>2\lambda$. Thus, we must have $\hat{Z}^{(1)}\neq\hat{Z}^{(2)}$. Note that by this claim and the argument proving this claim, we have already proved (D.5). We then prove (D.6). When $\|A^{(1)}-A^{(2)}\|_{\text{F}}>2\lambda$, by the claim above, we must have $\hat{Z}^{(1)}\neq\hat{Z}^{(2)}$. Then by (D.7)-(D.10), we have $\hat{Z}^{(1)}-A^{(1)}+\frac{\lambda}{\|\hat{Z}^{(1)}-\hat{Z}^{(2)}\|_{\text{F}}}\left(\hat{Z}^{(1)}-\hat{Z}^{(2)}\right)=0,$ (D.11) $\hat{Z}^{(2)}-A^{(2)}-\frac{\lambda}{\|\hat{Z}^{(1)}-\hat{Z}^{(2)}\|_{\text{F}}}\left(\hat{Z}^{(1)}-\hat{Z}^{(2)}\right)=0.$ (D.12) (D.11) and (D.12) implies that $\hat{Z}^{(1)}-\hat{Z}^{(2)}-\left(A^{(1)}-A^{(2)}\right)+\frac{2\lambda}{\|\hat{Z}^{(1)}-\hat{Z}^{(2)}\|_{\text{F}}}\left(\hat{Z}^{(1)}-\hat{Z}^{(2)}\right)=0,$ which implies that $\hat{Z}^{(1)}-\hat{Z}^{(2)}=\alpha\cdot\left(A^{(1)}-A^{(2)}\right),$ (D.13) where $\alpha$ is a constant. We then substitue (D.13) back to (D.11) and (D.12), we then have (D.6). [2mm] ###### Lemma 7 For a set of indices $\mathcal{G}=\\{G_{t}\\}_{t=1,\ldots,N_{\mathcal{G}}}$, suppose $|\cdot|_{1,2}$ is defined in (B.3). Then for any matrix $A\in{\mathbb{R}^{p^{2}M^{2}\times{p^{2}M^{2}}}}$ and $\theta\in{\mathbb{R}^{p^{2}M^{2}}}$ $|\theta^{\top}A\theta|\leq{M^{2}|A|_{\infty}|\theta|^{2}_{1,2}}.$ (D.14) Proof By direct calculation, we have $\displaystyle|\theta^{\top}A\theta|$ $\displaystyle=\left|\sum_{i}\sum_{j}A_{ij}\theta_{i}\theta_{j}\right|$ $\displaystyle\leq{\sum_{i}\sum_{j}|A_{ij}\theta_{i}\theta_{j}|}$ $\displaystyle\leq{|A|_{\infty}\left(\sum_{i}|\theta_{i}|\right)^{2}}$ $\displaystyle=|A|_{\infty}\left(\sum_{t=1}^{N_{\mathcal{G}}}\sum_{k\in{G_{t}}}|\theta_{k}|\right)^{2}$ $\displaystyle=|A|_{\infty}\left(\sum_{t=1}^{N_{\mathcal{G}}}|\theta_{G_{t}}|_{1}\right)^{2}$ $\displaystyle\leq{|A|_{\infty}\left(\sum_{t=1}^{N_{\mathcal{G}}}M|\theta_{G_{t}}|_{2}\right)^{2}}$ $\displaystyle=M^{2}|A|_{\infty}|\theta|^{2}_{1,2},$ where in the penultimate line, we use that for any vector $v\in{\mathbb{R}^{n}}$, $|v|_{1}\leq{\sqrt{n}|v|_{2}}$. [2mm] ###### Lemma 8 Suppose $\mathcal{M}$ is defined as in (B.1). For any $\theta\in{\mathcal{M}}$, we have $|\theta|_{1,2}\leq{\sqrt{s}}|\theta|_{2}$. Furthermore, for $\Psi(\mathcal{M})$ as defined in (B.5), we have $\Psi(\mathcal{M})=\sqrt{s}$. Proof By definition of $\mathcal{M}$ and $|\cdot|_{1,2}$, we have $\displaystyle|\theta|_{1,2}$ $\displaystyle=\sum_{t\in{S_{\mathcal{G}}}}|\theta_{G_{t}}|_{2}+\sum_{t\notin{S_{\mathcal{G}}}}|\theta_{G_{t}}|_{2}$ $\displaystyle=\sum_{t\in{S_{\mathcal{G}}}}|\theta_{G_{t}}|_{2}$ $\displaystyle\leq{\sqrt{s}}\left(\sum_{t\in{S_{\mathcal{G}}}}|\theta_{G_{t}}|^{2}_{2}\right)^{\frac{1}{2}}$ $\displaystyle=\sqrt{s}|\theta|_{2}.$ In the penultimate line, we appeal to the Cauchy-Schwartz inequality. To show $\Psi(\mathcal{M})=\sqrt{s}$, it suffices to show that the upper bound above can be achieved. Select $\theta\in{\mathbb{R}^{p^{2}M^{2}}}$ such that $|\theta_{G_{t}}|_{2}=c$, $\forall{t\in{S_{\mathcal{G}}}}$, where $c$ is some positive constant. This implies that $|\theta|_{1,2}=sc$ and $|\theta|_{2}=\sqrt{s}c$ so that $|\theta|_{1,2}=\sqrt{s}|\theta|_{2}$. Thus, $\Psi(\mathcal{M})=\sqrt{s}$. [2mm] ###### Lemma 9 For $\mathcal{R}(\cdot)$ norm defined in (B.3), its dual norm $\mathcal{R}^{*}(\cdot)$, defined in (B.4), is $\mathcal{R}^{*}(v)\;=\;\max_{t=1,\ldots,N_{\mathcal{G}}}|v_{G_{t}}|_{2}.$ (D.15) Proof For any $u:|u|_{1,2}\leq{1}$ and $v\in{\mathbb{R}^{p^{2}M^{2}}}$, we have $\displaystyle\langle{v,u}\rangle$ $\displaystyle=\sum_{t=1}^{N_{\mathcal{G}}}\langle{v_{G_{t}},u_{G_{t}}}\rangle$ $\displaystyle\leq{\sum_{t=1}^{N_{\mathcal{G}}}|v_{G_{t}}|_{2}|u_{G_{t}}|_{2}}$ $\displaystyle\leq\left(\max_{t=1,2,\cdots,N_{\mathcal{G}}}|v_{G_{t}}|_{2}\right)\sum_{t=1}^{N_{\mathcal{G}}}|u_{G_{t}}|_{2}$ $\displaystyle=\left(\max_{t=1,2,\cdots,N_{\mathcal{G}}}|v_{G_{t}}|_{2}\right)|u|_{1,2}$ $\displaystyle\leq{\max_{t=1,2,\cdots,N_{\mathcal{G}}}|v_{G_{t}}|_{2}}.$ To complete the proof, we to show that this upper bound can be obtained. Let $t^{*}=\operatorname*{arg\,max}_{t=1,2,\cdots,N_{\mathcal{G}}}|v_{G_{t}}|$, and select $u$ such that $\displaystyle u_{G_{t}}$ $\displaystyle=0$ $\displaystyle\qquad{\forall{t\neq{t^{*}}}},$ $\displaystyle u_{G_{t}}$ $\displaystyle=\frac{v_{G_{t^{*}}}}{|v_{G_{t^{*}}}|_{2}}$ $\displaystyle\qquad{t={t^{*}}}.$ It follows that $|u|_{1,2}=1$ and $\langle{v,u}\rangle=|v_{G_{t^{*}}}|_{2}=\max_{t=1,\ldots,N_{\mathcal{G}}}|v_{G_{t}}|_{2}$. [2mm] ###### Lemma 10 Given that $A1$-$A5$ hold, we have $\lvert I_{1}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof This directly follows the assumption that $A_{5}$ holds. [2mm] ###### Lemma 11 Given that $A1$-$A5$ hold, we have $\lvert I_{2}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$ hold, we then have $\displaystyle\lvert I_{2}\rvert$ $\displaystyle=\lvert\langle\frac{1}{n}\sum^{n}_{i=1}a_{ijk}(\hat{g}_{il}-g_{il}),\phi_{lm}\rangle\rvert$ $\displaystyle\leq\lVert\frac{1}{n}\sum^{n}_{i=1}a_{ijk}(\hat{g}_{il}-g_{il})\rVert$ $\displaystyle\overset{(i)}{\leq}\sqrt{\frac{1}{n}\sum^{n}_{i=1}a^{2}_{ijk}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}}$ $\displaystyle\overset{(ii)}{\leq}\delta_{1}\sqrt{\frac{1}{n}\sum^{n}_{i=1}a^{2}_{ijk}}$ $\displaystyle=\delta_{1}\lambda^{1/2}_{jk}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}}$ $\displaystyle\overset{(iii)}{\leq}\sqrt{\frac{3}{2}}\delta_{1}\lambda^{1/2}_{jk}$ $\displaystyle\leq\sqrt{\frac{3}{2}}\sqrt{d_{1}}\delta_{1}k^{-\beta/2}$ $\displaystyle\leq\sqrt{\frac{3}{2}}\sqrt{d_{1}}\delta_{1},$ where $(i)$ follows Lemma 26, $(ii)$ follows $A_{1}$, $(iii)$ follows $A_{3}$. Note the definition of $d_{0}$, we thus have $\lvert I_{2}\rvert\leq\sqrt{\frac{3}{2}}d_{0}\delta_{1}.$ (D.16) Since $\delta_{1}=\delta/\left(144d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}\right)\leq\delta/(8\sqrt{6}d_{0}),$ (D.17) we have $\sqrt{\frac{3}{2}}d_{0}\delta_{1}\leq\sqrt{\frac{3}{2}}d_{0}\cdot\frac{\delta}{8\sqrt{6}d_{0}}=\frac{\delta}{16}.$ (D.18) Thus, $\lvert I_{2}\rvert\leq\frac{\delta}{16}.$ [2mm] ###### Lemma 12 Given that $A1$-$A5$ hold, we have $\lvert I_{3}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$ hold, we then have $\displaystyle\lvert I_{3}\rvert$ $\displaystyle=\lvert\langle\frac{1}{n}\sum^{n}_{i=1}a_{ijk}g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$ $\displaystyle\leq\lVert\frac{1}{n}\sum^{n}_{i=1}a_{ijk}g_{il}\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$ $\displaystyle=\lambda^{1/2}_{jk}\lVert\frac{1}{n}\sum^{n}_{i=1}\xi_{ijk}g_{il}\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$ $\displaystyle\overset{(i)}{\leq}\lambda^{1/2}_{jk}\left(\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}\right)^{1/2}\left(\frac{1}{n}\sum^{n}_{i=1}\lVert g_{il}\rVert^{2}\right)^{1/2}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$ $\displaystyle\overset{(ii)}{\leq}\lambda^{1/2}_{jk}\left(\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}\right)^{1/2}\left(\frac{1}{n}\sum^{n}_{i=1}\lVert g_{il}\rVert^{2}\right)^{1/2}d_{lm}\lVert\hat{K}_{ll}-K_{ll}\rVert_{\text{HS}},$ where $(i)$ follows Lemma 26, and $(ii)$ follows Lemma 27. Note that $\lambda^{1/2}_{jk}\leq\sqrt{d_{1}}k^{-\beta/2}$, $d_{lm}\leq d_{2}m^{1+\beta}$ and $A_{2}$-$A_{4}$ hold, thus we have $\displaystyle\lvert I_{3}\rvert$ $\displaystyle\leq\sqrt{d_{1}}d_{2}k^{-\beta/2}m^{1+\beta}\sqrt{\frac{3}{2}}\sqrt{2\lambda_{0,\max}}\delta_{2}$ (D.19) $\displaystyle\leq d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}\delta_{2}.$ By definition of $\delta_{2}$, we have $d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}\delta_{2}\leq d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}\times\frac{\delta}{16d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}}=\frac{\delta}{16}.$ (D.20) Thus, $\lvert I_{3}\rvert\leq\frac{\delta}{16}.$ [2mm] ###### Lemma 13 Given that $A1$-$A5$ hold, we have $\lvert I_{4}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$ hold, we then have $\displaystyle\lvert I_{4}\rvert$ $\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}a_{ijk}\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$ $\displaystyle\leq\frac{1}{n}\lVert\sum^{n}_{i=1}a_{ijk}\left(\hat{g}_{il}-g_{il}\right)\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$ $\displaystyle=\leq\lambda^{1/2}_{jk}\frac{1}{n}\lVert\sum^{n}_{i=1}\xi_{ijk}\left(\hat{g}_{il}-g_{il}\right)\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$ $\displaystyle\overset{(i)}{\leq}\lambda^{1/2}_{jk}\left(\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}\right)^{1/2}\left(\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}\right)^{1/2}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$ $\displaystyle\overset{(ii)}{\leq}\lambda^{1/2}_{jk}d_{lm}\left(\frac{1}{n}\sum^{n}_{i=1}\xi^{2}_{ijk}\right)^{1/2}\left(\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}\right)^{1/2}\lVert\hat{K}_{ll}-K_{ll}\rVert_{\text{HS}},$ where $(i)$ follows Lemma 26, and $(ii)$ follows Lemma 27. Note that $\lambda^{1/2}_{jk}\leq\sqrt{d_{1}}k^{-\beta/2}$, $d_{lm}\leq d_{2}m^{1+\beta}$ and $A_{1}$-$A_{3}$ hold, thus we have $\displaystyle\lvert I_{4}\rvert$ $\displaystyle\leq\sqrt{\frac{3}{2}}\sqrt{d_{1}}d_{2}k^{-\beta/2}m^{1+\beta}\delta_{1}\delta_{2}$ $\displaystyle\leq\sqrt{\frac{3}{2}}d^{2}_{0}M^{1+\beta}\delta_{1}\delta_{2}$ $\displaystyle\overset{(iii)}{\leq}\frac{\delta}{16}\times\frac{\sqrt{\frac{3}{2}}d^{2}_{0}M^{1+\beta}\delta_{1}\delta_{2}}{\sqrt{\frac{3}{2}}d_{0}\delta_{1}}$ $\displaystyle\leq\frac{\delta}{16}\times d_{0}M^{1+\beta}\delta_{2}$ $\displaystyle\leq\frac{\delta}{16}\times d_{0}M^{1+\beta}\times\frac{\delta}{16d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}}$ $\displaystyle=\frac{\delta}{16}\times\frac{\delta}{16d_{0}\sqrt{3\lambda_{0,\max}}}$ $\displaystyle\leq\frac{\delta}{16},$ where $(iii)$ follows (D.18). [2mm] ###### Lemma 14 Given that $A1$-$A5$ hold, we have $\lvert I_{5}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof This proof is similar to the proof of Lemma 11, thus is omitted. [2mm] ###### Lemma 15 Given that $A1$-$A5$ hold, we have $\lvert I_{6}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$ hold, we then have $\displaystyle\lvert I_{6}\rvert$ $\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle\rvert$ $\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\rvert\lvert\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle\rvert$ $\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\rvert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{il}-g_{il},\phi_{lm}\rangle\rvert^{2}}$ $\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{ij}-g_{ij}\rVert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}}.$ By the assumption that $A_{1}$ holds, we thus have $\lvert I_{6}\rvert\leq\delta^{2}_{1}.$ (D.21) By (D.17),(D.18) and Lemma 11, we have $\displaystyle\delta^{2}_{1}$ $\displaystyle\leq\frac{\delta}{16}\times\frac{\delta^{2}_{1}}{\sqrt{\frac{3}{2}}d_{0}\delta_{1}}$ (D.22) $\displaystyle=\frac{\delta}{16}\times\frac{\delta_{1}}{\sqrt{\frac{3}{2}}d_{0}}$ (D.23) $\displaystyle\leq\frac{\delta}{16}\times\frac{1}{\sqrt{\frac{3}{2}}d_{0}}\times\frac{\delta}{8\sqrt{6}d_{0}}$ (D.24) $\displaystyle=\frac{\delta}{16}\times\frac{\delta}{24d^{2}_{0}}$ (D.25) $\displaystyle\leq\frac{\delta}{16},$ (D.26) and thus $\lvert I_{6}\rvert\leq\frac{\delta}{16}.$ [2mm] ###### Lemma 16 Given that $A1$-$A5$ hold, we have $\lvert I_{7}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$ hold, we then have $\displaystyle\lvert I_{7}\rvert$ $\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$ $\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$ $\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\rvert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert^{2}}$ $\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{ij}-g_{ij}\rVert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert g_{il}\rVert^{2}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert^{2}}$ $\displaystyle\overset{(i)}{\leq}\delta_{1}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert g_{il}\rVert^{2}}$ $\displaystyle\overset{(ii)}{\leq}\delta_{1}\sqrt{2\lambda_{0,\max}}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$ $\displaystyle\overset{(iii)}{\leq}\delta_{1}\sqrt{2\lambda_{0,\max}}d_{lm}\|\hat{K}_{ll}-K_{ll}\rVert_{\text{HS}}$ $\displaystyle\overset{(iv)}{\leq}\delta_{1}\delta_{2}\sqrt{2\lambda_{0,\max}}d_{lm}$ $\displaystyle\leq\delta_{1}\delta_{2}\sqrt{2\lambda_{0,\max}}d_{2}m^{1+\beta}$ $\displaystyle\leq d_{0}\sqrt{2\lambda_{0,\max}}M^{1+\beta}\delta_{1}\delta_{2},$ where $(i)$ follows the assumption that $A_{1}$ holds, $(ii)$ follows the assumption that $A_{4}$ holds, $(iii)$ follows Lemma 27, and $(iv)$ follows the assumption that $A_{2}$ holds. By (D.17) and (D.20), we have $\displaystyle\lvert I_{7}\rvert$ $\displaystyle\leq\frac{\delta}{16}\times\frac{d_{0}\sqrt{2\lambda_{0,\max}}M^{1+\beta}\delta_{1}\delta_{2}}{d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}\delta_{2}}$ $\displaystyle=\frac{\delta}{16}\times\sqrt{\frac{2}{3}}\times\frac{\delta_{1}}{d_{0}}$ $\displaystyle\leq\frac{\delta}{16}\times\sqrt{\frac{2}{3}}\times\frac{\delta}{8\sqrt{6}d^{2}_{0}}$ $\displaystyle=\frac{\delta}{16}\times\frac{\delta}{24\delta^{2}_{0}}$ $\displaystyle\leq\frac{\delta}{16}.$ [2mm] ###### Lemma 17 Given that $A1$-$A5$ hold, we have $\lvert I_{8}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$ hold, we then have $\displaystyle\lvert I_{8}\rvert$ $\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$ $\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\rvert\lvert\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$ $\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\phi_{jk}\rangle\rvert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert^{2}}$ $\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{ij}-g_{ij}\rVert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert^{2}}$ $\displaystyle\overset{(i)}{\leq}\delta^{2}_{1}\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$ $\displaystyle\overset{(ii)}{\leq}\delta^{2}_{1}d_{lm}\lVert\hat{K}_{ll}-K_{ll}\rVert_{\text{HS}}$ $\displaystyle\leq\delta^{2}_{1}d_{2}m^{1+\beta}\lVert\hat{K}_{ll}-K_{ll}\rVert_{\text{HS}}$ $\displaystyle\leq\delta^{2}_{1}d_{0}M^{1+\beta}\lVert\hat{K}_{ll}-K_{ll}\rVert_{\text{HS}}$ $\displaystyle\overset{(iii)}{\leq}d_{0}M^{1+\beta}\delta^{2}_{1}\delta_{2}$ where $(i)$ follows the assumption that $A_{1}$ holds, $(ii)$ follows the assumption that Lemma 27 holds, and $(iii)$ follows the assumption that $A_{2}$ holds. By (D.22), we have $\displaystyle\lvert I_{8}\rvert$ $\displaystyle\leq\frac{\delta}{16}\times\frac{d_{0}M^{1+\beta}\delta^{2}_{1}\delta_{2}}{\delta^{2}_{1}}$ $\displaystyle=\frac{\delta}{16}\times d_{0}M^{1+\beta}\delta_{2}$ $\displaystyle=\frac{\delta}{16}\times d_{0}M^{1+\beta}\times\frac{\delta}{16d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}}$ $\displaystyle=\frac{\delta}{16}\times\frac{\delta}{16d_{0}\sqrt{3\lambda_{0,\max}}}$ $\displaystyle\leq\frac{\delta}{16}.$ [2mm] ###### Lemma 18 Given that $A1$-$A5$ hold, we have $\lvert I_{9}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof This proof is similar to the proof of Lemma 12, thus is omitted. [2mm] ###### Lemma 19 Given that $A1$-$A5$ hold, we have $\lvert I_{10}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof This proof is similar to the proof of Lemma 16, thus is omitted. [2mm] ###### Lemma 20 Given that $A1$-$A5$ hold, we have $\lvert I_{11}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$ hold, we then have $\displaystyle\lvert I_{11}\rvert$ $\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$ $\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\lvert\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\rvert\lvert\langle g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$ $\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\rvert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert^{2}}$ $\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert g_{ij}\rVert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert g_{il}\rVert^{2}}\lVert\hat{\phi}_{jk}-\phi_{jk}\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$ $\displaystyle\overset{(i)}{\leq}2\lambda_{0,\max}\lVert\hat{\phi}_{jk}-\phi_{jk}\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$ $\displaystyle\overset{(ii)}{\leq}2\lambda_{0,\max}\delta^{2}_{2}d_{jk}d_{lm}$ $\displaystyle\leq 2\lambda_{0,\max}\delta^{2}_{2}d^{2}_{2}k^{1+\beta}m^{1+\beta},$ where $(i)$ follows because assumption $A_{4}$ holds, $(ii)$ follows Lemma 27. Then, we have $\lvert I_{11}\rvert\leq 2d^{2}_{0}\lambda_{0,\max}M^{2+2\beta}\delta^{2}_{2}.$ (D.27) Thus, by (D.20), we have $\displaystyle 2d^{2}_{0}\lambda_{0,\max}M^{2+2\beta}\delta^{2}_{2}$ $\displaystyle\leq\frac{\delta}{16}\times\frac{2d^{2}_{0}\lambda_{0,\max}M^{2+2\beta}\delta^{2}_{2}}{d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}\delta_{2}}$ (D.28) $\displaystyle=\frac{\delta}{16}\times\frac{2}{\sqrt{3}}M^{1+\beta}\sqrt{\lambda_{0,\max}}\delta_{2}$ (D.29) $\displaystyle=\frac{\delta}{16}\times\frac{2}{\sqrt{3}}M^{1+\beta}\sqrt{\lambda_{0,\max}}\times\frac{\delta}{16d^{2}_{0}M^{1+\beta}\sqrt{3\lambda_{0,\max}}}$ (D.30) $\displaystyle=\frac{\delta}{16}\times\frac{\delta}{24d^{2}_{0}}$ (D.31) $\displaystyle\leq\frac{\delta}{16},$ (D.32) which implies that $\lvert I_{11}\rvert\leq\frac{\delta}{16}.$ [2mm] ###### Lemma 21 Given that $A1$-$A5$ hold, we have $\lvert I_{12}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$ hold, we then have $\displaystyle\lvert I_{12}\rvert$ $\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$ $\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\lvert\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\rvert\lvert\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$ $\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\rvert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert^{2}}$ $\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert g_{ij}\rVert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}}\lVert\hat{\phi}_{jk}-\phi_{jk}\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$ $\displaystyle\overset{(i)}{\leq}\sqrt{2\lambda_{0,\max}}\delta_{1}\delta^{2}_{2}d_{jk}d_{lm}$ $\displaystyle\leq d^{2}_{2}\sqrt{2\lambda_{0,\max}}k^{1+\beta}m^{1+\beta}\delta_{1}\delta^{2}_{2},$ where $(i)$ follows the assumption that $A_{1}$-$A_{3}$ hold along with Lemma 27. Then, we have $\lvert I_{12}\rvert\leq d^{2}_{0}\sqrt{2\lambda_{0,\max}}M^{2+2\beta}\delta_{1}\delta^{2}_{2}.$ (D.33) By (D.17) and (D.28), we have $\displaystyle d^{2}_{0}\sqrt{2\lambda_{0,\max}}M^{2+2\beta}\delta_{1}\delta^{2}_{2}$ $\displaystyle\leq\frac{\delta}{16}\times\frac{d^{2}_{0}\sqrt{2\lambda_{0,\max}}M^{2+2\beta}\delta_{1}\delta^{2}_{2}}{2d^{2}_{0}\lambda_{0,\max}M^{2+2\beta}\delta^{2}_{2}}$ (D.34) $\displaystyle=\frac{\delta}{16}\times\frac{\delta_{1}}{\sqrt{2\lambda_{0,\max}}}$ (D.35) $\displaystyle\leq\frac{\delta}{16}\times\frac{1}{\sqrt{2\lambda_{0,\max}}}\times\frac{\delta}{8\sqrt{6}d_{0}}$ (D.36) $\displaystyle=\frac{\delta}{16}\times\frac{\delta}{16d_{0}\sqrt{3\lambda_{0,\max}}}$ (D.37) $\displaystyle\leq\frac{\delta}{16},$ (D.38) which implies that $I_{12}\leq\frac{\delta}{16}.$ [2mm] ###### Lemma 22 Given that $A1$-$A5$ hold, we have $\lvert I_{13}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof This proof is similar to the proof of Lemma 13, thus is omitted. [2mm] ###### Lemma 23 Given that $A1$-$A5$ hold, we have $\lvert I_{14}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof This proof is similar to the proof of Lemma 17, thus is omitted. [2mm] ###### Lemma 24 Given that $A1$-$A5$ hold, we have $\lvert I_{15}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof This proof is similar to the proof of Lemma 11, thus is omitted. [2mm] ###### Lemma 25 Given that $A1$-$A5$ hold, we have $\lvert I_{16}\rvert\leq\delta/16$ for all $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$. Proof For any $1\leq j,l\leq p$, $\ 1\leq k,m\leq M$, assume that $A1$-$A5$ hold, we then have $\displaystyle\lvert I_{16}\rvert$ $\displaystyle=\lvert\frac{1}{n}\sum^{n}_{i=1}\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$ $\displaystyle\leq\frac{1}{n}\sum^{n}_{i=1}\lvert\langle\hat{g}_{ij}-g_{ij},\hat{\phi}_{jk}-\phi_{jk}\rangle\rvert\lvert\langle\hat{g}_{il}-g_{il},\hat{\phi}_{lm}-\phi_{lm}\rangle\rvert$ $\displaystyle\leq\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{ij}-g_{ij}\rVert^{2}}\sqrt{\frac{1}{n}\sum^{n}_{i=1}\lVert\hat{g}_{il}-g_{il}\rVert^{2}}\lVert\hat{\phi}_{jk}-\phi_{jk}\rVert\lVert\hat{\phi}_{lm}-\phi_{lm}\rVert$ $\displaystyle\overset{(i)}{\leq}\delta^{2}_{1}d_{jk}d_{lm}\delta^{2}_{2}$ $\displaystyle\leq d^{2}_{2}k^{1+\beta}m^{1+\beta}\delta^{2}_{1}\delta^{2}_{2}$ $\displaystyle\leq d^{2}_{0}M^{2+2\beta}\delta^{2}_{1}\delta^{2}_{2},$ where $(i)$ follows the assumption that $A_{1}$, $A_{2}$ hold along with Lemma 27. Thus, by (D.18) and (D.34), we have $\displaystyle\lvert I_{16}\rvert$ $\displaystyle\leq\frac{\delta}{16}\times\frac{d^{2}_{0}M^{2+2\beta}\delta^{2}_{1}\delta^{2}_{2}}{d^{2}_{0}\sqrt{2\lambda_{0,\max}}M^{2+2\beta}\delta_{1}\delta^{2}_{2}}$ $\displaystyle=\frac{\delta}{16}\times\frac{\delta_{1}}{\sqrt{2\lambda_{0,\max}}}$ $\displaystyle\leq\frac{\delta}{16}\times\frac{1}{\sqrt{2\lambda_{0,\max}}}\times\frac{\delta}{8\sqrt{6}d_{0}}$ $\displaystyle=\frac{\delta}{16}\times\frac{\delta}{16d_{0}\sqrt{3\lambda_{0,\max}}}$ $\displaystyle\leq\frac{\delta}{16}.$ [2mm] ###### Lemma 26 Suppose $f_{1},f_{2},\dots,f_{n}\in\mathbb{H}$ and $v_{1},v_{2},\dots,v_{n}\in\mathbb{R}$, we have $\lVert\sum^{n}_{i=1}v_{i}f_{i}\rVert\leq\sqrt{\sum^{n}_{i=1}v^{2}_{i}}\sqrt{\sum^{n}_{i=1}\lVert f_{i}\rVert^{2}}$ Proof Note that $\displaystyle\lVert\sum^{n}_{i=1}v_{i}f_{i}\rVert^{2}$ $\displaystyle=\int\left(\sum^{n}_{i=1}v_{i}f_{i}(t)\right)^{2}dt$ $\displaystyle\overset{(i)}{\leq}\int\left(\sum^{n}_{i=1}v^{2}_{i}\right)\left(\sum^{n}_{i=1}f^{2}_{i}(t)\right)dt$ $\displaystyle=\left(\sum^{n}_{i=1}v^{2}_{i}\right)\left(\sum^{n}_{i=1}\lVert f_{i}\rVert^{2}\right),$ where $(i)$ follows Cauchy-Schwartz inequality, which directly implies the result. [2mm] ###### Lemma 27 Suppose that Assumption 3 holds. Denote $\tilde{\phi}_{jk}=\text{sgn}\left(\langle\hat{\phi}_{jk},\phi_{jk}\rangle\right)\phi_{jk}$, where $\text{sgn}(t)=1$ if $t\geq 0$ and $\text{sgn}(t)=-1$ if $t<0$. Then we have $\lVert\hat{\phi}_{jk}-\tilde{\phi}_{jk}\rVert\leq d_{jk}\lVert\hat{K}_{jj}-K_{jj}\rVert_{\text{HS}},$ where $d_{jk}=2\sqrt{2}\max\\{(\lambda_{j(k-1)}-\lambda_{jk})^{-1},(\lambda_{jk}-\lambda_{j(k+1)})^{-1}\\}$ if $k\geq 2$ and $d_{j1}=2\sqrt{2}(\lambda_{j1}-\lambda_{j2})^{-1}$. Proof This lemma can be found in Lemma 4.3 of Bosq (2000) and hence the proof is omitted. [2mm] ###### Lemma 28 For $z\sim N_{L}\left(0,I_{L}\right)$, then for any $\delta>0$, we have $P\left(\lVert z\rVert_{2}>\delta\right)\leq 2\exp\left(-\frac{\delta^{2}}{8L+2\sqrt{2L}\delta}\right).$ Proof Since $\mathbb{E}\left[\lVert z\rVert^{2k}_{2}\right]=\frac{\Gamma(\frac{L}{2}+k)}{\Gamma(\frac{L}{2})}\times 2^{k}\leq k!(2L)^{k},$ we have $\mathbb{E}\left[\lVert z\rVert^{k}_{2}\right]\leq\sqrt{\mathbb{E}\left[\lVert z\rVert^{2k}_{2}\right]}\leq\sqrt{k!}\left(\sqrt{2L}\right)^{k}\leq\frac{k!}{2}\cdot 4L\cdot(\sqrt{2L})^{k-2}$ for $k\geq 2$. Thus, by Lemma 29, we have proved the result. [2mm] ###### Lemma 29 Let $Z_{1},Z_{2},\dots,Z_{n}$ be independent random variables in a separable Hilbert space with norm $\lVert\cdot\rVert$. If $\mathbb{E}[Z_{i}]=0$ ($i=1,\ldots,n$) and $\sum^{n}_{i=1}\mathbb{E}\left[\lVert Z_{i}\rVert^{k}\right]\leq\frac{k!}{2}nL_{1}L^{k-2}_{2},k=2,3,\dots,$ for two positive constants $L_{1}$ and $L_{2}$, then for all $\delta>0$, $P\left(\lVert\sum^{n}_{i=1}Z_{i}\rVert\geq n\delta\right)\leq 2\exp\left(-\frac{n\delta^{2}}{2L_{1}+2L_{2}\delta}\right).$ Proof This lemma can be derived directly from Theorem 2.5 (2) of Bosq (2000) and hence its proof is omitted. [2mm] ###### Lemma 30 For a function $f(t)$ defined on $\mathcal{T}$, assuming that $f$ has continuous derivative, and let $D_{0,f}\coloneqq\sup_{t\in{\mathcal{T}}}\lvert f(t)\rvert$, $D_{1,f}\coloneqq\sup_{t\in{\mathcal{T}}}\lvert f^{\prime}(t)\rvert$, assume that $D_{0,f},D_{1,f}<\infty$. Let $\lvert\mathcal{T}\rvert$ denote the length of interval $\mathcal{T}$, and let $u_{1}<u_{2}<\dots<u_{T}\in\mathcal{T}$, we denote endpoints of $\mathcal{T}$ as $u_{0}$ and $u_{T+1}$. Assume that there is positive constant $\zeta_{0}$ such that $\max_{1\leq k\leq T+1}\left|\frac{u_{k}-u_{k-1}}{\lvert\mathcal{T}\rvert}-\frac{1}{T}\right|\leq\frac{\zeta_{0}}{T^{2}}$ (D.39) hold. Let $\zeta_{1}=\zeta_{0}+1$, then we have $\left|\frac{1}{T}\sum^{T}_{k=1}f(u_{k})-\frac{1}{\lvert\mathcal{T}\rvert}\int_{\mathcal{T}}f(t)dt\right|\leq\frac{D_{1,f}\zeta^{2}_{1}\lvert\mathcal{T}\rvert/2+D_{0,f}(\zeta_{1}+\zeta_{0})}{T}.$ Proof Since $\displaystyle\left|\frac{1}{T}\sum^{T}_{k=1}f(u_{k})-\frac{1}{\lvert\mathcal{T}\rvert}\int_{\mathcal{T}}f(t)dt\right|$ $\displaystyle\leq\left|\frac{1}{T}\sum^{T}_{k=1}f(u_{k})-\frac{1}{\lvert\mathcal{T}\rvert}\sum^{T}_{k=1}f(u_{k})(u_{k}-u_{k-1})\right|$ $\displaystyle+\left|\frac{1}{\lvert\mathcal{T}\rvert}\sum^{T}_{k=1}f(u_{k})(u_{k}-u_{k-1})-\frac{1}{\lvert\mathcal{T}\rvert}\int_{\mathcal{T}}f(t)dt\right|,$ we will first prove the first part is smaller than $D_{0,f}\zeta_{0}/T$, and then prove the second part is smaller than $(D_{1,f}\zeta^{2}_{1}\lvert\mathcal{T}\rvert/2+D_{0,f}\zeta_{1})/T$. For first part, we have $\displaystyle\left|\frac{1}{T}\sum^{T}_{k=1}f(u_{k})-\frac{1}{\lvert\mathcal{T}\rvert}\sum^{T}_{k=1}f(u_{k})(u_{k}-u_{k-1})\right|$ $\displaystyle=\left|\sum^{T}_{k=1}f(u_{k})\left(\frac{1}{T}-\frac{u_{k}-u_{k-1}}{\lvert\mathcal{T}\rvert}\right)\right|$ $\displaystyle\leq\sum^{T}_{k=1}\left|f(u_{k})\right|\left|\frac{1}{T}-\frac{u_{k}-u_{k-1}}{\lvert\mathcal{T}\rvert}\right|$ $\displaystyle\leq\max_{1\leq k\leq T}\left|\frac{u_{k}-u_{k-1}}{\lvert\mathcal{T}\rvert}-\frac{1}{T}\right|\sum^{T}_{k=1}\left|f(u_{k})\right|$ $\displaystyle\leq\frac{\zeta_{0}}{T^{2}}\times T\times D_{0,f}$ $\displaystyle=\frac{\zeta_{0}D_{0,f}}{T}.$ To prove second part, we first note that based on (D.39), we have $\max_{1\leq k\leq T+1}\lvert u_{k}-u_{k-1}\rvert\leq\frac{\zeta_{1}\lvert\mathcal{T}\rvert}{T}.$ Then, for any $t\in(u_{k},u_{k+1})$, by Taylor’s expansion, we have $f(t)=f(u_{k})+f^{\prime}(\bar{t})(t-u_{k}),$ where $\bar{t}\in(u_{k},t)$. Thus, $\lvert f(t)-f(u_{k})\rvert=\lvert f^{\prime}(\bar{t})\rvert(t-u_{k})\leq D_{1,f}(t-u_{k}).$ This way, we have $\displaystyle\left|\frac{1}{\lvert\mathcal{T}\rvert}\sum^{T}_{k=1}f(u_{k})(u_{k}-u_{k-1})-\frac{1}{\lvert\mathcal{T}\rvert}\int_{\mathcal{T}}f(t)dt\right|$ $\displaystyle\leq\frac{1}{\lvert\mathcal{T}\rvert}\sum^{T}_{k=1}\int^{u_{k}}_{u_{k-1}}\lvert f(u_{k})-f(t)\rvert dt+\frac{1}{\lvert\mathcal{T}\rvert}\int^{u_{T+1}}_{u_{T}}\lvert f(t)\rvert dt$ $\displaystyle\leq\frac{1}{\lvert\mathcal{T}\rvert}\times T\times D_{1,f}\times\int^{u_{k}}_{u_{k-1}}(t-u_{k})dt+\frac{1}{\lvert\mathcal{T}\rvert}\times D_{0,f}\times\frac{\zeta_{1}\lvert\mathcal{T}\rvert}{T}$ $\displaystyle=\frac{1}{\lvert\mathcal{T}\rvert}\times T\times D_{1,f}\times\frac{(u_{k+1}-u_{k})^{2}}{2}+\frac{1}{\lvert\mathcal{T}\rvert}\times D_{0,f}\times\frac{\zeta_{1}\lvert\mathcal{T}\rvert}{T}$ $\displaystyle\leq\frac{1}{\lvert\mathcal{T}\rvert}\times T\times\frac{D_{1,f}}{2}\times\left(\max_{1\leq k\leq T+1}\lvert u_{k+1}-u_{k}\rvert\right)^{2}+\frac{1}{\lvert\mathcal{T}\rvert}\times D_{0,f}\times\frac{\zeta_{1}\lvert\mathcal{T}\rvert}{T}$ $\displaystyle\leq\frac{1}{\lvert\mathcal{T}\rvert}\times T\times\frac{D_{1,f}}{2}\times\left(\frac{\zeta_{1}\lvert\mathcal{T}\rvert}{T}\right)^{2}+\frac{1}{\lvert\mathcal{T}\rvert}\times D_{0,f}\times\frac{\zeta_{1}\lvert\mathcal{T}\rvert}{T}$ $\displaystyle=\frac{D_{1,f}\zeta^{2}_{1}\lvert\mathcal{T}\rvert/2+D_{0,f}\zeta_{1}}{T}.$ Thus, combining part 1 and part 2, we have $\left|\frac{1}{T}\sum^{T}_{k=1}f(u_{k})-\frac{1}{\lvert\mathcal{T}\rvert}\int_{\mathcal{T}}f(t)dt\right|\leq\frac{D_{1,f}\zeta^{2}_{1}\lvert\mathcal{T}\rvert/2+D_{0,f}(\zeta_{1}+\zeta_{0})}{T}.$ [2mm] ###### Lemma 31 For Gaussian random function $g$ in Hilbert Space $\mathbb{H}$ with mean zero, that is, $\mathbb{E}[g]=0$, we have $\mathbb{E}\left[\|g\|^{2k}\right]\leq(2\lambda_{0})^{k}\cdot k!,$ where $\lambda_{0}=\mathbb{E}\left[\|g\|^{2}\right]$. Proof Let $\\{\phi_{m}\\}_{m\geq 1}$ be othornormal eigenfunctions of $g$, and $a_{m}=\langle g,\phi_{m}\rangle$, then $a_{m}\sim N(0,\lambda_{m})$ and $\lambda_{0}=\sum_{m\geq 1}\lambda_{m}$. Let $\xi_{m}=\lambda^{-1/2}_{m}a_{m}$, then we have $\xi_{m}\sim N(0,1)$ i.i.d.. By Karhunen–Loève theorem, we have $g=\sum^{\infty}_{m=1}\lambda_{m}^{1/2}\xi_{m}\phi_{m}.$ Thus, $\|g\|=\left(\sum_{m\geq 1}\lambda_{m}\xi^{2}_{m}\right)^{1/2}$, and $\|g\|^{2k}=\left(\sum_{m\geq 1}\lambda_{m}\xi^{2}_{m}\right)^{k}$. Recall Jensen’s inequality, for convex function $\psi(\cdot)$, and real numbers $x_{1},x_{2},\dots,x_{n}$ in its domain, and positive real numbers $a_{1},a_{2},\dots,a_{n}$, we have $\psi\left(\frac{\sum^{n}_{i=1}a_{i}x_{i}}{\sum^{n}_{i=1}a_{i}}\right)\leq\frac{\sum^{n}_{i=1}a_{i}\psi(x_{i})}{\sum^{n}_{i=1}a_{i}}.$ Here, let $\psi(t)=t^{k}$, and we then have $\displaystyle\|g\|^{2k}$ $\displaystyle=\left(\sum_{m\geq 1}\lambda_{m}\right)^{k}\cdot\left(\frac{\sum_{m\geq 1}\lambda_{m}\xi^{2}_{m}}{\sum_{m\geq 1}\lambda_{m}}\right)^{k}$ $\displaystyle\leq\left(\sum_{m\geq 1}\lambda_{m}\right)^{k}\cdot\frac{\sum_{m\geq 1}\lambda_{m}\xi^{2k}_{m}}{\sum_{m\geq 1}\lambda_{m}}$ $\displaystyle=\left(\sum_{m\geq 1}\lambda_{m}\right)^{k-1}\cdot\left(\sum_{m\geq 1}\lambda_{m}\xi^{2k}_{m}\right).$ Thus, $\displaystyle\mathbb{E}\left[\|g\|^{2k}\right]$ $\displaystyle\leq\left(\sum_{m\geq 1}\lambda_{m}\right)^{k-1}\cdot\left(\sum_{m\geq 1}\lambda_{m}\mathbb{E}\left[\xi^{2k}_{m}\right]\right)$ $\displaystyle=\left(\sum_{m\geq 1}\lambda_{m}\right)^{k}\mathbb{E}\left[\xi^{2k}_{1}\right]$ $\displaystyle=\left(\sum_{m\geq 1}\lambda_{m}\right)^{k}\cdot\pi^{-1/2}\cdot 2^{k}\cdot\Gamma(k+1/2)$ $\displaystyle\leq\left(\sum_{m\geq 1}\lambda_{m}\right)^{k}\cdot 2^{k}\cdot k!$ $\displaystyle=(2\lambda_{0})^{k}k!$ [2mm] ###### Lemma 32 For any $\delta>0$, we have $P\left(\left\|\frac{1}{n}\sum^{n}_{i=1}\left[g_{ij}(t)g_{ij}(s)-K_{jj}(s,t)\right]\right\|_{\text{HS}}>\delta\right)\leq 2\exp\left(-\frac{n\delta^{2}}{64\lambda^{2}_{0,\max}+8\lambda_{0,\max}\delta}\right)$ holding for any $j=1,\ldots,p$. Proof Since $g_{ij}(t)=\sum_{m\geq 1}\lambda^{1/2}_{jm}\xi_{ijm}\phi_{jm}(t)$, and $\xi_{ijm}\sim N(0,1)$ i.i.d. for $m\geq 1$, we have $g_{ij}(s)g_{ij}(t)=\sum_{m,m^{\prime}\geq 1}\lambda^{1/2}_{jm}\lambda^{1/2}_{jm^{\prime}}\xi_{ijm}\xi_{ijm^{\prime}}\phi_{jm}(s)\phi_{jm^{\prime}}(t)$, and $K_{jj}(s,t)=\mathbb{E}[g_{ij}(s)g_{ij}(t)]=\sum_{m,m^{\prime}\geq 1}\lambda^{1/2}_{jm}\lambda^{1/2}_{jm^{\prime}}\phi_{jm}(s)\phi_{jm^{\prime}}(t)\mathbbm{1}_{mm^{\prime}}$, where $\mathbbm{1}_{mm^{\prime}}=\mathbbm{1}(m=m^{\prime})=1$ if $m=m^{\prime}$ and $0$ if $m\neq m^{\prime}$. Thus, $\left\|g_{ij}(s)g_{ij}(t)-K_{jj}(s,t)\right\|^{2}_{\text{HS}}=\sum_{m,m^{\prime}\geq 1}\lambda_{jm}\lambda_{jm^{\prime}}(\xi_{ijm}\xi_{ijm^{\prime}}-\mathbbm{1}_{mm^{\prime}})^{2},$ and for any $k\geq 2$, we have $\displaystyle\mathbb{E}\left[\left\|g_{ij}(s)g_{ij}(t)-K_{jj}(s,t)\right\|^{k}_{\text{HS}}\right]$ $\displaystyle=\mathbb{E}\left[\left\\{\sum_{m,m^{\prime}\geq 1}\lambda_{jm}\lambda_{jm^{\prime}}(\xi_{ijm}\xi_{ijm^{\prime}}-\mathbbm{1}_{mm^{\prime}})^{2}\right\\}^{k/2}\right]$ $\displaystyle\overset{(i)}{\leq}\left(\sum_{m,m^{\prime}\geq 1}\lambda_{jm}\lambda_{jm^{\prime}}\right)^{k/2-1}\sum_{m,m^{\prime}\geq 1}\lambda_{jm}\lambda_{jm^{\prime}}\mathbb{E}\left[\left(\xi_{ijm}\xi_{ijm^{\prime}}-\mathbbm{1}_{mm^{\prime}}\right)^{k}\right],$ where $(i)$ follows Jensen’s inequality with convex function $\psi(x)=x^{k/2}$. 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2024-09-04T02:54:59.296596

2003.05425

arxiv-papers

# Gauge Equivariant Mesh CNNs Anisotropic convolutions on geometric graphs Pim de Haan Qualcomm AI Research University of Amsterdam &Maurice Weiler∗ QUVA Lab University of Amsterdam &Taco Cohen Qualcomm AI Research &Max Welling Qualcomm AI Research University of Amsterdam Equal ContributionQualcomm AI Research is an initiative of Qualcomm Technologies, Inc. ###### Abstract A common approach to define convolutions on meshes is to interpret them as a graph and apply graph convolutional networks (GCNs). Such GCNs utilize _isotropic_ kernels and are therefore insensitive to the relative orientation of vertices and thus to the geometry of the mesh as a whole. We propose Gauge Equivariant Mesh CNNs which generalize GCNs to apply _anisotropic_ gauge equivariant kernels. Since the resulting features carry orientation information, we introduce a geometric message passing scheme defined by parallel transporting features over mesh edges. Our experiments validate the significantly improved expressivity of the proposed model over conventional GCNs and other methods. ## 1 Introduction Convolutional neural networks (CNNs) have been established as the default method for many machine learning tasks like speech recognition or planar and volumetric image classification and segmentation. Most CNNs are restricted to flat or spherical geometries, where convolutions are easily defined and optimized implementations are available. The empirical success of CNNs on such spaces has generated interest to generalize convolutions to more general spaces like graphs or Riemannian manifolds, creating a field now known as geometric deep learning (Bronstein et al., 2017). A case of specific interest is convolution on _meshes_ , the discrete analog of 2-dimensional embedded Riemannian manifolds. Mesh CNNs can be applied to tasks such as detecting shapes, registering different poses of the same shape and shape segmentation. If we forget the positions of vertices, and which vertices form faces, a mesh $M$ can be represented by a graph ${\mathcal{G}}$. This allows for the application of _graph convolutional networks_ (GCNs) to processing signals on meshes. Figure 1: Two local neighbourhoods around vertices $p$ and their representations in the tangent planes $T_{p}M$. The distinct geometry of the neighbourhoods is reflected in the different angles $\theta_{pq_{i}}$ of incident edges from neighbours $q_{i}$. Graph convolutional networks apply isotropic kernels and can therefore not distinguish both neighbourhoods. Gauge Equivariant Mesh CNNs apply anisotropic kernels and are therefore sensitive to orientations. The arbitrariness of reference orientations, determined by a choice of neighbour $q_{0}$, is accounted for by the gauge equivariance of the model. However, when representing a mesh by a graph, we lose important geometrical information. In particular, in a graph there is no notion of angle between or ordering of two of a node’s incident edges (see figure 1). Hence, a GCNs output at a node $p$ is designed to be independent of relative angles and _invariant_ to any permutation of its neighbours $q_{i}\in\mathcal{N}(p)$. A graph convolution on a mesh graph therefore corresponds to applying an _isotropic_ convolution kernel. Isotropic filters are insensitive to the orientation of input patterns, so their features are strictly less expressive than those of orientation aware anisotropic filters. To address this limitation of graph networks we propose Gauge Equivariant Mesh CNNs (GEM-CNN)111Implementation at https://github.com/Qualcomm-AI- research/gauge-equivariant-mesh-cnn, which minimally modify GCNs such that they are able to use anisotropic filters while sharing weights across different positions and respecting the local geometry. One obstacle in sharing anisotropic kernels, which are functions of the angle $\theta_{pq}$ of neighbour $q$ with respect to vertex $p$, over multiple vertices of a mesh is that there is no unique way of selecting a reference neighbour $q_{0}$, which has the direction $\theta_{pq_{0}}=0$. The reference neighbour, and hence the orientation of the neighbours, needs to be chosen arbitrarily. In order to guarantee the equivalence of the features resulting from different choices of orientations, we adapt Gauge Equivariant (or coordinate independent) CNNs (Cohen et al., 2019b; Weiler et al., 2021) to general meshes. The kernels of our model are thus designed to be _equivariant under gauge transformations_ , that is, to guarantee that the responses for different kernel orientations are related by a prespecified transformation law. Such features are identified as geometric objects like scalars, vectors, tensors, etc., depending on the specific choice of transformation law. In order to compare such geometric features at neighbouring vertices, they need to be _parallel transported_ along the connecting edge. In our implementation we first specify the transformation laws of the feature spaces and compute a space of gauge equivariant kernels. Then we pick arbitrary reference orientations at each node, relative to which we compute neighbour orientations and compute the corresponding edge transporters. Given these quantities, we define the forward pass as a message passing step via edge transporters followed by a contraction with the equivariant kernels evaluated at the neighbour orientations. Algorithmically, Gauge Equivariant Mesh CNNs are therefore just GCNs with anisotropic, gauge equivariant kernels and message passing via parallel transporters. Conventional GCNs are covered in this framework for the specific choice of isotropic kernels and trivial edge transporters, given by identity maps. In Sec. 2, we will give an outline of our method, deferring details to Secs. 3 and 4. In Sec. 3.2, we describe how to compute general geometric quantities, not specific to our method, used for the computation of the convolution. In our experiments in Sec. 6.1, we find that the enhanced expressiveness of Gauge Equivariant Mesh CNNs enables them to outperform conventional GCNs and other prior work in a shape correspondence task. ## 2 Convolutions on Graphs with Geometry We consider the problem of processing signals on discrete 2-dimensional manifolds, or meshes $M$. Such meshes are described by a set ${\mathcal{V}}$ of vertices in $\mathbb{R}^{3}$ together with a set ${\mathcal{F}}$ of tuples, each consisting of the vertices at the corners of a face. For a mesh to describe a proper manifold, each edge needs to be connected to two faces, and the neighbourhood of each vertex needs to be homeomorphic to a disk. Mesh $M$ induces a graph ${\mathcal{G}}$ by forgetting the coordinates of the vertices while preserving the edges. A conventional graph convolution between kernel $K$ and signal $f$, evaluated at a vertex $p$, can be defined by $\displaystyle(K\star f)_{p}\ =\ K_{\text{self}}f_{p}\,+\sum\nolimits_{q\in{\mathcal{N}}_{p}}\\!\\!K_{\textup{neigh}}f_{q},$ (1) where ${\mathcal{N}}_{p}$ is the set of neighbours of $p$ in ${\mathcal{G}}$, and $K_{\text{self}}\in\mathbb{R}^{C_{\textup{in}}\times C_{\textup{out}}}$ and $K_{\textup{neigh}}\in\mathbb{R}^{C_{\textup{in}}\times C_{\textup{out}}}$ are two linear maps which model a self interaction and the neighbour contribution, respectively. Importantly, graph convolution does not distinguish different neighbours, because each feature vector $f_{q}$ is multiplied by the same matrix $K_{\textup{neigh}}$ and then summed. For this reason we say the kernel is _isotropic_. Algorithm 1 Gauge Equivariant Mesh CNN layer Input: mesh $M$, input/output feature types $\rho_{\textup{in}},\rho_{\textup{out}}$, reference neighbours $(q_{0}^{p}\in{\mathcal{N}}_{p})_{p\in M}$. Compute basis kernels $K^{i}_{\textup{self}},K^{i}_{\textup{neigh}}(\theta)$ $\rhd$ Sec. 3 Initialise weights $w_{\textup{self}}^{i}$ and $w^{i}_{\textup{neigh}}$. For each neighbour pair, $p\in M,q\in{\mathcal{N}}_{p}$: $\rhd$ App. A. compute neighbor angles $\theta_{pq}$ relative to reference neighbor compute parallel transporters $g_{q\to p}$ Forward$\big{(}$input features $(f_{p})_{p\in M}$, weights $w^{i}_{\textup{self}},w^{i}_{\textup{neigh}}$$\big{)}$: $f^{\prime}_{p}\leftarrow\sum_{i}w_{\textup{self}}^{i}K^{i}_{\textup{self}}f_{p}+\\!\\!\\!\sum_{i,q\in{\mathcal{N}}_{p}}\\!\\!w^{i}_{\textup{neigh}}K^{i}_{\textup{neigh}}(\theta_{pq})\rho_{\textup{in}}(g_{q\to p})f_{q}$ Consider the example in figure 1, where on the left and right, the neighbourhood of one vertex $p$, containing neighbours $q\in{\mathcal{N}}_{p}$, is visualized. An isotropic kernel would propagate the signal from the neighbours to $p$ in exactly the same way in both neighbourhoods, even though the neighbourhoods are geometrically distinct. For this reason, our method uses direction sensitive (_anisotropic_) kernels instead of isotropic kernels. Anisotropic kernels are inherently more expressive than isotropic ones which is why they are used universally in conventional planar CNNs. We propose the Gauge Equivariant Mesh Convolution, a minimal modification of graph convolution that allows for anisotropic kernels $K(\theta)$ whose value depends on an orientation $\theta\in[0,2\pi)$.222 In principle, the kernel could be made dependent on the radial distance of neighboring nodes, by $K_{\textup{neigh}}(r,\theta)=F(r)K_{\textup{neigh}}(\theta)$, where $F(r)$ is unconstrained and $K_{\textup{neigh}}(\theta)$ as presented in this paper. As this dependency did not improve the performance in our empirical evaluation, we omit it. To define the orientations $\theta_{pq}$ of neighbouring vertices $q\in{\mathcal{N}}_{p}$ of $p$, we first map them to the tangent plane $T_{p}M$ at $p$, as visualized in figure 1. We then pick an _arbitrary_ reference neighbour $q^{p}_{0}$ to determine a reference orientation333 Mathematically, this corresponds to a choice of _local reference frame_ or _gauge_. $\theta_{pq^{p}_{0}}:=0$, marked orange in figure 1. This induces a basis on the tangent plane, which, when expressed in polar coordinates, defines the angles $\theta_{pq}$ of the other neighbours. As we will motivate in the next section, features in a Gauge Equivariant CNN are coefficients of geometric quantities. For example, a tangent vector at vertex $p$ can be described either geometrically by a 3 dimensional vector orthogonal to the normal at $p$ or by two coefficients in the basis on the tangent plane. In order to perform convolution, geometric features at different vertices need to be linearly combined, for which it is required to first “parallel transport” the features to the same vertex. This is done by applying a matrix $\rho(g_{q\to p})\in\mathbb{R}^{C_{\textup{in}}\times C_{\textup{in}}}$ to the coefficients of the feature at $q$, in order to obtain the coefficients of the feature vector transported to $p$, which can be used for the convolution at $p$. The transporter depends on the geometric _type_ (group representation) of the feature, denoted by $\rho$ and described in more detail below. Details of how the tangent space is defined, how to compute the map to the tangent space, angles $\theta_{pq}$, and the parallel transporter are given in Appendix A. In combination, this leads to the GEM-CNN convolution $(K\star f)_{p}\ =\ K_{\textup{self}}f_{p}+\sum\nolimits_{q\in{\mathcal{N}}_{p}}K_{\textup{neigh}}(\theta_{pq})\rho(g_{q\to p})f_{q}$ (2) which differs from the conventional graph convolution, defined in Eq. 1 only by the use of an anisotropic kernel and the parallel transport message passing. We require the outcome of the convolution to be _equivalent_ for any choice of reference orientation. This is not the case for any anisotropic kernel but only for those which are _equivariant under changes of reference orientations_ (gauge transformations). Equivariance imposes a linear constraint on the kernels. We therefore solve for complete sets of “basis-kernels” $K_{\text{self}}^{i}$ and $K_{\text{neigh}}^{i}$ satisfying this constraint and linearly combine them with parameters $w_{\text{self}}^{i}$ and $w_{\text{neigh}}^{i}$ such that $K_{\text{self}}=\sum_{i}w_{\text{self}}^{i}K_{\text{self}}^{i}$ and $K_{\text{neigh}}=\sum_{i}w_{\text{neigh}}^{i}K_{\text{neigh}}^{i}$. Details on the computation of basis kernels are given in section 3. The full algorithm for initialisation and forward pass, which is of time and space complexity linear in the number of vertices, for a GEM-CNN layer are listed in algorithm 1. Gradients can be computed by automatic differentiation. The GEM-CNN is gauge equivariant, but furthermore satisfies two important properties. Firstly, it depends only on the intrinsic shape of the 2D mesh, not on the embedding of the mesh in $\mathbb{R}^{3}$. Secondly, whenever a map from the mesh to itself exists that preserves distances and orientation, the convolution is equivariant to moving the signal along such transformations. These properties are proven in Appendix D and empirically shown in Appendix F.2. $p$$q_{A}$$q_{B}$$q_{C}$$p$$q_{A}$$q_{B}$$q_{C}$$p$$q_{A}$$q_{B}$$q_{C}$$p$$q_{A}$$q_{B}$$q_{C}$pick gauge $q_{0}=q_{A}$map back to meshpick gauge $q_{0}=q_{B}$map back to meshgeometric convconv in gauge $A$conv in gauge $B$gauge transformation $A\\!\to\\!B$gauge transformation $A\\!\to\\!B$ (a) Convolution from scalar to scalar features. $p$$q_{A}$$q_{B}$$q_{C}$$p$$q_{A}$$q_{B}$$q_{C}$$p$$q_{A}$$q_{B}$$q_{C}$$p$$q_{A}$$q_{B}$$q_{C}$pick gauge $q_{0}=q_{A}$map back to meshpick gauge $q_{0}=q_{B}$map back to meshgeometric convconv in gauge $A$conv in gauge $B$gauge transfromation $A\\!\to\\!B$gauge transfromation $A\\!\to\\!B$ (b) Convolution from scalar to vector features. Figure 2: Visualization of the Gauge Equivariant Mesh Convolution in two configurations, scalar to scalar and scalar to vector. The convolution operates in a gauge, so that vectors are expressed in coefficients in a basis and neighbours have polar coordinates, but can also be seen as a _geometric convolution_ , a gauge-independent map from an input signal on the mesh to a output signal on the mesh. The convolution is equivariant if this geometric convolution does not depend on the intermediate chosen gauge, so if the diagram commutes. ## 3 Gauge Equivariance & Geometric Features On a general mesh, the choice of the reference neighbour, or gauge, which defines the orientation of the kernel, can only be made arbitrarily. However, this choice should not arbitrarily affect the outcome of the convolution, as this would impede the generalization between different locations and different meshes. Instead, Gauge Equivariant Mesh CNNs have the property that their output transforms according to a known rule as the gauge changes. Consider the left hand side of figure 2(a). Given a neighbourhood of vertex $p$, we want to express each neighbour $q$ in terms of its polar coordinates $(r_{q},\theta_{q})$ on the tangent plane, so that the kernel value at that neighbour $K_{\textup{neigh}}(\theta_{q})$ is well defined. This requires choosing a basis on the tangent plane, determined by picking a neighbour as reference neighbour (denoted $q_{0}$), which has the zero angle $\theta_{q_{0}}=0$. In the top path, we pick $q_{A}$ as reference neighbour. Let us call this gauge A, in which neighbours have angles $\theta^{A}_{q}$. In the bottom path, we instead pick neighbour $q_{B}$ as reference point and are in gauge B. We get a different basis for the tangent plane and different angles $\theta^{B}_{q}$ for each neighbour. Comparing the two gauges, we see that they are related by a rotation, so that $\theta^{B}_{q}=\theta^{A}_{q}-\theta^{A}_{q_{B}}$. This change of gauge is called a gauge transformation of angle $g:=\theta^{A}_{q_{B}}$. In figure 2(a), we illustrate a gauge equivariant convolution that takes input and output features such as gray scale image values on the mesh, which are called scalar features. The top path represents the convolution in gauge A, the bottom path in gauge B. In either case, the convolution can be interpreted as consisting of three steps. First, for each vertex $p$, the value of the scalar features on the mesh at each neighbouring vertex $q$, represented by colors, is mapped to the tangent plane at $p$ at angle $\theta_{q}$ defined by the gauge. Subsequently, the convolutional kernel sums for each neighbour $q$, the product of the feature at $q$ and kernel $K(\theta_{q})$. Finally the output is mapped back to the mesh. These three steps can be composed into a single step, which we could call a _geometric convolution_ , mapping from input features on the mesh to output features on the mesh. The convolution is _gauge equivariant_ if this geometric convolution does not depend on the gauge we pick in the interim, so in figure 2(a), if the convolution in the top path in gauge A has same result the convolution in the bottom path in gauge B, making the diagram commute. In this case, however, we see that the convolution output needs to be the same in both gauges, for the convolution to be equivariant. Hence, we must have that $K(\theta_{q})=K(\theta_{q}-g)$, as the orientations of the neighbours differ by some angle $g$, and the kernel must be isotropic. As we aim to design an anisotropic convolution, the output feature of the convolution at $p$ can, instead of a scalar, be two numbers $v\in\mathbb{R}^{2}$, which can be interpreted as coefficients of a tangent feature vector in the tangent space at $p$, visualized in figure 2(b). As shown on the right hand side, different gauges induce a different basis of the tangent plane, so that the _same tangent vector_ (shown on the middle right on the mesh), is represented by _different coefficients_ in the gauge (shown on the top and bottom on the right). This gauge equivariant convolution must be anisotropic: going from the top row to the bottom row, if we change orientations of the neighbours by $-g$, the coefficients of the output vector $v\in\mathbb{R}^{2}$ of the kernel must be also rotated by $-g$. This is written as $R(-g)v$, where $R(-g)\in\mathbb{R}^{2\times 2}$ is the matrix that rotates by angle $-g$. Vectors and scalars are not the only type of geometric features that can be inputs and outputs of a GEM-CNN layer. In general, the coefficients of a geometric feature of $C$ dimensions changes by an invertible linear transformation $\rho(-g)\in\mathbb{R}^{C\times C}$ if the gauge is rotated by angle $g$. The map $\rho:[0,2\pi)\to\mathbb{R}^{C\times C}$ is called the _type_ of the geometric quantity and is formally known as a group representation of the planar rotation group $\operatorname{SO}(2)$. Group representations have the property that $\rho(g+h)=\rho(g)\rho(h)$ (they are group homomorphisms), which implies in particular that $\rho(0)=\mathbbm{1}$ and $\rho(-g)=\rho(g)^{-1}$. For more background on group representation theory, we refer the reader to (Serre, 1977) and, specifically in the context of equivariant deep learning, to (Lang & Weiler, 2020). From the theory of group representations, we know that any feature type can be composed from “irreducible representations” (irreps). For $\operatorname{SO}(2)$, these are the one dimensional invariant scalar representation $\rho_{0}$ and for all $n\in{\mathbb{N}}_{>0}$, a two dimensional representation $\rho_{n}$, $\rho_{0}(g)=1,\quad\rho_{n}(g)=\begin{pmatrix}\cos ng&\shortminus\sin ng\\\ \sin ng&\phantom{\shortminus}\cos ng\end{pmatrix}.$ where we write, for example, $\rho=\rho_{0}\oplus\rho_{1}\oplus\rho_{1}$ to denote that representation $\rho(g)$ is the direct sum (i.e. block-diagonal stacking) of the matrices $\rho_{0}(g),\rho_{1}(g),\rho_{1}(g)$. Scalars and tangent vector features correspond to $\rho_{0}$ and $\rho_{1}$ respectively and we have $R(g)=\rho_{1}(g)$. The type of the feature at each layer in the network can thus be fully specified (up to a change of basis) by the number of copies of each irrep. Similar to the dimensionality in a conventional CNN, the choice of type is a hyperparameter that can be freely chosen to optimize performance. ### 3.1 Kernel Constraint Given an input type $\rho_{\textup{in}}$ and output type $\rho_{\textup{out}}$ of dimensions $C_{\textup{in}}$ and $C_{\textup{out}}$, the kernels are $K_{\textup{self}}\in\mathbb{R}^{C_{\textup{out}}\times C_{\textup{in}}}$ and $K_{\textup{neigh}}:[0,2\pi)\to\mathbb{R}^{C_{\textup{out}}\times C_{\textup{in}}}$. However, not all such kernels are equivariant. Consider again examples figure 2(a) and figure 2(b). If we map from a scalar to a scalar, we get that $K_{\textup{neigh}}(\theta-g)=K_{\textup{neigh}}(\theta)$ for all angles $\theta,g$ and the convolution is isotropic. If we map from a scalar to a vector, we get that rotating the angles $\theta_{q}$ results in the same tangent vector as rotating the output vector coefficients, so that $K_{\textup{neigh}}(\theta-g)=R(-g)K_{\textup{neigh}}(\theta)$. $\rho_{\textup{in}}\to\rho_{\textup{out}}$ | linearly independent solutions for $K_{\textup{neigh}}(\theta)$ ---|--- $\rho_{0}\to\rho_{0}$ | $(1)$ $\rho_{n}\to\rho_{0}$ | $\begin{pmatrix}\cos n\theta&\sin n\theta\end{pmatrix},\begin{pmatrix}\sin n\theta&\shortminus\cos n\theta\end{pmatrix}$ $\rho_{0}\to\rho_{m}$ | $\begin{pmatrix}\cos m\theta\\\ \sin m\theta\end{pmatrix},\begin{pmatrix}\phantom{\shortminus}\sin m\theta\\\ \shortminus\cos m\theta\end{pmatrix}$ $\rho_{n}\to\rho_{m}$ | ​​​ $\begin{pmatrix}c&\shortminus s\\\ s&\phantom{\shortminus}c\end{pmatrix}$, $\begin{pmatrix}\phantom{\shortminus}s&c\\\ \shortminus c&s\end{pmatrix}$, $\begin{pmatrix}c_{+}&\phantom{\shortminus}s_{+}\\\ s_{+}&\shortminus c_{+}\end{pmatrix}$, $\begin{pmatrix}\shortminus s_{+}&c_{+}\\\ \phantom{-}c_{+}&s_{+}\end{pmatrix}$ ​​​ $\rho_{\textup{in}}\to\rho_{\textup{out}}$ | linearly independent solutions for $K_{\textup{self}}$ $\rho_{0}\to\rho_{0}$ | $(1)$ $\rho_{n}\to\rho_{n}$ | $\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$, $\begin{pmatrix}\phantom{\shortminus}0&1\\\ \shortminus 1&0\end{pmatrix}$ Table 1: Solutions to the angular kernel constraint for kernels that map from $\rho_{n}$ to $\rho_{m}$. We denote ${c_{\pm}=\cos((m\pm n)\theta)}$ and ${s_{\pm}=\sin((m\pm n)\theta)}$. In general, as derived by Cohen et al. (2019b); Weiler et al. (2021) and in appendix B, the kernels must satisfy for any gauge transformation $g\in[0,2\pi)$ and angle $\theta\in[0,2\pi)$, that $\displaystyle\\!\\!K_{\textup{neigh}}(\theta-g)$ $\displaystyle=\rho_{\textup{out}}(-g)K_{\textup{neigh}}(\theta)\rho_{\textup{in}}(g),$ (3) $\displaystyle K_{\textup{self}}$ $\displaystyle=\rho_{\textup{out}}(-g)\;K_{\textup{self}}\;\rho_{\textup{in}}(g).$ (4) The kernel can be seen as consisting of multiple blocks, where each block takes as input one irrep and outputs one irrep. For example if $\rho_{\textup{in}}$ would be of type $\rho_{0}\oplus\rho_{1}\oplus\rho_{1}$ and $\rho_{\textup{out}}$ of type $\rho_{1}\oplus\rho_{3}$, we have $4\times 5$ matrix $\displaystyle K_{\textup{neigh}}(\theta)=\begin{pmatrix}K_{10}(\theta)&K_{11}(\theta)&K_{11}(\theta)\\\ K_{30}(\theta)&K_{31}(\theta)&K_{31}(\theta)\\\ \end{pmatrix}$ where e.g. $K_{31}(\theta)\in\mathbb{R}^{2\times 2}$ is a kernel that takes as input irrep $\rho_{1}$ and as output irrep $\rho_{3}$ and needs to satisfy Eq. 3. As derived by Weiler & Cesa (2019) and in Appendix C, the kernels $K_{\textup{neigh}}(\theta)$ and $K_{\textup{self}}$ mapping from irrep $\rho_{n}$ to irrep $\rho_{m}$ can be written as a linear combination of the basis kernels listed in Table 1. The table shows that equivariance requires the self-interaction to only map from one irrep to the same irrep. Hence, we have $K_{\textup{self}}=\begin{pmatrix}0&K_{11}&K_{11}\\\ 0&0&0\\\ \end{pmatrix}\in\mathbb{R}^{4\times 3}.$ All basis-kernels of all pairs of input irreps and output irreps can be linearly combined to form an arbitrary equivariant kernel from feature of type $\rho_{\textup{in}}$ to $\rho_{\textup{out}}$. In the above example, we have $2\times 2+4\times 4=20$ basis kernels for $K_{\textup{neigh}}$ and 4 basis kernels for $K_{\textup{self}}$. The layer thus has 24 parameters. As proven in (Weiler & Cesa, 2019) and (Lang & Weiler, 2020), this parameterization of the equivariant kernel space is _complete_ , that is, more general equivariant kernels do not exist. ### 3.2 Geometry and Parallel Transport In order to implement gauge equivariant mesh CNNs, we need to make the abstract notion of tangent spaces, gauges and transporters concrete. As the mesh is embedded in $\mathbb{R}^{3}$, a natural definition of the tangent spaces $T_{p}M$ is as two dimensional subspaces that are orthogonal to the normal vector at $p$. We follow the common definition of normal vectors at mesh vertices as the area weighted average of the adjacent faces’ normals. The Riemannian logarithm map $\log_{p}:{\mathcal{N}}_{p}\to T_{p}M$ represents the one-ring neighborhood of each point $p$ on their tangent spaces as visualized in figure 1. Specifically, neighbors $q\in{\mathcal{N}}_{p}$ are mapped to $\log_{p}(q)\in T_{p}M$ by first projecting them to $T_{p}M$ and then rescaling the projection such that the norm is preserved, i.e. $|\log_{p}(q)|=|q-p|$; see Eq. 9. A choice of reference neighbor $q_{p}\in{\mathcal{N}}$ uniquely determines a right handed, orthonormal reference frame $(e_{p,1},\,e_{p,2})$ of $T_{p}M$ by setting $e_{p,1}:=\log_{p}(q_{0})/|\log_{p}(q_{0})|$ and $e_{p,2}:=n\times e_{p,1}$. The polar angle $\theta_{pq}$ of any neighbor $q\in{\mathcal{N}}$ relative to the first frame axis is then given by $\theta_{pq}\ :=\ \operatorname{atan2}\big{(}e_{p,2}^{\top}\log_{p}(q),\ e_{p,1}^{\top}\log_{p}(q))\big{)}.$ Given the reference frame $(e_{p,1},e_{p,2})$, a 2-tuple of coefficients $(v_{1},v_{2})\in\mathbb{R}^{2}$ specifies an (embedded) tangent vector ${v_{1}e_{p,1}+v_{2}e_{p,2}}\in T_{p}M\subset\mathbb{R}^{3}$. This assignment is formally given by the _gauge map_ $E_{p}:\mathbb{R}^{2}\to T_{p}M\subset\mathbb{R}^{3}$ which is a vector space isomorphism. In our case, it can be identified with the matrix $\displaystyle E_{p}=\left[\begin{array}[]{cc}\rule[-2.15277pt]{0.5pt}{5.38193pt}&\rule[-2.15277pt]{0.5pt}{5.38193pt}\\\ e_{p,1}&e_{p,2}\\\ \rule[0.0pt]{0.5pt}{5.38193pt}&\rule[0.0pt]{0.5pt}{5.38193pt}\end{array}\right]\in\mathbb{R}^{3\times 2}.$ (8) Feature vectors $f_{p}$ and $f_{q}$ at neighboring (or any other) vertices $p\in M$ and $q\in{\mathcal{N}}_{p}\subseteq M$ live in different vector spaces and are expressed relative to independent gauges, which makes it invalid to sum them directly. Instead, they have to be parallel transported along the mesh edge that connects the two vertices. As explained above, this transport is given by group elements $g_{q\to p}\in[0,2\pi)$, which determine the transformation of tangent vector _coefficients_ as $v_{q}\mapsto R(g_{q\to p})v_{q}\in\mathbb{R}^{2}$ and, analogously, for feature vector coefficients as $f_{q}\mapsto\rho(g_{q\to p})f_{q}$. Figure 4 in the appendix visualizes the definition of edge transporters for flat spaces and meshes. On a flat space, tangent vectors are transported by keeping them parallel in the usual sense on Euclidean spaces. However, if the source and target frame orientations disagree, the vector coefficients relative to the source frame need to be transformed to the target frame. This coordinate transformation from polar angles $\varphi_{q}$ of $v$ to $\varphi_{p}$ of $R(g_{q\to p})v$ defines the transporter $g_{q\to p}=\varphi_{p}-\varphi_{q}$. On meshes, the source and target tangent spaces $T_{q}M$ and $T_{p}M$ are not longer parallel. It is therefore additionally necessary to rotate the source tangent space and its vectors parallel to the target space, before transforming between the frames. Since transporters effectively make up for differences in the source and target frames, the parallel transporters transform under gauge transformations $g_{p}$ and $g_{q}$ according to $g_{q\to p}\mapsto g_{p}+g_{q\to p}-g_{q}$. Note that this transformation law cancels with the transformation law of the coefficients at $q$ and lets the transported coefficients transform according to gauge transformations at $p$. It is therefore valid to sum vectors and features that are parallel transported into the same gauge at $p$. A more detailed discussion of the concepts presented in this section can be found in Appendix A. ## 4 Non-linearity Besides convolutional layers, the GEM-CNN contains non-linear layers, which also need to be gauge equivariant, for the entire network to be gauge equivariant. The coefficients of features built out of irreducible representaions, as described in section 3, do not commute with point-wise non- linearities (Worrall et al., 2017; Thomas et al., 2018; Weiler et al., 2018a; Kondor et al., 2018). Norm non-linearities and gated non-linearities (Weiler & Cesa, 2019) can be used with such features, but generally perform worse in practice compared to point-wise non-linearities (Weiler & Cesa, 2019). Hence, we propose the _RegularNonlinearity_ , which uses point-wise non-linearities and is approximately gauge equivariant. This non-linearity is built on Fourier transformations. Consider a continuous periodic signal, on which we perform a band-limited Fourier transform with band limit $b$, obtaining $2b+1$ Fourier coefficients. If this continuous signal is shifted by an arbitrary angle $g$, then the corresponding Fourier components transform with linear transformation $\rho_{0:b}(-g)$, for $2b+1$ dimensional representation $\rho_{0:b}:=\rho_{0}\oplus\rho_{1}\oplus...\oplus\rho_{b}$. It would be exactly equivariant to take a feature of type $\rho_{0:b}$, take a continuous inverse Fourier transform to a continuous periodic signal, then apply a point-wise non-linearity to that signal, and take the continuous Fourier transform, to recover a feature of type $\rho_{0:b}$. However, for implementation, we use $N$ intermediate samples and the discrete Fourier transform. This is exactly gauge equivariant for gauge transformation of angles multiple of $2\pi/N$, but only approximately equivariant for other angles. In App. G we prove that as $N\to\infty$, the non-linearity is exactly gauge equivariant. The run-time cost per vertex of the (inverse) Fourier transform implemented as a simple linear transformation is $\mathcal{O}(bN)$, which is what we use in our experiments. The pointwise non-linearity scales linearly with $N$, so the complexity of the RegularNonLineariy is also $\mathcal{O}(bN)$. However, one can also use a fast Fourier transform, achieving a complexity of $\mathcal{O}(N\log N)$. Concrete memory and run-time cost of varying $N$ are shown in appendix F.1. ## 5 Related Work Our method can be seen as a practical implementation of coordinate independent convolutions on triangulated surfaces, which generally rely on $G$-steerable kernels (Weiler et al., 2021). The irregular structure of meshes leads to a variety of approaches to define convolutions. Closely related to our method are graph based methods which are often based on variations of graph convolutional networks (Kipf & Welling, 2017; Defferrard et al., 2016). GCNs have been applied on spherical meshes (Perraudin et al., 2019) and cortical surfaces (Cucurull et al., 2018; Zhao et al., 2019a). Verma et al. (2018) augment GCNs with anisotropic kernels which are dynamically computed via an attention mechanism over graph neighbours. Instead of operating on the graph underlying a mesh, several approaches leverage its geometry by treating it as a discrete manifold. Convolution kernels can then be defined in geodesic polar coordinates which corresponds to a projection of kernels from the tangent space to the mesh via the exponential map. This allows for kernels that are larger than the immediate graph neighbourhood and message passing over faces but does not resolve the issue of ambiguous kernel orientation. Masci et al. (2015); Monti et al. (2016) and Sun et al. (2018) address this issue by restricting the network to orientation invariant features which are computed by applying anisotropic kernels in several orientations and pooling over the resulting responses. The models proposed in (Boscaini et al., 2016) and (Schonsheck et al., 2018) are explicitly gauge dependent with preferred orientations chosen via the principal curvature direction and the parallel transport of kernels, respectively. Poulenard & Ovsjanikov (2018) proposed a non-trivially gauge equivariant network based on geodesic convolutions, however, the model parallel transports only partial information of the feature vectors, corresponding to certain kernel orientations. In concurrent work, Wiersma et al. (2020) also define convolutions on surfaces equivariantly to the orientation of the kernel, but differ in that they use norm non-linearities instead of regular ones and that they apply the convolution along longer geodesics, which adds complexity to the geometric pre-computation - as partial differential equations need to be solved, but may result in less susceptibility to the particular discretisation of the manifold. A comprehensive review on such methods can be found in Section 12 of (Weiler et al., 2021). Another class of approaches defines spectral convolutions on meshes. However, as argued in (Bronstein et al., 2017), the Fourier spectrum of a mesh depends heavily on its geometry, which makes such methods instable under deformations and impedes the generalization between different meshes. Spectral convolutions further correspond to isotropic kernels. Kostrikov et al. (2018) overcomes isotropy of the Laplacian by decomposing it into two applications of the first-order Dirac operator. A construction based on toric covering maps of topologically spherical meshes was presented in (Maron et al., 2017). An entirely different approach to mesh convolutions is to apply a linear map to a spiral of neighbours (Bouritsas et al., 2019; Gong et al., 2019), which works well only for meshes with a similar graph structure. The above-mentioned methods operate on the intrinsic, $2$-dimensional geometry of the mesh. A popular alternative for embedded meshes is to define convolutions in the embedding space $\mathbb{R}^{3}$. This can for instance be done by voxelizing space and representing the mesh in terms of an occupancy grid (Wu et al., 2015; Tchapmi et al., 2017; Hanocka et al., 2018). A downside of this approach are the high memory and compute requirements of voxel representations. If the grid occupancy is low, this can partly be addressed by resorting to an inhomogeneous grid density (Riegler et al., 2017). Instead of voxelizing space, one may interpret the set of mesh vertices as a point cloud and run a convolution on those (Qi et al., 2017a; b). Point cloud based methods can be made equivariant w.r.t. the isometries of $\mathbb{R}^{3}$ (Zhao et al., 2019b; Thomas et al., 2018), which implies in particular the isometry equivariance on the embedded mesh. In general, geodesic distances within the manifold differ usually substantially from the distances in the embedding space. Which approach is more suitable depends on the particular application. On flat Euclidean spaces our method corresponds to Steerable CNNs (Cohen & Welling, 2017; Weiler et al., 2018a; Weiler & Cesa, 2019; Cohen et al., 2019a; Lang & Weiler, 2020). As our model, these networks process geometric feature fields of types $\rho$ and are equivariant under gauge transformations, however, due to the flat geometry, the parallel transporters become trivial. Jenner & Weiler (2021) extended the theory of steerable CNNs to include equivariant partial differential operators. Regular nonlinearities are on flat spaces used in group convolutional networks (Cohen & Welling, 2016; Weiler et al., 2018b; Hoogeboom et al., 2018; Bekkers et al., 2018; Winkels & Cohen, 2018; Worrall & Brostow, 2018; Worrall & Welling, 2019; Sosnovik et al., 2020). ## 6 Experiments ### 6.1 Embedded MNIST We first investigate how Gauge Equivariant Mesh CNNs perform on, and generalize between, different mesh geometries. For this purpose we conduct simple MNIST digit classification experiments on embedded rectangular meshes of $28\\!\times\\!28$ vertices. As a baseline geometry we consider a flat mesh as visualized in figure 5(a). A second type of geometry is defined as different _isometric_ embeddings of the flat mesh, see figure 5(b). Note that this implies that the _intrinsic_ geometry of these isometrically embedded meshes is indistinguishable from that of the flat mesh. To generate geometries which are intrinsically curved, we add random normal displacements to the flat mesh. We control the amount of curvature by smoothing the resulting displacement fields with Gaussian kernels of different widths $\sigma$ and define the roughness of the resulting mesh as $3-\sigma$. Figures 5(c)-5(h) show the results for roughnesses of 0.5, 1, 1.5, 2, 2.25 and 2.5. For each of the considered settings we generate $32$ different train and $32$ test geometries. To test the performance on, and generalization between, different geometries, we train equivalent GEM-CNN models on a flat mesh and meshes with a roughness of 1, 1.5, 2, 2.25 and 2.5. Each model is tested individually on each of the considered test geometries, which are the flat mesh, isometric embeddings and curved embeddings with a roughness of 0.5, 1, 1.25, 1.5, 1.75, 2, 2.25 and 2.5. Figure 3 shows the test errors of the GEM-CNNs on the different train geometries (different curves) for all test geometries (shown on the x-axis). Since our model is purely defined in terms of the intrinsic geometry of a mesh, it is expected to be insensitive to isometric changes in the embeddings. This is empirically confirmed by the fact that the test performances on flat and isometric embeddings are exactly equal. As expected, the test error increases for most models with the surface roughness. Models trained on more rough surfaces are hereby more robust to deformations. The models generalize well from a rough training to smooth test geometry up to a training roughness of 1.5. Beyond that point, the test performances on smooth meshes degrades up to the point of random guessing at a training roughness of 2.5. As a baseline, we build an _isotropic_ graph CNN with the same network topology and number of parameters ($\approx 163k$). This model is insensitive to the mesh geometry and therefore performs exactly equal on all surfaces. While this enhances its robustness on very rough meshes, its test error of $19.80\pm 3.43\%$ is an extremely bad result on MNIST. In contrast, the use of anisotropic filters of GEM-CNN allows it to reach a test error of only $0.60\pm 0.05\%$ on the flat geometry. It is therefore competitive with conventional CNNs on pixel grids, which apply anisotropic kernels as well. More details on the datasets, models and further experimental setup are given in appendix E.1. ### 6.2 Shape Correspondence As a second experiment, we perform non-rigid shape correspondence on the FAUST dataset (Bogo et al., 2014), following Masci et al. (2015) 444These experiments were executed on QUVA machines. . The data consists of 100 meshes of human bodies in various positions, split into 80 train and 20 test meshes. The vertices are registered, such that vertices on the same position on the body, such as the tip of the left thumb, have the same identifier on all meshes. All meshes have $6890$ vertices, making this a $6890$-class segmentation problem. The architecture transforms the vertices’ ${XYZ}$ coordinates (of type $3\rho_{0}$), via 6 convolutional layers to features $64\rho_{0}$, with intermediate features $16(\rho_{0}\oplus\rho_{1}\oplus\rho_{2})$, with residual connections and the RegularNonlinearity with $N=5$ samples. Afterwards, we use two $1\\!\times\\!1$ convolutions with ReLU to map first to 256 and then 6980 channels, after which a softmax predicts the registration probabilities. The $1\\!\times\\!1$ convolutions use a dropout of 50% and 1E-4 weight decay. The network is trained with a cross entropy loss with an initial learning rate of 0.01, which is halved when training loss reaches a plateau. As all meshes in the FAUST dataset share the same topology, breaking the gauge equivariance in higher layers can actually be beneficial. As shown in (Weiler & Cesa, 2019), symmetry can be broken by treating non-invariant features as invariant features as input to the final $1\\!\times\\!1$ convolution. As baselines, we compare to various models, some of which use more complicated pipelines, such as (1) the computation of geodesics over the mesh, which requires solving partial differential equations, (2) pooling, which requires finding a uniform sub-selection of vertices, (3) the pre-computation of SHOT features which locally describe the geometry (Tombari et al., 2010), or (4) post-processing refinement of the predictions. The GEM-CNN requires none of these additional steps. In addition, we compare to SpiralNet++ (Gong et al., 2019), which requires all inputs to be similarly meshed. Finally, we compare to an isotropic version of the GEM-CNN, which reduces to a conventional graph CNN, as well as a non-gauge-equivariant CNN based on SHOT frames. The results in table 2 show that the GEM-CNN outperforms prior works and a non-gauge- equivariant CNN, that isotropic graph CNNs are unable to solve the task and that for this data set breaking gauge symmetry in the final layers of the network is beneficial. More experimental details are given in appendix E.2. Figure 3: Test errors for MNIST digit classification on embedded meshes. Different lines denote train geometries, x-axis shows test geometries. Regions are standard errors of the means over 6 runs. Model | Features | Accuracy (%) ---|---|--- ACNN (Boscaini et al., 2016) | SHOT | 62.4 Geodesic CNN (Masci et al., 2015) | SHOT | 65.4 MoNet (Monti et al., 2016) | SHOT | 73.8 FeaStNet (Verma et al., 2018) | XYZ | 98.7 ZerNet (Sun et al., 2018) | XYZ | 96.9 SpiralNet++ (Gong et al., 2019) | XYZ | 99.8 Graph CNN | XYZ | 1.40$\pm$0.5 Graph CNN | SHOT | 23.80$\pm$8 Non-equiv. CNN (SHOT frames) | XYZ | 73.00$\pm$4.0 Non-equiv. CNN (SHOT frames) | SHOT | 75.11$\pm$2.4 GEM-CNN | XYZ | 99.73$\pm$0.04 GEM-CNN (broken symmetry) | XYZ | 99.89$\pm$0.02 Table 2: Results of FAUST shape correspondence. Statistics are means and standard errors of the mean of over three runs. All cited results are from their respective papers. ## 7 Conclusions Convolutions on meshes are commonly performed as a convolution on their underlying graph, by forgetting geometry, such as orientation of neighbouring vertices. 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H., Shen, D., and Li, G. Spherical u-net on cortical surfaces: Methods and applications. _CoRR_ , abs/1904.00906, 2019a. URL http://arxiv.org/abs/1904.00906. * Zhao et al. (2019b) Zhao, Y., Birdal, T., Lenssen, J. E., Menegatti, E., Guibas, L., and Tombari, F. Quaternion equivariant capsule networks for 3d point clouds. _arXiv preprint arXiv:1912.12098_ , 2019b. ## Appendix A Geometry & Parallel Transport A gauge, or choice of reference neighbor at each vertex, fully determines the neighbor orientations $\theta_{pq}$ and the parallel transporters $g_{q\to p}$ along edges. The following two subsections give details on how to compute these quantities. ### A.1 Local neighborhood geometry Neighbours $q$ of vertex $p$ can be mapped uniquely to the tangent plane at $p$ using a map called the Riemannnian logarithmic map, visualized in figure 1. A choice of reference neighbor then determines a reference frame in the tangent space which assigns polar coordinates to all other neighbors. The neighbour orientations $\theta_{pq}$ are the angular components of each neighbor in this polar coordinate system. We define the tangent space $T_{p}M$ at vertex $p$ as that two dimensional subspace of $\mathbb{R}^{3}$, which is determined by a normal vector $n$ given by the area weighted average of the normal vectors of the adjacent mesh faces. While the tangent spaces are two dimensional, we implement them as being embedded in the ambient space $\mathbb{R}^{3}$ and therefore represent their elements as three dimensional vectors. The reference frame corresponding to the chosen gauge, defined below, allows to identify these 3-vectors by their coefficient 2-vectors. Each neighbor $q$ is represented in the tangent space by the vector $\log_{p}(q)\in T_{p}M$ which is computed via the discrete analog of the Riemannian logarithm map. We define this map $\log_{p}:{\mathcal{N}}_{p}\to T_{p}M$ for neighbouring nodes as the projection of the edge vector $q-p$ on the tangent plane, followed by a rescaling such that the norm $|\log_{p}(q)|=|q-p|$ is preserved. Writing the projection operator on the tangent plane as $(\mathbbm{1}-nn^{\top})$, the logarithmic map is thus given by: $\displaystyle\log_{p}(q)\ :=\ |q-p|\frac{(\mathbbm{1}-nn^{\top})(q-p)}{|(\mathbbm{1}-nn^{\top})(q-p)|}$ (9) Geometrically, this map can be seen as “folding” each edge up to the tangent plane, and therefore encodes the orientation of edges and preserves their lengths. The normalized reference edge vector $\log_{p}(q_{0})$ uniquely determines a right handed, orthonormal reference frame $(e_{p,1},\,e_{p,2})$ of $T_{p}M$ by setting $e_{p,1}:=\log_{p}(q_{0})/|\log_{p}(q_{0})|$ and $e_{p,2}:=n\times e_{p,1}$. The angle $\theta_{pq}$ is then defined as the angle of $\log_{p}(q)$ in polar coordinates corresponding to this reference frame. Numerically, it can be computed by $\theta_{pq}\ :=\ \operatorname{atan2}\big{(}e_{p,2}^{\top}\log_{p}(q),\ e_{p,1}^{\top}\log_{p}(q))\big{)}.$ Given the reference frame $(e_{p,1},e_{p,2})$, a 2-tuple of coefficients $(v_{1},v_{2})\in\mathbb{R}^{2}$ specifies an (embedded) tangent vector ${v_{1}e_{p,1}+v_{2}e_{p,2}}\in T_{p}M\subset\mathbb{R}^{3}$. This assignment is formally given by the _gauge map_ $E_{p}:\mathbb{R}^{2}\to T_{p}M\subset\mathbb{R}^{3}$ which is a vector space isomorphism. In our case, it can be identified with the matrix $\displaystyle E_{p}=\left[\begin{array}[]{cc}\rule[-2.15277pt]{0.5pt}{5.38193pt}&\rule[-2.15277pt]{0.5pt}{5.38193pt}\\\ e_{p,1}&e_{p,2}\\\ \rule[0.0pt]{0.5pt}{5.38193pt}&\rule[0.0pt]{0.5pt}{5.38193pt}\end{array}\right]\in\mathbb{R}^{3\times 2}.$ (13) ### A.2 Parallel edge transporters (a) Parallel transport on a flat mesh. (b) Parallel transport along an edge of a general mesh. Figure 4: Parallel transport of tangent vectors $v\in T_{q}M$ at $q$ to $R(g_{q\to p})v\in T_{p}M$ at $p$ on meshes. On a flat mesh, visualized in figure 4(a), parallel transport moves a vector such that it stays parallel in the usual sense on flat spaces. The parallel transporter $g_{q\to p}=\varphi_{p}-\varphi_{q}$ corrects the transported vector _coefficients_ for differing gauges at $q$ and $p$. When transporting along the edge of a general mesh, the tangent spaces at $q$ and $p$ might not be aligned, see figure 4(b). Before correcting for the relative frame orientation via $g_{q\to p}$, the tangent space $T_{q}M$, and thus $v\in T_{q}M$, is rotated by an angle $\alpha$ around $n_{q}\\!\times\\!n_{p}$ such that its normal $n_{q}$ coincides with that of $n_{p}$. On curved meshes, feature vectors $f_{q}$ and $f_{p}$ at different locations $q$ and $p$ are expressed in different gauges, which makes it geometrically invalid to accumulate their information directly. Instead, when computing a new feature at $p$, the neighboring feature vectors at $q\in{\mathcal{N}}_{p}$ first have to be parallel transported into the feature space at $p$ before they can be processed. The parallel transport along the edges of a mesh is determined by the (discrete) Levi-Civita connection corresponding to the metric induced by the ambient space $\mathbb{R}^{3}$. This connection is given by parallel transporters $g_{q\to p}\in[0,2\pi)$ on the mesh edges which map tangent vectors $v_{q}\in T_{q}M$ at $q$ to tangent vectors $R(g_{q\to p})v_{q}\in T_{p}M$ at $p$. Feature vectors $f_{q}$ of type $\rho$ are similarly transported to $\rho(g_{q\to p})f_{q}$ by applying the corresponding feature vector transporter $\rho(g_{q\to p})$. In order to build some intuition, it is illustrative to first consider transporters on a planar mesh. In this case the parallel transport can be thought of as moving a vector along an edge without rotating it. The resulting abstract vector is then parallel to the original vector in the usual sense on flat spaces, see figure 4(a). However, if the (transported) source frame at $q$ disagrees with the target frame at $p$, the _coefficients_ of the transported vector have to be transformed to the target coordinates. This coordinate transformation from polar angles $\varphi_{q}$ of $v$ to $\varphi_{p}$ of $R(g_{q\to p})v$ defines the transporter $g_{q\to p}=\varphi_{p}-\varphi_{q}$. On general meshes one additionally has to account for the fact that the tangent spaces $T_{q}M\subset\mathbb{R}^{3}$ and $T_{p}M\subset\mathbb{R}^{3}$ are usually not parallel in the ambient space $\mathbb{R}^{3}$. The parallel transport therefore includes the additional step of first aligning the tangent space at $q$ to be parallel to that at $p$, before translating a vector between them, see figure 4(b). In particular, given the normals $n_{q}$ and $n_{p}$ of the source and target tangent spaces $T_{q}M$ and $T_{p}M$, the source space is being aligned by rotating it via $R_{\alpha}\in\operatorname{SO}(3)$ by an angle $\alpha=\arccos(n_{q}^{\top}n_{p})$ around the axis $n_{q}\\!\times\\!n_{p}$ in the ambient space. Denote the rotated source frame by $(R_{\alpha}e_{q,1},R_{\alpha}e_{q,2})$ and the target frame by $(e_{p,1},e_{p,2})$. The angle to account for the parallel transport between the two frames, defining the discrete Levi-Civita connection on mesh edges, is then found by computing $g_{q\to p}\ =\ \operatorname{atan2}\big{(}(R_{\alpha}e_{q,2})^{\\!\top}\\!e_{p,1},\;(R_{\alpha}e_{q,1})^{\\!\top}\\!e_{p,1}\big{)}.$ (14) In practice we precompute these connections before training a model. Under gauge transformations by angles $g_{p}$ at $p$ and $g_{q}$ at $q$ the parallel transporters transform according to $g_{q\to p}\ \mapsto\ g_{p}+g_{q\to p}-g_{q}\,.$ (15) Intuitively, this transformation states that a transporter in a transformed gauge is given by a gauge transformation back to the original gauge via $-g_{q}$ followed by the original transport by $g_{q\to p}$ and a transformation back to the new gauge via $g_{p}$. For more details on discrete connections and transporters, extending to arbitrary paths e.g. over faces, we refer to (Lai et al., 2009; Crane et al., 2010; 2013). ## Appendix B Deriving the Kernel Constraint Given an input type $\rho_{\textup{in}}$, corresponding to vector space $V_{\textup{in}}$ of dimension $C_{\textup{in}}$ and output type $\rho_{\textup{out}}$, corresponding to vector space $V_{\textup{out}}$ of dimension $C_{\textup{out}}$, we have kernels $K_{\textup{self}}\in\mathbb{R}^{C_{\textup{out}}\times C_{\textup{in}}}$ and $K_{\textup{neigh}}:[0,2\pi)\to\mathbb{R}^{C_{\textup{out}}\times C_{\textup{in}}}$. Following Cohen et al. (2019b); Weiler et al. (2021), we can derive a constraint on these kernels such that the convolution is invariant. First, note that for vertex $p\in M$ and neighbour $q\in{\mathcal{N}}_{p}$, the coefficients of a feature vector $f_{p}$ at $p$ of type $\rho$ transforms under gauge transformation $f_{p}\mapsto\rho(-g)f_{p}$. The angle $\theta_{pq}$ gauge transforms to $\theta_{pq}-g$. Next, note that $\hat{f}_{q}:=\rho_{\textup{in}}(g_{q\to p})f_{q}$ is the input feature at $q$ parallel transported to $p$. Hence, it transforms as a vector at $p$. The output of the convolution $f^{\prime}_{p}$ is also a feature at $p$, transforming as $\rho_{\textup{out}}(-g)f^{\prime}_{p}$. The convolution then simply becomes: $f^{\prime}_{p}=K_{\textup{self}}f_{p}+\sum_{q}K_{\textup{neigh}}(\theta_{pq})\hat{f_{q}}$ Gauge transforming the left and right hand side, and substituting the equation in the left hand side, we obtain: $\displaystyle\rho_{\textup{out}}(-g)f_{p}^{\prime}=$ $\displaystyle\rho_{\textup{out}}(-g)\left(K_{\textup{self}}f_{p}+\sum_{q}K_{\textup{neigh}}(\theta_{pq})\hat{f_{q}}\right)=$ $\displaystyle K_{\textup{self}}\rho_{\textup{in}}(-g)f_{p}+\sum_{q}K_{\textup{neigh}}(\theta_{pq}-g)\rho_{\textup{in}}(-g)\hat{f_{q}}$ Which is true for any features, if $\forall g\in[0,2\pi),\theta\in[0,2\pi)$: $\displaystyle K_{\textup{neigh}}(\theta-g)$ $\displaystyle=\rho_{\textup{out}}(-g)\;K_{\textup{neigh}}(\theta)\;\rho_{\textup{in}}(g),$ (16) $\displaystyle K_{\textup{self}}$ $\displaystyle=\rho_{\textup{out}}(-g)\;K_{\textup{self}}\;\rho_{\textup{in}}(g).$ (17) where we used the orthogonality of the representations $\rho(-g)=\rho(g)^{-1}$. ## Appendix C Solving the Kernel Constraint As also derived in (Weiler & Cesa, 2019; Lang & Weiler, 2020), we find all angle-parametrized linear maps between $C_{\textup{in}}$ dimensional feature vector of type $\rho_{\textup{in}}$ to a $C_{\textup{out}}$ dimensional feature vector of type $\rho_{\textup{out}}$, that is, $K:S^{1}\to\mathbb{R}^{C_{\textup{out}}\times C_{\textup{in}}}$, such that the above equivariance constraint holds. We will solve for $K_{\textup{neigh}}(\theta)$ and discuss $K_{\textup{self}}$ afterwards. The irreducible real representations (irreps) of $\operatorname{SO}(2)$ are the one dimensional trivial representation $\rho_{0}(g)=1$ of order zero and $\forall n\in{\mathbb{N}}$ the two dimensional representations of order n: $\rho_{n}:\operatorname{SO}(2)\to\operatorname{GL}(2,\mathbb{R}):g\mapsto\begin{pmatrix}\cos ng&-\sin ng\\\ \sin ng&\phantom{-}\cos ng\end{pmatrix}.$ Any representation $\rho$ of $\operatorname{SO}(2)$ of $D$ dimensions can be written as a direct sum of irreducible representations $\displaystyle\rho$ $\displaystyle\cong\rho_{l_{1}}\oplus\rho_{l_{2}}\oplus...$ $\displaystyle\rho(g)$ $\displaystyle=A(\rho_{l_{1}}\oplus\rho_{l_{2}}\oplus...)(g)A^{-1}.$ where $l_{i}$ denotes the order of the irrep, $A\in\mathbb{R}^{D\times D}$ is some invertible matrix and the direct sum $\oplus$ is the block diagonal concatenations of the one or two dimensional irreps. Hence, if we solve the kernel constraint for all irrep pairs for the in and out representations, the solution for arbitrary representations, can be constructed. We let the input representation be irrep $\rho_{n}$ and the output representation be irrep $\rho_{m}$. Note that $K(g^{-1}\theta)=(\rho_{\textup{reg}}(g)[K])(\theta)$ for the infinite dimensional regular representation of $\operatorname{SO}(2)$, which by the Peter-Weyl theorem is equal to the infinite direct sum $\rho_{0}\oplus\rho_{1}\oplus...$. Using the fact that all $\operatorname{SO}(2)$ irreps are orthogonal, and using that we can solve for $\theta=0$ and from the kernel constraints we can obtain $K(\theta)$, we see that Eq. 16 is equivalent to $\hat{\rho}(g)K:=(\rho_{\textup{reg}}\otimes\rho_{n}\otimes\rho_{m})(g)K=K$ where $\otimes$ denotes the tensor product, we write $K:=K(\theta)$ and filled in $\rho_{\textup{out}}=\rho_{m},\;\rho_{\textup{in}}=\rho_{n}$. This constraint implies that the space of equivariant kernels is exactly the trivial subrepresentation of $\hat{\rho}$. The representation $\hat{\rho}$ is infinite dimensional, though, and the subspace can not be immediately computed. For $\operatorname{SO}(2)$, we have that for $n\geq 0$, $\rho_{n}\otimes\rho_{0}=\rho_{n}$, and for $n,m>0$, $\rho_{n}\otimes\rho_{m}\cong\rho_{n+m}\oplus\rho_{|n-m|}$. Hence, the trivial subrepresentation of $\hat{\rho}$ is a subrepresentation of the finite representation $\tilde{\rho}:=(\rho_{n+m}\oplus\rho_{|n-m|})\otimes\rho_{n}\otimes\rho_{m}$, itself a subrepresentation of $\hat{\rho}$. As $\operatorname{SO}(2)$ is a connected Lie group, any $g\in\operatorname{SO}(2)$ can be written as $g=\exp tX$ for $t\in\mathbb{R}$, $X\in\mathfrak{so}(2)$, the Lie algebra of $\operatorname{SO}(2)$, and $\exp:\mathfrak{so}(2)\to\operatorname{SO}(2)$ the Lie exponential map. We can now find the trivial subrepresentation of $\tilde{\rho}$ looking infinitesimally, finding $\displaystyle\tilde{\rho}(\exp tX)K$ $\displaystyle=K$ $\displaystyle\Longleftrightarrow d\tilde{\rho}(X)K:=\frac{\partial}{\partial t}\tilde{\rho}(\exp tX)|_{t=0}K$ $\displaystyle=0$ where we denote $d\tilde{\rho}$ the Lie algebra representation corresponding to Lie group representation $\tilde{\rho}$. $\operatorname{SO}(2)$ is one dimensional, so for any single $X\in\mathfrak{so}(2)$, $K$ is an equivariant map from $\rho_{m}$ to $\rho_{n}$, if it is in the null space of matrix $d\tilde{\rho}(X)$. The null space can be easily found using a computer algebra system or numerically, leading to the results in table 1. ## Appendix D Equivariance The GEM-CNN is by construction equivariant to gauge transformations, but additionally satisfies two important properties. Firstly, it only depends on the intrinsic shape of the 2D mesh, not how the mesh vertices are embedded in $\mathbb{R}^{3}$, since the geometric quantities like angles $\theta_{pq}$ and parallel transporters depend solely on the intrinsic properties of the mesh. This means that a simultaneous rotation or translation of all vertex coordinates, with the input signal _moving along_ with the vertices, will leave the convolution output at the vertices unaffected. The second property is that if a mesh has an orientation-preserving mesh isometry, meaning that we can map between the vertices preserving the mesh structure, orientations and all distances between vertices, the GEM-CNN is equivariant with respect to moving the signal along such a transformation. An (infinite) 2D grid graph is an example of a mesh with orientation-preserving isometries, which are the translations and rotations by 90 degrees. Thus a GEM-CNN applied to such a grid has the same equivariance properties a G-CNN (Cohen & Welling, 2016) applied to the grid. ### D.1 Proof of Mesh Isometry Equivariance Throughout this section, we denote $p^{\prime}=\phi(p),q^{\prime}=\phi(q)$. An orientation-preserving mesh isometry is a bijection of mesh vertices $\phi:{\mathcal{V}}\to{\mathcal{V}}$, such that: * • Mesh faces are one-to-one mapped to mesh faces. As an implication, edges are one-to-one mapped to edges and neighbourhoods to neighbourhoods. * • For each point $p$, the differential $d\phi_{p}:T_{p}M\to T_{p^{\prime}}M$ is orthogonal and orientation preserving, meaning that for two vectors $v_{1},v_{2}\in T_{p}M$, the tuple $(v_{1},v_{2})$ forms a right-handed basis of $T_{p}M$, then $(d\phi_{p}(v_{1}),d\phi_{p}(v_{2}))$ forms a right-handed basis of $T_{p^{\prime}}M$. ###### Lemma D.1. Given a orientation-preserving isometry $\phi$ on mesh $M$, with on each vertex a chosen reference neighbour $q^{p}_{0}$, defining a frame on the tangent plane, so that the log-map $\log_{p}q$ has polar angle $\theta^{p}_{q}$ in that frame. For each vertex $p$, let $g_{p}=\theta^{p^{\prime}}_{\phi(q_{0}^{p})}$. Then for each neighbour $q\in{\mathcal{N}}_{p}$, we have $\theta^{p^{\prime}}_{q^{\prime}}=\theta^{p}_{q}+g_{p}$. Furthermore, we have for parallel transporters that $g_{q^{\prime}\to p^{\prime}}=g_{q\to p}-g_{p}+g_{q}$. ###### Proof. For any $v\in T_{p}M$, we have that $\phi(\exp_{p}(v))=\exp_{p^{\prime}}(d\phi_{p}(v))$ (Tu, 2017, Theorem 15.2). Thus $\phi(\exp_{p}(\log_{p}q))=q^{\prime}=\exp_{p^{\prime}}(d\phi_{p}(\log_{p}q))$. Taking the log-map at $p^{\prime}$ on the second and third expression and expressing in polar coordinates in the gauges, we get $(r^{p^{\prime}}_{q^{\prime}},\theta^{p^{\prime}}_{q^{\prime}})=d\phi_{p}(r^{p}_{q},\theta^{p}_{q})$. As $\phi$ is an orientation-preserving isometry, $d\phi_{p}$ is a special orthogonal linear map $\mathbb{R}^{2}\to\mathbb{R}^{2}$ when expressed in the gauges. Hence $d\phi_{p}(r,\theta)=(r,\theta+z_{p})$ for some angle $z_{p}$. Filling in $\theta^{p}_{q^{p}_{0}}=0$, we find $z_{p}=g_{p}$, proving the first statement. The second statement follows directly from the fact that parallel transport $q\to p$, then push-forward along $\phi$ to $p^{\prime}$ yields the same first pushing forward from $q$ to $q^{\prime}$ along $\phi$, then parallel transporting $q^{\prime}\to p^{\prime}$ (Gallier & Quaintance, 2020, Theorem 18.3 (2)). ∎ For any feature $f$ of type $\rho$, we can define a push-forward along $\phi$ as $\phi_{*}(f)_{p^{\prime}}=\rho(-g_{p})f_{p}$. ###### Theorem D.1. Given GEM-CNN convolution $K\star\cdot$ from a feature of type $\rho_{\textup{in}}$ to a feature of type $\rho_{\textup{out}}$, we have that $K\star\phi_{*}(f)=\phi_{*}(K\star f)$. ###### Proof. $\displaystyle\phi_{*}(K\star f)_{p^{\prime}}$ $\displaystyle=\rho_{\textup{out}}(-g_{p})\left(K_{\textup{self}}f_{p}+\sum\nolimits_{q\in{\mathcal{N}}_{p}}K_{\textup{neigh}}(\theta_{pq})\rho_{\textup{in}}(g_{q\to p})f_{q}\right)$ $\displaystyle=\rho_{\textup{out}}(-g_{p})\left(K_{\textup{self}}f_{p}+\sum\nolimits_{q^{\prime}\in{\mathcal{N}}_{p^{\prime}}}K_{\textup{neigh}}(\theta_{p^{\prime}q^{\prime}}-g_{p})\rho_{\textup{in}}(g_{q^{\prime}\to p^{\prime}}+g_{p}-g_{q})f_{q}\right)$ $\displaystyle=\rho_{\textup{out}}(-g_{p})\left(K_{\textup{self}}f_{p}+\sum\nolimits_{q^{\prime}\in{\mathcal{N}}_{p^{\prime}}}K_{\textup{neigh}}(\theta_{p^{\prime}q^{\prime}}-g_{p})\rho_{\textup{in}}(g_{p})\rho_{\textup{in}}(g_{q^{\prime}\to p^{\prime}})\rho_{\textup{in}}(-g_{q})f_{q}\right)$ $\displaystyle=K_{\textup{self}}\rho_{\textup{in}}(-g_{p})f_{p}+\sum\nolimits_{q^{\prime}\in{\mathcal{N}}_{p^{\prime}}}K_{\textup{neigh}}(\theta_{p^{\prime}q^{\prime}})\rho_{\textup{in}}(g_{q^{\prime}\to p^{\prime}})\rho_{\textup{in}}(-g_{q})f_{q}$ $\displaystyle=(K\star\phi_{*}(f))_{p^{\prime}}$ where in the second line we apply lemma D.1 and the fact that $\phi$ gives a bijection of neighbourhoods of $p$, in the third line we use the functoriality of $\rho$ and in the fourth line we apply the kernel constraints on $K_{\textup{self}}$ and $K_{\textup{neigh}}$. ∎ ## Appendix E Additional details on the experiments ### E.1 Embedded MNIST (a) Flat embedding (b) Isometric embedding (c) Curved, roughness = 0.5 (d) Curved, roughness = 1 (e) Curved, roughness = 1.5 (f) Curved, roughness = 2 (g) Curved, roughness = 2.25 (h) Curved, roughness = 2.5 Figure 5: Examples of different grid geometries on which the MNIST dataset is evaluated. All grids have $28\\!\times\\!28$ vertices but are embedded differently in the ambient space. Figure 5(a) shows a flat embedding, corresponding to the usual pixel grid. The grid in Figure 5(b) is isometric to the flat embedding, its internal geometry is indistinguishable from that of the flat embedding. Figures 5(c)-5(h) show curved geometries which are not isometric to the flat grid. They are produced by a random displacement of each vertex in its normal direction, followed by a smoothing of displacements. To create the intrinsically curved grids we start off with the flat, rectangular grid, shown in figure 5(a), which is embedded in the $XY$-plane. An independent displacement for each vertex in $Z$-direction is drawn from a uniform distribution. A subsequent smoothing step of the normal displacements with a Gaussian kernel of width $\sigma$ yields geometries with different levels of curvature. Figures 5(c)-5(h) show the results for standard deviations of 2.5, 2, 1.5, 1, 0.75 and 0.5 pixels, which are denoted by their roughness $3-\sigma$ as 0.5, 1, 1.5, 2, 2.25 and 2.5. In order to facilitate the generalization between different geometries we normalize the resulting average edge lengths. The same GEM-CNN is used on all geometries. It consists of seven convolution blocks, each of which applies a convolution, followed by a RegularNonlinearity with $N=7$ orientations, batch normalization and dropout of 0.1. This depth is chosen since GEM-CNNs propagate information only between direct neighbors in each layer, such that the field of view after 7 layers is $2\times 7+1=15$ pixel. The input and output types of the network are scalar fields of multiplicity 1 and 64, respectively, which transform under the trivial representation and ensure a gauge invariant prediction. All intermediate layers use feature spaces of types $M\rho_{0}\oplus M\rho_{1}\oplus M\rho_{2}\oplus M\rho_{3})$ with $M=4,\ 8,\ 12,\ 16,\ 24,\ 32$. After a spatial max pooling, a final linear layer maps the 64 resulting features to 10 neurons, on which a softmax function is applied. The model has 163k parameters. A baseline GCN, applying by isotropic kernels, is defined by replacing the irreps $\rho_{i}$ of orders $i\geq 1$ with trivial irreps $\rho_{0}$ and rescaling the width of the model such that the number of parameters is preserved. All models are trained for 20 epochs with a weight decay of 1E-5 and an initial learning rate of 1E-2. The learning rate is automatically decayed by a factor of 2 when the validation loss did not improve for 3 epochs. The experiments were run on a single TitanX GPU. ### E.2 Shape Correspondence experiment All experiments were ran on single RTX 2080TI GPUs, requiring 3 seconds / epoch. The non-gauge-equivariant CNN uses as gauges the SHOT local reference frames (Tombari et al., 2010). For one input and output channel, it has features $f_{p}\in\mathbb{R}$ convolution and weights $w\in\mathbb{R}^{2B+2}$, for $B\in{\mathbb{N}}$. The convolution is: $(K\star f)_{p}=w_{0}f_{p}+\sum_{q\in{\mathcal{N}}_{p}}\left(w_{1}+\sum_{n=1}^{B}(w_{2n}\cos(n\theta_{pq})+w_{2n+1}\sin(n\theta_{pq})\right)f_{q}.$ (18) This convolution kernel is an unconstranied band-limited spherical function. This is then done for $C_{\textup{in}}$ input channels and $C_{\textup{out}}$ output channels, giving $(2B+2)C_{\textup{in}}C_{\textup{out}}$ parameters per layer. In our experiments, we use $B=2$ and 7 layers, with ReLU non- linearities and batch-norm, just as for the gauge equivariant convolution. After hyperparameter search in $\\{16,32,64,128,256\\}$, we found 128 channels to perform best. ## Appendix F Additional experiments ### F.1 RegularNonLinearity computational cost Number of samples | Time / epoch (s) | Memory (GB) ---|---|--- none | 21.2 | 1.22 1 | 21.9 | 1.22 5 | 21.6 | 1.23 10 | 21.5 | 1.24 20 | 22.0 | 1.27 50 | 21.7 | 1.35 Table 3: Run-time of one epoch training and validation and max memory usage of FAUST model without RegularNonLinearity of with varying number of samples used in the non-linearity. The hyperparameters are modified to have batch size 1. In table 3, we show the computational cost of the RegularNonLinearity, computed by training and computing validation errors for 10 epochs. The run- time is not significantly affected, but memory usage is. ### F.2 Equivariance Errors In this experiment, we evaluate empirically equivariance to three kinds of transformations: gauge transformations, transformations of the vertex coordinates and transformations under isometries of the mesh, as introduced above in appenndix D. We do this on two data sets: the icosahedron, a platonic solid of 12 vertices, referred to in the plots as ’Ico’; and the deformed icosahedron, in which the vertices have been moved away from the origin by a factor of sampled from ${\mathcal{N}}(1,0.01)$, referred to in the plots as ’Def. Ico’. We evaluate this on the GEM-CNN (7 layers, 101 regular samples, unless otherwise noted in the plots) and the Non-Equivariance CNN based on SHOT frames introduced above in Eq. 18 (7 layers unless otherwise noted in the plots). Both models have 16 channels input and 16 channels output. The equivariance model has scalar features as input and output and intermediate activations with band limit 2 with multiplicity 16. The non-equivariant model has hidden activations of 16 dimensions. If not for the finite samples of the RegularNonLinearity, the equivariant model should be exactly gauge invariant and invariant to isometries. Both models use batchnorm, in order to evaluate deeper models. #### F.2.1 Gauge Equivariance Figure 6: Equivariance error to gauge transformation. We evaluate gauge equivariance by randomly initialising a model, randomly sampling input features. We also sample 16 random gauge transformations at each point. We compare the outputs of the model based on the different gauges. As the input and output features of the equivariant model are scalars, the outputs should coincide. This process is repeated 10 times. For the non- equivariant model, we compute frames based on SHOT and then randomly rotate these. The equivariance error is quantified by as: $\sqrt{\frac{\mathbb{E}_{\Phi,f}\mathbb{E}_{p,c}\mathrm{Var}_{g}(\Phi_{g}(f)_{p,c})}{\mathrm{Var}_{\Phi,f,p,c}(\Phi_{g_{0}}(f)_{p,c})}}$ (19) where $\Phi_{g}(f)_{c}$ denotes the model $\Phi$ with gauge transformed by $g$ applied to input $f$ then taken the $c$-th channel, $\mathbb{E}_{\Phi,f}$ denotes the expectation over model initialisations and random inputs, for which we take 10 samples, $\mathbb{E}_{p,c}$ denotes averaging over the 12 vertices and 16 output channels, $\mathrm{Var}_{g}$ denotes the variance over the different gauge transformations, $\mathrm{Var}_{\Phi,f,c}$ takes the variance over the models, inputs and channels, and $g_{0}$ denotes one of the sampled gauge transformations. This quantity indicates how much the gauge transformation affects the output, normalized by how much the model initialisations and initial parameters affect the output. Results are shown in Figure 6. As expected, the non-equivariant model is not equivariant to gauge transformations. The equivariant model approaches gauge equivariance as the number of samples of the Regular NonLinearity increases. As expected, the error to gauge equivariance accumulate as the number of layers increases. The icosahedron and deformed icosahedron behave the same. #### F.2.2 Ambient Equivariance Figure 7: Equivariance error to ambient transformations of the vertex coordinates. In this experiment, we measure whether the output is invariant to when all vertex coordinates are jointly transformed under rotations and translations. We perform the experiment as above, but sample as transformations $g$ 300 translations and rotations of the ambient space $\mathbb{R}^{3}$. We evaluate again using Eq 19, where $g$ now denotes a ambient transformation. Results are shown in Figure 7. We see that the equivariant GEM-CNN is invariant to these ambient transformations. Somewhat unexpectedly, we see that the non-equivariant model based on SHOT frames is not invariant. This is because of an significant failure mode of SHOT frames in particular and heuristically chosen gauges with a non-gauge-equivariant methods in general. On some meshes, the heuristic is unable to select a canonical frame, because the mesh is locally symmetric under (discrete subgroups of) planar rotations. This is the case for the icosahedron. Hence, SHOT can not disambiguate the X from the Y axis. The reason this happens in the SHOT local reference frame selection (Tombari et al., 2010) is the first two singular values of the $M$ matrix are equal, making a choice between the first and second singular vectors ambiguous. This ambiguity breaks ambient invariance. For the non- symmetric deformed icosahedron, this problem for the non-equivariant method disappears. #### F.2.3 Isometry equivariance Figure 8: Equivariance error to isometry transformation. The icosahedron has 60 orientation-preserving isometries. We evaluate equivariance using: $\sqrt{\frac{\mathbb{E}_{\Phi,f}\mathbb{E}_{p,c}(\Phi(g(f)_{p,c}-\Phi(f)_{g(p),c})^{2}}{\mathrm{Var}_{\Phi,f,p,c}(\Phi(f)_{p,c})}}$ where $g:M\to M$ is an orientation-preserving isometry, sampled uniformly from all 60 and $g(f)$ is the transformation of a scalar input feature $f:M\to\mathbb{R}^{C_{\textup{in}}}$ by pre-composing with $g^{-1}$. As expected, the non-equivariant model is not equivariant to isometries. The GEM-CNN is not equivariant to the icosahedral isometries on the deformed icosahedron, as the deformation removes the symmetry. As the number of Regular NonLinearity samples increases, the GEM-CNN becomes more equivariant. Interestingly, the GEM-CNN is equivariant whenever the number of samples is a multiple of 5. This is because the stabilizer subgroup of the icosahedron at the vertices is the cyclic group of order 5. Whenever the RegularNonLinearity has a multiple of 5 samples, it is exactly equivariant to these transformations. ## Appendix G Equivariance Error Bounds on Regular Non-Linearity The regular non-linearity acts on each point on the sphere in the following way. For simplicity, we assume that the representation is $U$ copies of $\rho_{0}\oplus\rho_{1}\oplus...\oplus\rho_{M}$. One such copy can be treated as the discrete Fourier modes of a circular signal with band limit $M$. We map these Fourier modes to $N$ spatial samples with an inverse Discrete Fourier Transform (DFT) matrix. Then apply to those samples a point-wise non- linearity, like ReLU, and map back to the Fourier modes with a Discrete Fourier Transform Matrix. This procedure is exactly equivariant for gauge transformation with angles multiple of $2\pi/N$, but approximately equivariant for small rotations in between. In equations, we start with Fourier modes $x_{0},(x_{\alpha}(m),x_{\beta}(m))_{m=1}^{B}$ at some point on the sphere and result in Fourier modes $z_{0},(z_{\alpha}(m),z_{\beta}(m))_{m=1}^{B}$. We let $t=0,...,N-1$ index the spatial samples. $\displaystyle x(t)$ $\displaystyle=x_{0}+\sum_{m}x_{\alpha}(m)\cos\left(\dfrac{2\pi}{N}mt\right)+\ldots$ (20) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sum_{m}x_{\beta}(m)\sin\left(\dfrac{2\pi}{N}mt\right)$ $\displaystyle y(t)$ $\displaystyle=f(x(t))$ $\displaystyle z_{0}$ $\displaystyle=\dfrac{1}{N}\sum_{t}y(t)$ $\displaystyle z_{\alpha}(m)$ $\displaystyle=\dfrac{2}{N}\sum_{t}\cos\left(\dfrac{2\pi}{N}mt\right)y(t)$ $\displaystyle z_{\beta}(m)$ $\displaystyle=\dfrac{2}{N}\sum_{t}\sin\left(\dfrac{2\pi}{N}mt\right)y(t)$ Note that Nyquist’s sampling theorem requires us to pick $N\geq 2B+1$, as otherwise information is always lost. The normalization is chosen so that $z_{\alpha}(m)=x_{\alpha}(m)$ if $f$ is the identity. Now we are interested in the equivariance error between the following two terms, for small rotation $\delta\in[0,1)$. Any larger rotation can be expressed in a rotation by a multiple of $2\pi/N$, which is exactly equivariant, followed by a smaller rotation. We let $z_{\alpha}^{FT}(m)$ be the resulting Fourier mode if first the input is gauge-transformed and then the regular non-linearity is applied, and let $z_{\alpha}^{TF}(m)$ be the result of first applying the regular non-linearity, followed by the gauge transformation. $\displaystyle z_{\alpha}^{FT}(m)$ $\displaystyle=\dfrac{2}{N}\sum_{t}\cos\left(\dfrac{2\pi}{N}mt\right)y(t+\delta)$ $\displaystyle=\dfrac{2}{N}\sum_{t}c_{m}(t)y(t+\delta)$ $\displaystyle z_{\alpha}^{TF}(m)$ $\displaystyle=\dfrac{2}{N}\sum_{t}\cos\left(\dfrac{2\pi}{N}m(t-\delta)\right)y(t)$ $\displaystyle=\dfrac{2}{N}\sum_{t}c_{m}(t-\delta)y(t)$ where we defined for convenience $c_{m}(t)=\cos(2\pi mt/N)$. We define norms $||x||_{1}=|x_{0}|+\sum_{m}(|x_{\alpha}(m)|+|x_{\beta}(m)|)$ and $||\partial x||_{1}=\sum_{m}m(|x_{\alpha}(m)|+|x_{\beta}(m)|)$. ###### Theorem G.1. If the input $x$ is band limited by $B$, the output $z$ is band limited by $B^{\prime}$, $N$ samples are used and the non-linearity has Lipschitz constant $L_{f}$, then the error to the gauge equivariance of the regular non- linearity bounded by: $\displaystyle||z^{FT}-z^{TF}||_{1}\leq\dfrac{4\pi L_{f}}{N}\left((2B^{\prime}+\dfrac{1}{2})||\partial x||_{1}+B^{\prime}(B^{\prime}+1)||x||_{1}\right)$ which goes to zero as $N\to\infty$. ###### Proof. First, we note, since the Lipschitz constant of the cosine and sine is 1: $\displaystyle|c_{m}(t-\delta)-c_{m}(t)|$ $\displaystyle\leq\dfrac{2\pi m\delta}{N}\leq\dfrac{2\pi m}{N}$ $\displaystyle|x(t+\delta)-x(t)|$ $\displaystyle\leq\dfrac{2\pi}{N}\sum_{m}m(|x_{\alpha}(m)|+|x_{\beta}(m)|)$ $\displaystyle\leq\dfrac{2\pi}{N}||\partial x||_{1}$ $\displaystyle|y(t+\delta)-y(t)|$ $\displaystyle\leq L_{f}\dfrac{2\pi}{N}||\partial x||_{1}$ $\displaystyle|c_{m}(t)|$ $\displaystyle\leq 1$ $\displaystyle|x(t)|$ $\displaystyle\leq|x_{0}|+\sum_{m}(|x_{\alpha}(m)|+|x_{\beta}(m)|)$ $\displaystyle\leq||x||_{1}$ $\displaystyle|y(t)|$ $\displaystyle\leq L_{f}||x||_{1}$ Then: $\displaystyle|c_{m}(t)y(t+\delta)-c_{m}(t-\delta)y(t)|$ $\displaystyle=$ $\displaystyle|c_{m}(t)\left[y(t+\delta)-y(t)\right]-y(t)\left[c_{m}(t-\delta)-c_{m}(t)\right]|$ $\displaystyle\leq$ $\displaystyle|c_{m}(t)||y(t+\delta)-y(t)|+|y(t)||c_{m}(t-\delta)-c_{m}(t)|$ $\displaystyle\leq$ $\displaystyle L_{f}\dfrac{2\pi}{N}||\partial x||_{1}+L_{f}||x||_{1}\dfrac{2\pi m}{N}$ $\displaystyle=$ $\displaystyle\dfrac{2\pi L_{f}}{N}\left(||\partial x||_{1}+m||x||_{1}\right)$ So that finally: $\displaystyle|z_{\alpha}^{FT}(m)-z_{\alpha}^{TF}(m)|$ $\displaystyle\leq$ $\displaystyle\dfrac{2}{N}\sum_{t}|c_{m}(t)y(t+\delta)-c_{m}(t-\delta)y(t)|$ $\displaystyle\leq$ $\displaystyle\dfrac{4\pi L_{f}}{N}\left(||\partial x||_{1}+m||x||_{1}\right)$ The sinus component $|z_{\beta}^{FT}(m)-z_{\beta}^{TF}(m)|$ has the same bound, while $|z_{0}^{FT}-z_{0}^{TF}|=|y(t+\delta)-y(t)|$, which is derived above. So if $z$ is band-limited by $B^{\prime}$: $\displaystyle||z^{FT}-z^{TF}||_{1}$ $\displaystyle=|z_{0}^{FT}-z_{0}^{TF}|+$ $\displaystyle\sum_{m=1}^{B^{\prime}}|z_{\alpha}^{FT}(m)-z_{\alpha}^{TF}(m)|+|z_{\beta}^{FT}(m)-z_{\beta}^{TF}(m)|$ $\displaystyle\leq\dfrac{4\pi L_{f}}{N}\left((2B^{\prime}+\dfrac{1}{2})||\partial x||_{1}+\sum_{m=1}^{B^{\prime}}2m||x||_{1}\right)$ $\displaystyle=\dfrac{4\pi L_{f}}{N}\left((2B^{\prime}+\dfrac{1}{2})||\partial x||_{1}+B^{\prime}(B^{\prime}+1)||x||_{1}\right)$ Since $||\partial x||_{1}=\mathcal{O}(B||x||_{1})$, we get $||z^{FT}-z^{TF}||_{1}=\mathcal{O}(\frac{BB^{\prime}+{B^{\prime}}^{2}}{N}||x||_{1})$, which obviously vanishes as $N\to\infty$. ∎

2024-09-04T02:54:59.310286

2003.05428

arxiv-papers

# Template Matching Route Classification Mitchell Kinney University of Minnesota - Twin Cities ###### Abstract This paper details a route classification method for American football using a template matching scheme that is quick and does not require manual labeling. Pre-defined routes from a standard receiver route tree are aligned closely with game routes in order to determine the closest match. Based on a test game with manually labeled routes, the method achieves moderate success with an overall accuracy of 72% of the 232 routes labeled correctly. Keywords— Route Classification, Template Matching, Unsupervised Setting ## 1 Introduction The world of sports is fully embracing the power of data driven decision making and analytics. Recently, NFL players have started to wear RFID chips in their shoulder pads to be able to track their movements on the field during play. While there are many avenues of exploration with this new wave of data, in this article automatic labeling of receiver routes is the focus. When executing a passing play it is important for quarterbacks and receivers to be coordinated so the ball can arrive on time and safely for a completed catch. Many pass plays happen every game, and it is of interest when studying game film to know what routes work well. Automatically labeling the types of routes these receivers are running would be a major help in understanding what determines success in a passing play. This paper uses a template matching scheme to try and minimize the distance between game routes run by the players and pre-defined routes from a typical route tree. The important aspects of this method are scaling and translating the pre-defined routes to match closely with the game routes. Closely matched routes will imply they are of the same type and the label of the pre-defined route can be assigned to the game route. The data used in this paper comes from the NFL’s Big Data Bowl which provided player tracking data from part of the 2017 NFL season. ## 2 Related work Previous work in route classification has two main approaches. The first is to manually label a portion of routes and then use those labels to train a model to identify the remaining routes. This was done previously in [2] and [4]. In [2] route characteristics such as the depth of the route before turning, the direction of the turn and the length of the route after turning were recorded. Then a training set was built from manually labeled routes and these characteristics as well as their labels were fed to various models to train classifiers. In [4] the author made use of a Convolutional Neural Network to learn the hidden features of the routes after labeling approximately 1,000 routes by hand. The other approach is to use hierarchical clustering to guarantee similar looking routes share the same label. This has been used in [1] and [6]. In [1] the authors use a expectation-maximization (EM) algorithm in a likelihood based approach. The authors assume each receiver trajectory comes from a distribution with distinct parameters based on known distinct route features. They attempt to tune the number of these distinct route types to best separate the routes. In [6] features of the route were extracted and hierarchical clustering was used. There were two methods shown useful in the paper. The first used the beginning and ending of the routes as features and the second used the length of the route before turning, the angle of the turn and the length of the route after turning as features. The main weakness of these methods is the amount of time that is needed to tune/ train the models and the need in each to manually label routes at the beginning or manually label clusters at the end. While these methods require manual labeling of routes which the method proposed in this paper does not require. A method which also uses pre-defined curves to compare routes was introduced in [3]. The authors used a belief network, which is similar to a naive Bayes classifier, to classify offensive plays. The pre-defined curves were used as priors in the network. This method is similar to the one proposed in this paper because it requires no manual labeling of routes. The goal of the method in [3] is oriented more towards classifying a whole play rather than individual routes though. Figure 1: Empirical cumulative distribution of route cutoff times in seconds ## 3 Wrangling the routes The data is from the Big Data Bowl competition which released tracking data from games during the 2017 season. Each player was equipped with a tracking device and their movements were recorded every 100 milliseconds. For this method the positions were filtered to only include wide receivers, tight ends, and running backs that were split out. It was found that running backs in the backfield do not adhere to the same route tree when running routes because they have to navigate the offensive and defensive lines some of the time. The routes begin at the snap so no pre-snap motion was included. Once the ball was caught by the receiver, incomplete, or intercepted the trajectory collection was stopped. Also to avoid any creative deviations by players to possibly get open on a broken play each route was cut off if the route was run for at least 5 seconds, (as was done in [2]). Therefore, the recorded routes ended at the minimum time between when the pass outcome happened and 5 seconds after the ball was snapped. A visualization of times in seconds that routes were cut off in an example game is shown in Figure 1. There is relatively few routes that reached the full 5 seconds, around 15%. The vast majority fell between 3 and 4 seconds. Finally, the routes were rotated and mirrored as if they were run from the left side of the ball. There was no distinction between running from the left or right side of the ball or the direction up or down the field. Figure 2: Basic receiver route tree used to classify routes Figure 3: Game route classified as a ‘post’ run by Bennie Fowler while a Denver Bronco ## 4 Route Tree Routes in the NFL are differentiated based on direction changes made when running up or down the field. Routes are an integral part of a passing play in the NFL. Only the offense knows where a receiver will run, and these routes help the quarterback know where a receiver will be, which allows a pass completion to be made. In the route tree in Figure 3, the difference between an out route and a dig route is the direction the receiver runs after running forwards a few yards. Turning towards the center of the field will be a dig route and turning towards the closest side line will be an out route. Another difference is the length of the field the receiver runs before changing direction. In a post route the receiver runs up the field before turning slightly towards the center of the field and running towards the goal post. While in a slant route, the receiver runs towards the center of field much sooner, sometimes without running up the field first. As seen in Bennie Fowler’s 20 yard post route run in the Denver Broncos versus Los Angeles Chargers game in 2017 in Figure 3, routes run in a real NFL game have turns that are not as crisp as seen in the route tree in Figure 3 and the length the receivers run down the field before turning is not uniform. Therefore, a method to classify routes must be adjustable to the angle of direction change and the distance run down the field. The method proposed in this paper captures this flexibility by scaling pre-defined routes to match closely to game routes. Figure 4: Bounding box of the post route from Figure 3 Figure 5: Aligned bounding boxes of an in-game post route (blue) and a manually created post route (red) Figure 6: Scaled bounding box of a manually created post route (red) from Figure 6 ## 5 Scaling Pre-Defined Routes Scaling to match routes is a crucial first step in the method proposed, because a distance metric is used to classify routes. Each proposed pre- defined route should be overlaid in such a way that if a pre-defined route label should be assigned to a game route the distance between the pre-defined route and the game route is minimal. The only routes that are changed when scaling/ transforming are the pre-defined routes to align closely with the game routes. All pre-defined routes have been manually given coordinates that match the shapes given in Figure 3. The calculations to scale any pre-defined route only requires the bounding box of the pre-defined route and game route. The bounding box of a set of two dimensional coordinates is the smallest rectangle that captures all of the points. A route $R$ is defined as a set of $(x,y)$ coordinate pairs with cardinality $|T|$ such that: $R=\\{(x_{1},y_{1}),(x_{2},y_{2}),\dots,(x_{T},y_{T})\\},$ (1) where $T$ is the number of timesteps the receiver’s position was recorded. The bounding box of the set $R$ is a four dimensional tuple defined as $\big{(}\min\limits_{x}R,\min\limits_{y}R,\max\limits_{x}R,\max\limits_{y}R\;\big{)}$. An example of a bounding box for a route is shown in Figure 6. The scaling approach used was to find the largest difference between the width and height ratios of the bounding boxes of the pre-defined route and the game route and then scale each coordinate in the pre-defined route so the largest difference would match instead. Let the set of game route coordinates be $R_{\text{game}}$ and the pre-defined route coordinates be $R_{\text{pre- defined}}$. The first step in scaling is to translate all the coordinates in both routes so the minimum $x$ and $y$ coordinates are $(0,0)$ and $\displaystyle\min\limits_{x}R_{\text{game}}$ $\displaystyle=\min\limits_{x}R_{\text{pre-defined}}=0$ (2) $\displaystyle\min\limits_{y}R_{\text{game}}$ $\displaystyle=\min\limits_{y}R_{\text{pre-defined}}=0$ (3) This will align the bottom and left sides of the bounding boxes of the routes and allow for the correct calculation of the ratio between the horizontal and vertical direction of the bounding boxes. An example can be seen in Figure 6. After aligning the bounding boxes the horizontal ratio $r_{h}$ and vertical ratio $r_{v}$ can be calculated by $\displaystyle r_{h}$ $\displaystyle=\dfrac{\max\limits_{x}R_{\text{pre- defined}}}{\max\limits_{x}R_{\text{game}}}$ (4) $\displaystyle r_{v}$ $\displaystyle=\dfrac{\max\limits_{y}R_{\text{pre- defined}}}{\max\limits_{y}R_{\text{game}}}.$ (5) The larger ratio is used to scale each coordinate in the pre-defined route. For instance, if $r_{h}>r_{v}$, then each $(x,y)$ coordinate in the pre- defined route is multiplied by $r_{h}^{-1}$. Let $R_{\text{scaled}}$ be the pre-defined route coordinates multiplied by the proper ratio. $\displaystyle R_{\text{scaled}}=\min(r_{h}^{-1},r_{v}^{-1})\cdot R_{\text{pre-defined}}.$ (6) A Scaled dig route B Scaled post route Figure 7: Routes scaled using an exact match bounding box (red) over an in game route run by Emmanuel Sanders while a Denver Bronco (blue) Figure 8: Out route scaled using a smaller discrepancy bounding box (red) over an in game route run by Emmanuel Sanders while a Denver Bronco (blue) An example is shown in Figure 6. This approach was chosen because it maintains the aspect ratio of the bounding box of the pre-defined route and reduces the possibility of a pre-defined route not scaling “reasonably.” The aspect ratio is the ratio between the height and width of the bounding box. Maintaining the aspect ratio is critical to differentiating routes since direction changes are what separates routes, for example, separating outs from corners and corners from streaks. Allowing the aspect ratio to change could change the angles at the direction changes in the routes so much that two types routes can become almost indistinguishable. In Figure 7 an example of changing the aspect ratio when scaling, is shown where a dig game route is closer to a post route than the correct dig route because the direction change of the scaled pre-defined post route conforms to the game route. Changing the aspect ratio occurs when the bounding box of the pre-defined route is scaled to exactly match the game route as in Figure 7. This angle manipulation is most problematic in routes such as corners or posts and flats or slants. Matching bounding boxes of the pre-defined routes and the game route would allow for “perfect” matches in the ideal scenario but will also change the aspect ratio, sometimes drastically. The other scaling approach is to minimize the smaller discrepancy which is simply using the smaller ratio to scale the pre-defined route instead of the larger ratio. The issue that arises with this approach mainly comes from how the pre-defined routes are initially plotted. The coordinates of the pre- defined routes were made much larger than necessary compared to game routes to guarantee that the pre-defined routes would always scale down (both $x$ and $y$ coordinates get smaller). This saves an additional logic step that would be needed to possibly scale pre-defined routes up or down. The large size of the pre-defined route in comparison to the game route is true of all pre- defined routes. Matching the larger discrepancy implies that the pre-defined route bounding box will be shrunk to always be contained within the game route bounding box as in Figures 6 and 6. This way after scaling, all pre-defined routes will be approximately similar sizes, whereas scaling to match the smaller discrepancy might cause erratic behavior. This can be seen in Figure 8 where the scaled pre-defined route is unrealistically large compared to the game route. Figure 9: Example of a grid search which shifts the pre-defined route’s bounding box incrementally upward over an in game route run by Bennie Fowler while a Denver Bronco ## 6 Route Classification To classify the game routes, a simple Euclidean distance is used between the game route and the scaled pre-defined routes after shifting the scaled pre- defined route to align as closely as possible. Then the label of the scaled pre-defined route that is the minimum distance from the game route is used to also label the game route. To match up the game route and scaled route as closely as possible, a shift is used on the scaled route. Recall from the Section 5 that the method of scaling chosen was to minimize the largest discrepancy. This implies that the bounding box of the scaled route will be completely contained within the bounding box of the game route each time. Therefore, a grid search for the optimal position of the scaled pre-defined route can be done within the bounding box of the game route. An example of a grid search with a scaled pre-defined out route is shown in Figure 9. Note the scaled and game route bounding boxes will still be aligned on their bottom and left sides. If the scaled route used $r_{h}^{-1}$ then the right side of the bounding boxes will be aligned, and similarly if $r_{v}^{-1}$ was used the top side of the bounding boxes will be aligned. Therefore the shift on the grid will either be exclusively vertical or horizontal to keep the scaled route bounding box within the game route bounding box. The chosen shift step size was half a yard for this method. The distance that the scaled route can be shifted is equal to $w$ defined as $\displaystyle w=\max\big{(}|\max\limits_{x}R_{\text{scaled}}-\max\limits_{x}R_{\text{game}}|,|\max\limits_{y}R_{\text{scaled}}-\max\limits_{y}R_{\text{game}}|\big{)}.$ (7) Figure 10: Representation of $w$ which is the distance between the remaining discrepancy between the bounding boxes of the in game route (blue) and scaled pre-defined route (red) Figure 11: Visual of how the original pre-defined route (left) gets points added (right) to equal the number of points of the game route One of the two elements in the max will be zero, so $w$ is equal to whichever is positive. The number of steps taken is equal to the ceiling of $\;\dfrac{w}{0.5}$. If the tops of the bounding boxes are aligned, then the distances between the game and scaled routes will be measured after shifting the $x$-coordinate of the scaled route $\\{0,0.5,\dots,w_{0.5}\\}$, where $w_{0.5}$ is $w$ rounded down to the nearest $0.5$ increment. If the right sides of the bounding boxes are aligned, the $y$-coordinate of the scaled route will be shifted. A visual of $w$ is shown in Figure 11. The distance at each step is calculated by measuring the distance between every coordinate in the game route to the closest point on the line of the scaled route and adding the distance between every coordinate in the scaled route to the closest point on the line of the game route. Let $\ell_{i,i+1}$ be the line segment between coordinates $(x_{i},y_{i})$ and $(x_{i+1},y_{i+1})$ for $i\in 1,\dots,T-1$, where $T$ is still the number of coordinates recorded in the game route. Points are artificially added to the scaled routes until $R_{\text{scaled}}$ has the same cardinality as $R_{\text{game}}$; $|T|$. These points are placed evenly on the route as shown in Figure 11. These added points do not affect the bounding box of the scaled route. Then the collection of line segments that make up a route is defined as $\displaystyle L=\\{\ell_{1,2},\ell_{2,3},\dots,\ell_{T-1,T}\\}.$ (8) Let $\delta\big{(}(x,y),L\big{)}$ be defined as the minimum distance between the point $(x,y)$ and $L$. Then the distance measurement $D_{\text{game}}$ is found by summing the minimum distance to the line $L_{\text{scaled}}$ over the points in $R_{\text{game}}$. $\displaystyle D_{\text{game}}=\sum_{(x,y)\in R_{\text{game}}}\delta\big{(}(x,y),L_{\text{scaled}}\big{)}.$ (9) To avoid misclassification when the game route is close to only part of the scaled route, as in Figure 13, the same measurement is taken between coordinates of the scaled route and the line of the game route. $\displaystyle D_{\text{scaled}}=\sum_{(x,y)\in R_{\text{scaled}}}\delta\big{(}(x,y),L_{\text{game}}\big{)}.$ (10) Figure 12: Partial overlap of routes showing the necessity to calculate the closest distances between both routes Figure 13: Distance measured in $D_{\text{scaled}}$ with route from Figure 9 An example of the distance being measured for each coordinate in $D_{\text{scaled}}$ can be found in Figure 13 using the routes from the earlier grid search example in Figure 9. The total distance between the scaled and game routes is summed with a weight $\gamma$ on $D_{\text{scaled}}$. $\displaystyle D_{\text{route}}=D_{\text{game}}+\gamma D_{\text{scaled}}.$ (11) Here $D_{\text{route}}$ is the measurement of distance between the game route and one of the named routes from the route tree in Figure 3. The weight $\gamma\leq 1$ and is designed to help balance the distance measurements since the scaled route will necessarily be smaller than the game route because of the scaling strategy. The weight $\gamma$ is to represent $D_{\text{game}}$ being more important since it is possible for the game route to extend further than the scaled route while the scaled route lines up extremely closely with only part of the game route. Weighting $D_{\text{scaled}}$ down will help with this problem by making the distances calculated from game route coordinates overlapping with the scaled route line more important. The route name with the minimum distance among the entire route tree and all shifts will be assigned to the game route. For each classification the same pre-defined route tree is used initially. There is no attempt made to incorporate labeled routes into future predictions through a process such as active learning where labeling is done while learning the coefficient space. This is done to prevent the routes used for labeling from drifting too far from the known truth as described in [5]. This is a phenomenon seen when the input distribution changes, especially in semi- supervised problems. An example is in correlation matching in images. The template being used to match within the image can start to drift away from the truth if updated regularly. Especially in cases like route labeling when there is little supervision, it is preferred to guarantee the templates reflect the truth at all times rather than attempt to leverage game routes that have already been labeled. This avoids treating a wrong label as truth. Route | Precision | Recall | Count ---|---|---|--- Corner | 0.36 | 0.76 | 21 Dig | 0.75 | 0.27 | 45 Flat | 0.64 | 0.78 | 23 Out | 0.75 | 0.20 | 30 Post | 0.33 | 0.50 | 22 Slant | 0.67 | 0.76 | 38 Sluggo | 0.33 | 1.0 | 1 Streak | 0.54 | 0.36 | 36 Wheel | 0.33 | 0.29 | 7 Overall | 0.48 | 0.48 | 234 Route | Precision | Recall | Count ---|---|---|--- Corner | 0.54 | 0.95 | 21 Dig | 0.77 | 0.60 | 45 Flat | 0.70 | 0.91 | 23 Out | 0.95 | 0.60 | 30 Post | 0.53 | 0.86 | 22 Slant | 0.76 | 0.82 | 38 Sluggo | 0.25 | 1.0 | 1 Streak | 0.87 | 0.75 | 36 Wheel | 0.67 | 0.29 | 7 Overall | 0.72 | 0.71 | 234 Table 1: Precision and recall of labeled routes for cutoff times of 3 (left) and 5 (right) seconds from the 2017 season game between the Denver Broncos and Los Angeles Chargers ## 7 Performance Overall this method performed well and was able to distinguish between routes based on their fundamental characteristics. The method was able to handle the direction differences (e.g. out route versus dig route) and was able to differentiate routes based on the angle at which the receivers initially turned. Examples of routes labeled post, corner, out, dig, slant, flat, streak, sluggo, and wheel can be seen in the Appendix. What stands out is the difference in when receivers are making their breaks which differentiates the short routes such as slants and flats, the intermediate routes such as digs and corners, and the deep routes such as posts and corners. In the flats and slants many of these captured routes have little to no angle as a break, the digs and outs show a very sharp turn to the center or sideline, while the posts and corners show a more gradual turn. This method tends to over categorize corner and post routes. It can be seen in the Appendix that there are some dig and out routes that are mislabeled as post and corner routes respectively because of their more gradual turns. Instead, the method should be taking into account the ending turn which is much sharper than what would be expected of a post or corner route. Figure 14: Normalized confusion matrix of accuracy of routes A more quantitative way to assess the performance of this method is similar to the analysis performed in [2], to measure precision and recall. Precision and recall are calculated for each route label using the number of true positives ($t_{p}$) correctly identified routes of the current label, the number of false positives ($f_{p}$) other routes mislabeled as the current label, and false negatives ($f_{n}$) routes of the current label that were misclassified. They are defined as Precision $\displaystyle=\dfrac{t_{p}}{t_{p}+f_{p}}\;,$ (12) Recall $\displaystyle=\dfrac{t_{p}}{t_{p}+f_{n}}\;.$ (13) Table 1 shows individual precision and recall scores for each route based on a manually labeled game. The true labels were gathered by systematically watching and recording a best guess for each route run during the Denver Broncos versus the Los Angeles Chargers game in the 2017 season. The overall score shows a moderate success at labeling. Of note is that even though there were curl and comeback routes in the game ($\leq 5$ of both) there were no curl or comeback routes predicted. This method will struggle with these routes because many times when these routes are run the receiver is doubling over onto the route which does not distinguish itself in this classification method. Better techniques for classifying these specific routes are left for future work. Also receivers that were labeled to be either blocking or waiting for a bubble screen were classified as such and included in the accuracy measurements but not shown. Blocking or waiting for a bubble screen is indistinguishable with this method. Receivers were classified as blocking or waiting for a bubble screen if they did not move more than 4 yards during the play. The overall accuracy for this game was 72%. The confusion matrix in Figure 14 shows the normalized categorization probabilities for classified routes. The greatest mislabeling was outs erroneously labeled as flats. In Figure 15 the distribution of routes can be seen which shows that tight end routes were dominated by slants and flats while wide receivers seemed to run an approximately equal amount of streaks, slants, posts, digs and corners. These observations align closely with work done in [1]. Figure 15: Distribution of routes by position Cutting off routes at 3 seconds was also considered because this is a more natural amount of time a receiver would develop their routes fully. This resulted in a sharp degradation of performance as can be seen in Table 1. Recall in Figure 1 this cuts off many routes before the play was complete. Another possibility would be to cut off the routes at 4 seconds but this too showed poor performance compared to cutting off routes at 5 seconds. ## 8 Conclusion This paper presented an unsupervised template matching method that allows for routes to be classified using a simple distance metric. This method’s main benefits are the overall speed and no manual labeling of routes is required. After pre-processing the game routes, the three main parts of this method are scaling, translating and measuring distances. Each of these operations have $\mathcal{O}(T)$ complexity where $T$ is the number of points in the game route. This $T$ is actually capped by the amount of maximum time allowed for each route, which for this method is 5 seconds or 50 points. When labeling a full game with 252 routes this method took 303 seconds or approximately 5 minutes. The other benefit is that labeling is done for each route without having to manually label clusters afterwards or labeling routes beforehand to use as a training set. Other methods require labeling at some time by humans, but template matching assigns a label without any human intervention. Converting raw coordinates of players to route labels is a step in trying to glean more information from NFL games. The next step is to use these labels with more standard statistical methods to understand what routes work well in different situations: Namely how do certain route combinations work against certain defensive coverage schemes or how does a specific player’s routes work against various coverage types. This involves labeling individual defensive players as zone or man, then using that information to imply an overall coverage scheme. Producing summary statistics about these matchups will follow. The github url github.com/kinne174 is where this project and others are stored. In this paper a template based search criterion was used to automatically classify routes run by receivers. It was shown that moderate success is achieved through an appropriate scaling method and translations of the route to align the pre-defined routes as closely as possible with the game route. ## References * [1] Chu, Dani, et al. “Route Identification in the National Football League.” arXiv preprint arXiv:1908.02423 (2019). * [2] Hochstedler and Gagnon. “American Football Route Identification Using Supervised Machine Learning.” MIT Sloan Sports Analytics Conference 2017. * [3] Intille, Stephen S., and Aaron F. Bobick. “A framework for recognizing multi-agent action from visual evidence.” AAAI/IAAI 99.518-525 (1999): 2. * [4] Sterken, Nathan. “RouteNet: a Convolutional Nueral Network for Classifying Routes.” NFL Big Data Bowl (2019). * [5] Quionero-Candela, Joaquin, et al. Dataset shift in machine learning. The MIT Press, 2009. * [6] Vonder Haar, Adam. “Exploratoy Data Analysis of Passing Plays using NFL Tracking Data.” NFL Big Data bowl (2019). ## Appendix Examples of game routes that were labeled the same in the Denver Broncos versus Los Angeles Chargers game during the 2017 season. The magenta line represents the median route of the group showing this method is able to partition essential qualities of common routes. A Corner B Post A Out B Dig A Flat B Slant A Wheel B Sluggo

2024-09-04T02:54:59.326223

2003.05465

arxiv-papers

[1]Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, Australia # Characterization of solvable spin models via graph invariants Adrian Chapman<EMAIL_ADDRESS>Steven T. Flammia (May 27, 2020) ###### Abstract Exactly solvable models are essential in physics. For many-body spin-$\mathbf{\nicefrac{{1}}{{2}}}$ systems, an important class of such models consists of those that can be mapped to free fermions hopping on a graph. We provide a complete characterization of models which can be solved this way. Specifically, we reduce the problem of recognizing such spin models to the graph-theoretic problem of recognizing line graphs, which has been solved optimally. A corollary of our result is a complete set of constant-sized commutation structures that constitute the obstructions to a free-fermion solution. We find that symmetries are tightly constrained in these models. Pauli symmetries correspond to either: (i) cycles on the fermion hopping graph, (ii) the fermion parity operator, or (iii) logically encoded qubits. Clifford symmetries within one of these symmetry sectors, with three exceptions, must be symmetries of the free-fermion model itself. We demonstrate how several exact free-fermion solutions from the literature fit into our formalism and give an explicit example of a new model previously unknown to be solvable by free fermions. ## 1 Introduction Exactly solvable models provide fundamental insight into physics without the need for difficult numerical methods or perturbation theory. In the particular setting of many-body spin-$\nicefrac{{1}}{{2}}$ systems, a remarkable method for producing exact solutions involves finding an effective description of the system by noninteracting fermions. This reduces the problem of solving the $n$-spin system over its full $2^{n}$-dimensional Hilbert space to one of solving a single-particle system hopping on a lattice of $O(n)$ sites. The paradigmatic example of this method is the exact solution for the XY model [1], where the Jordan-Wigner transformation [2] is employed to describe the model in terms of free fermions propagating in one spatial dimension. These fermions are resolved as nonlocal Pauli operators in the spin picture, and the nonlocal nature of this mapping may suggest that finding generalizations to this mapping for more complicated spin systems is a daunting task. Of the many generalizations that have since been proposed [3, 4, 5, 6, 7, 8, 9, 10, 11], a particularly interesting solution to this problem is demonstrated in the exact solution of a 2-d spin model on a honeycomb lattice introduced by Kitaev [12]. For this model, the transformation to free-fermions can be made locality- preserving over a fixed subspace through the use of local symmetries. The dynamics of free-fermion systems are generated by Gaussian-fermionic Hamiltonians and correspond to the class of so-called matchgate circuits. This circuit class coincides with the group of free-fermion propagators generated by arbitrarily time-dependent single-particle Hamiltonians [13, 14] and has an extensive complexity-theoretic characterization. In general, matchgate circuits can be efficiently simulated classically with arbitrary single-qubit product-state inputs and measurement [15, 16]. However, they become universal for quantum computation with the introduction of non-matchgates such as the $\mathrm{SWAP}$ gate [17, 18], certain measurements and resource inputs [19, 20], and when acting on nontrivial circuit geometries [21]. Furthermore, these circuits share an interesting connection to the problem of counting the number of perfect matchings in a graph, which is the context in which they were first developed [22, 23, 24, 25]. This problem is known to be very hard computationally (it is #P-complete [26]), but is efficiently solved for planar graphs using the so-called Fisher-Kasteleyn-Temperley algorithm [27, 28]. In this work, we develop a distinct connection between free-fermion systems and graph theory by using tools from quantum information science. The central object of our formalism is the _frustration graph_. This is a network quantifying the anticommutation structure of terms in the spin Hamiltonian when it is expanded in the basis of Pauli operators [29]. This graph has been invoked previously in the setting of variational quantum eigensolvers [30, 31, 32, 33, 34, 35, 36, 37], commonly under the name “anti-compatibility graph". We show that the problem of recognizing whether a given spin model admits a free-fermion solution is equivalent to that of recognizing whether its frustration graph is a _line graph_ , which can be performed optimally in linear time [38, 39, 40]. From the definition of a line graph, it will be clear that such a condition is necessary, but we will show that it is also sufficient. When the condition is met, we provide an explicit solution to the model. Line graphs have recently emerged as the natural structures describing the effective tight-binding models for superconducting waveguide networks [41, 42, 43]. In this setting, the line graph corresponds to the physical hopping graph of photons in the network. We will see how this scenario is a kind of “inverse problem" to the one we consider, wherein fermions are hopping on the _root_ of the line graph. It is clear from both scenarios that the topological connectivity structure of many-body systems plays a central role in their behavior, and it is remarkable that this is already being observed in experiments. We expect that further investigation of the graph structure of many-body Hamiltonians will continue to yield important insights into their physics. ### 1.1 Summary of Main Results Here we give a brief summary of the main results. We first define the frustration graph of a Hamiltonian, given in the Pauli basis, as the graph with nonzero Pauli terms as vertices and an edge between two vertices if their corresponding terms anticommute. A line graph $G$ of a graph $R$ is the intersection graph of the edges of $R$ as two-element subsets of the vertices of $R$. With these simple definitions, we can informally state our first main result, which we call our “fundamental theorem:" ###### Result 1 (Existence of free-fermion solution; Informal version of Thm. 1). Given an $n$-qubit Hamiltonian in the Pauli basis for which the frustration graph $G$ is the line graph of another graph $R$, then there exists a free- fermion description of $H$. From this description, an exact solution for the spectrum and eigenstates of $H$ can be constructed. This theorem illustrates a novel connection between the physics of quantum many-body systems and graph theory with some surprising implications. First, it gives the exact correspondence between the spatial structure of a spin Hamiltonian and that of its effective free-fermion description. As we will see through several examples, this relationship is not guaranteed to be straightforward. Second, the theorem gives an exact condition by which a spin model can _fail_ to have a free-fermion solution, the culprit being the presence of forbidden anticommutation structures in the frustration graph of $H$. Some caveats to Result 1 (that are given precisely in the formal statement, Theorem 1) involve cases in which this mapping between Pauli terms in $H$ and fermion hopping terms is not one-to-one. In particular, if we are given a Hamiltonian whose frustration graph is not a line graph, then a free-fermion solution may still be possible via a non-injective mapping over a subspace defined by fixing stabilizer degrees of freedom. Additionally, it is possible for a given spin Hamiltonian to describe multiple free-fermion models simultaneously, each generating dynamics over an independent stabilizer subspace of the full Hilbert space as for the Kitaev honeycomb model [12]. These symmetries are sometimes referred to as gauge degrees of freedom, though we will reserve this term for freedoms which cannot affect the physics of the free-fermion model. Finally, it may be the case that the free-fermion model contains states which are nonphysical in the spin-Hamiltonian picture, and so these must be removed by fixing a symmetry as well. Luckily, all of these cases manifest as structures in the frustration graph of $H$. The first, regarding when a non-injective free-fermion solution is required, is signified by the presence of so-called twin vertices, or vertices with the same neighborhood. We deal with this case in our first lemma. The next two cases are covered by our second theorem: ###### Result 2 (Graphical symmetries; Informal version of Thm. 2). Given an $n$-qubit Hamiltonian in the Pauli basis for which the frustration graph $G$ is the line graph of another graph $R$, then Pauli symmetries of $H$ correspond to either: 1. (i) Cycles of $R$; 2. (ii) A T-join of $R$, associated to the fermion-parity operator; 3. (iii) Logically encoded qubits; and these symmetries generate an abelian group. We then prove that we can always fix all of the cycle symmetries by choosing an orientation of the root graph $R$. Our results also relate the more general class of Clifford symmetries to the symmetries of the single-particle free- fermion Hamiltonian. We show that with exactly three exceptions, Clifford symmetries of the spin model, in a subspace defined by fixing the symmetries listed above, must also be symmetries of the single-particle Hamiltonian (see Corollary 1.2 for a precise statement). Finally, we illustrate these ideas with several examples: small systems of up to 3 qubits, the 1-dimensional anisotropic $XY$ model in a transverse field and its nearest-neighbor solvable generalization, the Kitaev honeycomb model, the 3-dimensional frustrated hexagonal gauge color code [44], and the Sierpinski-Hanoi model. To the best of our knowledge, this last model was previously not known to be solvable. The remainder of the paper is organized as follows. In Section 2, we will introduce notation and give some background on the formalism of free-fermions and frustration graphs. In Section 3, we will formally state Theorem 1 and some general implications thereof. In Section 4, we elaborate on the structure of symmetries which can be present in our class of solvable models. In Section 4.1, we will use the theorems of the previous two sections to outline an explicit solution method. We close by demonstrating how the examples of free- fermion solutions listed above fit into this formalism in Section 5. ## 2 Background ### 2.1 Frustration Graphs The models we consider are spin-$\nicefrac{{1}}{{2}}$ (qubit) Hamiltonians written in the Pauli basis $\displaystyle H=\sum_{\boldsymbol{j}\in V}h_{\boldsymbol{j}}\sigma^{\boldsymbol{j}}\mathrm{,}$ (1) where $\boldsymbol{j}\equiv(\boldsymbol{a},\boldsymbol{b})$, with $\boldsymbol{a}$, $\boldsymbol{b}\in\\{0,1\\}^{\times n}$ labeling an $n$-qubit Pauli operator as $\displaystyle\sigma^{\boldsymbol{j}}=i^{\boldsymbol{a}\cdot\boldsymbol{b}}\left(\bigotimes_{k=1}^{n}X_{k}^{a_{k}}\right)\left(\bigotimes_{k=1}^{n}Z_{k}^{b_{k}}\right)\mathrm{.}$ (2) The exponent of the phase factor, $\boldsymbol{a}\cdot\boldsymbol{b}$, is the _Euclidean inner product_ between $\boldsymbol{a}$ and $\boldsymbol{b}$. This phase is chosen such that the overall operator is Hermitian, and such that $a_{k}=b_{k}=1$ means that $\sigma^{\boldsymbol{j}}$ acts on qubit $k$ by a Pauli-$Y$ operator. We denote the full $n$-qubit Pauli group by $\mathcal{P}$, and $V\subseteq\mathcal{P}$ is the set of Pauli terms in $H$ (i.e. $h_{\boldsymbol{j}}=0$ for all $\boldsymbol{j}\notin V$). Let the Pauli subgroup generated by this set be denoted $\mathcal{P}_{H}$. For our purposes, what is important is not the explicit Pauli description of the Hamiltonian, but rather the commutation relations between its terms. As Pauli operators only either commute or anticommute, a useful quantity is their _scalar commutator_ $[\\![\cdot,\cdot]\\!]$, which we define implicitly as $\displaystyle\sigma^{\boldsymbol{j}}\sigma^{\boldsymbol{k}}=[\\![\sigma^{\boldsymbol{j}},\sigma^{\boldsymbol{k}}]\\!]\sigma^{\boldsymbol{k}}\sigma^{\boldsymbol{j}}\mathrm{.}$ (3) The scalar commutator thus only takes the values $\pm 1$. Additionally, the scalar commutator distributes over multiplication in each argument, e.g. $\displaystyle[\\![\sigma^{\boldsymbol{j}},\sigma^{\boldsymbol{k}}\sigma^{\boldsymbol{l}}]\\!]=[\\![\sigma^{\boldsymbol{j}},\sigma^{\boldsymbol{k}}]\\!][\\![\sigma^{\boldsymbol{j}},\sigma^{\boldsymbol{l}}]\\!].$ (4) For $n$-qubit Paulis, the scalar commutator can thus be read off from the Pauli labels as $\displaystyle[\\![\sigma^{\boldsymbol{j}},\sigma^{\boldsymbol{k}}]\\!]=(-1)^{\langle\boldsymbol{j},\boldsymbol{k}\rangle}$ (5) Here, $\langle\boldsymbol{j},\boldsymbol{k}\rangle$ is the _symplectic inner product_ $\displaystyle\langle\boldsymbol{j},\boldsymbol{k}\rangle\equiv\begin{pmatrix}\boldsymbol{a}_{j}&\boldsymbol{b}_{j}\end{pmatrix}\begin{pmatrix}\mathbf{0}_{n}&\mathbf{I}_{n}\\\ -\mathbf{I}_{n}&\mathbf{0}_{n}\end{pmatrix}\begin{pmatrix}\boldsymbol{a}_{k}\\\ \boldsymbol{b}_{k}\end{pmatrix}\mathrm{,}$ (6) where naturally $\boldsymbol{j}\equiv(\boldsymbol{a}_{j},\boldsymbol{b}_{j})$ and $\boldsymbol{k}\equiv(\boldsymbol{a}_{k},\boldsymbol{b}_{k})$. $\mathbf{0}_{n}$ is the $n\times n$ all-zeros matrix, and $\mathbf{I}_{n}$ is the $n\times n$ identity matrix. Eq. (5) captures the fact that a factor of $-1$ is included in the scalar commutator for each qubit where the operators $\sigma^{\boldsymbol{j}}$ and $\sigma^{\boldsymbol{k}}$ differ and neither acts trivially. Since the inner product appears as the exponent of a sign factor, without loss of generality, we can replace it with the _binary symplectic inner product_ $\displaystyle\langle\boldsymbol{j},\boldsymbol{k}\rangle_{2}\equiv\langle\boldsymbol{j},\boldsymbol{k}\rangle\bmod 2.$ (7) $H$ | $\sum\limits_{j\in\\{x,y,z\\}}h_{j}\sigma^{j}$ | $\sum\limits_{\begin{subarray}{c}\boldsymbol{j}\in\\{0,x,y,z\\}^{\times 2}\\\ \boldsymbol{j}\neq(0,0)\end{subarray}}h_{\boldsymbol{j}}\sigma^{\boldsymbol{j}}$ ---|---|--- $G(H)$ $\simeq L(R)$ | | $R$ | or | Table 1: Example frustration graphs for general Hamiltonians on small (1- and 2-qubit) systems. (Left column) For general single-qubit Hamiltonians, the frustration graph is the complete graph on three vertices, $K_{3}$. By the Whitney isomorphism theorem [45], $K_{3}$ is the only graph which is not the line graph of a unique graph, but rather is the line graph of both $K_{3}$ and the ‘claw’ graph, $K_{1,3}$. This implies the existence of two distinct free- fermion solutions of single-qubit Hamiltonians. (Right column) For general two-qubit Hamiltonians, the frustration graph is the line graph of the complete graph on six vertices $K_{6}$ [29, 46]. Colored are the size-five cliques corresponding to the degree-five vertices of the root graph. This mapping implies the existence of a free-fermion solution for general two-qubit Hamiltonians by six fermions, reflecting the accidental Lie-algebra isomorphism $\mathfrak{su}(4)\simeq\mathfrak{spin}(6)$ (see Section 5.1). Through the binary symplectic inner product, the scalar commutator defines a symmetric binary relation between terms in the Hamiltonian, to which we associate the adjacency matrix of a graph. Denote the _frustration graph_ for a Hamiltonian of the form in Eq. (1) by $G(H)\equiv(V,E)$ with vertex set given by the Pauli terms appearing in $H$, and edge set $\displaystyle E\equiv\\{(\boldsymbol{j},\boldsymbol{k})|\langle\boldsymbol{j},\boldsymbol{k}\rangle_{2}=1\\}$ (8) That is, two Pauli terms correspond to neighboring vertices in $G(H)$ if and only if they anticommute. Without loss of generality, we can assume that $G(H)$ is connected, as disconnected components of this graph correspond to commuting collections of terms in the Hamiltonian and can thus be independently treated. As such, we will further assume that $H$ has no identity component in the expansion (1)—rendering it traceless—since this will only contribute an overall energy shift to the system with no effect on dynamics. ### 2.2 Majorana Fermions A related set of Hermitian operators which only either commute or anticommute is that of the Majorana fermion modes $\\{\gamma_{\mu}\\}_{\mu}$, which satisfy the canonical anticommutation relations $\displaystyle\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2\delta_{\mu\nu}I\mathrm{,}$ (9) and for which $\gamma_{\mu}^{\dagger}=\gamma_{\mu}$. A familiar way of realizing these operators in terms of $n$-qubit Pauli observables is through the Jordan-Wigner transformation $\displaystyle\gamma_{2j-1}=\bigotimes_{k=1}^{j-1}Z_{k}\otimes X_{j}\mbox{\hskip 28.45274pt}\gamma_{2j}=\bigotimes_{k=1}^{j-1}Z_{k}\otimes Y_{j}\mathrm{.}$ (10) The Pauli operators on the right can easily be verified to constitute $2n$ operators satisfying Eq. (9). Of course, we will explore the full set of generalizations to this transformation in this work. We seek to identify those qubit Hamiltonians which can be expressed as quadratic in the Majorana modes. Such _free-fermion_ Hamiltonians are written as $\displaystyle\widetilde{H}=i\boldsymbol{\gamma}\cdot\mathbf{h}\cdot\boldsymbol{\gamma}^{\mathrm{T}}\equiv 2i\sum_{(j,k)\in\widetilde{E}}h_{jk}\gamma_{j}\gamma_{k}$ (11) where $\boldsymbol{\gamma}$ is a row-vector of the Majorana operators, and $\mathbf{h}$ is the _single-particle Hamiltonian_. Without loss of generality, $\mathbf{h}$ can be taken as a real antisymmetric matrix, as we can similarly assume $\widetilde{H}$ is traceless, and the canonical anticommutation relations Eq. (9) guarantee that any symmetric component of $\mathbf{h}$ will not contribute to $\widetilde{H}$. $\widetilde{E}$ is the edge-set of the _fermion-hopping graph_ $R\equiv(\widetilde{V},\widetilde{E})$ on the fermion modes $\widetilde{V}$. That is, $h_{jk}=0$ for those pairs $(j,k)\notin\widetilde{E}$, and the factor of two in the rightmost expression accounts for the fact that each edge in $\widetilde{E}$ is included only once in the sum. As a result of the canonical anticommutation relations (9), the individual Majorana modes transform covariantly under the time evolution generated by $\widetilde{H}$ $\displaystyle\mathrm{e}^{i\widetilde{H}t}\gamma_{\mu}\mathrm{e}^{-i\widetilde{H}t}=\sum_{\nu\in\widetilde{V}}\left(\mathrm{e}^{4\mathbf{h}t}\right)_{\mu\nu}\gamma_{\nu}$ (12) since $\displaystyle[\bm{\gamma}\cdot\mathbf{h}\cdot\boldsymbol{\gamma}^{\mathrm{T}},\gamma_{\mu}]=-4(\mathbf{h}\cdot\boldsymbol{\gamma}^{\mathrm{T}})_{\mu}\,.$ (13) Since $\mathbf{h}$ is antisymmetric and real, $\mathrm{e}^{4\mathbf{h}t}\in\mathrm{SO}(2n,\mathds{R})$. Thus, $\mathbf{h}$ can be block-diagonalized via a real orthogonal matrix, $\mathbf{W}\in\mathrm{SO}(2n,\mathds{R})$, as $\displaystyle\mathbf{W}^{\mathrm{T}}\cdot\mathbf{h}\cdot\mathbf{W}=\bigoplus_{j=1}^{n}\begin{pmatrix}0&-\lambda_{j}\\\ \lambda_{j}&0\\\ \end{pmatrix}$ (14) We can represent $\mathbf{W}$ as the exponential of a quadratic Majorana fermion operator as well, by defining $\displaystyle\mathbf{W}\equiv\mathrm{e}^{4\mathbf{w}}\mathrm{,}$ (15) $\widetilde{H}$ is therefore diagonalized as $\displaystyle\mathrm{e}^{-\boldsymbol{\gamma}\cdot\mathbf{w}\cdot\boldsymbol{\gamma}^{\mathrm{T}}}\widetilde{H}\mathrm{e}^{\boldsymbol{\gamma}\cdot\mathbf{w}\cdot\boldsymbol{\gamma}^{\mathrm{T}}}$ $\displaystyle=i\boldsymbol{\gamma}\cdot\left(\mathbf{W}^{\mathrm{T}}\cdot\mathbf{h}\cdot\mathbf{W}\right)\cdot\boldsymbol{\gamma}^{\mathrm{T}}$ (16) $\displaystyle=-2i\sum_{j=1}^{n}\lambda_{j}\gamma_{2j-1}\gamma_{2j}$ (17) $\displaystyle\mathrm{e}^{-\boldsymbol{\gamma}\cdot\mathbf{w}\cdot\boldsymbol{\gamma}^{\mathrm{T}}}\widetilde{H}\mathrm{e}^{\boldsymbol{\gamma}\cdot\mathbf{w}\cdot\boldsymbol{\gamma}^{\mathrm{T}}}$ $\displaystyle=2\sum_{j=1}^{n}\lambda_{j}Z_{j}$ (18) Note that the exact diagonalization can be performed with reference to the _quadratics_ in the Majorana fermion modes only. To completely solve the system, it is only necessary to diagonalize $\mathbf{h}$ classically, find a generating matrix $\mathbf{w}$, and diagonalize $\widetilde{H}$ using an exponential of quadratics with regard to some fermionization like Eq. (10). Eigenstates of $\widetilde{H}$ can be found by acting $\mathrm{e}^{\boldsymbol{\gamma}\cdot\mathbf{w}\cdot\boldsymbol{\gamma}^{\mathrm{T}}}$ on a computational basis state $\left|\mathbf{x}\right\rangle$ for $\mathbf{x}\in\\{0,1\\}^{\times n}$. The associated eigenvalue is $\displaystyle E_{\mathbf{x}}=2\sum_{j=1}^{n}(-1)^{x_{j}}\lambda_{j}$ (19) Therefore, systems of the form in Eq. (11) may be considered _exactly solvable_ classically, since their exact diagonalization is reduced to exact diagonalization on a _poly_ $(n)$-sized matrix $\mathbf{h}$. ## 3 Fundamental Theorem As mentioned previously, we seek to characterize the full set of Jordan- Wigner-like transformations, generalizing Eq. (10). To be more precise, we ask for the conditions under which there exists a mapping $\phi:V\mapsto\widetilde{V}^{\times 2}$, for some set $\widetilde{V}$ (the fermion modes), effecting $\displaystyle\sigma^{\boldsymbol{j}}\mapsto i\gamma_{\phi_{1}(\boldsymbol{j})}\gamma_{\phi_{2}(\boldsymbol{j})}\mathrm{,}$ (20) for $\phi_{1}(\boldsymbol{j})$, $\phi_{2}(\boldsymbol{j})\in\widetilde{V}$, and such that $\displaystyle[\\![\sigma^{\boldsymbol{j}},\sigma^{\boldsymbol{k}}]\\!]=[\\![\gamma_{\phi_{1}(\boldsymbol{j})}\gamma_{\phi_{2}(\boldsymbol{j})},\gamma_{\phi_{1}(\boldsymbol{k})}\gamma_{\phi_{2}(\boldsymbol{k})}]\\!]$ (21) for all pairs, $\boldsymbol{j}$ and $\boldsymbol{k}$. Such a mapping induces a term-by-term _free-fermionization_ of the Hamiltonian (1) to one of the form (11) such that $\displaystyle G(H)\simeq G(\widetilde{H})\mathrm{.}$ (22) Again, $G(H)$ is the frustration graph of $H$. From the canonical anticommutation relations, Eq. (9), and the distribution rule Eq. (4), we see that scalar commutators between quadratic Majorana- fermion operators are given by $\displaystyle[\\![\gamma_{\mu}\gamma_{\nu},\gamma_{\alpha}\gamma_{\beta}]\\!]=(-1)^{|(\mu,\nu)\cap(\alpha,\beta)|}$ (23) Eqs. (22) and (23) can be restated graph theoretically as saying that $G(H)$ is the graph whose vertex set is the edge set of the fermion hopping graph $R$, and vertices of $G(H)$ are neighboring if and only if the associated edges of $R$ share exactly one vertex. Such a graph is called the _line graph_ of $R$. ###### Definition 1 (Line Graphs). The line graph $L(R)\equiv(E,F)$ of a root graph $R\equiv(V,E)$ is the graph whose vertex set is the edge set of $R$ and whose edge set is given by $\displaystyle F\equiv\\{(e_{1},e_{2})\ |\ e_{1},e_{2}\in E,\ |e_{1}\cap e_{2}|=1\\}$ (24) That is, vertices are neighboring in $L(R)$ if the corresponding edges in $R$ are incident at a vertex. Notice that if $L(R)$ is connected if and only if $R$ is. With these definitions in-hand, our first main result can be stated simply as ###### Theorem 1 (Existence of free-fermion solution). An injective map $\phi$ as defined in Eq. (20) and Eq. (21) exists for the Hamiltonian $H$ as defined in Eq. (1) if and only if there exists a root graph $R$ such that $\displaystyle G(H)\simeq L(R),$ (25) where R is the hopping graph of the free-fermion solution. | With Twins | Without twins ---|---|--- Forbidden Graphs | (a) | (d) Twin-Free, $L(R)$ | (b) | Root $R$ | (c) | (e) Table 2: A graph is a line graph if and only if it does not contain any of the nine forbidden graphs in (a), (d), and (e) as an induced subgraph [47]. Of these nine graphs, the three in (a) contain twin vertices, highlighted. If these three graphs are induced subgraphs of a frustration graph such that these highlighted vertices are twins in the larger graph, then the twins can be removed by restricting onto a fixed mutual eigenspace of their products, which correspond to constants of motion of the Hamiltonian. (b) The twin-free restrictions of the graphs in (a), with all but one highlighted vertex from (a) removed. These graphs are the line graphs of the graphs in (c). In Ref. [48], it was shown that only five graphs contain the forbidden subgraphs in (e) and none of those in (a) or (d). Finally, this set was further refined in Ref. [49] to a set of three forbidden subgraphs for 3-connected line graphs of minimum degree at least seven, though we do not display these graphs here. ###### Proof. The proof can be found in Section 6.1. ∎ The intuition for this result is that the root graph $R$ is the graph where the vertices are fermions and the edges are the bilinears that appear in the Hamiltonian $H$. The result reveals a correspondence between a characterization of line graphs and a characterization of free-fermion spin models, as not every graph can be expressed as the line graph of some root. We must however note that, strictly speaking, the existence of this mapping alone does not guarantee a free-fermion solution, since the “Lie-homomorphism" constraint, Eq. (21), does not fix the _sign_ of the terms in the free-fermion Hamiltonian. Choosing a sign for each term is equivalent to _orienting_ the root graph, since multiplying by a sign is equivalent to making the exchange $\phi_{1}(\boldsymbol{j})\leftrightarrow\phi_{2}(\boldsymbol{j})$ in Eq. (20). Different orientations may not faithfully reproduce the properties of $H$, but we will see that such an orientation can always be chosen. The line graph condition in Eq. (25) is therefore necessary and sufficient for a free-fermion solution to exist. Before turning to further implications of Theorem 1, let us first detail some properties of line graphs. Line graphs are closely related to so-called _intersection graphs_ , originally studied by Erdős [50] and others (see, for example, Ref. [51]). An intersection graph $G\equiv(V,E)$ is a graph whose vertex set, $V\subseteq 2^{S}$, consists of distinct subsets of some set $S$. Two vertices, $u$ and $v$, are neighboring in $G$ if their intersection is nonempty ($|v\cap w|\neq 0$). A line graph is a special case of an intersection graph where every vertex corresponds to a subset of size at most two. When we specify that $\phi$ be injective, we are requiring that no distinct vertices have identical subsets, and our definition of a free-fermion solution Eq. (25) identically coincides with that of a line graph. Since terms in $H$ can thus intersect by at most one Majorana mode, collections of terms containing a given mode are all neighboring in $G(H)$, so this mode corresponds to a _clique_ , or complete subgraph, of $G(H)$. This characterization of line graphs was first given by Krausz [52] and bears stating formally. ###### Definition 2 (Krausz decomposition of line graphs). Given a line graph $G\simeq L(R)$, there exists a partition of the edges of $G$ into cliques such that every vertex appears in at most two cliques. Cliques in $G(H)$ can therefore be identified with the individual Majorana modes in a free-fermion solution of $H$. If a term belongs to only one clique, we can ensure our resulting fermion Hamiltonian is quadratic by taking the second clique for that term to be a clique of no edges, as we will see in several examples below. The existence of a Krausz decomposition is utilized in a linear-time algorithm to recognize line graphs by Roussopoulos [38], though the earliest such algorithm for line-graph recognition was given by Lehot [39]. A dynamic solution was later given by Degiorgi and Simon [40]. These algorithms are optimal and constructive, and so can be applied to a given spin model to provide an exact free-fermion solution. We next turn to the _hereditary property_ of line graphs, for which we require the following definition: ###### Definition 3 (Induced subgraphs). Given a graph $G\equiv(V,E)$, an induced subgraph of $G$ by a subset of vertices $V^{\prime}\subset V$, is a graph $G[V^{\prime}]\equiv(V^{\prime},E^{\prime})$ such that for any pair of vertices $u$, $v\in V^{\prime}$, $(u,v)\in E^{\prime}$ if and only if $(u,v)\in E$ in $G$. An induced subgraph of $G$ can be constructed by removing the subset of vertices $V/V^{\prime}$ from $G$, together with all edges incident to any vertex in this subset. Line graphs are a _hereditary class_ of graphs in the sense that any induced subgraph of a line graph is also a line graph. This coincides with our intuition that removing a term from a free-fermion Hamiltonian does not change its free-fermion solvability. Conversely, Hamiltonians for which no free-fermion solution exists are accompanied by “pathological" structures in their frustration graphs, which obstruct a free- fermion description no matter how we try to impose one. This is captured by the forbidden subgraph characterization of Beineke [47] and later refined by others [48, 49]. ###### Corollary 1.1 (Beineke no-go theorem). A given spin Hamiltonian $H$ has a free-fermion solution if and only if its frustration graph $G(H)$ does not contain any of nine forbidden subgraphs, shown in Table 2, (a) (d) (e), as an induced subgraph. These forbidden subgraphs above can be interpreted as collections of “frustrating" terms. At least one of the terms must be assigned to a fermion interaction in every possible assignment from Pauli operators to fermions. Correspondingly, ignoring these terms by removing their corresponding vertices from the frustration graph may remove a forbidden subgraph and cause the Hamiltonian to become solvable. The terms which we need to remove in this way need not be unique. In the next section, we discuss one such strategy for removing vertices such that our solution will remain faithful to the original spin Hamiltonian by exploiting symmetries. ## 4 Symmetries An important class of symmetries involves twin vertices in the frustration graph. ###### Definition 4 (Twin Vertices). Given a graph $G\equiv(V,E)$, vertices $u$, $v\in V$ are twin vertices if, for every vertex $w\in V$, $(u,w)\in E$ if and only if $(v,w)\in E$. Twin vertices have exactly the same neighborhood, and are thus never neighbors in a frustration graph, which contains no self edges due to the fact that every operator commutes with itself. Sets of twin vertices are the subject of our first lemma. ###### Lemma 1 (Twin vertices are constants of motion). Suppose a pair of terms $\sigma^{\boldsymbol{j}}$ and $\sigma^{\boldsymbol{k}}$ in $H$ correspond to twin vertices in $G(H)$, then the product $\sigma^{\boldsymbol{j}}\sigma^{\boldsymbol{k}}$ is a nontrivial Pauli operator commuting with every term in the Hamiltonian. Distinct such products therefore commute with each other. ###### Proof. The statement follows straightforwardly from the definition of twin vertices: every term in $H$ (including $\sigma^{\boldsymbol{j}}$ and $\sigma^{\boldsymbol{k}}$ themselves) either commutes with both $\sigma^{\boldsymbol{j}}$ and $\sigma^{\boldsymbol{k}}$ or anticommutes with both of these operators. Terms in $H$ therefore always commute with the product $\sigma^{\boldsymbol{j}}\sigma^{\boldsymbol{k}}$. This product is furthermore a nontrivial Pauli operator, for if $\sigma^{\boldsymbol{j}}\sigma^{\boldsymbol{k}}=I$, then $\boldsymbol{j}=\boldsymbol{k}$, and we would not identify these Paulis with distinct vertices in $G(H)$. Constants of motion generated this way must commute with one another, since they commute with every term in the Hamiltonian and are themselves products of Hamiltonian terms. They therefore generate an abelian subgroup of the symmetry group of the Hamiltonian. ∎ Let the symmetry subgroup generated by products of twin vertices in this way be denoted $\mathcal{S}$. We can leverage these symmetries to remove twin vertices from the frustration graph $G(H)$. To do this, choose a minimal generating set $\\{\sigma^{\boldsymbol{s}}\\}$ of Pauli operators for $\mathcal{S}$ and choose a $\pm 1$ eigenspace for each. Let $(-1)^{x_{\boldsymbol{s}}}$ be the eigenvalue associated to the generator $\sigma^{\boldsymbol{s}}\in\mathcal{S}$, for $x_{\boldsymbol{s}}\in\\{0,1\\}$. We restrict to the subspace defined as the mutual $+1$ eigenspace of the stabilizer group $\displaystyle\mathcal{S}_{\boldsymbol{x}}=\langle(-1)^{x_{\boldsymbol{s}}}\sigma^{\boldsymbol{s}}\rangle$ (26) For a pair of twin vertices corresponding to Hamiltonian terms $\sigma^{\boldsymbol{j}}$ and $\sigma^{\boldsymbol{k}}$, we let $\displaystyle\sigma^{\boldsymbol{j}}\sigma^{\boldsymbol{k}}\equiv(-1)^{d_{\boldsymbol{j},\boldsymbol{k}}}\left[\prod_{\boldsymbol{s}\in S_{\boldsymbol{j},\boldsymbol{k}}}(-1)^{x_{\boldsymbol{s}}}\sigma^{\boldsymbol{s}}\right]$ (27) where $d_{\boldsymbol{j},\boldsymbol{k}}\in\\{0,1\\}$ specifies the appropriate sign factor, and $S_{\boldsymbol{j},\boldsymbol{k}}$ is the subset of generators of $\mathcal{S}$ such that $\displaystyle\bigoplus_{\boldsymbol{s}\in S_{\boldsymbol{j},\boldsymbol{k}}}\boldsymbol{s}=\boldsymbol{j}\oplus\boldsymbol{k}$ (28) where “$\oplus$” denotes addition modulo 2 here. In the stabilizer subspace of $\mathcal{S}_{\boldsymbol{x}}$, we can make the substitution $\displaystyle\sigma^{\boldsymbol{k}}\rightarrow(-1)^{d_{\boldsymbol{j},\boldsymbol{k}}}\sigma^{\boldsymbol{j}}$ (29) effectively removing the vertex $\boldsymbol{k}$ from $G(H)$. Twin vertices capture the cases where a free-fermion solution for $H$ exists, but is necessarily non-injective. Indeed, note that we are careful in our statement of Theorem 1 to specify that our condition Eq. (25) is necessary and sufficient when $\phi$ is injective. If we instead relax our requirement that vertices of a line graph correspond to distinct subsets of size two in our earlier discussion of intersection graphs, then we are allowing for line graphs of graphs with multiple edges, or _multigraphs_. However, our definition of $G(\widetilde{H})$ will differ from the line graph of a multigraph for pairs of vertices corresponding to identical edges, which must be adjacent in the line graph of a multigraph, but will be nonadjacent in $G(\widetilde{H})$ from Eq. (23). Such vertices will nevertheless be twin vertices in $G(\widetilde{H})$ due to the graph-isomorphism constraint, Eq. (22). Therefore, if no _injective_ mapping $\phi$ satisfying Theorem 1 exists, a many-to-one free-fermion solution exists only when twin vertices are present. Lemma 1 allows us to deal with this non-injective case by removing twin vertices until we obtain the line graph of a simple graph when possible. The particular way we choose to perform this removal cannot affect the overall solvability of the model, since the frustration graph with all twin vertices removed is an induced subgraph of any frustration graph with only a proper subset of such vertices removed. A model which is solvable by free fermions this way is therefore solvable in all of its symmetry sectors. Finally, we can see when a non-injective free-fermion solution may be possible from the forbidden subgraph characterization, Corollary 1.1. As seen in Table 2, some of the forbidden subgraphs shown in (a) themselves contain twin vertices. If these forbidden subgraphs are connected to the global frustration graph such that their twin vertices remain twins in the larger graph, then they may be removed, possibly allowing for a solution of the full Hamiltonian by free fermions. When the twins are removed from the forbidden subgraphs, they become line graphs as shown in Table 2 (b) and (c). An example of a model which can be solved this way is the Heisenberg-Ising model introduced in Ref. [1]. We next proceed to identify the remaining Pauli symmetries for a Hamiltonian satisfying Theorem 1. For this, we invoke the natural partition of the Pauli group $\mathcal{P}$ into the subgroup $\mathcal{P}_{H}$, again defined as that generated by Hamiltonian terms $\\{\sigma^{\boldsymbol{j}}\\}_{\boldsymbol{j}\in V}$, and the Pauli operators outside this subgroup, $\mathcal{P}_{\perp}\equiv\mathcal{P}/\mathcal{P}_{H}$. Note that the latter set does not form a group in general, as for example, single-qubit Paulis may be outside of $\mathcal{P}_{H}$ yet may be multiplied to operators in $\mathcal{P}_{H}$. A subgroup of the symmetries of the Hamiltonian is the center $\mathcal{Z}(\mathcal{P}_{H})$ of $\mathcal{P}_{H}$, the set of $n$-qubit Pauli operators in $\mathcal{P}_{H}$ which commute with every element of $\mathcal{P}_{H}$ and therefore with every term in the Hamiltonian. To characterize this group, we need two more definitions. ###### Definition 5 (Cycle subgroup). A cycle of a graph $G\equiv(V,E)$ is a subset of its edges, $Y\subseteq E$, such that every vertex contains an even number of incident edges from the subset. If a Pauli Hamiltonian satisfies Eq. (25) for some root graph $R$, we define its cycle subgroup $Z_{H}\subseteq\mathcal{P}_{H}$ as the abelian Pauli subgroup generated by the cycles $\\{Y_{i}\\}_{i}$ of $R$, $\displaystyle Z_{H}=\bigl{\langle}\Pi_{\\{\boldsymbol{j}|\phi(\boldsymbol{j})\in Y_{i}\\}}\sigma^{\boldsymbol{j}}\bigr{\rangle}_{i}.$ (30) Since $\displaystyle\prod_{\\{\boldsymbol{j}|\phi(\boldsymbol{j})\in Y_{i}\\}}\gamma_{\phi_{1}(\boldsymbol{j})}\gamma_{\phi_{2}(\boldsymbol{j})}=\pm I$ (31) we have, from Eq. (21) and the definition of $\phi$, that the elements of $Z_{H}$ commute with every term in the Hamiltonian and thus with each other (since they are products of Hamiltonian terms). That is, $Z_{H}\subseteq\mathcal{Z}(\mathcal{P}_{H})$. Notice that the definition of the generators for $Z_{H}$ in Eq. (30) may sometimes yield operators proportional to identity. A familiar symmetry of free-fermion Hamiltonians is the _parity operator_ $\displaystyle P\equiv i^{\frac{1}{2}|\widetilde{V}|(|\widetilde{V}|-1)}\prod_{k\in\widetilde{V}}\gamma_{k}$ (32) which commutes with every term in the Hamiltonian since each term is quadratic in the Majorana modes. The phase factor is chosen such that $P$ is Hermitian. Here, we define this operator in terms of Pauli Hamiltonian terms through a combinatorial structure known as a T-join. ###### Definition 6 (Parity operator). A T-join of a graph $G\equiv(V,E)$ is a subset of edges, $T\subseteq E$, such that an odd number of edges from $T$ is incident to every vertex in $V$. If a Pauli Hamiltonian satisfies Eq. (25) for some root graph $R$ such that the number of vertices in $R$ is even, we define the parity operator as $\displaystyle P\equiv i^{d}\prod_{\boldsymbol{j}\in T}\sigma^{\boldsymbol{j}}$ (33) where the product is taken over a T-join of $R$, and $d\in\\{0,1,2,3\\}$ specifies the phase necessary to agree with Eq. (32). Here we have $\displaystyle i^{d}\prod_{\boldsymbol{j}\in T}i\gamma_{\phi_{1}(\boldsymbol{j})}\gamma_{\phi_{2}(\boldsymbol{j})}$ $\displaystyle=i^{\frac{1}{2}|\widetilde{V}|(|\widetilde{V}|-1)}\prod_{\mu\in\widetilde{V}}\gamma_{\mu}=P\mathrm{,}$ (34) since every fermion mode will be hit an odd number of times in the T-join. Unlike with the cycle subgroup, $P$ is never proportional to the identity in the fermion description, though it may still be proportional to the identity in the Pauli description (up to stabilizer equivalences). In this case, only solutions for the free-fermion Hamiltonian in a fixed-parity subspace will be physical. We will see several examples of this in the next section. When no T-join exists, we cannot form $P$ as a product of Hamiltonian terms. In fact, $P\in\mathcal{Z}(\mathcal{P}_{H})$ only when $|\widetilde{V}|$ is even. Now with these definitions in hand, we are ready to state our second theorem. ###### Theorem 2 (Symmetries are cycles and parity). Given a Hamiltonian satisfying Eq. (25) such that the number of vertices $|\widetilde{V}|$ in the root graph is odd, then we have $\displaystyle\mathcal{Z}(\mathcal{P}_{H})=Z_{H}.$ (35) If the number of vertices in the root graph is even, then we have $\displaystyle\mathcal{Z}(\mathcal{P}_{H})=\left\langle Z_{H},P\right\rangle.$ (36) ###### Proof. The proof can be found in Section 6.2. ∎ The Pauli symmetries of the Hamiltonian outside of $\mathcal{Z}(\mathcal{P}_{H})$ may be thought of as “logical" or “gauge" qubits, and this characterization allows for a simple accounting of these qubits. Suppose we express a spin Hamiltonian $H$ on $n$ qubits as a free- fermion Hamiltonian on the hopping graph $R=(\widetilde{V},\widetilde{E})$, and let $|\mathcal{Z}(\mathcal{P}_{H})|$ be the number of independent generators of $\mathcal{Z}(\mathcal{P}_{H})$. The number of logical qubits $n_{L}$ of the model is given by $\displaystyle n_{L}\equiv\begin{cases}n-\left[\frac{1}{2}(|\widetilde{V}|-1)+|\mathcal{Z}(\mathcal{P}_{H})|\right]&|\widetilde{V}|\ \mathrm{odd}\\\ n-\left[\frac{1}{2}(|\widetilde{V}|-2)+|\mathcal{Z}(\mathcal{P}_{H})|\right]&|\widetilde{V}|\ \mathrm{even}\end{cases}.$ (37) This follows from the fact that the $\mathds{F}_{2}$-rank of the adjacency matrix of $G(H)$ is twice the number of qubits spanned by the fermionic degrees of freedom in the model, and also the number of vertices of the root graph $R$ up to a constant shift. $R$ | $L(R)$ ---|--- | | | Table 3: The Whitney isomorphism theorem [45] guarantees that the edge automorphisms exchanging $e$ and $e^{\prime}$ in the graphs $R$ in the left column, or corresponding vertices in their line graphs on the right, which cannot be realized by any vertex automorphism of $R$, are the only such cases. Finally, we note that there may be additional symmetries, such as translation invariance, if the coefficients $h_{\boldsymbol{j}}$ themselves satisfy a symmetry. Our characterization will allow us to say something about this situation when the associated symmetry transformation is a Clifford operator—that is, a unitary operator in the normalizer of the Pauli group—commuting with the Pauli symmetries in $\mathcal{Z}(\mathcal{P}_{H})$, such as, e.g., a spatial translation. The following statement follows from a theorem by Whitney [45] (and extended to infinite graphs in [53]). ###### Corollary 1.2 (Clifford Symmetries and Whitney Isomorphism). Let $\widetilde{H}=i\boldsymbol{\gamma}\cdot\mathbf{h}\cdot\boldsymbol{\gamma}^{\mathrm{T}}$ be a free-fermion Hamiltonian with single-particle Hamiltonian $\mathbf{h}$ in a fixed symmetry sector of $\mathcal{Z}(\mathcal{P}_{H})$. Then any unitary Clifford symmetry $U$ such that $U^{\dagger}\widetilde{H}U=\widetilde{H}$ induces a signed permutation symmetry $\mathbf{u}$ such that $\mathbf{h}=\mathbf{u}^{\mathrm{T}}\cdot\mathbf{h}\cdot\mathbf{u}$, except for when $U$ induces one of the three edge isomorphisms shown in Table 3. ###### Proof. This follows from the Whitney isomorphism theorem: except for the three cases shown in Table 3, any adjacency-preserving permutation of the vertices of $G(\widetilde{H})$ is induced by an adjacency-preserving permutation of the vertices of $R$. A Clifford symmetry $U$ acts as a signed permutation of the Hamiltonian terms which preserves $\widetilde{H}$. Suppose the associated unsigned permutation is not one of the exceptional cases, and so is induced by a permutation $\pi$ on the vertices of $R$. This gives $\displaystyle\widetilde{H}$ $\displaystyle=U^{\dagger}\widetilde{H}U$ (38) $\displaystyle=i\sum_{(j,k)\in\widetilde{E}}h_{jk}\left(U^{\dagger}\gamma_{j}U\right)\left(U^{\dagger}\gamma_{k}U\right)$ (39) $\displaystyle=i\sum_{(j,k)\in\widetilde{E}}(-1)^{x_{j}+x_{k}}h_{jk}\gamma_{\pi(j)}\gamma_{\pi(k)}$ (40) where $x_{j}\in\\{0,1\\}$ designates the sign associated to the permutation of vertex $j\in\widetilde{V}$. By unitarity, this sign must depend on $j$ alone, since $U^{\dagger}\gamma_{j}U$ can only depend on $j$. Let $\mathbf{u}$ be a single-particle transition matrix defined as $\displaystyle u_{jk}=(-1)^{x_{j}}\delta_{k\pi(j)}.$ (41) Then we can reinterpret Eq. (40) in the single-particle picture as $\displaystyle i\boldsymbol{\gamma}\cdot\mathbf{h}\cdot\boldsymbol{\gamma}^{\mathrm{T}}$ $\displaystyle=i\boldsymbol{\gamma}\cdot\left(\mathbf{u}^{\mathrm{T}}\cdot\mathbf{h}\cdot\mathbf{u}\right)\cdot\boldsymbol{\gamma}^{\mathrm{T}}.$ (42) By linear independence, Eq. (42) therefore implies $\displaystyle\mathbf{h}=\mathbf{u}^{\mathrm{T}}\cdot\mathbf{h}\cdot\mathbf{u}$ (43) and the claim follows. ∎ See section 5.1 for a simple example of an exceptional Hamiltonian realizing a frustration graph shown in Table 3. We now complete our characterization of free-fermion solutions by choosing an orientation for every edge in the root graph over a restricted subspace determined by the constants of motion. ### 4.1 Orientation and Full Solution As discussed previously, the Lie-homomorphism condition Eq. (21) does not fully constrain the free-fermion solution of a given Pauli Hamiltonian. This is because we are free to choose a direction to each edge in the root graph by exchanging $\phi_{1}(\boldsymbol{j})\leftrightarrow\phi_{2}(\boldsymbol{j})$, which is equivalent to changing the sign of the term $i\gamma_{\phi_{1}(\boldsymbol{j})}\gamma_{\phi_{2}(\boldsymbol{j})}$ in $\widetilde{H}$ corresponding to $\sigma^{\boldsymbol{j}}$ in $H$. A related ambiguity corresponds to the cycle symmetry subgroup $Z_{H}$: we are free to choose a symmetry sector over which to solve the Hamiltonian $H$ by choosing a mutual $\pm 1$-eigenspace of independent nontrivial generators of this group. It will turn out that both ambiguities are resolved simultaneously. First suppose we have a Hamiltonian $H$ satisfying Eq. (25) for some root graph $R\equiv(\widetilde{V},\widetilde{E})$. Construct a spanning tree $\Upsilon\equiv(\widetilde{V},\widetilde{E}^{\prime})$ of $R$, defined as: ###### Definition 7 (Spanning Tree). Given a connected graph $G\equiv(V,E)$, a spanning tree $\Upsilon\equiv(V,E^{\prime})\subseteq G$ is a connected subgraph of $G$ such that $E^{\prime}$ contains no cycles. This can be performed in linear time in $|\widetilde{V}|$. Designate a particular vertex $v\in\widetilde{V}$ as the root of this tree. Each vertex $u\in\widetilde{V}$ has a unique path $p(u,v)\subseteq\widetilde{E}$ in $\Upsilon$ to the root, the path $p(v,v)$ being empty. Choose an arbitrary direction for each edge in $\widetilde{E}^{\prime}$ (we will see shortly to what extent this choice is important). Our choice of spanning tree determines a basis of _fundamental cycles_ for the binary cycle space of $R$ and thus a generating set of Paulis for the cycle subgroup $Z_{H}$. To see this, note that for each edge $\phi(\boldsymbol{j})\in\widetilde{E}/\widetilde{E}^{\prime}$, there is a unique cycle of $R$ given by $\displaystyle Y_{\boldsymbol{j}}\equiv p[\phi_{1}(\boldsymbol{j}),v]\cup p[\phi_{2}(\boldsymbol{j}),v]\cup\phi(\boldsymbol{j}).$ (44) Let $\sigma^{\boldsymbol{y}(\boldsymbol{j})}$ be the cycle subgroup generator associated to $Y_{\boldsymbol{j}}$, defined by $\displaystyle\boldsymbol{y}(\boldsymbol{j})=\bigoplus_{\\{\boldsymbol{z}|\phi(\boldsymbol{z})\in Y_{\boldsymbol{j}}\\}}\boldsymbol{z}$ (45) such that $\displaystyle\sigma^{\boldsymbol{y}(\boldsymbol{j})}=i^{d}\prod_{\\{\boldsymbol{z}|\phi(\boldsymbol{z})\in Y_{\boldsymbol{j}}\\}}\sigma^{\boldsymbol{z}}$ (46) where $d\in\\{0,1,2,3\\}$ again designates the appropriate phase. The set of such $Y_{\boldsymbol{j}}$ contains $|\widetilde{E}|-|\widetilde{V}|+1$ cycles and forms an independent generating set for all the cycles of $R$ under symmetric difference. The corresponding set of $\sigma^{\boldsymbol{y}(\boldsymbol{j})}$ is therefore an independent generating set of the cycle subgroup up to signs, since individual Pauli operators either commute or anticommute and square to the identity. In a similar fashion as with twin-vertex symmetries, we restrict to a mutual $\pm 1$ eigenspace of the cycle-subgroup generators, designated by a binary string $\boldsymbol{x}\in\\{0,1\\}^{\times|\widetilde{E}|-|\widetilde{V}|+1}$ over the $\boldsymbol{j}$ such that $\phi(\boldsymbol{j})\in\widetilde{E}/\widetilde{E}^{\prime}$. That is, we restrict to the mutual $+1$ eigenspace of the stabilizer group $\displaystyle Z_{H,\boldsymbol{x}}\equiv\bigl{\langle}(-1)^{x_{\boldsymbol{j}}}\sigma^{\boldsymbol{y}(\boldsymbol{j})}\bigr{\rangle}.$ (47) If Eq. (45) gives $\boldsymbol{y}(\boldsymbol{j})=\mathbf{0}$ for any $\boldsymbol{j}$, then we take the corresponding $x_{\boldsymbol{j}}=0$. We then simply choose the direction for the edge $\phi(\boldsymbol{j})$ such that $\displaystyle(-1)^{x_{\boldsymbol{j}}}i^{d}\left[\prod_{\\{\boldsymbol{z}|\phi(\boldsymbol{z})\in Y_{\boldsymbol{j}}\\}}i\gamma_{\phi_{1}(\boldsymbol{z})}\gamma_{\phi_{2}(\boldsymbol{z})}\right]=+I$ (48) where $d$ is as defined in Eq. (46). This ensures that the product of Majorana hopping terms around a fundamental cycle $Y_{\boldsymbol{j}}$ agrees with the corresponding Pauli product over the restricted subspace (i.e. up to equivalencies by stabilizers in the group $Z_{H,\boldsymbol{x}}$). By the Lie- homomorphism constraint Eq. (21), all products of Majorana hopping terms around a cycle of $R$ therefore agree with their corresponding Pauli products over this subspace, and so the multiplication relations of the Paulis are respected by their associated fermion hopping terms up to stabilizer equivalencies. Since we have exactly as many elements $Y_{\boldsymbol{j}}$ in our fundamental cycle basis as undirected edges $\phi(\boldsymbol{j})$, such an orientation can always be chosen. $\sigma^{i}_{1}$\$\sigma^{j}_{2}$ | $I$ | $X$ | $Y$ | $Z$ | 2-qubit frustration graph $L(K_{6})$ ---|---|---|---|---|--- $I$ | $P\equiv i\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}\gamma_{5}\gamma_{6}$ | $i\gamma_{3}\gamma_{5}$ | $i\gamma_{2}\gamma_{5}$ | $i\gamma_{2}\gamma_{3}$ | $X$ | $i\gamma_{4}\gamma_{6}$ | $i\gamma_{1}\gamma_{2}$ | $-i\gamma_{1}\gamma_{3}$ | $i\gamma_{1}\gamma_{5}$ | $Y$ | $-i\gamma_{1}\gamma_{6}$ | $-i\gamma_{2}\gamma_{4}$ | $i\gamma_{3}\gamma_{4}$ | $i\gamma_{4}\gamma_{5}$ | $Z$ | $-i\gamma_{1}\gamma_{4}$ | $i\gamma_{2}\gamma_{6}$ | $-i\gamma_{3}\gamma_{6}$ | $i\gamma_{5}\gamma_{6}$ | Table 4: (Left) Fermionization of the two-qubit Pauli algebra $\mathcal{P}_{2}=\\{\sigma^{i}_{1}\otimes\sigma^{j}_{2}\\}_{(i,j)\neq(0,0)}$ by six fermion modes. The graph isomorphism $G(\mathcal{P}_{2})\simeq L(K_{6})$ reflects the Lie-algebra isomorphism between $\mathfrak{su}(4)$ and $\mathfrak{spin}(6)$. Though the scalar-commutation relations are reproduced by quadratics in $\\{\gamma_{\mu}\\}_{\mu=1}^{6}$, the one-sided multiplication relations are only recovered upon projecting onto the $+1$ eigenspace of $P\equiv i\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}\gamma_{5}\gamma_{6}$ on the fermion side of the mapping. (Right) The graph $L(K_{6})$, with vertices labeled by a particular satisfying Pauli assignment. Edges are colored to identify the six $K_{5}$ subgraphs in the Krausz decomposition of this graph, corresponding to the six fermion modes. Each vertex belongs to exactly two such subgraphs, as must be the case for a line graph. This graphical correspondence was first observed in Ref. [46] in the language of the Dirac algebra. Why are we free then, to choose an arbitrary sign for each free-fermion term in our original spanning tree? This choice is actually equivalent to a choice of signs on the definitions of the individual Majorana modes themselves and so amounts to a choice of orientation for the coordinate basis in which we write $\mathbf{h}$. To see this, choose a fiducial orientation for $R$ satisfying Eq. (48), and suppose our particular free-fermion solution—not necessarily oriented this way—corresponds to the mapping $\displaystyle\sigma^{\boldsymbol{j}}\mapsto i(-1)^{x_{\boldsymbol{j}}}\gamma_{\phi_{1}(\boldsymbol{j})}\gamma_{\phi_{2}(\boldsymbol{j})}$ (49) for $\phi(\boldsymbol{j})\in\widetilde{E}^{\prime}$ and with $x_{\boldsymbol{j}}\in\\{0,1\\}$ designating the edge-direction of $\phi(\boldsymbol{j})$ relative to the fiducial orientation. We have $\displaystyle(-1)^{x_{\boldsymbol{j}}}$ $\displaystyle=(-1)^{r_{\phi_{1}(\boldsymbol{j})}+r_{\phi_{2}(\boldsymbol{j})}}$ (50) where $\displaystyle r_{u}=\sum_{\\{\boldsymbol{k}|\phi(\boldsymbol{k})\in p(u,v)\\}}x_{\boldsymbol{k}}.$ (51) Since the symmetric difference of $p[\phi_{1}(\boldsymbol{j}),v]$ and $p[\phi_{2}(\boldsymbol{j}),v]$ is the edge $\phi(\boldsymbol{j})\in\widetilde{E}^{\prime}$, all sign factors on the right side of Eq. (50) cancel except for $(-1)^{x_{\boldsymbol{j}}}$. We can then absorb $(-1)^{r_{\phi_{1}(\boldsymbol{j})}}$ and $(-1)^{r_{\phi_{2}(\boldsymbol{j})}}$ onto the definitions of $\gamma_{\phi_{1}(\boldsymbol{j})}$ and $\gamma_{\phi_{2}(\boldsymbol{j})}$, respectively. We furthermore see that imposing Eq. (48) gives an edge- direction for $\phi(\boldsymbol{j})\in\widetilde{E}/\widetilde{E}^{\prime}$ that differs from that of the fiducial orientation by the associated sign factor $(-1)^{r_{\phi_{1}(\boldsymbol{j})}+r_{\phi_{2}(\boldsymbol{j})}}$, which remains consistent with a redefinition of the signs on the individual Majorana modes. Letting $\mathbf{h}$ be the single-particle Hamiltonian for the fiducial orientation, such a redefinition corresponds to conjugating $\mathbf{h}$ by a $\pm 1$ diagonal matrix. As no scalar quantity of $\mathbf{h}$ can depend on this choice, this redefinition corresponds to a gauge freedom. Proceeding this way, we can solve the effective Hamiltonian $\displaystyle H_{\boldsymbol{x}}\equiv H\prod_{\\{\boldsymbol{j}|\phi(\boldsymbol{j})\in\widetilde{E}/\widetilde{E}^{\prime}\\}}\left(\frac{I+(-1)^{x_{\boldsymbol{j}}}\sigma^{\boldsymbol{y}(\boldsymbol{j})}}{2}\right)$ (52) sector-by-sector over each stabilizer eigenspace designated by $\boldsymbol{x}$. If we also need to remove twin vertices from $G(H)$ before it is a line graph, we project onto the mutual $+1$ eigenspace of the stabilizer group $\mathcal{S}_{\boldsymbol{x}}$ defined previously in Section 4 as well. Finally, if the parity operator $P$ is trivial in the Pauli description, then only a fixed-parity eigenspace in the fermion description will be physical. In the next section, we will see how known free-fermion solutions fit into this characterization and demonstrate how our method can be used to find new free-fermion solvable models, for which we give an example. ## 5 Examples ### 5.1 Small Systems The frustration graph of single-qubit Paulis ${X,Y,Z}$ is $K_{3}$, the complete graph on three vertices. This graph is the line graph of not one, but two non-isomorphic graphs: the so-called ‘claw’ graph $K_{1,3}$, and $K_{3}$ itself (see Table 1). By the Whitney isomorphism theorem [45], $K_{3}$ is the only graph which is not the line graph of a unique graph. This ambiguity results in the existence of two distinct free-fermion solutions of a single qubit Hamiltonian, which we will hereafter refer to as “even" (labeled “0") and “odd" (labeled “1") fermionizations $\displaystyle\begin{cases}X_{0}=i\gamma_{0}\gamma_{1}&X_{1}=i\gamma_{2}\gamma_{3}\\\ Y_{0}=i\gamma_{1}\gamma_{2}&Y_{1}=i\gamma_{0}\gamma_{3}\\\ Z_{0}=i\gamma_{0}\gamma_{2}&Z_{1}=-i\gamma_{1}\gamma_{3}\end{cases}\mathrm{.}$ (53) In the even fermionization, no T-join of the root graph $K_{3}$ exists since there are only three fermion modes $\\{\gamma_{0},\gamma_{1},\gamma_{2}\\}$. The orientation of the root graph is constrained by the identity $XYZ=iI$. In the odd fermionization, there are four fermion modes $\\{\gamma_{0},\gamma_{1},\gamma_{2},\gamma_{3}\\}$, and so a T-join does exist for the root graph $K_{1,3}$. It is the set of all edges of this graph. The parity operator is trivial in the Pauli description however, and so the constraint $XYZ=iI$ is enforced by restricting to the $+1$ eigenspace of $P\equiv-\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}$ in the fermion description. We are free to choose the orientation of the root graph $K_{1,3}$ however we like in this case, since it contains no cycles, though this choice will affect what we call the physical eigenspace of $P$. By virtue of the line-graph construction and our choice of orientation, both fermionizations respect the single-qubit Pauli multiplication relations, up to stabilizer equivalencies in some cases. We have made the choice to label the Paulis in the two fermionizations in a compatible way, such that $\displaystyle\sigma^{j}_{0}=P\sigma^{j}_{1}$ (54) This gives $\displaystyle[\sigma^{j}_{m},\sigma^{k}_{m}]=2i\varepsilon_{jk\ell}\sigma^{\ell}_{0}\mathrm{.}$ (55) where $m\in\\{0,1\\}$ and $\varepsilon$ is the Levi-Civita tensor. Since an even number of parity-operator factors appear on the left side of Eq. (55), the commutator between two Paulis in either fermionization is always a Pauli in the even fermionization. We can additionally write multiplication relations between the two fermionizations concisely, as $\displaystyle\sigma^{j}_{p}\sigma^{k}_{q}=\delta_{jk}P^{p\oplus q}+i(1-\delta_{jk})\varepsilon_{jk\ell}\sigma_{p\oplus q}^{\ell}\mathrm{.}$ (56) Another exceptional situation arises for 2 qubits, for which the full frustration graph is the line graph of $K_{6}$, again depicted graphically in Table 1. This reveals a free-fermion solution for all 2-qubit Hamiltonians by six fermion modes, listed explicitly in Table 4. We again choose our orientation by picking a spanning tree of the root graph (for example all terms containing the mode $\gamma_{5}$), choosing an arbitrary orientation on this tree, and choosing the remaining orientations by enforcing the condition in Eq. (48). Since $K_{6}$ has an even number of vertices, there exists a T-join for this graph, e.g. the terms $\\{XX,YY,ZZ\\}$. The associated parity operator is trivial however, as $(XX)(YY)(ZZ)=-I$, and so only the $+1$ eigenspace of $P$ in the fermion description will be physical. This solution reflects the exceptional Lie algebra isomorphism, $\mathfrak{su}(4)\simeq\mathfrak{spin}(6)$. Finally, we give an example of a three-qubit Hamiltonian with an exceptional _symmetry_ , namely $\displaystyle H=XII+YII+ZXX+ZZZ$ (57) This Hamiltonian has the frustration graph shown in the top right entry of Table 3, and thus is an exceptional case to Corollary 1.2. A symmetry transformation exchanging $e$ and $e^{\prime}$ for this Hamiltonian is the Hadamard gate applied to the second and third qubits, which exchanges the third and fourth terms, but cannot be realized as any permutation of the individual Majorana modes in its free-fermion description. Figure 1: Frustration graph for the general XY model and its root graph, shown below. Cliques are colored to show the Krausz decomposition, which is the image of the model under the Jordan-Wigner transform. Vertices in the root graph are correspondingly colored, and a spanning tree is highlighted. ### 5.2 1-dimensional chains Shown in Figure 1 is the frustration graph $G(H)$ for the most general nearest-neighbor Pauli Hamiltonian in 1-d (on open boundary conditions) which is mapped to a free-fermion Hamiltonian under the Jordan-Wigner transformation, $\displaystyle H=\sum_{j=1}^{n-1}\sum_{\alpha,\beta\in\\{x,y\\}}\mu^{j}_{\alpha\beta}\sigma_{j}^{\alpha}\otimes\sigma_{j+1}^{\beta}+\sum_{j=1}^{n}\nu_{j}Z_{j}.$ (58) Cliques are colored according to the Krausz decomposition of this graph, which is easily seen by the free-fermion description. The fermion hopping graph, $R$, is shown below. Note that the cycle symmetry subgroup $Z_{H}$ for this model is trivial, as every product of Hamiltonian terms along a cycle in $R$ is the identity. Since the number of vertices in $R$ is even, a T-join does exist, and the parity operator is in-fact $P=Z^{\otimes n}$. Therefore, we have $|\mathcal{Z}(\mathcal{P}_{H})|=1$. An example spanning tree for the root graph is highlighted, taken simply to be the path along edges $(j,j+1)$ from $\gamma_{1}$ to $\gamma_{2n}$. Including any additional edge in this tree will form a cycle. A natural orientation for this tree is to direct every edge from vertex $j+1$ to vertex $j$. Note that we can recover the Jordan-Wigner transformation from this graphical description alone. We first adjoin a single fictitious qubit and a single coupling term to the Hamiltonian, as $\displaystyle H^{\prime}=\mu_{xx}^{0}X_{0}X_{1}+H.$ (59) Since the remaining qubits only couple to qubit “0" along the $X$-direction, all operators in $\mathcal{P}_{H}$ commute on this qubit. Furthermore, this new term adds one vertex to the black clique at the left boundary of the chain in Fig. 1. It thus extends the spanning tree of $R$ by one vertex – which we label $\gamma_{0}$ – due to the fact that this new term only belongs to one clique (so we take its additional Majorana mode to be a clique of size zero). It can be easily verified that products of Hamiltonian terms from this new vertex to any vertex along the chosen spanning tree have the form $\displaystyle\begin{cases}i\gamma_{0}\gamma_{2j-1}\equiv X_{0}\bigotimes_{k=1}^{j-1}Z_{k}\otimes X_{j}&\\\ i\gamma_{0}\gamma_{2j}\equiv X_{0}\bigotimes_{k=1}^{j-1}Z_{k}\otimes Y_{j}&\\\ \end{cases}$ (60) All such operators share $\gamma_{0}$, so their commutation relations are unchanged by truncating $\gamma_{0}$. Furthermore, since all operators in $\mathcal{P}_{H}$ commute on qubit-0, we may truncate this qubit as well without changing the commutation relations of the operators above to obtain the Jordan-Wigner transformation $\displaystyle\begin{cases}\gamma_{2j-1}\equiv\bigotimes_{k=1}^{j-1}Z_{k}\otimes X_{j}&j\ \mathrm{odd}\\\ \gamma_{2j}\equiv\bigotimes_{k=1}^{j-1}Z_{k}\otimes Y_{j}&j\ \mathrm{odd}\\\ \end{cases}$ (61) In principle, a similar trick would work in general, but we find it generally simpler to define Majorana quadratic operators to avoid truncating at a boundary. Our method is especially convenient when considering the case of periodic boundary conditions on this model, wherein we add the boundary term $\displaystyle H_{\mathrm{boundary}}=\sum_{\alpha,\beta\in\\{x,y\\}}\mu^{n}_{\alpha\beta}\sigma_{1}^{\alpha}\otimes\sigma_{n}^{\beta}$ (62) to the Hamiltonian in Eq. (58). With this term included, the model has a nontrivial cycle symmetry given by taking the product of fermion bilinears around the periodic boundary, and this product is also proportional to the parity operator $P=Z^{\otimes n}$ in the spin picture. Adding a boundary term therefore does not change $|\mathcal{Z}(\mathcal{P}_{H})|$, though it does require that we solve the model over each of the eigenspaces of the cycle symmetry independently by choosing the sign of the additional terms in the fermion picture as described in Section 4.1. Let the eigenspace of $Z^{\otimes n}$ be specified by the eigenvalue $(-1)^{p}$. For each associated free- fermion model solution, we must then restrict to the $+1$ eigenspace of the parity operator in the spin picture $\displaystyle\prod_{j=1}^{n}X_{j}X_{j+1}\mapsto(-i)^{n}(-1)^{p}\prod_{k=1}^{2n}\gamma_{k}\mathrm{.}$ (63) where index addition is taken modulo $n$. This ensures that our free-fermionic solution respects the constraint Figure 2: Frustration graph for the Kitaev honeycomb model (left) and its root graph (right). Cliques are colored to show the Krausz decomposition. Interestingly, this model’s root graph is the same as its interaction graph. A spanning tree of the root is again highlighted. $\displaystyle\prod_{j=1}^{n}X_{j}X_{j+1}=I\mathrm{.}$ (64) Notice that solving the two free fermion models together (one for each eigenspace of $Z^{\otimes n}$) gives $2^{n+1}$ eigenstates, yet restricting to a fixed-parity sector in each keeps only $2^{n}$ of them, as required. Finally, we see that this model contains no logical qubits via $\displaystyle n_{L}=n-\left[\frac{1}{2}(2n-2)+1\right]=0$ (65) as we might expect. ### 5.3 The Kitaev honeycomb model Next we consider the Kitaev honeycomb model in two dimensions [12]. This model has the Hamiltonian $\displaystyle H=\sum_{\alpha\in\\{x,y,z\\}}\sum_{\alpha-\mathrm{links}\ j}J^{j}_{\alpha}\sigma^{\alpha}_{j}\sigma^{\alpha}_{j+\hat{\alpha}}$ (66) where each of the $\alpha$ links correspond to one of the compass directions of the edges of a honeycomb lattice. Once again, the frustration graph with shaded cliques according to the Krausz decomposition is shown in Fig. 2. Interestingly, the root graph of this model’s frustration graph is again the honeycomb lattice. By going backwards, we can see that indeed, any free- fermion model with trivalent hopping graph can be embedded in a 2-body qubit Hamiltonian with the same interaction graph. This is because we can find a set of Pauli operators satisfying any frustration graph whose edges can be partitioned into triangles by assigning a different single-qubit Pauli to each of the vertices of every triangle. A term in the Hamiltonian is then the tensor product of all of the Pauli operators from the triangles to which its vertex in $G(H)$ belongs. Unlike in the one-dimensional example, the cycle subgroup of this model, $Z_{H}$, is nontrivial. This subgroup is generated by the products of Hamiltonian terms around a hexagonal plaquette of the honeycomb lattice, denoted $W_{p}$ for plaquette $p$. These cycles are not independent, however, with constraints between them depending on the boundary conditions of the lattice. In particular, if the model is on a torus of dimension $L_{x}$ by $L_{y}$, then the product of all Hamiltonian terms is trivial $\displaystyle\prod_{\boldsymbol{j}\in V}\sigma^{\boldsymbol{j}}=(-1)^{L_{x}L_{y}}I$ (67) In this case, the cycles of the honeycomb lattice are not independent, since they similarly multiply to the identity. There are thus $L_{x}L_{y}-1$ independent plaquettes on the lattice. There are additionally two homotopically nontrivial cycles, which are independent as well. Notice that the edges of the honeycomb lattice itself form a T-join, and so the above constraint is also the statement that $P$ is furthermore trivial. Therefore, we have $\displaystyle|\mathcal{Z}(\mathcal{P}_{H})|=L_{x}L_{y}-1+2=L_{x}L_{y}+1$ (68) and once again (as first computed in Ref. [54]) $\displaystyle n_{L}=2L_{x}L_{y}-\left[\frac{1}{2}\left(2L_{x}L_{y}-2\right)+L_{x}L_{y}+1\right]=0.$ (69) This example also illustrates that quite a large number of symmetries could be present, and in general this will complicate finding, e.g., the symmetry sector that contains the ground state. Figure 3: The frustrated hexagonal gauge 3d color code, proposed in Ref. [44]. This model is based on the 3d gauge color code, whose qubits live on the vertices of the lattice shown. Gauge generators for the 3d gauge color code consist of Pauli-$Z$ and Pauli-$X$ operators around both the square and hexagonal faces of the lattice. Stabilizers of the 3d gauge color code consist of Pauli-$Z$ and Pauli-$X$ operators on both the cube and “ball" cells. The frustrated hexagonal gauge 3D color code is given by taking the stabilizers of the gauge color code together with the hexagonal gauge generators, which commute with the stabilizers, but not with each other. We see from the colored hexagonal faces above that the frustration graph of these gauge generators is a set of disconnected path graphs. Every hexagonal plaquette term anticommutes with exactly two others—the plaquette terms of the other Pauli type intersecting it at exactly one qubit—and commutes with all other terms in the Hamiltonian. ### 5.4 Frustrated Hexagonal Gauge 3D Color Code Figure 4: The Sierpinski-Hanoi model (left) with its frustration graph, highlighted, and its root graph (right) with a spanning tree highlighted, for $k=5$ and local fields absent. Hamiltonian terms are 3-qubit operators acting on qubits at the vertices of the Sierpinski sieve graph, highlighted in blue. Cliques of the frustration graph are colored to show the graph’s Krausz decomposition. Green and orange cells depict generators for the model’s logical Pauli group. At the interior triangular cells of the lattice are the 3-body generators shown in green. At the interior and exterior edges of the model are 2-body generators shown in orange. These are obtained from their adjoining Hamiltonian terms by reflecting the action on the intersection of their supports (so these generators act differently depending on which edge they act on). The frustration graph of this model is the Hanoi graph $H_{3}^{k-1}$. The vertices of this graph are in correspondence to the states of the towers of Hanoi problem with three towers and $k-1$ discs. The root graph of $H_{3}^{k-1}$ contains $H_{3}^{k-2}$ as a topological minor. The frustrated hexagonal gauge 3d color code is a noncommuting Hamiltonian whose terms consist of the stabilizer generators and a subset of the gauge generators from the gauge color code. The gauge color code [55, 56, 57, 58] has a Hamiltonian that is defined in terms of a natural set of gauge generators as $\displaystyle H=-\sum_{S\in\square,\varhexagon}\left(J_{x}\bigotimes_{j\in S}X_{j}+J_{Z}\bigotimes_{j\in S}Z_{j}\right)$ $\displaystyle+\ \text{(boundary terms)}$ (70) where “$\square$" and “$\varhexagon$" denote the sets of square and hexagonal faces on the lattice in Fig. 3, respectively (see Ref. [59] for a detailed description of this lattice). Here the qubits live on the vertices of the lattice. Nontrivial boundary conditions are required to restrict the logical space of this code to a single qubit. We will ignore these boundary conditions and consider only gauge generators in the bulk of the lattice. The stabilizers for this model are given by products of $X$ or $Z$ around every elementary cell, either a cube or a “ball". We consider a model where we partially restore some of these symmetries. Namely, we will consider the cube and balls to be “restored” symmetries of the model, and we will remove the square generators. This leads to the following gauge Hamiltonian that sums over only hexagonal faces, balls, and cubes, $\displaystyle H=-\sum_{S\in\varhexagon,\text{\mancube},{\mathchoice{\includegraphics[height=4.82224pt]{BallCell.png}}{\includegraphics[height=4.82224pt]{BallCell.png}}{\includegraphics[height=3.61664pt]{BallCell.png}}{\includegraphics[height=2.71246pt]{BallCell.png}}}}\left(J_{X}\bigotimes_{j\in S}X_{j}+J_{Z}\bigotimes_{j\in S}Z_{j}\right).$ (71) The cube and ball terms commute with all of the hexagon terms, and so constitute symmetries of the model. Once we fix a sector for these terms, we can solve the remaining model by mapping to free-fermions as follows [44]. In Figure 3, we represent a subsection of the qubit lattice of this code, where qubits live at the tetravalent vertices. Because the cube and ball terms commute with everything, the frustration graph depends only on the hexagonal faces, several of which are colored in Figure 3. We see that some of these faces intersect at exactly one vertex, and so the $X$\- and $Z$-type gauge generators will anticommute on the associated qubit. These intersection patterns only occur in 1D chains along the cardinal axes of the lattice. In particular, every hexagonal face only overlaps with two other hexagonal faces along these chains and otherwise intersects the other faces at an even number of qubits. The frustration graph of this model thus decouples into a set of disconnected paths, which are line graphs, and in fact they are the frustration graph of the XY-model [1] and the 1-d Kitaev wire [60]. A free- fermion mapping therefore exists for this model, and this demonstrates an example of how one might construct subsystem codes with a free-fermion solution to obtain desired spectral properties. In particular, when $|J_{x}|\not=|J_{z}|$ the model in Eq. (71) is gapped. We note that this observation was made previously in Ref. [44] in the context of quantum error correcting codes. Figure 5: Single-Particle spectrum of the Sierpinski-Hanoi model for $k=5$ with an additional local field term present in the symmetry sector for which all cycles are $+1$. Circled are two critical points where excited bands become degenerate. ### 5.5 Sierpinski-Hanoi model Finally, we introduce our own example of a solvable spin model, which was previously unknown to the best of our knowledge. This model consists of 3-body $XYZ$-interaction terms on the shaded cells of the Sierpinski triangle, all with the same orientation, as depicted in Fig. 4. Explicitly, the Hamiltonian for this model is given by $\displaystyle H=\sum_{(i,j,k)\in\color[rgb]{0.63,0.79,0.95}\mbox{\normalsize$\blacktriangle$}}X_{i}Y_{j}Z_{k}+JH_{\text{local}}$ (72) where $(i,j,k)$ is an ordered triple of qubits belonging to a particular shaded cell on the lattice and we will define additional on-site terms $H_{\text{local}}$ in Eq. (77). An instance of the model is parameterized by $k$, the fractal recursion depth of the underlying Sierpinski lattice, where $k=1$ is taken to be a single 3-qubit interaction. Let us first consider the simplified model where $J=0$. Then the frustration graph of this model is the so-called _Hanoi graph_ $H_{3}^{k-1}$. The vertices of this graph are labeled by states of the towers of Hanoi problem with $k-1$ discs, and two vertices are neighboring if transitioning between the corresponding states is an allowed move in the problem. Perhaps surprisingly, this graph is a line graph, with root and highlighted spanning tree shown in Fig. 4. Furthermore, the root graph contains $H_{3}^{k-2}$ as a topological minor, obtained by removing the vertices of degree one and contracting the vertices of degree two each along one of their two edges. This model contains $\displaystyle n=\frac{3}{2}(3^{k-1}+1)$ (73) physical qubits, and its root graph contains $\displaystyle|\widetilde{V}|=\begin{cases}2&k=1\\\ \frac{1}{2}\left[5\times 3^{k-2}+3\right]&k>1\end{cases}$ (74) vertices. A T-join for this graph therefore exists only for even $k$ and $k=1$, and the parity operator is never trivial when a T-join exists. The root graph also contains $\displaystyle|Z_{H}|=\sum_{j=0}^{k-3}3^{j}=\begin{cases}0&k\leq 2\\\ \frac{1}{2}\left(3^{k-2}-1\right)&k>2\end{cases}$ (75) fundamental cycles. None of the generators of the cycle subgroup are trivial since every qubit is acted upon by at most two anti-commuting operators. The number of logical qubits in this model is therefore $\displaystyle n_{L}=\begin{cases}2&k=1\\\ \frac{1}{4}\left[11\times 3^{k-2}+8+(-1)^{k}\right]&k>1,\end{cases}$ (76) and so this Hamiltonian encodes logical qubits at a constant rate of $\frac{11}{18}$ in the infinite $k$ limit. Perhaps unsurprisingly, these logical qubits live on the boundaries of the fractal, and we can obtain a set of generators for the logical Pauli group of this model as shown in Fig. 4. We can encode logical quantum information in this model by picking symplectic pairs of generators from this group, which anticommute with one another yet commute with the remaining generators in the group. The remaining such generators can be used as gauge qubits for the logical qubit we wish to protect, and the free-fermion Hamiltonian of the model can be used for error suppression. We are also free to add an anisotropic local-field term to a subset of the qubits without breaking solvability $\displaystyle H_{\mathrm{local}}=\sum_{i\in\color[rgb]{0.63,0.79,0.95}\mbox{\normalsize$\blacktriangle$}\mathbf{-}\color[rgb]{0.63,0.79,0.95}\mbox{\normalsize$\blacktriangle$}}\sigma_{i}^{j_{i}}$ (77) The sum is taken over all qubits corresponding to _black_ edges connecting two shaded cells in Fig. 4. $j_{i}$ is the third Pauli type from the two Paulis acting on qubit $i$ by the interaction terms. The effect of these local-field terms is to couple every black vertex in the root graph shown in Fig. 4, except for those at the three corners, to a dedicated fermion mode. We do not depict these additional modes to avoid cluttering the figure. These terms also do not affect the symmetries of the model, except possibly to add the parity operator to $\mathcal{Z}(\mathcal{P}_{H})$ when the number of vertices in the original graph was odd, as the number of vertices in the graph with local field terms present will always be even. The parity operator can then be constructed as the product of all of the Hamiltonian terms, since the root graph only has vertices of degrees 1 and 3. In Fig. 5, we display the single-particle spectrum of the Sierpinski-Hanoi model as a function of the local field $J$ for $k=5$ in the sector for which all of the cycle symmetries are in their mutual $+1$ eigenspace. We highlight two critical points where excited energy levels become degenerate to within our numerical precision. We observe that the locations of these points are not system-size-independent, but rather asymptotically approach $J=0$ as the system size is increased. We conjecture that this is connected to the emergence of scale symmetry, which the model possesses in the thermodynamic limit, yet not for any finite size. It would be intriguing if certain physical features of this symmetry could be realized at the critical points at finite size, potentially opening the door to simulating scale-invariant systems on a finite-sized quantum computer. ## 6 Proofs of Main Theorems ### 6.1 Proof of Theorem 1 We restate Theorem 1 for convenience. ###### Theorem 1, restated (Existence of free-fermion solution). An injective map $\phi$ as defined in Eq. (20) and Eq. (21) exists for the Hamiltonian $H$ as defined in Eq. (1) if and only if there exists a root graph $R$ such that $\displaystyle G(H)\simeq L(R),$ (78) where R is the hopping graph of the free-fermion solution. ###### Proof. If $\phi$ exists, define $R=(V,E)$, where $E\equiv\\{(\phi_{1}(\boldsymbol{j}),\phi_{2}(\boldsymbol{j}))|\phi_{1}(\boldsymbol{j}),\phi_{2}(\boldsymbol{j})\in V,\boldsymbol{j}\in E\\}$. If and only if $|(\phi_{1}(\boldsymbol{j}),\phi_{2}(\boldsymbol{j}))\cap(\phi_{1}(\boldsymbol{k}),\phi_{2}(\boldsymbol{k}))|=1$, then the vertices corresponding to $\boldsymbol{j}$ and $\boldsymbol{k}$ are neighboring in $G$ by Eq. (21). Thus, $G(H)\simeq L(R)$ and a mapping $\phi$ exists only if $R$ does. If there exists a graph $R\equiv(V,E)$ such that $G\simeq L(R)$, take the Krausz decomposition of $G(H)$. Namely, partition the edges of $G(H)$ as $F=\\{C_{1},\dots,C_{|V|}\\}$, where each $C_{i}$ constitutes a clique in $G$ and such that every vertex in $G$ appears in at most two $C_{i}$. The cliques in this partitioning correspond to the vertices $V$ of $R$. For each vertex $\boldsymbol{j}$, define $\phi(\boldsymbol{j})$ to be the pair of cliques in which $\boldsymbol{j}$ appears. Since the cliques partition the edges of $G$, then if vertices $\boldsymbol{j}$ and $\boldsymbol{k}$ are neighboring in $G$, they must appear in exactly one clique together, and thus $|(\phi_{1}(\boldsymbol{j}),\phi_{2}(\boldsymbol{j}))\cap(\phi_{1}(\boldsymbol{k}),\phi_{2}(\boldsymbol{k}))|=1$. Thus, $\phi$ satisfies Eq. (21). Furthermore, $\phi$ is injective, since if there are two vertices $\boldsymbol{j}$, $\boldsymbol{k}\in G$ such that $\phi(\boldsymbol{j})=\phi(\boldsymbol{k})$, then $\boldsymbol{j}$ and $\boldsymbol{k}$ appear in the same two cliques, but since the Krausz decomposition is a partition of the edges, this would require that $\boldsymbol{j}$ and $\boldsymbol{k}$ neighbor by two edges. However, the definition of $G$ guarantees that pairs of vertices can only neighbor by at most one edge, and so this is impossible. Therefore $\phi$ is injective. ∎ ### 6.2 Proof of Theorem 2 Once again, we restate our theorem for convenience ###### Theorem 2, restated (Symmetries are cycles and parity). Given a Hamiltonian satisfying Eq. (78) such that the number of vertices $|\widetilde{V}|$ in the root graph is odd, then we have $\displaystyle\mathcal{Z}(\mathcal{P}_{H})=Z_{H}.$ (79) If the number of vertices in the root graph is even, then we have $\displaystyle\mathcal{Z}(\mathcal{P}_{H})=\left\langle Z_{H},P\right\rangle.$ (80) _Proof._ Let $G\equiv(E,F)\simeq L(R)$ be the connected line graph of a connected root graph $R=(V,E)$, and let $G$ have adjacency matrix $\mathbf{A}$. We will need the following well-known factorization of a line graph adjacency matrix $\mathbf{A}$ $\displaystyle\mathbf{A}=\mathbf{B}\mathbf{B}^{\mathrm{T}}\ \mathrm{(mod\ 2)}$ (81) where $\mathbf{B}$ is the edge-vertex incidence matrix of $R$. That is, $\mathbf{B}$ is a $|E|\times|V|$ matrix such that $\displaystyle B_{\boldsymbol{j}l}=\begin{cases}1&l\in\boldsymbol{j}\\\ 0&\mathrm{otherwise}\end{cases}$ (82) for all $\boldsymbol{j}\in E$ and $l\in V$. We can interpret $\mathbf{B}$ as defining the map $\phi$ via $\displaystyle\phi:\sigma^{\boldsymbol{j}}\mapsto\prod_{l\in V}\gamma_{l}^{B_{\boldsymbol{j}l}}$ (83) That is, $\phi_{1}(\boldsymbol{j})$ and $\phi_{2}(\boldsymbol{j})$ are the indices of the nonzero elements in the row labeled by $\boldsymbol{j}$ in $\mathbf{B}$. This then defines the adjacency matrix $\mathbf{A}$ through the scalar commutator as $\displaystyle[\\![\prod_{l\in V}\gamma_{l}^{B_{\boldsymbol{j}l}},\prod_{m\in V}\gamma_{m}^{B_{\boldsymbol{k}l}}]\\!]$ $\displaystyle=\prod_{l,m\in V}[\\![\gamma_{l}^{B_{\boldsymbol{j}l}},\gamma_{m}^{B_{\boldsymbol{k}m}}]\\!]$ (84) $\displaystyle=\prod_{l,m\in V}(-1)^{(1-\delta_{lm})B_{\boldsymbol{j}l}B_{\boldsymbol{k}m}}$ (85) $\displaystyle[\\![\prod_{l\in V}\gamma_{l}^{B_{\boldsymbol{j}l}},\prod_{m\in V}\gamma_{m}^{B_{\boldsymbol{k}l}}]\\!]$ $\displaystyle=(-1)^{(\mathbf{B}\mathbf{B}^{\mathrm{T}})_{\boldsymbol{j}\boldsymbol{k}}+\left(\sum_{l}B_{\boldsymbol{j}l}\right)\left(\sum_{l}B_{\boldsymbol{k}l}\right)}$ (86) $\displaystyle(-1)^{A_{\boldsymbol{j}\boldsymbol{k}}}$ $\displaystyle=(-1)^{(\mathbf{B}\mathbf{B}^{\mathrm{T}})_{\boldsymbol{j}\boldsymbol{k}}}$ (87) From the third to the fourth line, we replaced the left-hand side with the definition of $\mathbf{A}$ and used the fact that the rows of $\mathbf{B}$ have exactly two nonzero elements. By the distributive property of the scalar commutator Eq. (4), we can extend the above equation to products of Hamiltonian terms $\displaystyle\prod_{\boldsymbol{j}\in E}\left(\prod_{l\in V}\gamma_{l}^{B_{\boldsymbol{j}l}}\right)^{v_{\boldsymbol{j}}}=\pm\prod_{l\in V}\gamma_{l}^{\left(\mathbf{B}^{\mathrm{T}}\cdot\mathbf{v}\right)_{l}}\mathrm{,}$ (88) where $\mathbf{v}\in\\{0,1\\}^{\times|E|}$, as $\displaystyle[\\![\prod_{l\in V}\gamma_{l}^{B_{\boldsymbol{j}l}},\prod_{\boldsymbol{k}\in E}\left(\prod_{m\in V}\gamma_{m}^{B_{\boldsymbol{k}m}}\right)^{v_{\boldsymbol{k}}}]\\!]=(-1)^{\left(\mathbf{B}\mathbf{B}^{\mathrm{T}}\cdot\mathbf{v}\right)_{\boldsymbol{j}}}$ (89) since linear combinations of rows of $\mathbf{B}$ over $\mathds{F}_{2}$ will have even-many ones. Every element of $\mathcal{P}_{H}$ is a (non-unique) linear combination of rows of $\mathbf{B}$ over $\mathds{F}_{2}$, and so to characterize the elements of $\mathcal{Z}(\mathcal{P}_{H})$, it is sufficient to find a spanning set of the kernel of $\mathbf{A}$, $\displaystyle\mathbf{A}\cdot\mathbf{v}=\mathbf{B}\mathbf{B}^{\mathrm{T}}\cdot\mathbf{v}=\mathbf{0}\ \mathrm{(mod\ 2)}$ (90) It is again well-known that the $\mathds{F}_{2}$-kernel of $\mathbf{B}^{\mathrm{T}}$ is the cycle space of $R$, and this specifies the cycle subgroup $Z_{H}$ as being contained in $\mathcal{Z}(\mathcal{P}_{H})$. All that is left is therefore to find all $\mathbf{v}$ such that $\mathbf{B}^{\mathrm{T}}\cdot\mathbf{v}$ is in the kernel of $\mathbf{B}$. Since we have assumed $G$ is connected, it is easy to see that the only element in this kernel is $\mathbf{1}$, the all-ones vector. Thus, $\mathbf{v}$ will also be in the kernel of $\mathbf{A}$ if it defines a T-join of $G$. If $|V|$ is even, then we can construct a T-join by first pairing the vertices along paths of $G$. We can then ensure that each edge appears at most once in the T-join by taking the symmetric difference of all paths. If $|V|$ is odd, then no T-join exists. Indeed, assume that a T-join $T$ does exist for $|V|$ odd, and let $\widetilde{G}=(V,T)\subseteq G$ be the subgraph of $G$ containing exactly the edges from the T-join. By construction $\widetilde{G}$ contains all the vertices of $G$ and has odd degree for every vertex, though it may no longer be connected. Let these degrees be $\\{d_{j}\\}_{j\in V}$, then by the handshaking lemma $\displaystyle\sum_{j=1}^{|V|}d_{j}=2|T|\mathrm{.}$ (91) However, the left side must be odd since we have assumed the degree of every vertex in $\widetilde{G}$ is odd, and the number of vertices is also odd, and so we have a contradiction. ∎ ## 7 Discussion We have seen how the tools of graph theory can be leveraged to solve a wide class of spin models via mapping to free fermions, and given an explicit procedure for constructing the free-fermion solution when one exists. A major remaining open question, however, concerns the characterization of free- fermion solutions beyond the generator-to-generator mappings we consider here. That is, if $G(H)$ is not a line graph and no removal of twin vertices will make it so, then it may _still_ be possible for a free-fermion solution for $H$ to exist thanks to the continuum of locally equivalent Pauli-bases into which $H$ may be expanded. Our fundamental theorem does not rule out the possibility that special such bases may exist. The problem of finding such bases is equivalent to finding specific unitary rotations of $H$ for which the $G(H)$ again becomes a line graph. These rotations must be outside of the Clifford group, since the frustration graph is a Clifford invariant. Their existence may therefore depend on specific algebraic relationships between the Pauli coefficients $h_{\boldsymbol{j}}$ in the Hamiltonian, since the existence of a free-fermionization is a spectral invariant. We expect such transformations will be hard to find in general, though perhaps progress can be made for single-qubit rotations on 2-local Hamiltonians in a similar vein as in Ref. [61] for stoquasticity. Recently, a local spin-$\nicefrac{{1}}{{2}}$ model with a free-fermion solution – despite no such generator-to-generator solution existing – has been found in Ref. [62]. An investigation of models which may be fermionized by these more general transformations is therefore an interesting subject of future work. It is natural to ask whether our results could have implications for simulating quantum systems and quantum computation. We expect our characterization to shed some light on the inverse problem of finding fermion- to-qubit mappings, such as the Bravyi-Kitaev superfast encoding [63], Bravyi- Kitaev transform [64], and generalized superfast encoding [65]. It is possible to achieve further encodings by introducing ancillary fermion modes, as seen in the Verstraete-Cirac mapping [7] and the contemporaneous mapping introduced by Ball [66]. Encodings can be further improved through tailoring to specific symmetries [67], connectivity structures [68, 69], and through the application of Fenwick trees [70]. Recently, a treelike mapping was shown to achieve optimal average-case Pauli-weight in Ref [71]. In their “Discussion” section, the authors remark that an interesting future direction for their work would involve introducing ancillary qubits to their mapping. We expect our classification of the symmetries of free-fermion spin models to help guide this investigation, though further work is required to fully characterize the logical symmetry groups which can be realized by these models. Finally, our characterization highlights the possibility of a “free-fermion rank" for Hamiltonians as an important measure of classical simulability. Namely, if there is no free-fermion solution for a given Hamiltonian, we can still group terms into collections such that each collection independently has such a solution. 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2024-09-04T02:54:59.341499

2003.05485

arxiv-papers

# A Non-Iterative Transformation Method for a Class of Free Boundary Value Problems Governed by ODEs Riccardo Fazio111Corresponding author: e-mail<EMAIL_ADDRESS>home-page: http://mat521.unime.it/fazio and Salvatore Iacono Department of Mathematics and Computer Science University of Messina Viale F. Stagno D’Alcontres 31, 98166 Messina, Italy ###### Abstract The aim of this work is to point out that the class of free boundary problems governed by second order autonomous ordinary differential equations can be transformed to initial value problems. Interest in the numerical solution of free boundary problems arises because these are always nonlinear problems. The theoretical content of this paper is original: results already available in literature are related to the invariance properties of scaling or spiral groups of point transformations but here we show how it is also possible to use e invariance properties of a translation group. We test the proposed algorithm by solving three problems: a problem describing a rope configuration against an obstacle, a dynamical problem with a nonlinear force, and a problem related to the optimal length estimate for tubular flow reactors. Key Words. Ordinary differential equations, free boundary problems, initial value methods, non-iterative transformation method, translation group of point transformations. AMS Subject Classifications. 65L10, 65L99, 34B15, 34B99. ## 1 Introduction Free boundary value problems (BVPs) occur in all branches of applied mathematics and science. The oldest problem of this type was formulated by Isaac Newton, in book II of his great “Principia Mathematica” of 1687, by considering the optimal nose-cone shape for the motion of a projectile subject to air resistance, see Edwards [1] or Fazio [2]. In the classical numerical treatment of a free BVP a preliminary reduction to a BVP is introduced by considering a new independent variable; see, Stoer and Bulirsch [3, p. 468], Ascher and Russell [4], or Ascher, Mattheij and Russell [5, p. 471]. By rewriting a free BVP as a BVP it becomes evident that the former is always a nonlinear problem; the first to point out the nonlinearity of free BVPs was Landau [6]. Therefore, in that way free BVPs are BVPs. In this paper we show that free BVPs invariant with respect to a translation group can be solved non-iteratively by solving a related initial value problem (IVP). Therefore in this way those free BVPs are indeed IVPs. Moreover, we are able to characterize a class of free BVPs that can be solved non-iteratively by solving related IVPs. The non-iterative numerical solution of BVPs is a subject of past and current research. Several different strategies are available in literature for the non-iterative solution of BVPs: superposition [5, pp. 135-145], chasing [7, pp. 30-51], and adjoint operators method [7, pp. 52-69] that can be applied only to linear models; parameter differentiation [7, pp. 233-288] and invariant imbedding [8] can be applied also to nonlinear problems. In this context transformation methods (TMs) are founded on group invariance theory, see Bluman and Cole [9], Dresner [10], or Bluman and Kumei [11]. These methods are initial value methods because they achieve the numerical solution of BVPs through the solution of related IVPs. The first application of a non-iterative TM was given by Töpfer in [12] for the Blasius problem, without any consideration of group invariance theory. Töpfer’s algorithm is quoted in several books on fluid dynamics, see, for instance, Goldstein [13, pp. 135-136]. Acrivos, Shah and Petersen [14] first and Klamkin [15] later extended Töpfer’s method respectively to a more general problem and to a class of problems. Along the lines of the work of Klamkin, for a given problem Na [16, 17] showed the relation between the invariance properties, with respect to a linear group of transformation (the scaling group), and the applicability of a non-iterative TM. Na and Tang [18] proposed a non-iterative TM based on the spiral group and applied it to a non-linear heat generation model. Belford [19] first, and Ames and Adams [20, 21] later defined non-iterative TMs for eigenvalue problems. A review paper was written by Klamkin [22]. Extensions of non-iterative TM, by requiring the invariance of one and of two or more physical parameters when they are involved in the mathematical model, were respectively proposed by Na [23] and by Scott, Rinschler and Na [24]; see also Na [7, Chapters 8 and 9]. A survey book, written by Na [7, Chs 7-9] on the numerical solution of BVP, devoted three chapters to numerical TMs. As far as free BVPs are concerned, non-iterative and iterative TMs were proposed by Fazio and Evans [25]. Fazio [26] has shown that we can extend the applicability of non-iterative TMs by rewriting a given free BVP using a variables transformation obtained by linking two different invariant groups. However, non-iterative TMs are applicable only to particular classes of BVPs so that they have been considered as ad hoc methods, see Meyer [8, pp. 35-36], Na [7, p. 137] or Sachdev [27, p. 218]. The transformation of BVPs to IVPs has also a theoretical relevance. In fact, existence and uniqueness results can be obtained as a consequence of the invariance properties. For instance, for the Blasius problem, a simple existence and uniqueness theorem was given by J. Serrin [28] as reported by Meyer [29, pp. 104-105] or Hastings and McLeod [30, pp. 151-153]. Moreover, using scaling invariance properties the error analysis of the truncated boundary formulation of the Blasius problem was developed by Rubel [31]. On this topic a first application of a numerical test, defined within group invariance theory, to verify the existence and uniqueness of the solution of a free BVPs was considered by Fazio in [32]. A formal definition of the mentioned numerical test can be found in [33]. In this paper we consider the class of free BVPs governed by second order autonomous differential equations, and define, for these problems, a non- iterative TM using the invariance properties of a translation group. As far as applications of the proposed algorithm are concerned, we solve three problems. First we test our method with a problem describing a rope configuration against an obstacle, where we compare the obtained numerical results with the exact solution. Then we solve a dynamical problem with a nonlinear force, and a problem related to the optimal length estimate for tubular flow reactors, where in both cases our results are compared to numerical data available in literature. Finally, the last section is concerned with concluding remarks pointing out limitations and possible extensions of the proposed approach. ## 2 The non-iterative TM Let us consider the class of second order free BVPs given by $\displaystyle\frac{d^{2}u}{dx^{2}}=\Omega\left(u,\frac{du}{dx}\right)\ ,\qquad x\in(0,s)$ (2.1) $\displaystyle A_{1}u(0)+A_{2}\frac{du}{dx}(0)=A_{3}\ ,$ (2.2) $\displaystyle u(s)=B\quad,\quad\frac{du}{dx}(s)=C\ ,$ (2.3) where $A_{i}$, for $i=1,2,3$, $B$ and $C$ are arbitrary constants, and $s>0$ is an unknown free boundary. The differential equation (2.1) and the two free boundary conditions (2.3) are invariant with respect to the translation group $\displaystyle x^{*}=x+\mu\quad;\quad s^{*}=s+\mu\quad;\quad u^{*}=u\ .$ (2.4) By using this invariance, we can define the following non-iterative algorithm for the numerical solution of (2.1)-(2.3): * • we fix freely a value of $s^{*}$; * • we integrate backwards from $s^{*}$ to $x_{0}^{*}$ the following auxiliary IVP $\displaystyle\frac{d^{2}u^{*}}{dx^{*2}}=\Omega\left(u^{*},\frac{du^{*}}{dx^{*}}\right)$ $\displaystyle u^{*}(s^{*})=B\ ,\qquad\frac{du^{*}}{dx^{*}}(s^{*})=C\ ,$ using an event locator in order to find $x_{0}^{*}$ such that $A_{1}u^{*}(x_{0}^{*})+A_{2}\frac{du^{*}}{dx^{*}}(x_{0}^{*})=A_{3}\ ;$ (2.6) * • finally, through the invariance property, we can deduce the similarity parameter $\mu=x_{0}^{*}\ ,$ (2.7) from which we get the unknown free boundary $s=s^{*}-\mu\ .$ (2.8) The missing initial conditions are given by $u(0)=u^{*}(x_{0}^{*})\ ,\qquad\frac{du}{dx}(0)=\frac{du^{*}}{dx^{*}}(x_{0}^{*})\ .$ (2.9) Let us define now a simple event locator suited to the class of problems (2.1)-(2.3). We consider first the case where $A_{1}u^{*}(s^{*})+A_{2}\frac{du^{*}}{dx^{*}}(s^{*})<A_{3}\ .$ (2.10) We can integrate the auxiliary IVP (• ‣ 2) with a constant step size $\Delta x^{*}$ until at a given mesh point $x_{k}^{*}$ we get $A_{1}u^{*}(x_{k}^{*})+A_{2}\frac{du^{*}}{dx^{*}}(x_{k}^{*})>A_{3}\ ,$ (2.11) and repeat the last step with the smaller step size $\Delta x_{0}^{*}=\Delta x^{*}\frac{\displaystyle A_{3}-A_{1}u^{*}(x_{k-1}^{*})-A_{2}\frac{du^{*}}{dx^{*}}(x_{k-1}^{*})}{\displaystyle A_{1}u^{*}(x_{k}^{*})+A_{2}\frac{du^{*}}{dx^{*}}(x_{k}^{*})-A_{1}u^{*}(x_{k-1}^{*})-A_{2}\frac{du^{*}}{dx^{*}}(x_{k-1}^{*})}\ .$ (2.12) In defining the last step size in equation (2.12) we use a linear interpolation. As a consequence, we have that $x_{0}^{*}\approx x_{k}^{*}-\Delta x_{0}^{*}$. We notice that the condition imposed by this event locator converges to the correct condition (2.6) as the step size goes to zero, cf. the second column of table 2. The other case $A_{1}u^{*}(s^{*})+A_{2}\frac{du^{*}}{dx^{*}}(s^{*})>A_{3}\ ,$ (2.13) can be treated in a similar way. Of course, also in this second case the last step size is smaller than the previous ones. In the next section we apply the proposed non-iterative TM to three problems. The reported numerical results were computed by the classical fourth-order Runge-Kutta’s method, reported by Butcher [34, p. 166], coupled with the event locator defined above. ## 3 The obstacle problem on a string For the obstacle problem on a string, depicted on figure 1 within the $(x,u)$-plane where the $x$ axis is taken overlying to the obstacle, we have to consider the following mathematical model, see Collatz [35] or Glashoff and Werner, [36] $\displaystyle\frac{d^{2}u}{dx^{2}}=\theta\left[1+\left(\frac{du}{dx}\right)^{2}\right]^{1/2}\ ,\qquad x\in(0,s)$ $\displaystyle u(0)=u_{0}\ ,\qquad u(s)=\frac{du}{dx}(s)=0\ ,$ where the positive value of $\theta$ depends on the string properties. In this problem we have to find the position of a uniform string of finite length $L$ under the action of gravity. The string has fixed end points, say $(0,u_{0})$ and $(b,0)$, where $u_{0}>0$ and $b>0$. Furthermore, we assume that the condition $L^{2}>\left(u_{0}^{2}+b^{2}\right)$ is fulfilled; this condition allows us to define a free boundary $s$ for this problem, where $s$ is the detached rope position from the obstacle. The free BVP (3) was solved by the first author in [37] by iterative methods, namely a shooting method and the iterative extension of the TM derived by using the invariance with respect to a scaling group. The exact solution of the free BVP (3) is given by $\displaystyle u(x)=\theta^{-1}\left[\cosh\left(\theta\left(x-s\right)\right)-1\right]\ ,$ (3.2) $\displaystyle s=\theta^{-1}\ln\left[\theta u_{0}+1+\left(\left(\theta u_{0}+1\right)^{2}-1\right)^{1/2}\right]\ ,$ from this we easily find $\frac{du}{dx}(0)=\sinh(-\theta s)=\frac{1}{2}\left(e^{-\theta s}-e^{\theta s}\right)\ ,$ (3.3) and, therefore, for $\theta=0.1$ and $u_{0}=1$ from equations (3)-(3.3) we get the values $s=4.356825433\ ,\qquad\frac{du}{dx}(0)=-0.458257569\ ,$ (3.4) that are correct to the ninth decimals. Table 1: Convergence of numerical results for $\theta=0.1$. $\Delta x$ | $\frac{du}{dx}(0)$ | $e_{r}$ | $s$ | $e_{r}$ ---|---|---|---|--- $-0.1$ | $-0.458227362$ | $6.59\mbox{D}-05$ | $4.435407932$ | $6.19\mbox{D}-05$ $-0.05$ | $-0.458250809$ | $1.47\mbox{D}-05$ | $4.435621088$ | $1.39\mbox{D}-05$ $-0.025$ | $-0.458255551$ | $4.40\mbox{D}-06$ | $4.435664194$ | $4.14\mbox{D}-06$ $-0.0125$ | $-0.458257313$ | $5.59\mbox{D}-07$ | $4.435680211$ | $5.26\mbox{D}-07$ $-0.00625$ | $-0.458257463$ | $2.31\mbox{D}-07$ | $4.435681576$ | $2.18\mbox{D}-07$ $-0.003125$ | $-0.458257538$ | $6.74\mbox{D}-08$ | $4.435682258$ | $6.43\mbox{D}-08$ $-0.0015625$ | $-0.458257565$ | $8.52\mbox{D}-09$ | $4.435682504$ | $9.05\mbox{D}-09$ Let us consider a convergence numerical test for our non-iterative TM. Table 1 reports the obtained numerical results for the missing initial condition and the free boundary value for the free BVP (3) with $\theta=0.1$ and $u_{0}=1$, as well as the corresponding relative errors denoted by $e_{r}$. An example of the numerical solutions is shown in figure 1. Figure 1: Picture of the numerical solution obtained for $\theta=0.1$, $u_{0}=1$, and $b=4.5$; the obstacle, is superimposed to the $x$ axis, and is displayed by a black solid line. For this numerical solution, we applied a large step size in order to show the mesh used and to empathize how our event locator reduces the last step. ## 4 A dynamical free BVP Suppose a particle of unitary mass is moving against a nonlinear force, given by $-1-u-\left(\frac{du}{dx}\right)^{2}$, from the origin $u=0$ to a final position $u=1$, our goal is to determine the duration of motion $s$ and the initial velocity that assures that the particle is momentarily at rest at $u=1$; see Meyer [8, pp. 97-99]. This problem can be formulated as follows $\displaystyle\frac{d^{2}u}{dx^{2}}=-1-u-\left(\frac{du}{dx}\right)^{2}\ ,\qquad x\in(0,s)$ $\displaystyle u(0)=0\ ,\qquad u(s)=1\ ,\qquad\frac{du}{dx}(s)=0\ ,$ where $u$ and $x$ are the particle position and the time variable, respectively, on the right hand side of the governing differential equation we have the nonlinear force acting on the particle and $s$ is the free boundary. In table 2 we propose a numerical convergence test for decreasing values of the step size. Table 2: Convergence of numerical results for the free BVP dynamical model. $\Delta x$ | $u(0)$ | $\frac{du}{dx}(0)$ | $s$ ---|---|---|--- $-0.1$ | $1.16\mbox{D}-02$ | $3.212263787$ | $0.867662139$ $-0.05$ | $3.54\mbox{D}-03$ | $3.240676696$ | $0.870143219$ $-0.025$ | $4.61\mbox{D}-04$ | $3.2516023692$ | $0.871089372$ $-0.0125$ | $1.90\mbox{D}-04$ | $3.252564659$ | $0.871172452$ $-0.00625$ | $5.42\mbox{D}-05$ | $3.253049203$ | $0.871214290$ $-0.003125$ | $9.25\mbox{D}-06$ | $3.253209165$ | $0.871228100$ $-0.0015625$ | $3.43\mbox{D}-06$ | $3.253229900$ | $0.871229890$ $-0.00078125$ | $5.12\mbox{D}-07$ | $3.253240276$ | $0.871230785$ $-0.000390625$ | $2.01\mbox{D}-07$ | $3.253241381$ | $0.871230881$ $-0.0001953125$ | $4.62\mbox{D}-08$ | $3.253241934$ | $0.871230929$ The obtained results can be contrasted with those reported by Meyer [8, pp. 97-99] where, by using the invariant imbedding method, he found $s=1.2651$ but a value of $u(0)=0.0163$ instead of $u(0)=0$ as prescribed by the free BVP (4). The behaviour of the solution can be seen in the figure 2. Figure 2: Numerical solution for the dynamical free BVP (4) obtained with $\Delta x=0.0125$. Again we applied a large step size in order to show how our event locator reduces the last step. A free BVP similar to (4) was considered by Na [7, p. 88] where the nonlinear force was replaced by $-u\exp(-u)$. However, in this case it is possible to prove [33], using the conservation of energy principle, that the free BVP has countable infinite many solutions, with the missing initial conditions given by $\frac{du}{dx}(0)=\pm 0.726967811\ .$ (4.2) ## 5 Length estimation for tubular flow reactors Roughly speaking, a tubular flow chemical reactor can be seen as a device where on one side it is introduced a material A that along its passage inside the reactor undergoes a chemical reaction so that at the exit we get a product B plus a residual part of A; see figure 3. A $n$th order chemical reactor is usually indicated with the notation A${}^{n}\rightarrow$ B. Figure 3: Schematic set-up of a tubular flow reactor. A free BVP for a tubular reactor can be formulated as $\displaystyle\frac{d^{2}u}{dx^{2}}=N_{Pe}\left(\frac{du}{dx}+Ru^{n}\right)\ ,$ $\displaystyle u(0)-\frac{1}{N_{Pe}}\frac{du}{dx}(0)=1\ ,\qquad u(s)=\tau\ ,\quad\frac{du}{dx}(s)=0\ ,$ where $u(x)$ is the ratio between the concentration of the reactant A at a distance $x$ and the concentration of it at $x=0$, $N_{Pe}$, $R$, $n$ and $\tau$ are, the Peclet group, the reaction rate group, the order of the chemical reaction and the residual fraction of reactant A at exit, respectively. Moreover, $N_{Pe}$ and $R$ are both greater than zero. Finally, for the free BVP (5), the free boundary $s$ is the length of the flow reactor we are trying to estimate. This is an engineering problem that consists in determining the optimal length of a tubular flow chemical reactor with axial missing and has been already treated by Fazio in [32], through an iterative TM, whereas Fazio in [38] made a comparison between the results obtained with a shooting method and the upper bound of the free boundary value obtained by a non-iterative TM. Here, for the sake of comparing the numerical results, we fix the parameters as follows: $N_{Pe}=6$, $R=2$, $n=2$, and $\tau=0.1$. We apply the algorithm outlined above to the numerical solution of the free BVP (5). Table 3 shows a numerical convergence test for decreasing values of the step size. Table 3: Convergence of numerical results. $N_{pe}=6$, $R=2$, $n=2$, and $\tau=0.1$. $\Delta x$ | $u(0)$ | $\frac{du}{dx}(0)$ | $s$ ---|---|---|--- $-0.1$ | $0.829314641$ | $-1.008175212$ | $5.117905669$ $-0.05$ | $0.830537187$ | $-1.010745699$ | $5.119104349$ $-0.025$ | $0.831147822$ | $-1.012077034$ | $5.119707352$ $-0.0125$ | $0.831227636$ | $-1.012251496$ | $5.119786158$ $-0.00625$ | $0.831267467$ | $-1.012338738$ | $5.119825502$ $-0.003125$ | $0.831271635$ | $-1.012347868$ | $5.119829619$ $-0.0015625$ | $0.831273719$ | $-1.012352436$ | $5.119831678$ $-0.00078125$ | $0.831274182$ | $-1.012353449$ | $5.119832135$ $-0.000390625$ | $0.831274327$ | $-1.012353767$ | $5.119832278$ $-0.0001953125$ | $0.831274348$ | $-1.012353814$ | $5.119832299$ The obtained numerical results are reported on table 4 and compared with numerical results available in literature. Table 4: Comparison of numerical results for the tubular flow reactor model. | $u(0)$ | $\frac{du}{dx}(0)$ | $s$ ---|---|---|--- iterative TM [32] | $0.831280$ | $-1.012298$ | $5.121648$ shooting method [38] | $0.831274$ | $-1.012354$ | $5.119832$ non-iterative TM | $0.831274$ | $-1.012354$ | $5.119832$ As it is easily seen the computed values are in good agreement with the ones found in [32] and [38]. The behaviour of the solution can be seen in figure 4. Figure 4: Numerical solution for length estimation of a tubular flow reactor obtained with $\Delta x=0.1$. Once again, we used a large step size to make clear how our event locator reduces the last step size. ## 6 Conclusion In closing, we can outline some further implications coming out from this work. First of all, the algorithm proposed in this paper can be extended to free BVPs governed by a system of first order autonomous differential equations belonging to the general class of problems $\displaystyle{\displaystyle\frac{d{\bf u}}{dx}}={\bf q}\left({\bf u}\right)\ ,\quad x\in[0,\infty)\ ,$ $\displaystyle u_{j}(0)=u_{j0}\ ,\qquad{\bf u}(s)={\bf u}_{s}\ ,$ where ${\bf u}(x),{\bf u}_{s}\in\hbox{I\kern-1.99997pt\hbox{R}}^{d}$, ${\bf q}:\hbox{I\kern-1.99997pt\hbox{R}}^{d}\rightarrow~{}\hbox{I\kern-1.99997pt\hbox{R}}^{d}$, with $d\geq 1$, $j\in\\{1,\dots,d\\}$, $u_{j0}$ and all components of ${\bf u}_{s}$ are given constants and $s$ is the free boundary. Moreover, our algorithm can be applied by using an integrator from the MATLAB ODE suite written by Samphine and Reichelt [39], and available with the latest releases of MATLAB, with the event locator option command set in options = odeset(’Events’,@name) where “name” is an external, problem dependent, event function. As mentioned in the introduction, the first application of a non-iterative TM was defined by Töpfer in [12] more than a century ago. In this paper, by considering the invariance with respect to a translation group, we have investigated a possible way to solve a large class of free BVPs by a non- iterative TM. However, it is a simple matter to show a differential equation not admitting any group of transformations: e.g. the differential equation considered by Bianchi [40, pp. 470-475]. Consequently, it is easy to realize that non- iterative TMs cannot be extended to every BVPs. Therefore, non-iterative TMs are ad hoc methods. Their applicability depends on the invariance properties of the governing differential equation and the given boundary conditions. On the other hand, free BVPs governed by the most general second order differential equation, in normal form, can be solved iteratively by extending a scaling group via the introduction of a numerical parameter so as to recover the original problem as the introduced parameter goes to one, see Fazio [25, 26, 33, 41]. The extension of this iterative TM to problems in boundary layer theory has been considered in [42, 43, 44, 45]. Moreover, a further extension to the sequence of free BVPs obtained by a semi-discretization of parabolic moving boundary problems was repoted in [46]. ## References * [1] C.H. Edwards. Newton’s nose-cone problem. Mathematica J., 7:64–71, 1997. * [2] R. Fazio. A non-iterative transformation method for newton’s free boundary problem. Int. J. Non-Linear Mech., 59:23–27, 2014. * [3] J. Stoer and R. Bulirsch. Introduction to Numerical Analysis. Springer-Verlag, Berlin, 1980. * [4] U. M. Ascher and R. D. Russell. 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2024-09-04T02:54:59.350369

2003.05489

arxiv-papers

SWIFT: Scalable Ultra-Wideband Sub-Nanosecond Wavelength Switching for Data Centre Networks Thomas Gerard*, Christopher Parsonson, Zacharaya Shabka, Polina Bayvel, Domaniç Lavery and Georgios Zervas Optical Networks Group, Dept. of Electronic and Electrical Engineering, University College London, London, UK, WC1E 7JE. <EMAIL_ADDRESS> ###### Abstract We propose a time-multiplexed DS-DBR/SOA-gated system to deliver low-power fast tuning across S-/C-/L-bands. Sub-ns switching is demonstrated, supporting 122$\times$50 GHz channels over 6.05 THz using AI techniques. OCIS codes: 140.3600 Lasers, tunable, 060.6718 Switching, circuit, 060.1155 All-optical networks ## 1 Introduction The most common data center network (DCN) packet length is $<$256 bytes which translates to 20 ns slots in 100G links [1]. Optical circuit switching (OCS) aims to transform data centre networks (DCNs) but needs to operate at packet speed and granularity [2]. Recent breakthroughs have brought OCS closer to reality. A hardware-based OCS scheduling algorithm has demonstrated synchronous scheduling of up to 32,768 nodes within 2.3 ns [2]. A clock phase caching method has enabled clock and data recovery in less than 625 ps, supporting 10,000 remote nodes [1]. Yet, energy-efficient, sub-ns, many- channel optical switching remains a challenge. Wideband fast tuneable lasers have demonstrated switching on ns timescales [4, 5], and as low as 500 ps but over limited bandwidths [6]. Static laser diodes (LDs) gated by semiconductor optical amplifiers (SOAs) have achieved 912 ps 10-90% rise-fall times with $\sim$2 ns settling time ($\pm 5\%$ of the target value) [3]; however, the power consumption and device count limit the scalability of this approach. A similar method used an optical comb where each wavelength was filtered then gated by an SOA [7]; the power consumption and device count therefore also increase linearly with number of channels, limiting scalability (see Fig. 1(b)). In this paper, we introduce SWIFT: a modular system with Scalable Wideband Interconnects for Fast Tuning. SWIFT combines pairs of optimised widely tuneable lasers (TLs), multiplexing their wavelength reconfiguration on packet timescales. The lasers are gated by pairs of fast switching SOAs, resulting in wideband, sub-ns switching. The modular design of SWIFT (Fig. 1(a)) shows that just two lasers and two SOAs cover each optical transmission band. SWIFT power consumption is, therefore, practically independent of channel count; Fig. 1(b) shows that SWIFT becomes more power efficient than alternative sub-ns switching sources beyond 8$\times$50 GHz spaced channels. | | | ---|---|---|--- (a) | (b) | (c) | (d) Fig. 1: (a) Modular SWIFT architecture across S-, C- and L-bands. (b) Power consumption comparison of laser switch designs vs. no. of channels, using data reported in [8]. (c) PULSE DCN architecture with SWIFT modules (in red). (d) Comparison of switching times (reported rise (solid) and estimated settling (faded)) against no of channels for different switch systems. The SWIFT modules can be deployed as transmitters in DCN architectures such as PULSE [8], as shown in Fig. 1(c). In this architecture, each node has $x$ SWIFT transmitters (highlighted in red), each local pod has $N$ nodes, and $x^{2}$ star couplers enable there to be $x$ source and $x$ destination pods. Thus, PULSE network’s number of end-points scales with $N\times x$, where $N$ is limited by the number of wavelength channels. The large number of channels supported by SWIFT therefore allows for significant scalability in the PULSE DCN [2]. The concept of time-multiplexed, fast tuneable lasers was proposed in [8, 9], but faced the challenge of optimising multiple lasers and SOAs for reliable fast tuning. SWIFT overcomes this by applying artificial intelligence (AI) techniques to the devices, enabling autonomous optimisation. This has allowed us to demonstrate, for the first time, a time-multiplexed, gated laser tuning system that can tune over 6.05 THz of bandwidth and consistently switch in 547 ps or better to support 20 ns timeslots. SWIFT outperforms other fast switching systems in terms of rise time, settling time and channel count, as shown in Fig 1(d). ## 2 Experimental Setup The setup used to demonstrate SWIFT is shown in Fig. 2(a). A pair of commercial Oclaro (now Lumentum) digital-supermode distributed Bragg reflector (DS-DBR) lasers were driven by 250 MS/s arbitrary waveform generators (AWGs) with 125 MHz bandwidth. Detailed IV measurements were used to map supplied voltage to desired current. Each laser was connected to a commercial InPhenix SOA, supporting 69 nm of bandwidth with typical characteristics of 7 dB noise figure, 20 dB gain, and 10 dBm saturation power. Each SOA was driven with a 45 mA current source modulated by a 12 GS/s AWG with $\pm$0.5 V output and amplified to $\pm$4 V using an electrical amplifier. All four optical devices were held at 25∘C using temperature controllers. The SOAs were coupled together and passed to a digital coherent receiver (50 GS/s, 22 GHz bandwidth) and a digital sampling oscilloscope (50 GS/s, 30 GHz bandwidth), which provided optimisation feedback to the DS-DBR lasers and to the SOAs respectively. \begin{overpic}[scale={0.45}]{Figures/setup_small_labels.png} \put(-5.0,55.0){(a)} \end{overpic} | \begin{overpic}[scale={0.14}]{Figures/fo.png} \put(12.0,50.0){(b)} \end{overpic} | \begin{overpic}[scale={0.14}]{Figures/dsdbr_cdf.png} \put(12.0,50.0){(c)} \end{overpic} ---|---|--- Fig. 2: (a) Experimental setup of time-multiplexed SWIFT tuneable lasers (TL) gated by SOAs. (b) TL frequency offset (FO) of worst-case current swing w/ & w/o optimiser. (c) CDF of all worst-case laser switch combinations w/ & w/o optimiser. ## 3 Results and Discussion ### 3.1 Regression optimised laser switching Fast wavelength switching can be achieved by applying ‘pre-emphasis’ to the drive sections of an integrated semiconductor laser. Until recently, pre- emphasis values had to be carefully tuned by hand for select samples then extrapolated [4]. Here, we apply a linear regression optimiser to automatically calculate the pre-emphasis values for reliable fast tuning. We measured the output of the DS-DBR laser during a switching event using the coherent receiver, then used the instantaneous frequency response as the error term within a linear regression optimiser to iteratively update the applied pre-emphasis values [5]. Fig 2(b) shows an example of the laser’s switching response before and after application of the optimiser. We applied this optimiser to 21 of the 122 supported channels, testing the extremes of lasing frequency and drive current, covering 462 any-to-any switching events across 6.05 THz (1524.11-1572.48 nm). Fig. 2(c) shows the cumulative distribution of the time taken to reach $\pm$5 GHz of the target wavelength. We measure a worst case switch time of 14.7 ns, and a worst case frequency offset after 20 ns of $-$4.5 GHz. This indicates that SWIFT is potentially suitable for burst mode coherent detection, as 28 GBd dual-polarisation quadrature phase shift keying is tolerant of frequency offsets up to $\pm$7 GHz [10]. ### 3.2 Particle swarm optimised SOA switching SOA driving signals must also be optimised to approach their theoretical rise/fall times of $\sim 100$ ps. Previous optimisation attempts did not consider settling times nor the ability to automate the optimisation of driving conditions for 1,000s of different SOAs in real DCNs [11, 12]. To solve this, PSO (a population-based metaheuristic for optimising continuous nonlinear functions by combining swarm theory with evolutionary programming) was used in this work to optimise the SOA driving signals. PSO has previously been applied to proportional-integral-derivative (PID) tuning in control theory [13], but has not yet been used as an autonomous method for optical switch control. In the optimisation, $n=160$ particles (driving signals) were initialised in an $m=240$ (number of points in the signal) hyperdimensional search space and iteratively ‘flown’ through the space by evaluating each particle’s position with a fitness function $f$, defined as the mean squared error between the drive signals’ corresponding optical outputs (recorded on the oscilloscope) and an ideal target ‘set point’ (SP) with 0 overshoot, settling time and rise time. As shown in Fig. 3(a) and (b), the $\pm$ 5% settling time (effective switching time) of the SOA was reduced from 3.72 ns (when driven by a simple square driving signal) to 547 ps, with the 10-90% rise time also reduced from 697 ps to 454 ps. The PSO routine required no prior knowledge of the SOA, therefore provides a flexible, automated and scalable method for optimising SOA gating. \begin{overpic}[scale={0.15}]{Figures/soa_outputs_annotated.png} \put(15.0,52.0){(a)} \end{overpic} | \begin{overpic}[scale={0.15}]{Figures/soa_outputs_zoomed_annotated.png} \put(12.0,52.0){(b)} \end{overpic} | \begin{overpic}[scale={0.03}]{Figures/1572p48_1524p11_1565p5_1530p72_2_channel_gating.png} \put(5.0,40.0){(c)} \end{overpic} ---|---|--- \begin{overpic}[scale={0.22}]{Figures/transitions.png} \put(3.0,45.0){(d)} \end{overpic} | \begin{overpic}[scale={0.15}]{Figures/osa_outputs_all_pairs.png} \put(12.0,50.0){(e)} \end{overpic} | \begin{overpic}[scale={0.037}]{Figures/burst_fo_data.png} \put(-1.0,45.0){(f)} \end{overpic} Fig. 3: SOA outputs showing (a) step & (b) PSO rise & settling times, (c) SWIFT system output, (d) $\lambda$-to-$\lambda$ 90-90% switching times, (e) optical spectrum of the 21 worst-case channels, (f) frequency offset (FO) of DSDBR (top) & SWIFT (bottom). ### 3.3 SWIFT module demonstration After optimisation, the DS-DBR lasers were driven with with 12.5 MHz pre- emphasised square waves, resulting in $\leq$40 ns bursts on each wavelength. The lasers were driven 20 ns out of phase, so that one lased while the other reconfigured. The SOAs were driven by 25 MHz PSO-optimised signals, resulting in 20 ns gates, and aligned to block the first 15 ns and last 5 ns of each laser burst, yielding four wavelength bursts of 20 ns each (see Fig. 2(a)). Fig. 3(c) shows the oscilloscope output for the most difficult switching instance, where DS-DBR laser 1 switched from 1572.48 nm to 1524.11 nm, incurring a large rear current swing of 45 mA. The oscilloscope shows a flat intensity response across each wavelength for 20 ns bursts, thereby providing twice the granularity reported in [3]. Packet-to-packet power variations are due to slight variations in laser wavelength power; these can be addressed by applying slot specific SOA drive currents (not possible in our setup). Measuring switch time by the 90-90% transition time, we report switch times for the four transitions of 771, 812, 521, and 792 ps, respectively. These are shown in Fig. 3(d). Furthermore, Fig. 3(f) shows the coherent receiver output of the four wavelength slots with and without gating. The observed frequency ripples are a result of the low sample rate of our 250 MS/s AWG that introduce Fourier components to the driving square wave; these can be easily suppressed by using a higher sample rate. Despite this, each slot stays within 5 GHz of its target. We repeated this process for each of the channels under test. Fig. 3(e) shows the optical spectrum for all channels, all undergoing gated switching. We measured a worst case value for the side mode suppression ratio of 35 dB, optical power output of 0.8 dBm for a single wavelength (at 1572.48 nm) and corresponding extinction ratio of 22 dB. The fully time-multiplexed optical output power of SWIFT was $>$6 dBm. This represents the largest number of sub- ns switching channels from a single sub-system ever reported, supporting 122$\times{}$50 GHz spaced channels. In conclusion, we propose a scalable, low power, tuneable wavelength subsystem capable of sub-ns switching. Using pairs of time-multiplexed tuneable lasers, gated by SOAs, we have experimentally demonstrated switching times of less than 900 ps for 122 x 50 GHz channels. Reliable and fast tuning was achieved for each laser and SOA using regression and particle swarm optimisation AI techniques. This enables automated, device-specific optimisation and represents a critically important technology in OCS architectures, potentially transforming DCN architectures. This work is supported by EPSRC (EP/R035342/1), IPES CDT, iCASE and Microsoft Research. ## References * [1] K. Clark, et al., “Sub-Nanosecond Clock and Data Recovery in an Optically-Switched Data Centre Network”, ECOC, pdp, 2018. * [2] J. Benjamin, et al., “Scaling PULSE Datacenter Network Architecture and Scheduling Optical Circuits in Sub-$\mu$seconds”, OFC, W1F.3, 2020. * [3] K. Shi, et al., “System Demonstration of Nanosecond Wavelength Switching with Burst-mode PAM4 Transceiver,” ECOC, pdp, 2019. * [4] J. Simsarian, et al., “Less than 5-ns wavelength switching with an SG-DBR laser”, PTL, 18(4), 2006. * [5] T. Gerard et al., “Packet Timescale Wavelength Switching Enabled by Regression Optimisation,” arXiv:2002.11640v1 [eess.SP] 2020. * [6] Y. Ueda, et al., “Electro-Optically Tunable Laser with $<$10-mW Tuning Power Dissipation and High-Speed $\lambda$-Switching for Coherent Networks”, ECOC, pdp, 2019. * [7] S. Lange et al., “Sub-Nanosecond Optical Switching Using Chip-Based Soliton Microcombs,” in _OFC_ , W2A.4, 2020. * [8] J. Benjamin, et al., “PULSE: Optical Circuit Switched Data Center Architecture Operating at Nanosecond Timescales”, arXiv:2002.04077v1 [cs.N1], 2020. * [9] N. Ryan, et al., “A 10Gbps Optical Burst Switching Network Incorporating Ultra-fast (5ns) Wavelength Switched Tunable Laser”, ICSO, 2008. * [10] J. Simsarian, “Fast-Tuning Coherent Burst-Mode Receiver for Metropolitan Networks”, PTL, 26(8), 2014. * [11] C. Gallep and E. Conforti, “Reduction of semiconductor optical amplifier switching times by preimpulse step-injected current technique,”, _PTL_ , 14(7), 2002. * [12] R. C. Figueiredo et al., “Hundred-Picoseconds Electro-Optical Switching With Semiconductor Optical Amplifiers Using Multi-Impulse Step Injection Current,”, _JLT_ , 13(1), 2015. * [13] D. H. Kusuma et al., “The comparison of optimization for active steering control on a vehicle using PID controller based on artificial intelligence techniques,” in _International Seminar on Applications for Technology of Information and Communication_ , 2016.

2024-09-04T02:54:59.358682

2003.05492

arxiv-papers

1.87cm1.87cm1.87cm1.87cm capbtabboxtable[][] # Lifted samplers for partially ordered discrete state-spaces Philippe Gagnon 1, Florian Maire 1 ###### Abstract A technique called lifting is employed in practice for avoiding that the Markov chains simulated for sampling backtrack too often. It consists in lifting the state-space to include direction variables for guiding these chains. Its implementation is direct when the probability mass function targeted is defined on a totally ordered set, such as that of a univariate random variable taking values on the integers. In this paper, we adapt this technique to the situation where only a partial order can be established and explore its benefits. Important applications include simulation of systems formed from binary variables, such as those described by the Ising model, and variable selection when the marginal model probabilities can be evaluated, up to a normalising constant. To accommodate for the situation where one does not have access to these marginal model probabilities, a lifted trans-dimensional sampler for partially ordered model spaces is introduced. We show through theoretical analyses and empirical experiments that the lifted samplers outperform their non-lifted counterparts in some situations, and this at no extra computational cost. The code to reproduce all experiments is available online.111See the ArXiv page of this paper. 1Department of Mathematics and Statistics, Université de Montréal. Keywords: Bayesian statistics; binary random variables; Ising model; Markov chain Monte Carlo methods; Peskun-Tierney ordering; trans-dimensional samplers; variable selection. ## 1 Introduction ### 1.1 Partially ordered state-spaces A partially ordered set $\bm{\mathcal{X}}$ is such that there exists a reflexive, antisymmetric, and transitive binary relation defined through a set $\bm{\mathcal{R}}$ in which it is possible to establish that $\mathbf{x}\leq\mathbf{y}$ and $\mathbf{y}\geq\mathbf{x}$ as a consequence of $(\mathbf{x},\mathbf{y})\in\bm{\mathcal{R}}$ for $\mathbf{x},\mathbf{y}\in\bm{\mathcal{X}}$. In such a set, pairs are comparable when either $\mathbf{x}<\mathbf{y}$ or $\mathbf{y}<\mathbf{x}$, and are incomparable when neither $\mathbf{x}<\mathbf{y}$ nor $\mathbf{y}<\mathbf{x}$. This represents the difference with totally ordered sets such as $\operatorname{\mathbb{R}}$ or $\mathbb{N}$ for which every pair of different elements is comparable. We refer the reader to Trotter (1992) for more details on partially ordered sets. An important example of such sets is when any $\mathbf{x}\in\bm{\mathcal{X}}$ can be written as a vector $\mathbf{x}:=(x_{1},\ldots,x_{n})$ for which each component $x_{i}$ can be of two types, say Type A or Type B, denoted by $x_{i}\in\\{\text{A},\text{B}\\}$. In this case, $\bm{\mathcal{R}}$ can be set to $\bm{\mathcal{R}}:=\\{(\mathbf{x},\mathbf{y})\in\bm{\mathcal{X}}\times\bm{\mathcal{X}}:n_{\text{A}}(\mathbf{x})\leq n_{\text{A}}(\mathbf{y})\\},$ where $n_{\text{A}}(\mathbf{x})$ is the number of components of Type A in $\mathbf{x}$: $n_{\text{A}}(\mathbf{x})=\sum_{i=1}^{n}\mathds{1}_{x_{i}=\text{A}}$. The function $n_{\text{B}}(\mathbf{x})$ is defined analogously. Note that $n=n_{\text{A}}(\mathbf{x})+n_{\text{B}}(\mathbf{x})$ for all $\mathbf{x}$ and that the order can be symmetrically established by instead considering $n_{\text{B}}(\mathbf{x})$ and $n_{\text{B}}(\mathbf{y})$. Two main statistical problems fit within this framework: simulation of binary random variables such as graphs or networks and variable selection. Indeed, for the former, $\bm{\mathcal{X}}$ can be parameterized such that $\bm{\mathcal{X}}=\\{-1,+1\\}^{n}$, where for example for an Ising model, $x_{i}\in\\{-1,+1\\}$ represents the state of a spin, or, for networks, $\bm{\mathcal{X}}=\mathcal{M}_{n}(\\{0,1\\})$ where $x_{i,j}\in\\{0,1\\}$ indicates whether nodes $i$ and $j$ are connected or not. For variable selection, $\bm{\mathcal{X}}=\\{0,1\\}^{n}$ and $x_{i}\in\\{0,1\\}$ indicates whether or not the $i$th covariate is included in the model characterised by the vector $\mathbf{x}\in\bm{\mathcal{X}}$. ### 1.2 Sampling on partially ordered state-spaces Let $\pi$ be a probability distribution defined on a measurable space $(\bm{\mathcal{X}},\bm{\mathsf{X}})$ where $\bm{\mathcal{X}}$ is a partially ordered set and $\bm{\mathsf{X}}$ a sigma algebra on $\bm{\mathcal{X}}$ and consider the problem of sampling from $\pi$. We assume that it is not possible to generate independent draws from $\pi$. Given the complex dependency structure of modern statistical models such as graphical models and the intractable nature of some distributions that arise, for instance, in Bayesian statistics, this is a realistic assumption. We turn to Markov chains based sampling methods which typically rely on an ergodic stochastic process $\\{\mathbf{X}(m)\,:\,m\in\mathbb{N}\\}$ whose limiting distribution is $\pi$. A typical Markov chain based sampler, such as the Glauber dynamics for graphical models or the tie-no-tie sampler for network models, selects uniformly at random one of the coordinates of $\mathbf{x}$, say $x_{i}$, and proposes to flip its value from B to A (if $x_{i}=\text{A}$), and accept or reject this move according to a prescribed probability that guarantees that the Markov chain limiting distribution is $\pi$. Such moves are often rejected when the mass concentrates on a manifold of the ambient space. Zanella (2019) recently proposed a locally informed generic approach for which the probability to select the $i$th coordinate depends on the relative mass of the resulting proposal, i.e. $\pi(\mathbf{y})/\pi(\mathbf{x})$, aiming at proposing less naive moves. Yet, the sampler is of reversible Metropolis–Hastings (MH, (Metropolis et al., 1953; Hastings, 1970)) type, implying that the chain may often go back to recently visited states. When this is the case, the state-space is explored through a random walk, a process exhibiting a diffusive behaviour. The lifting technique, which can be traced back to Gustafson (1998) and even to Horowitz (1991), aim at producing Markov chains which do not suffer from such a behaviour. This is achieved by introducing a momentum variable $\nu\in\\{-1,+1\\}$ which provides the process $\\{\mathbf{X}(m)\,:\,m\in\mathbb{N}\\}$ with some memory of its past in order to avoid backtracking. To remain Markovian, the process is thus enlarged to $\\{(\mathbf{X},\nu)(m)\,:\,m\in\mathbb{N}\\}$ and the momentum variable is flipped at random times according to a prescribed dynamic which guarantees that, marginally, the process $\\{\mathbf{X}(m)\,:\,m\in\mathbb{N}\\}$ retains its limiting distribution. Lifted Markov chains have been quantitatively studied in Chen et al. (1999) and Diaconis et al. (2000) and have been shown to reduce, sometimes dramatically, the mixing time of random walks on groups. Over the past few years, there has been a renewed interest for lifted techniques in the computational statistics community: in addition to speeding- up the mixing time of random walk Markov chains they are also suspected to reduce asymptotic variances of empirical averages of observables of interests, see Andrieu and Livingstone (2019) for some precise results. We also refer to Gagnon and Doucet (2019), Syed et al. (2019) and Neal (2020) for examples where popular Markov chain Monte Carlo (MCMC) algorithms such as reversible jump (Green, 1995), parallel tempering (Geyer, 1991) and slice sampling (Neal, 2003) have seen their performance improved in some situations by considering their lifted version. Remarkably, those lifted samplers are implemented at no additional computational cost over their non-lifted counterparts, and also with no additional implementation difficulty. All these successful applications of the lifting technique have been carried out in contexts where $\bm{\mathcal{X}}$ is one-dimensional (Gustafson, 1998) or exhibits a one-dimensional parameter which plays a central role in the sampling scheme: the annealing parameter in Syed et al. (2019), the model indicator reflecting the size/complexity of the nested models in Gagnon and Doucet (2019), and the height of the level-lines $\\{\pi(\mathbf{X}(m))\,:\,m\in\mathbb{N}\\}$ in Neal (2020). When such a one- dimensional feature does not exist, there does not exist a straightforward way of lifting the state-space without facing issues of reducibility or obtaining inefficient samplers. A possibility, deemed as naive in Gustafson (1998), is to lift each marginal component of the state-space and update them one at the time. When the state-space is uncountable, it is possible to construct a persistent walk by introducing bounces at random event times which change the direction of propagation (see Vanetti et al. (2017)). However, when the state- space is countable and partially ordered, such an approach is infeasible. The objective of this paper is to present and analyse generic methods based on the lifting technique to sample from a given probability mass function (PMF) with a partially ordered countable support. In particular, we break free from the requirement of having a one-dimensional parameter by exploiting the local one-dimensional neighborhood structure induced by the partial order on $\bm{\mathcal{X}}$: the neighbourhood of $\mathbf{x}$, denoted by $\mathcal{N}(\mathbf{x})$, is separated into two parts where one comprises states with $\mathbf{y}>\mathbf{x}$, denoted by $\mathcal{N}_{+1}(\mathbf{x})$, and the other one comprises states with $\mathbf{y}<\mathbf{x}$, denoted by $\mathcal{N}_{-1}(\mathbf{x})$ (considering that $\mathcal{N}(\mathbf{x})$ is only composed of states that can be compared with $\mathbf{x}$). Looking for instance at the variable selection example, the partial order is defined by mean of the model sizes, or in other words, the number of covariates included in the models. If the momentum is $\nu=+1$, the chain is forced to attempt visiting models with more variables until a move is rejected, then $\nu$ is flipped to $\nu=-1$. As a consequence, the momentum variable remains one-dimensional while the Markov chain is often irreducible and efficiently explore the state-space. An illustration showing the benefit of this approach is provided at Figure 1. Again, we stress that the typical lifted sampler is implemented at no additional computational cost over its non-lifted counterpart, and also with no additional implementation difficulty. $\begin{array}[]{cc}\textbf{Random walk behaviour}&\textbf{Persistent movement}\cr\vspace{-1mm}\textbf{ESS = 0.12 per it.}&\textbf{ESS = 0.33 per it.}\cr\includegraphics[width=173.44534pt]{Fig_1_a.pdf}&\includegraphics[width=173.44534pt]{Fig_1_b.pdf}\end{array}$ Figure 1: Trace plots for the statistic of number of covariates included in the model for a MH sampler with a locally informed proposal distribution (discussed in more details in Section 3.2) and its lifted counterpart, when applied to solve a real variable selection problem (presented in Section 4.2); ESS stands for effective sample size ### 1.3 Overview of our contributions In this paper, we focus on the simulation of two-dimensional Ising models and variable selection problems, without restricting ourselves to these examples of applications when we present the samplers and analyse them. For these examples, a generic sampler that we study corresponds to the discrete-time version of a specific sampler independently developed in Power and Goldman (2019), a paper in which the focus is rather on exploiting any structure of $\bm{\mathcal{X}}$ identified by users. The structure identified here is, in a sense, that $\bm{\mathcal{X}}$ exhibits a partial order. We consequently do not claim originality for the samplers that will be presented. Our contributions are the following: * • statement of the sampling problem within the specific framework of partially ordered discrete state-spaces (so that it becomes straightforward to implement a sampler using the lifting technique in this framework); * • identification of situations in which the lifted samplers are expected to outperform their non-lifted counterparts, based on theoretical analyses and numerical experiments; * • introduction of a trans-dimensional lifted sampler useful, among others, for variable selection when it is not possible to integrate out the parameters. ### 1.4 Organisation of the paper The generic algorithm is first presented in Section 2. We next analyse in Section 3 two important versions with uniform proposal distributions and locally informed ones, allowing to establish that they can outperform their non-lifted counterparts under some assumptions. In Section 4, we show the difference in empirical performance for a Ising model (Section 4.1) and real variable selection problem (Section 4.2). In Section 5, we consider that $\bm{\mathcal{X}}$ is a model space and propose a lifted trans-dimensional sampler allowing to simultaneously achieve model selection and parameter estimation. The paper finishes in Section 6 with retrospective comments and possible directions for future research. ## 2 Generic algorithm The sampler that we present is a MCMC algorithm that generates proposals belonging to a subset of $\mathcal{N}(\mathbf{x})$ chosen according to a “direction” $\nu\in\\{-1,+1\\}$, when the current state is $\mathbf{x}\in\bm{\mathcal{X}}$. In particular, the proposals belong to $\mathcal{N}_{+1}(\mathbf{x}):=\\{\mathbf{y}\in\mathcal{N}(\mathbf{x}):\mathbf{y}>\mathbf{x}\\}\subseteq\mathcal{N}(\mathbf{x})$ when $\nu=+1$ or $\mathcal{N}_{-1}(\mathbf{x}):=\\{\mathbf{y}\in\mathcal{N}(\mathbf{x}):\mathbf{y}<\mathbf{x}\\}\subseteq\mathcal{N}(\mathbf{x})$ when $\nu=-1$, where $\mathcal{N}_{-1}(\mathbf{x})$ and $\mathcal{N}_{+1}(\mathbf{x})$ thus denote two subsets of $\mathcal{N}(\mathbf{x})$ such that $\mathcal{N}_{-1}(\mathbf{x})\cup\mathcal{N}_{+1}(\mathbf{x})=\mathcal{N}(\mathbf{x})$. More formally, assuming that the Markov chain state is $\mathbf{x}\in\bm{\mathcal{X}}$, the proposal distribution $q_{\mathbf{x},\nu}$ has its support restricted to $\mathcal{N}_{\nu}(\mathbf{x})$. There exist natural candidates for such distributions, as will be explained in the following. This makes the implementation of the proposed sampler almost straightforward; once the neighbourhood structure has been specified. In our framework, the partial ordering induces a natural neighbourhood structure. The sampler is based on the well known technique of lifting: the state-space $\bm{\mathcal{X}}$ is lifted to an extended state-space $\bm{\mathcal{X}}\times\\{-1,+1\\}$ such that the marginal and the conditional distributions of the direction variable $\nu$ is the uniform distribution on $\\{-1,+1\\}$. The algorithm, which bares a strong resemblance with the guided walk (Gustafson, 1998), is now presented in Algorithm 1. Algorithm 1 A lifted sampler for partially ordered discrete state-spaces 1. 1. Generate $\mathbf{y}\sim q_{\mathbf{x},\nu}$ and $u\sim\mathcal{[}0,1]$. 2. 2. If $\displaystyle u\leq\alpha_{\nu}(\mathbf{x},\mathbf{y}):=1\wedge\frac{\pi(\mathbf{y})\,q_{\mathbf{y},-\nu}(\mathbf{x})}{\pi(\mathbf{x})\,q_{\mathbf{x},\nu}(\mathbf{y})},$ (1) set the next state of the chain to $(\mathbf{y},\nu)$. Otherwise, set it to $(\mathbf{x},-\nu)$. 3. 3. Go to Step 1. If $\bm{\mathcal{X}}$ is finite, there exists a boundary, in the sense that there may be no mass beyond a state $\mathbf{x}$ when the direction followed is $\nu$. For instance in variable selection, the posterior probability of a model with more covariates than the maximum number is zero. Algorithm 1 may thus seem incomplete, in the sense that it is not explicitly specified what to do on the boundary. We in fact consider that $q_{\mathbf{x},\nu}$ is defined even on the boundary, and that it is defined to generate a point outside of $\bm{\mathcal{X}}$. This point will be automatically rejected (because its mass is 0) and the chain will remain at $\mathbf{x}$ and the direction will be reversed. Note that this is a technical requirement. In practice, one can simply skip Step 1 in this case and set the next state to $(\mathbf{x},-\nu)$. It is possible to establish the correctness of the algorithm through that of a more general version based on the lifted algorithm presented in Andrieu and Livingstone (2019). Before presenting this more general version which has interesting features, we introduce necessary notation. Let $\rho_{\nu}:\bm{\mathcal{X}}\to[0,1]$, for $\nu\in\\{-1,+1\\}$, be a user- defined function for which we require that: $\displaystyle 0\leq\rho_{\nu}(\mathbf{x})\leq 1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}}),$ (2) $\displaystyle\rho_{\nu}(\mathbf{x})-\rho_{-\nu}(\mathbf{x})=T_{-\nu}(\mathbf{x},\bm{\mathcal{X}})-T_{\nu}(\mathbf{x},\bm{\mathcal{X}}),$ (3) where we have set, for all $(\mathbf{x},\nu)\in\bm{\mathcal{X}}\times\\{-1,+1\\}$, $\displaystyle T_{\nu}(\mathbf{x},\bm{\mathcal{X}}):=\sum_{\mathbf{x}^{\prime}\in\bm{\mathcal{X}}}q_{\mathbf{x},\nu}(\mathbf{x}^{\prime})\,\alpha_{\nu}(\mathbf{x},\mathbf{x}^{\prime})=\sum_{\mathbf{x}^{\prime}\in\mathcal{N}_{\nu}(\mathbf{x})}q_{\mathbf{x},\nu}(\mathbf{x}^{\prime})\,\alpha_{\nu}(\mathbf{x},\mathbf{x}^{\prime}).$ (4) These conditions make the algorithm valid and are thus considered satisfied in the sequel. Finally, let $Q_{\mathbf{x},\nu}$ be a PMF such that $Q_{\mathbf{x},\nu}(\mathbf{x}^{\prime})\propto q_{\mathbf{x},\nu}(\mathbf{x}^{\prime})\,\alpha_{\nu}(\mathbf{x},\mathbf{x}^{\prime})$. The more general algorithm is now presented in Algorithm 2. Algorithm 2 A more general lifted sampler for partially ordered discrete state-spaces 1. 1. Generate $u\sim\mathcal{U}[0,1]$. 1. (i) If $u\leq T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$, generate $\mathbf{y}\sim Q_{\mathbf{x},\nu}$ and set the next state of the chain to $(\mathbf{y},\nu)$; 2. (ii) if $T_{\nu}(\mathbf{x},\bm{\mathcal{X}})<u\leq T_{\nu}(\mathbf{x},\bm{\mathcal{X}})+\rho_{\nu}(\mathbf{x})$, set the next state of the chain to $(\mathbf{x},-\nu)$; 3. (iii) if $u>T_{\nu}(\mathbf{x},\bm{\mathcal{X}})+\rho_{\nu}(\mathbf{x})$, set the next state of the chain to $(\mathbf{x},\nu)$. 2. 2. Go to Step 1. ###### Proposition 1. The transition kernel of the Markov chain $\\{(\mathbf{X},\nu)(m):m\in\operatorname{\mathbb{N}}\\}$ simulated by Algorithm 2 admits $\pi\otimes\mathcal{U}\\{-1,1\\}$ as invariant distribution. ###### Proof. See Section 7. ∎ It is interesting to notice that $T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$ represents the probability to accept a proposal when the current state is $(\mathbf{x},\nu)$. In Algorithm 2, we thus first decide if we accept a proposal or not, and if it is the case, in Step 1.(i), we randomly select the value of the proposal (using the conditional distribution). It can be readily checked that choices for $\rho_{\nu}$ include $\rho_{\nu}(\mathbf{x})=1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$ and $\rho_{\nu}(\mathbf{x})=\max\\{0,T_{-\nu}(\mathbf{x},\bm{\mathcal{X}})-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})\\}$. If $\rho_{\nu}(\mathbf{x})=1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$, the condition for Case (iii) of Step 1 is never satisfied, and the algorithm either accepts the proposal and keeps the same direction, or the proposal is rejected and the direction is reversed. In this case, one can show that Algorithm 2 corresponds to Algorithm 1, which is what allows ensuring the correctness of Algorithm 1 as well. Setting $\rho_{\nu}(\mathbf{x})$ otherwise than $\rho_{\nu}(\mathbf{x})=1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$ allows in Case (iii) of Step 1 to keep following the same direction, even when the proposal is rejected. Intuitively, this is desirable when the rejection is due to “bad luck”, and not because there is no mass in the direction followed. The function $\rho_{\nu}(\mathbf{x})$ aims at incorporating this possibility in the sampler. In the typical MCMC framework, when one wants to sample from a probability density function, it is not possible to directly compute $T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$ as it requires computing an integral with respect to this density function. In such a case, it is therefore usually not possible to set $\rho_{\nu}(\mathbf{x})$ otherwise than $1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$. This contrasts with our discrete state-space framework in which it is often possible to directly compute $T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$. A corollary of Theorem 3.15 in Andrieu and Livingstone (2019) states that the best choice of function $\rho_{\nu}$ in terms of asymptotic variance is $\displaystyle\rho_{\nu}^{*}(\mathbf{x}):=\max(0,T_{-\nu}(\mathbf{x},\bm{\mathcal{X}})-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})),$ (5) and that the worst choice is $\rho_{\nu}^{\text{w}}(\mathbf{x}):=1-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$. ###### Corollary 1. If $\bm{\mathcal{X}}$ is finite, then for any function $\rho_{\nu}$ and $f\in\mathcal{L}_{2}^{*}(\bar{\pi}):=\left\\{f:\int d\bar{\pi}f^{2}<\infty\text{ and }f(\mathbf{x},-1)=\right.$ $\left.f(\mathbf{x},+1)\text{ for all }\mathbf{x}\right\\}$, $\mathrm{var}(f,P_{\rho^{*}})\leq\mathrm{var}(f,P_{\rho})\leq\mathrm{var}(f,P_{\rho^{\text{w}}}),$ where $\bar{\pi}:=\pi\otimes\mathcal{U}\\{-1,+1\\}$, $\mathrm{var}(f,P_{\rho}):=\mathbb{V}\mathrm{ar}f(\mathbf{X},\nu)+2\sum_{k>0}\left\langle f,P_{\rho}^{k}f\right\rangle$ and $P_{\rho}$ is the transition kernel simulated by Algorithm 2, $\left\langle f,P_{\rho}^{k}f\right\rangle$ being the inner product, i.e. $\left\langle f,P_{\rho}^{k}f\right\rangle:=\int d\bar{\pi}fP_{\rho}^{k}f$. ###### Proof. See Section 7. ∎ The price to pay for using $\rho_{\nu}^{*}$ instead of $\rho_{\nu}^{\text{w}}$, for instance, is that the algorithm is more complicated to implement because it is required to systematically compute $T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$ at each iteration (it is also sometimes required to compute $T_{-\nu}(\mathbf{x},\bm{\mathcal{X}})$). Using $\rho_{\nu}^{*}$ also comes with an additional computation cost (which is discussed in Section 3.2). In our numerical experiments, it is seen that the gain in efficiency of using Algorithm 2 with $\rho_{\nu}^{*}$ over Algorithm 1 is not significant. One may thus opt for simplicity and implement Algorithm 1. ## 3 Analysis of specific samplers In the previous section, we presented generic algorithms with conditions ensuring that they are valid. A necessary input to implement them is the proposal distribution $q_{\mathbf{x},\nu}$ to be used. In this section, we explore two natural choices and analyse their asymptotic variances. We start in Section 3.1 with the common situation where the proposal is picked uniformly at random. As mentioned in the introduction, this choice may lead to poor mixing. We thus go on in Section 3.2 to a distribution incorporating information about the target. This section finishes with a brief discussion in Section 3.3 on the computational costs associated to these choices in regular MH samplers and their lifted counterparts. ### 3.1 Uninformed uniform proposal In the reversible MH sampler, it is common to set the proposal distribution, denoted by $q_{\mathbf{x}}$ for this algorithm, to $q_{\mathbf{x}}:=\mathcal{U}\\{\mathcal{N}(\mathbf{x})\\}$. In our framework, the analogous proposal distribution is naturally defined as $q_{\mathbf{x},\nu}:=\mathcal{U}\\{\mathcal{N}_{\nu}(\mathbf{x})\\}$. In this case, the acceptance probability (1) of a proposal becomes $\alpha_{\nu}(\mathbf{x},\mathbf{y})=1\wedge\frac{\pi(\mathbf{y})\,|\mathcal{N}_{\nu}(\mathbf{x})|}{\pi(\mathbf{x})\,|\mathcal{N}_{-\nu}(\mathbf{y})|}.$ For ease of presentation of the analysis, consider again the important example described in Section 1.1 where each component $x_{i}$ of $\mathbf{x}=(x_{1},\ldots,x_{n})$ can be of two types. We in this section highlight the dependency on $n$ (the dimension) of the state-space and target because it will be relevant in our analysis. We thus write $\pi_{n}$ for the target and $\bm{\mathcal{X}}_{n}$ for the state-space, where each state is of the form $\mathbf{x}:=(x_{1},\ldots,x_{n})$ with $x_{i}\in\\{\text{A},\text{B}\\}$. For now on, consider for that $\text{A}=-1$ and $\text{B}=+1$, corresponding to the case of Ising model. We note that there is no loss of generality of considering this special case within the important example. In a MH sampler, one sets $\mathcal{N}(\mathbf{x}):=\\{\mathbf{y}\in\bm{\mathcal{X}}_{n}:\sum_{i}|x_{i}-y_{i}|=2\\}=\\{\mathbf{y}\in\bm{\mathcal{X}}_{n}:\exists j\text{ such that }y_{j}=-x_{j}\\}$, so that the algorithm proposes to flip a single bit at each iteration. It thus chooses uniformly at random which bit to flip. Therefore, the size of the neighbourhoods in this sampler is constant for any $\mathbf{x}$ and is given by $n$. This implies that the acceptance probability in this sampler, denoted by $\alpha(\mathbf{x},\mathbf{y})$, reduces to $\alpha(\mathbf{x},\mathbf{y})=1\wedge\pi(\mathbf{y})/\pi(\mathbf{x})$. In the lifted case, the acceptance probability can be rewritten as $\displaystyle\alpha_{\nu}(\mathbf{x},\mathbf{y})=1\wedge\frac{\pi(\mathbf{y})\,n_{-\nu}(\mathbf{x})}{\pi(\mathbf{x})\,n_{\nu}(\mathbf{y})}.$ (6) Indeed, $\mathcal{N}_{\nu}(\mathbf{x}):=\\{\mathbf{y}\in\bm{\mathcal{X}}_{n}:\exists j\text{ such that }y_{j}=-x_{j}=\nu\\}$, which implies that $|\mathcal{N}_{\nu}(\mathbf{x})|=n_{-\nu}(\mathbf{x})$ (with the analogous implication for $\mathcal{N}_{-\nu}(\mathbf{y})$). The acceptance probability $\alpha_{\nu}$ thus depends on an additional term $n_{-\nu}(\mathbf{x})/n_{\nu}(\mathbf{y})$ compared to that in the MH sampler. This term may have an negative impact by decreasing the acceptance probability. This represents in fact the price to pay for using the lifted sampler Algorithm 2 (including Algorithm 1 as a special case): the reversible sampler is allowed to backtrack, which makes the sizes of the neighbourhoods constant, whereas it is the opposite for Algorithm 2. The size of the neighbourhoods diminishes in the lifted sampler as the chain moves further in a direction (making the neighbourhoods in the reverse direction bigger and bigger). The impact is alleviated when $n$ is large and the mass of $\pi_{n}$ concentrates in the interior of $\bm{\mathcal{X}}_{n}$ in an area where $|n_{-\nu}(\mathbf{x})-n/2|\leq\kappa$, where $\kappa$ is a positive integer. Indeed, $n_{\nu}(\mathbf{y})=n_{\nu}(\mathbf{x})+1=n-n_{-\nu}(\mathbf{x})+1$, which implies that $\alpha_{\nu}(\mathbf{x},\mathbf{y})\approx 1\wedge\pi(\mathbf{y})/\pi(\mathbf{x})=\alpha(\mathbf{x},\mathbf{y})$. In fact, in an ideal situation where $|\mathcal{N}_{-1}(\mathbf{x})|=|\mathcal{N}_{+1}(\mathbf{x})|=|\mathcal{N}(\mathbf{x})|/2$ for all $\mathbf{x}\in\bm{\mathcal{X}}_{n}$ (which is impossible for the important example with types A and B), a corollary of Theorem 3.17 in Andrieu and Livingstone (2019) establishes that Algorithm 2 with any function $\rho_{\nu}$ dominates the MH algorithm in terms of asymptotic variances. In particular, Algorithm 1 dominates the MH algorithm. Before presenting this corollary, we define the transition kernel simulated by Algorithm 2 with $q_{\mathbf{x},\nu}:=\mathcal{U}\\{\mathcal{N}_{\nu}(\mathbf{x})\\},\mathbf{x}\in\bm{\mathcal{X}}_{n},\nu\in\\{-1,+1\\}$, as $P_{\rho,n}$ and that simulated by the MH sampler with $q_{\mathbf{x}}:=\mathcal{U}\\{\mathcal{N}(\mathbf{x})\\},\mathbf{x}\in\bm{\mathcal{X}}_{n}$, as $P_{\text{MH},n}$. ###### Corollary 2. If (a) $\bm{\mathcal{X}}_{n}$ is finite, (b) $|\mathcal{N}_{-1}(\mathbf{x})|=|\mathcal{N}_{+1}(\mathbf{x})|=|\mathcal{N}(\mathbf{x})|/2=n^{*}/2$ for all $\mathbf{x}\in\bm{\mathcal{X}}_{n}$, where $n^{*}$ is a positive integer that does not depend on $\mathbf{x}$ (but that may depend on $n$), then for any $f\in\mathcal{L}_{2}^{*}(\bar{\pi})$ and $n$, $\mathrm{var}(f,P_{\rho,n})\leq\mathrm{var}(f,P_{\text{MH},n}).$ ###### Proof. See Section 7. ∎ In light of the above, one might expect the inequality to (approximately) hold (up to an error term) when the mass is highly concentrated on the points $\mathbf{x}$ that are not too far from the center of the domain, where here the notion of centrality is defined in terms of the distance between $n_{-1}(\mathbf{x})$ or $n_{+1}(\mathbf{x})$ to $n/2$, suggesting that the lifted sampler outperforms the reversible MH algorithm in this situation. The rest of the section is dedicated to the introduction of conditions under which this statement is true. The key argument in proving Corollary 2 is to show that $\displaystyle q_{\mathbf{x}}(\mathbf{y})\,\alpha(\mathbf{x},\mathbf{y})=(1/2)\,q_{\mathbf{x},+1}(\mathbf{y})\,\alpha_{+1}(\mathbf{x},\mathbf{y})+(1/2)\,q_{\mathbf{x},-1}(\mathbf{y})\,\alpha_{-1}(\mathbf{x},\mathbf{y}),$ (7) for all $\mathbf{x}$ and $\mathbf{y}$. Indeed, once this is done, Theorem 3.17 in Andrieu and Livingstone (2019) can be directly applied, which yields the result. This in fact implies that once one has designed a lifted sampler, it is possible to identify its non-lifted counterpart through (7), and to establish that the latter is inferior. In particular, we can establish that the sampler that flips a coin at each iteration to next decide which PMF to use between $q_{\mathbf{x},+1}$ and $q_{\mathbf{x},-1}$ to generate a proposal $\mathbf{y}$ that will be subject to approval using $\alpha_{+1}(\mathbf{x},\mathbf{y})$ or $\alpha_{-1}(\mathbf{x},\mathbf{y})$ is inferior. Denote by $P_{\text{rev.},n}$ the Markov kernel simulated by this algorithm. Now, what we would ideally do is to show a Peskun-Tierney ordering (Peskun, 1973; Tierney, 1998) between $P_{\text{rev.},n}$ and $P_{\text{MH},n}$ to establish the domination of the lifted sampler over the reversible MH algorithm. Such an ordering is difficult to obtain as one needs to show that for any pair $(\mathbf{x},\mathbf{y})$ such that $\mathbf{x}\neq\mathbf{y}$, $P_{\text{rev.},n}(\mathbf{x},\mathbf{y})\geq P_{\text{MH},n}(\mathbf{x},\mathbf{y})$. A more general ordering is presented in Zanella (2019): if $P_{\text{rev.},n}(\mathbf{x},\mathbf{y})\geq\omega P_{\text{MH},n}(\mathbf{x},\mathbf{y})$ for all $\mathbf{x}\neq\mathbf{y}$, then $\mathrm{var}(f,P_{\text{rev.},n})\leq\mathrm{var}(f,P_{\text{MH},n})/\omega+((1-\omega)/\omega)\mathbb{V}\mathrm{ar}f(\mathbf{X})$, where $\omega$ is a positive constant. Note that $f$ is a function of $\nu$ as well, but because of the restriction $f(\mathbf{x},-1)=f(\mathbf{x},+1)$, $\nu$ can be treated as a constant. We show in this section that if $P_{\text{rev.},n}(\mathbf{x},\mathbf{y})\geq\omega P_{\text{MH},n}(\mathbf{x},\mathbf{y})$ with $\omega$ close to 1 for all $\mathbf{x}\neq\mathbf{y}$ belonging to a specific set having a high probability, then $\mathrm{var}(f,P_{\text{rev.},n})\leq\mathrm{var}(f,P_{\text{MH},n})+\text{small error term}$ (under regularity conditions). We start by defining this set: $\displaystyle\bm{\mathcal{X}}_{\varphi(n)}:=\\{\mathbf{x}\in\bm{\mathcal{X}}_{n}:n/2-\varphi(n)\leq n_{-1}(\mathbf{x}),n_{+1}(\mathbf{x})\leq n/2+\varphi(n)\\},$ (8) where $\varphi$ is a function such that $0\leq\varphi(n)<n/2$ and, for $\epsilon>0$, there exists $N>0$ such that for all $n\geq N$, we have that $\varphi(n)/n<\epsilon$. For $\mathbf{x}\neq\mathbf{y}$ belonging to $\bm{\mathcal{X}}_{\varphi(n)}$, if we consider $\omega_{n}$ now a function of $n$, it is possible to establish that: for $\epsilon>0$, there exists a $N>0$ such that for all $n\geq N$, $P_{\text{rev.},n}(\mathbf{x},\mathbf{y})\geq\omega_{n}P_{\text{MH},n}(\mathbf{x},\mathbf{y})$ with $1-\omega_{n}<\epsilon$. The explicit form of $\omega_{n}$ is $\displaystyle\omega_{n}:=\left(1+\frac{\varphi(n)}{n/2}\right)^{-2}\left(1-\frac{\varphi(n)}{n/2}\right).$ (9) One can imagine that if: the mass concentrates on $\bm{\mathcal{X}}_{\varphi(n)}$, the chains do not often leave this set, and when they do they do not take too much time to come back, then the asymptotic variances should not be too different to those of chains with stationary distribution $\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$, which assigns null mass on $\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}$ and is thus the normalised version of $\pi_{n}$ on $\bm{\mathcal{X}}_{\varphi(n)}$: $\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}(\mathbf{x}):=\begin{cases}\pi_{n}(\mathbf{x})\big{/}\sum_{\mathbf{x}^{\prime}\in\bm{\mathcal{X}}_{\varphi(n)}}\pi_{n}(\mathbf{x}^{\prime})&\text{if}\quad\mathbf{x}\in\bm{\mathcal{X}}_{\varphi(n)},\cr 0&\text{if}\quad\mathbf{x}\notin\bm{\mathcal{X}}_{\varphi(n)}.\end{cases}$ This is what we obtain assuming such a behaviour for the chains generated by $P_{\text{rev.},n}$ and $P_{\text{MH},n}$. We also require that the chains generated by the Markov kernels with the same proposal mechanisms as $P_{\text{rev.},n}$ and $P_{\text{MH},n}$, but whose stationary distribution is $\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$, mix sufficiently well (in a sense that will be made precise). We denote by $\tilde{P}_{\text{rev.},n}$ and $\tilde{P}_{\text{MH},n}$ these Markov kernels. ###### Theorem 1. Pick $f\in\mathcal{L}_{2}^{*}(\bar{\pi})$ and consider that all states $\mathbf{x}\in\bm{\mathcal{X}}_{n}$ are such that $\mathbf{x}=(x_{1},\ldots,x_{n})$ with $x_{i}\in\\{-1,+1\\}$. If, for $\epsilon>0$, there exists $N>0$ such that for all $n\geq N$ it is possible to choose a constant that may depend on $n$, $\varrho(n)$, with (a) $\sum_{k=\varrho(n)+1}^{\infty}\left\langle f,P^{k}f\right\rangle<\epsilon$, for all $P\in\\{P_{\text{rev.},n},\tilde{P}_{\text{rev.},n},P_{\text{MH},n},\tilde{P}_{\text{MH},n}\\}$, (b) $\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cup_{m=0}^{k}A_{m}^{\mathsf{c}}(\bm{\mathcal{X}}_{\varphi(n)-1})}]<\epsilon$, for all $P\in\\{P_{\text{rev.},n},P_{\text{MH},n}\\}$, where it is considered that the chain starts at stationarity and evolves using $P$ (and $\mathbb{E}[f(\mathbf{X}(k))]=0$ without loss of generality) and $A_{m}(\bm{\mathcal{X}}_{\varphi(n)-1}):=\\{\mathbf{X}(m)\in\bm{\mathcal{X}}_{\varphi(n)-1}\\}$ (see (8) for the definitions of $\bm{\mathcal{X}}_{\varphi(n)-1}$ and $\varphi(n)$), (c) $(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cap_{m=0}^{k}A_{m}(\bm{\mathcal{X}}_{\varphi(n)-1})}]<\epsilon$, for all $P\in\\{P_{\text{rev.},n},P_{\text{MH},n}\\}$, (d) $(1/\omega_{n}-1)\mathrm{var}(f,\tilde{P}_{\text{MH},n})<\epsilon$ and $((1-\omega_{n})/\omega_{n})\mathbb{V}\mathrm{ar}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}f(\mathbf{X})<\epsilon$, where $\omega_{n}$ is defined in (9) and $\mathbb{V}\mathrm{ar}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}$ denotes a variance computed with respect to $\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$, then there exists a positive constant $\kappa$ such that $\mathrm{var}(f,P_{\rho,n})\leq\mathrm{var}(f,P_{\text{rev.},n})\leq\mathrm{var}(f,P_{\text{MH},n})+\kappa\epsilon.$ ###### Proof. See Section 7. ∎ There are several assumptions involved in Theorem 1. But, to put this into perspective, Assumptions (c) and (d) are automatically verified if $\mathrm{var}(f,\tilde{P}_{\text{MH},n})$ and $\mathbb{V}\mathrm{ar}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}f(\mathbf{X})$ are bounded by constants that do not depend on $n$. Assumptions (a) and (b) are really the crucial ones. ### 3.2 Locally informed proposal In this section, we discuss and analyse the use of locally-balanced proposal distributions, as defined by Zanella (2019) in the reversible MH framework as $\displaystyle q_{\mathbf{x}}(\mathbf{y}):=g\left(\frac{\pi_{n}(\mathbf{y})}{\pi_{n}(\mathbf{x})}\right)\bigg{/}c_{n}(\mathbf{x}),\quad\mathbf{y}\in\mathcal{N}(\mathbf{x}),$ (10) where $c_{n}(\mathbf{x})$ represents the normalising constant, i.e. $c_{n}(\mathbf{x}):=\sum_{\mathbf{x}^{\prime}\in\mathcal{N}(\mathbf{x})}g\left(\pi_{n}(\mathbf{x}^{\prime})/\pi_{n}(\mathbf{x})\right)$, and $g$ is a strictly positive continuous function such that $g(x)/g(1/x)=x$. Note that we used the same notation as in Section 3.1 for the proposal distribution; this is to simplify. We will use the same notation as in Section 3.1 for the proposal distribution of the lifted sampler and for the Markov kernels as well. In this section, it will be implicit that the proposal distributions are informed proposals and the Markov kernels are those induced by these informed proposals. Note also that we again highlight the dependencies on $n$ of some terms that will appear in our analysis. Such a function $g$ defined in (10) implies that the acceptance probability in the MH algorithm is given by $\alpha(\mathbf{x},\mathbf{y})=1\wedge\frac{\pi_{n}(\mathbf{y})\,q_{\mathbf{y}}(\mathbf{x})}{\pi_{n}(\mathbf{x})\,q_{\mathbf{x}}(\mathbf{y})}=1\wedge\frac{c_{n}(\mathbf{x})}{c_{n}(\mathbf{y})}.$ Zanella (2019) shows that $c_{n}(\mathbf{x})/c_{n}(\mathbf{y})\longrightarrow 1$ as $n\longrightarrow\infty$ under some assumptions. More precisely, the author defines $\mathbf{x}:=(x_{1},\ldots,x_{n})$ and considers that at any given iteration, only a small fraction of the $n$ components is proposed to change values. The result holds when the random variables $(X_{1},\ldots,X_{n})$ exhibit a structure of conditional independence, which implies that the normalising constants $c_{n}(\mathbf{x})$ and $c_{n}(\mathbf{y})$ share a lot of terms. This is again a consequence of the backtracking of the reversible sampler and is thus in contrast with what we observe for the lifted algorithm. Two choices for $g$ are $g(x)=\sqrt{x}$ and $g(x)=x/(1+x)$, the latter being called the Barker proposal in reference to Barker (1965)’s acceptance probability choice. The advantage of the latter choice is that it is a bounded function of $x$, which stabilises the normalising constants and thus the acceptance probability. This is shown in Zanella (2019) and in the continuous random variable case in Livingstone and Zanella (2019). We use this function in our numerical analyses. The proposal distribution in the lifted sampler is given by $q_{\mathbf{x,\nu}}(\mathbf{y}):=g\left(\frac{\pi_{n}(\mathbf{y})}{\pi_{n}(\mathbf{x})}\right)\bigg{/}c_{n,\nu}(\mathbf{x}),\quad\mathbf{y}\in\mathcal{N}_{\nu}(\mathbf{x}),$ where $c_{n,\nu}(\mathbf{x})$ is the normalising constant. In this case, $\displaystyle\alpha_{\nu}(\mathbf{x},\mathbf{y}):=1\wedge\frac{\pi_{n}(\mathbf{y})\,q_{\mathbf{y},-\nu}(\mathbf{x})}{\pi_{n}(\mathbf{x})\,q_{\mathbf{x},\nu}(\mathbf{y})}=1\wedge\frac{c_{n,\nu}(\mathbf{x})}{c_{n,-\nu}(\mathbf{y})}.$ (11) There are two main differences with the reversible sampler. Firstly, the normalising constants $c_{n,\nu}(\mathbf{x})$ and $c_{n,-\nu}(\mathbf{y})$ are sums with (in general) not the same number of terms. Consider again, for ease of presentation of the analysis, the important example described in Section 1.1 where each component $x_{i}$ of $\mathbf{x}=(x_{1},\ldots,x_{n})$ can be of two types and more specifically the special case of Ising model. We know that in this case $c_{n,\nu}(\mathbf{x})$ is a sum of $n_{-\nu}(\mathbf{x})$ terms (see (6)), whereas $c_{n,-\nu}(\mathbf{y})$ is a sum of $n_{\nu}(\mathbf{y})$ terms. The second main difference is that, in the MH sampler, it is proposed to flip one of the $n_{-1}(\mathbf{x})$ components to $+1$ or one of the $n_{+1}(\mathbf{x})$ components to $-1$, and $c_{n}(\mathbf{x})$ is formed from these proposals. The constant $c_{n}(\mathbf{y})$ is also formed from proposals to flip components to $+1$ or $-1$ with $n_{-1}(\mathbf{y})$ and $n_{+1}(\mathbf{y})$ close to $n_{-1}(\mathbf{x})$ and $n_{+1}(\mathbf{x})$ (and this is why the ratio of the two constants converges to 1 provided that there exists a structure of conditional independence). In contrast, $c_{n,\nu}(\mathbf{x})$ is formed from proposals to flip one of the $n_{-\nu}(\mathbf{x})$ components to $\nu$ and $c_{n,-\nu}(\mathbf{y})$ from proposals to flip one of the $n_{\nu}(\mathbf{y})=n_{\nu}(\mathbf{x})+1$ components to $-\nu$; the compositions of these constants are thus fundamentally opposite. There is therefore no guarantee that $c_{n,\nu}(\mathbf{x})/c_{n,-\nu}(\mathbf{y})\longrightarrow 1$ even under the conditions stated in Zanella (2019). Nevertheless, there exists as in the previous section an ideal situation in which the lifter sampler outperforms the reversible MH algorithm. ###### Corollary 3. If (a) $\bm{\mathcal{X}}_{n}$ is finite, (b) $c_{n,-1}(\mathbf{x})=c_{n,+1}(\mathbf{x})=c_{n}(\mathbf{x})/2=c_{n}^{*}/2$ for all $\mathbf{x}\in\bm{\mathcal{X}}_{n}$, where $c_{n}^{*}$ is a positive constant that does not depend on $\mathbf{x}$ (but that may depend on $n$), then for any $f\in\mathcal{L}_{2}^{*}(\bar{\pi})$ and $n$, $\mathrm{var}(f,P_{\rho,n})\leq\mathrm{var}(f,P_{\text{MH},n}).$ ###### Proof. Analogous to that of Corollary 2. ∎ A sufficient condition for Assumption (b) to be verified is: $|\mathcal{N}_{-1}(\mathbf{x})|=|\mathcal{N}_{+1}(\mathbf{x})|=|\mathcal{N}(\mathbf{x})|/2=n^{*}/2$ (Assumption (b) in Corollary 2) and $\frac{1}{n^{*}/2}\sum_{\mathbf{x}^{\prime}\in\mathcal{N}_{-1}(\mathbf{x})}g\left(\frac{\pi_{n}(\mathbf{x}^{\prime})}{\pi_{n}(\mathbf{x})}\right)=\frac{1}{n^{*}/2}\sum_{\mathbf{x}^{\prime}\in\mathcal{N}_{+1}(\mathbf{x})}g\left(\frac{\pi_{n}(\mathbf{x}^{\prime})}{\pi_{n}(\mathbf{x})}\right)=\frac{1}{n^{*}}\sum_{\mathbf{x}^{\prime}\in\mathcal{N}(\mathbf{x})}g\left(\frac{\pi_{n}(\mathbf{x}^{\prime})}{\pi_{n}(\mathbf{x})}\right)=\mu,$ for all $\mathbf{x}\in\bm{\mathcal{X}}_{n}$, where $\mu$ is a positive constant. We thus notice that the acceptance probabilities can be directly rewritten in terms of averages when $|\mathcal{N}_{-1}(\mathbf{x})|=|\mathcal{N}_{+1}(\mathbf{x})|=|\mathcal{N}(\mathbf{x})|/2=n^{*}/2$ and that an additional condition to Assumption (b) in Corollary 2 is sufficient in the locally informed case for ordering the asymptotic variances. This thus allows establishing a connection with the uniform case. As in the previous section, it is possible to derive conditions under which the inequality in Corollary 3 holds approximately. They are based as before on the definition of a set, which in this case involves states $\mathbf{x}$ that are such that $c_{n,-1}(\mathbf{x})$ and $c_{n,+1}(\mathbf{x})$ are close to $c_{n}^{*}/2$. These states do not have to be in an area such that $n_{-1}(\mathbf{x})$ and $n_{+1}(\mathbf{x})$ are close to $n/2$, but in return the mass have to be (in some sense) evenly spread out in the area to which they belong. We now define this set: $\displaystyle\bm{\mathcal{X}}_{\varphi(n)}:=\\{\mathbf{x}\in\bm{\mathcal{X}}_{n}:c_{n}^{*}/2-\varphi(n)\leq c_{n,-1}(\mathbf{x}),c_{n,+1}(\mathbf{x}),c_{n}(\mathbf{x})/2\leq c_{n}^{*}/2+\varphi(n)\\},$ (12) where $\varphi$ is in this section a function such that $0\leq\varphi(n)<c_{n}^{*}/2$ and, for $\epsilon>0$, there exists $N>0$ such that for all $n\geq N$, we have that $\varphi(n)/c_{n}^{*}<\epsilon$. We consider to simplify that for any $\mathbf{x},\mathbf{y}\in\bm{\mathcal{X}}_{\varphi(n)}$ there exists a probable path from $\mathbf{x}$ to $\mathbf{y}$ generated by $P_{\text{MH},n}$ (and marginally for $P_{\rho,n}$) with all intermediate states belonging to $\bm{\mathcal{X}}_{\varphi(n)}$ as well. We now define a restricted version of $\bm{\mathcal{X}}_{\varphi(n)}$ for which from any state $\mathbf{x}\in\bm{\mathcal{X}}_{\varphi(n)}$, all the possible proposals $\mathbf{y}$ belong to $\bm{\mathcal{X}}_{\varphi(n)}$ as well; denote this set by $\bm{\mathcal{X}}_{\varphi(n)}^{0}$. For $\mathbf{x}\neq\mathbf{y}$ belonging to $\bm{\mathcal{X}}_{\varphi(n)}$, it is possible to establish that: for $\epsilon>0$, there exists a $N>0$ such that for all $n\geq N$, $P_{\text{rev.},n}(\mathbf{x},\mathbf{y})\geq\omega_{n}P_{\text{MH},n}(\mathbf{x},\mathbf{y})$ with $1-\omega_{n}<\epsilon$. The explicit form of $\omega_{n}$ is $\displaystyle\omega_{n}:=\left(1+\frac{\varphi(n)}{c_{n}^{*}/2}\right)^{-3}\left(1-\frac{\varphi(n)}{c_{n}^{*}/2}\right)^{3}.$ (13) We are now ready to present the analogous result to Theorem 1, in which here $\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$ is the normalised version of $\pi_{n}$ on $\bm{\mathcal{X}}_{\varphi(n)}$ defined in (12). ###### Theorem 2. Pick $f\in\mathcal{L}_{2}^{*}(\bar{\pi})$ and consider that all states $\mathbf{x}\in\bm{\mathcal{X}}_{n}$ are such that $\mathbf{x}=(x_{1},\ldots,x_{n})$ with $x_{i}\in\\{-1,+1\\}$. If, for $\epsilon>0$, there exists $N>0$ such that for all $n\geq N$ it is possible to choose a constant that may depend on $n$, $\varrho(n)$, with (a) $\sum_{k=\varrho(n)+1}^{\infty}\left\langle f,P^{k}f\right\rangle<\epsilon$, for all $P\in\\{P_{\text{rev.},n},\tilde{P}_{\text{rev.},n},P_{\text{MH},n},\tilde{P}_{\text{MH},n}\\}$, (b) $\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cup_{m=0}^{k}A_{m}^{\mathsf{c}}(\bm{\mathcal{X}}_{\varphi(n)}^{0})}]<\epsilon$, for all $P\in\\{P_{\text{rev.},n},P_{\text{MH},n}\\}$, where it is considered that the chain starts at stationarity and evolves using $P$ (and $\mathbb{E}[f(\mathbf{X}(k))]=0$ without loss of generality) and $A_{m}(\bm{\mathcal{X}}_{\varphi(n)}^{0}):=\\{\mathbf{X}(m)\in\bm{\mathcal{X}}_{\varphi(n)}^{0}\\}$, (c) $(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cap_{m=0}^{k}A_{m}(\bm{\mathcal{X}}_{\varphi(n)}^{0})}]<\epsilon$, for all $P\in\\{P_{\text{rev.},n},P_{\text{MH},n}\\}$, (d) $(1/\omega_{n}-1)\mathrm{var}(f,\tilde{P}_{\text{MH},n})<\epsilon$ and $((1-\omega_{n})/\omega_{n})\mathbb{V}\mathrm{ar}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}f(\mathbf{X})<\epsilon$, where $\omega_{n}$ is defined in (13), then there exists a positive constant $\kappa$ such that $\mathrm{var}(f,P_{\rho,n})\leq\mathrm{var}(f,P_{\text{rev.},n})\leq\mathrm{var}(f,P_{\text{MH},n})+\kappa\epsilon.$ ###### Proof. Analogous to that of Theorem 1. ∎ It is possible to establish a connection with Theorem 1 as we did for Corollary 3 with Corollary 2. Consider indeed that the set $\bm{\mathcal{X}}_{\varphi(n)}$ is comprised of states $\mathbf{x}$ with $n_{-1}(\mathbf{x})$ and $n_{+1}(\mathbf{x})$ close to $n/2$ and $(1/n_{+1}(\mathbf{x}))\sum_{\mathbf{x}^{\prime}\in\mathcal{N}_{-1}(\mathbf{x})}g(\pi(\mathbf{x}^{\prime})/\pi(\mathbf{x}))$, $(1/n_{-1}(\mathbf{x}))\sum_{\mathbf{x}^{\prime}\in\mathcal{N}_{+1}(\mathbf{x})}g(\pi(\mathbf{x}^{\prime})/\pi(\mathbf{x}))$ and $(1/n)\sum_{\mathbf{x}^{\prime}\in\mathcal{N}(\mathbf{x})}g(\pi(\mathbf{x}^{\prime})/\pi(\mathbf{x}))$ all close to a positive constant $\mu$. We thus notice that under a precise sense of what we mean by close to, this special case fits within the definition of $\bm{\mathcal{X}}_{\varphi(n)}$ (12), under an additional condition compared to the definition of the set in the previous section (8). ### 3.3 Computational costs We provide in this section an overview of the computational costs associated to using the proposal distributions described in Sections 3.1 and 3.2. The uniform distribution is the least expensive: at each iteration, one has to generate from a uniform and then evaluate the acceptance probability which requires the computation of a ratio $\pi(\mathbf{y})/\pi(\mathbf{x})$. Consider that the cost of the latter is the important one, in the sense that all the other costs are comparatively negligible. The approach of Zanella (2019) thus costs twice as much, if we assume that the cost of computing any ratio is the same and that the ratios $\pi(\mathbf{x}^{\prime})/\pi(\mathbf{x})$ for all $\mathbf{x}^{\prime}$ in the neighbourhood are all computed in parallel. Indeed, these ratios are necessary to generate the proposal $\mathbf{y}$, but once the latter has been generated, the process has to be repeated for the reverse move. This is true for the reversible MH sampler and Algorithm 1. Therefore, if the informed proposal leads to a sampler at least twice as effective (in terms of ESS for instance), then it is beneficial. It is the case in all our numerical experiments. Note that in light of the above, implementing the reversible MH sampler or Algorithm 1 costs essentially the same. Algorithm 2 is more costly. When used with a uniform distribution and $\rho:=\rho^{*}$ (5), at each iteration, $|\mathcal{N}_{\nu}(\mathbf{x})|$ ratio evaluations are required to compute $T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$ (4), and it is afterwards required to compute $T_{-\nu}(\mathbf{x},\bm{\mathcal{X}})$ from time to time. This makes the implementation cost somewhere in between that of Algorithm 1 with a uniform and Algorithm 1 using the approach of Zanella (2019). When Algorithm 2 is used with an informed proposal and $\rho:=\rho^{*}$ (5), the cost may explode. Consider that parallel computing is used to compute any normalising constant $c_{\nu}(\mathbf{x})$, but that the normalising constants are computed sequentially, then at each iteration it is required to compute at least $1+|\mathcal{N}_{\nu}(\mathbf{x})|$ normalising constants compared with two in Algorithm 1. The computation time per iteration will thus roughly be at least $(1+|\mathcal{N}_{\nu}(\mathbf{x})|)/2$ times larger. ## 4 Numerical experiments We first consider in Section 4.1 simulation of an Ising model and use this as a toy example for which we can control the dimension, the roughness of the target and where the mass concentrates to show how the performance of the lifted and non-lifted samplers varies when these parameters change. In Section 4.2, we evaluate their performance when employed to solve a real variable selection problem. ### 4.1 Ising model For the two-dimensional model, the ambiant space $(V_{\eta},E_{\eta})$ is a $\eta\times\eta$ square lattice regarded here as a square matrix in which each element takes either the value $-1$ or $+1$. We write each state as a vector as before: $\mathbf{x}=(x_{1},\ldots,x_{n})$, where $n=\eta^{2}$. The states can be encoded as for instance: the values of the components on the first line are $x_{1},\ldots,x_{\eta}$, those on the second line $x_{\eta+1},\ldots,x_{2\eta}$, and so on. The PMF is given by $\pi(\mathbf{x})=\frac{1}{Z}\exp\left(\sum_{i}\alpha_{i}x_{i}+\lambda\sum_{\langle ij\rangle}x_{i}x_{j}\right),$ where $\alpha_{i}\in\operatorname{\mathbb{R}}$ and $\lambda>0$ are known parameters, $Z$ is the normalising constant and the notation $\langle ij\rangle$ indicates that sites i and j are nearest neighbours. We first consider a base target distribution for which $n=50^{2}$, the spatial correlation is moderate and more precisely $\lambda:=0.5$, and which has the external field (the values for the $\alpha_{i}$’s) presented in Figure 2. Figure 2: External field of the base target We generated the $\alpha_{i}$ independently as follows: $\alpha_{i}=-\mu+\epsilon_{i}$ if the column index is smaller than or equal to $\ell=\lfloor\eta/2\rfloor$ and $\alpha_{i}=\mu+\epsilon_{i}$ otherwise, where $\mu=1$, the $\epsilon_{i}$ are independent uniform random variables on the interval $(-0.1,+0.1)$ and $\lfloor\cdot\rfloor$ is the floor function. In the simulation study, while keeping the other parameters fixed, we gradually increase $\eta$ from 50 to 500 to observe the impact of dealing with larger systems. Next, while keeping the other parameters fixed (with $\eta=50$), we gradually increase the value of $\mu$ from 1 to 3. This has for effect of increasing the contrast so that it is clearer which values the $x_{i}$ should take, thus making the target rougher and concentrated on fewer configurations. One could vary $\lambda$ as well, but this would also make the target rougher and concentrated on fewer configurations. We observe the impact on Algorithm 1 with uniform and informed proposal distributions, and their non-lifted MH counterparts. For such a simulation study, it would be simply too long to obtain the results for Algorithm 2 with $\rho_{\nu}^{*}$ (5). We tried to vary the value of $\ell$ to observe what happens with the acceptance rates for the uniform lifted sampler when the ratios $n_{-\nu}(\mathbf{x})/n_{\nu}(\mathbf{y})$ (see (6)) are far from 1. The impact in this example is however not the one expected: the acceptance rates increase instead of deteriorating. To see why, consider for instance that $\ell=5$. The ratios $n_{-\nu}(\mathbf{x})/n_{\nu}(\mathbf{y})$ with $\nu=-1$ are on average around 9 (45 columns with $+1$’s and 5 with $-1$’s). With $\nu=-1$, it is tried to flip a bit from $+1$ to $-1$ and $\pi(\mathbf{y})/\pi(\mathbf{x})$ is thus multiplied by a factor of around 9. It is likely that this bit is on the yellow side (see Figure 2). For such a move, $\pi(\mathbf{y})/\pi(\mathbf{x})$ is often around $\exp(-2(1+0.5\times 4))=0.002$. Therefore, it is more likely to accept this move compared to in the reversible MH sampler (in the MH sampler $\pi(\mathbf{y})/\pi(\mathbf{x})$ is not multiplied by 9). When $\nu=+1$, the multiplicative factor is thus around $1/9$, but it is relatively likely that the proposal will be to flip a bit from $-1$ to $+1$ on the yellow side, because there are some bits with the value $-1$ on this larger side, and this move is often automatically accepted (because $\pi(\mathbf{y})/\pi(\mathbf{x})$ is often around $1/0.002)$. Note that these conclusions are not in contradiction with our analysis of Section 3.1, because what we observed here is specific to the Ising model. We do not present the results because the graph is uninteresting: the performance is essentially constant for the informed samplers and that of the uniform ones is so low that we do not see the ESS vary. We present the other results in Figure 3. They are based on 1,000 independent runs of 100,000 iterations for each algorithm and each value of $\mu$ and $\eta$, with burn-ins of 10,000. For each run, an ESS per iteration is computed for $f(\mathbf{x},-1)=f(\mathbf{x},+1)=\sum_{i}x_{i}$ and then the results are averaged out. This function is proportional to the magnetisation. Monitoring such a statistic is relevant as a quicker variation of its value (leading to a higher ESS) indicates that the whole state-space is explored quicker. For the base target (represented by the starting points on the left of the graphs in Figure 3), the mass is concentrated on a manifold of several configurations, which allows for persistent movement for informed samplers. The lifted one indeed takes advantage of this; it is approximately 7 times more efficient than its non-lifted counterpart. The gap widens as $\eta$ increases; it is approximately 20 and 70 times more efficient when $\eta$ is 3.2 and 10 times larger (i.e. when $n$ is 10 and 100 times larger), respectively. We observed that the ratio of ESSs increases linearly with $\eta$. The non-informed samplers both perform poorly (the lines are on top of each other). As $\mu$ increases, the target becomes rougher and concentrated on fewer configurations. When the roughness and concentration level are too severe the performance of the lifted informed sampler stagnates, whereas that of the non- lifted MH sampler continues to improve. There are two reasons for this. Firstly, the acceptance rates deteriorate more rapidly for the lifted than the non-lifted sampler (as a consequence of the difference in the acceptance probability, see (11)). Secondly, when the mass is concentrated on few configurations, it leaves not much room for persistent movement for the lifted sampler. The latter thus loses its avantage. Again, the non-informed samplers both perform poorly (the lines are on top of each other). $\begin{array}[]{cc}\includegraphics[width=216.81pt]{Fig_Ising_n.pdf}&\includegraphics[width=216.81pt]{Fig_Ising_mu.pdf}\cr\textbf{(a)}&\textbf{(b)}\end{array}$ Figure 3: ESS per iteration of $f(\mathbf{x},-1)=f(\mathbf{x},+1)=\sum_{i}x_{i}$ for Algorithm 1 with uniform and informed proposal distributions and their non-lifted MH counterparts when: (a) $\eta$ increases from 50 to 500 and the other parameters are kept fixed ($\mu=1$, $\lambda=0.5$ and $\ell=25$); (b) $\mu$ increases from 1 to 3 and the other parameters are kept fixed ($\eta=50$, $\lambda=0.5$ and $\ell=25$) ### 4.2 Variable selection: US crime data A study of crime rate was first presented in Erhlich (1973) and then an expended and corrected version appeared in Vandaele (1978) in which corrected data were provided; the topic was more precisely on the connection between crime rate and 15 covariates (some were added by Vandaele (1978)) such as percentage of males of age between 14 and 23 and mean years of schooling in a given state. The data were indeed aggregated by state and were from 47 U.S. states in 1960. They were analysed in several statistics papers (see, e.g., Raftery et al. (1997)) and are available in the R package MASS. The data are modelled using the usual linear regression with normal errors. Here we set the prior distribution of the regression coefficients and scaling of the errors to be, conditionally on a model, the non-informative Jeffreys prior. It is proved in Gagnon (2019) that a simple modification to the uniform prior on the model random variable (represented here by $\mathbf{X}$) prevents the Jeffreys–Lindley (Lindley, 1957; Jeffreys, 1967) paradox from arising. With the resulting likelihood function and prior density on the parameters, the latter can be integrated out. It is thus possible to evaluate the exact marginal posterior probability of any of the $2^{15}=$ 32,768 models, up to a normalising constants. This allows us to implement the MH sampler with the locally informed proposal distribution of Zanella (2019) and its lifted counterparts (Algorithm 1 and Algorithm 2 with $\rho_{\nu}^{*}$ (5)). In the previous statistical studies, it was noticed that the mass is diffused over several models, so that we expect the lifted chains to exhibit persistent movement (as seen in Figure 1) and to perform well. To simplify the presentation, we do not show the performance of the uniform samplers because, as in the previous section, it is very poor. The performances of the algorithms are summarised in Figure 4. The results are based on 1,000 independent runs of 10,000 iterations for each algorithm, with burn-ins of 1,000. Each run is started from a distribution which approximates the target. On average, Algorithm 1 and Algorithm 2 with $\rho_{\nu}^{*}$ are $2.7$ and $3.3$ times more efficient than their non-lifted counterpart, respectively. The benefits of persistent movement thus compensate for a decrease in acceptance rates; the rate indeed decreases from 0.92 for the MH sampler to 0.71 for Algorithm 1 and Algorithm 2 with $\rho_{\nu}^{*}$ (5). Figure 4: ESS per iteration for $f(\mathbf{x},-1)=f(\mathbf{x},+1)=\sum_{i}x_{i}$ of 1,000 independent runs for the MH sampler with the locally informed proposal distribution and its lifted counterparts (Algorithm 1 and Algorithm 2 with $\rho_{\nu}^{*}$) ## 5 Lifted trans-dimensional sampler In this section, we introduce a trans-dimensional algorithm which is a non- reversible version of the popular reversible jump (RJ) algorithms introduced by Green (1995). We consider that $\bm{\mathcal{X}}$ is a model space and $\mathbf{X}$ a model indicator. The latter indicates, for instance, with 0’s and 1’s which covariates are included in the model in variable selection contexts as in Section 4.2. Such an algorithm is useful when it is not possible to integrate out the parameters, contrarily to the linear regression with normal errors and suitable priors. Examples of such situations include analyses based on linear regression with super heavy-tailed errors ensuring whole robustness (Gagnon et al., 2018) and generalised linear models and generalised linear mixed models (Forster et al., 2012). The parameters of a given model $\mathbf{x}$ are denoted by $\bm{\theta}_{\mathbf{x}}\in\bm{\Theta}_{\mathbf{x}}$. Trans-dimensional algorithms are MCMC methods that one uses to sample from a target distribution $\pi$ defined on a union of sets $\cup_{\mathbf{x}\in\bm{\mathcal{X}}}\\{\mathbf{x}\\}\times\bm{\Theta}_{\mathbf{x}}$, which corresponds in Bayesian statistics to the joint posterior of the model indicator $\mathbf{X}$ and the parameters of Model $\mathbf{X}$, $\bm{\theta}_{\mathbf{X}}$. Such a posterior allows to jointly infer about $(\mathbf{X},\bm{\theta}_{\mathbf{X}})$, or in other words, simultaneously achieve model selection and parameter estimation. In this section, we assume for simplicity that the parameters of all models are continuous random variables. Given the current state of the Markov chain $(\mathbf{x},\bm{\theta}_{\mathbf{x}})$, a trans-dimensional sampler generates the next state by first proposing a model candidate $\mathbf{y}\sim q_{\mathbf{x}}(\mathbf{y})$ and then a proposal for its corresponding parameter values. When $\mathbf{y}=\mathbf{x}$, we say that a parameter update is proposed, whereas we say that a model switch is proposed when $\mathbf{y}\neq\mathbf{x}$. Note that $\mathbf{x}\in\mathcal{N}(\mathbf{x})$, contrarily to the previous sections. This is to allow parameter updates. When a parameter update is proposed, we allow any fixed-dimensional methods to be used; we only require that the Markov kernels leave the conditional distributions $\pi(\,\cdot\mid\mathbf{x})$ invariant. When a model switch is proposed, a vector of auxiliary variables $\mathbf{u}_{\mathbf{x}\mapsto\mathbf{y}}$ is typically generated and this is followed by a proposal mechanism leading to $(\bm{\theta}^{\prime}_{\mathbf{y}},\mathbf{u}_{\mathbf{y}\mapsto\mathbf{x}})$, where $\bm{\theta}^{\prime}_{\mathbf{y}}$ is the proposal for the parameter values in Model $\mathbf{y}$. We require the whole proposal mechanism for $\bm{\theta}^{\prime}_{\mathbf{y}}$ to be valid in a RJ framework, in the sense that the model switch transitions are reversible in this framework. The non-reversibility in the lifted trans-dimensional sampler lies in the transitions for the $\mathbf{x}$ variable during model switches. More precisely, $\mathbf{y}$ is generated from $q_{\mathbf{x},\nu}(\mathbf{y})$ instead, but the proposal mechanism for $\bm{\theta}^{\prime}_{\mathbf{y}}$ during model switches is the same. We consider that the acceptance probability of these model switches in RJ is given by $\displaystyle\alpha_{\text{RJ}}((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}})):=1\wedge\frac{q_{\mathbf{y}}(\mathbf{x})}{q_{\mathbf{x}}(\mathbf{y})}\,r((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}})),$ (14) where the function $r$ depends on the method and may depend on several other variables. The algorithm is now presented in Algorithm 3. In it, we consider that the current model $\mathbf{x}$ always belongs to the neighbourhood $\mathcal{N}_{\nu}(\mathbf{x})$, regardless of the current direction $\nu$, and that $q_{\mathbf{x},-1}(\mathbf{x})=q_{\mathbf{x},+1}(\mathbf{x})$, which will typically be the case in practice. Algorithm 3 A lifted trans-dimensional sampler for partially ordered model spaces 1. 1. Generate $\mathbf{y}\sim q_{\mathbf{x},\nu}$, a PMF with support restricted to $\mathcal{N}_{\nu}(\mathbf{x})$. 2. 2.(a) If $\mathbf{y}=\mathbf{x}$, attempt a parameter update using a MCMC kernel of invariant distribution $\pi(\,\cdot\mid\mathbf{x})$ while keeping the current value of the model indicator $\mathbf{x}$ and direction $\nu$ fixed. 3. 2.(b) If $\mathbf{y}\neq\mathbf{x}$, attempt a model switch from Model $\mathbf{x}$ to Model $\mathbf{y}$. Generate $\bm{\theta}^{\prime}_{\mathbf{y}}$ using a method which is valid in RJ and $u\sim\mathcal{U}[0,1]$. If $\displaystyle u\leq\alpha_{\text{NRJ}}((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}})):=1\wedge\frac{q_{\mathbf{y},-\nu}(\mathbf{x})}{q_{\mathbf{x},\nu}(\mathbf{y})}\,r((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}})),$ (15) set the next state of the chain to $(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}},\nu)$. Otherwise, set it to $(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)$. 4. 3. Go to Step 1. ###### Proposition 2. The transition kernel of the Markov chain $\\{(\mathbf{X},\bm{\theta}_{\mathbf{X}},\nu)(m):m\in\operatorname{\mathbb{N}}\\}$ simulated by Algorithm 3 admits $\pi\otimes\mathcal{U}\\{-1,1\\}$ as invariant distribution. ###### Proof. See Section 7. ∎ The main difficulty with the implementation of trans-dimensional samplers is the construction of efficient proposal schemes for $(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}})$ during model switches. Gagnon (2019) discusses this in depth in the RJ framework. The author proposes a scheme and proves that it is possible to arbitrarily approach an asymptotic regime in which one is able to generate $\bm{\theta}^{\prime}_{\mathbf{y}}$ from $\pi(\,\cdot\mid\mathbf{y})$ (the correct conditional distribution) and evaluate exactly the ratios of marginal probabilities $\pi(\mathbf{y})/\pi(\mathbf{x})$ (and is therefore able to adequately construct $q_{\mathbf{x},\nu}$). In particular, for this scheme, $r((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}}))$ is a consistent estimator of $\pi(\mathbf{y})/\pi(\mathbf{x})$. We refer the reader to that paper for the details. We thus conclude that the marginal behaviour of $\\{(\mathbf{X},\nu)(m):m\in\operatorname{\mathbb{N}}\\}$ is the same as that of the stochastic process generated by Algorithm 1 in the asymptotic regime and considering only iterations for which model switches are proposed. All conclusions previously drawn thus hold, at least approximatively. In particular, one may analyse the same data as in Section 4.2, but using the super heavy-tailed regression of Gagnon et al. (2018) for robust inference and outlier detection. The results would be essentially the same because, as mentioned in Raftery et al. (1997), “standard diagnostic checking (see, e.g., Draper and Smith (1981)) did not reveal any gross violations of the assumptions underlying normal linear regression” and the robust method is designed for leading to similar results in the absence of outliers. We thus omit further analysis of Algorithm 3 and we do not illustrate how it performs for brevity. We nevertheless highlight that it is important for $r((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}^{\prime}_{\mathbf{y}}))$ to be a low variance estimator of $\pi(\mathbf{y})/\pi(\mathbf{x})$ as persistent movement may be interrupted otherwise, as shown in Gagnon and Doucet (2019). ## 6 Discussion In this paper, we presented and analysed generic algorithms allowing straightforward sampling from any PMF $\pi$ with a support $\bm{\mathcal{X}}$ on which a partial order can be established. The algorithms rely on the technique of lifting. We showed that these are expected to perform well when the shape of target (the level of concentration of the mass) allows for persistent movement. This is true even when the target concentrates on a manifold of the ambient space in the case where the lifting technique is combined with locally informed proposal distributions (provided that the shape of the manifold allows for persistent movement). The samplers are in particular useful for the simulation of binary random variables and variable selection. Algorithm 1 can be directly employed for the latter when the parameters of the models can be integrated out. A lifted trans-dimensional sampler for partially ordered model spaces have been introduced in Section 5 for, among others, variable selection when it is not possible to integrate out the parameters. We believe it would be interesting to continue this line of research by taking steps towards automatic generic samplers using the technique of lifting for any discrete state-space. ## References * Andrieu and Livingstone (2019) Andrieu, C. and Livingstone, S. (2019) Peskun-Tierney ordering for Markov chain and process Monte Carlo: beyond the reversible scenario. arXiv:1906.06197. * Barker (1965) Barker, A. A. (1965) Monte Carlo calculations of the radial distribution functions for a proton-electron plasma. Austral. J. Phys., 18, 119–134. * Chen et al. 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(2017) Piecewise-deterministic Markov chain Monte Carlo. arXiv:1707.05296. * Zanella (2019) Zanella, G. (2019) Informed proposals for local MCMC in discrete spaces. To appear in J. Amer. Statist. Assoc. ## 7 Proofs ###### Proof of Proposition 1. It suffices to prove that the probability to reach the state $(\mathbf{y},\nu^{\prime})$ in one step is equal to the probability of this state under the target: $\displaystyle\sum_{\mathbf{x},\nu}\pi(\mathbf{x})\,(1/2)\,P((\mathbf{x},\nu),(\mathbf{y},\nu^{\prime}))=\pi(\mathbf{y})\,(1/2).$ (16) where $P$ is the transition kernel. The probability to reach the state $(\mathbf{y},\nu^{\prime})$ from some $(\mathbf{x},\nu)$ is given by: $\displaystyle P((\mathbf{x},\nu),(\mathbf{y},\nu^{\prime}))$ $\displaystyle=T_{\nu}(\mathbf{x},\bm{\mathcal{X}})\,Q_{\mathbf{x},\nu}(\mathbf{y})\,\mathds{1}(\nu=\nu^{\prime})$ $\displaystyle\qquad+\mathds{1}(\nu=-\nu^{\prime},\mathbf{x}=\mathbf{y})\left[(\rho_{\nu}(\mathbf{x})+T_{\nu}(\mathbf{x},\bm{\mathcal{X}}))-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})\right]$ $\displaystyle\qquad+\mathds{1}(\nu=\nu^{\prime},\mathbf{x}=\mathbf{y})\left[1-\rho_{\nu}(\mathbf{x})-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})\right]$ $\displaystyle=q_{\mathbf{x},\nu}(\mathbf{y})\,\alpha_{\nu}(\mathbf{x},\mathbf{y})\,\mathds{1}(\nu=\nu)$ $\displaystyle\qquad+\mathds{1}(\nu=-\nu^{\prime},\mathbf{x}=\mathbf{y})\,\rho_{\nu}(\mathbf{x})$ $\displaystyle\qquad+\mathds{1}(\nu=\nu^{\prime},\mathbf{x}=\mathbf{y})\left[1-\rho_{\nu}(\mathbf{x})-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})\right].$ We have that $\displaystyle\pi(\mathbf{x})\,(1/2)\,P((\mathbf{x},\nu),(\mathbf{y},\nu^{\prime}))$ $\displaystyle=(1/2)\,\pi(\mathbf{y})\,q_{\mathbf{y},-\nu^{\prime}}(\mathbf{x})\,\alpha_{-\nu^{\prime}}(\mathbf{y},\mathbf{x})\,\mathds{1}(-\nu^{\prime}=-\nu)$ $\displaystyle\qquad+(1/2)\,\pi(\mathbf{y})\,\mathds{1}(-\nu^{\prime}=\nu,\mathbf{y}=\mathbf{x})\,\rho_{-\nu^{\prime}}(\mathbf{y})$ $\displaystyle\qquad+(1/2)\,\pi(\mathbf{y})\mathds{1}(-\nu^{\prime}=-\nu,\mathbf{y}=\mathbf{x})\left[1-\rho_{-\nu^{\prime}}(\mathbf{y})-T_{-\nu^{\prime}}(\mathbf{y},\bm{\mathcal{X}})\right]$ $\displaystyle=(1/2)\,\pi(\mathbf{y})\,T_{-\nu^{\prime}}(\mathbf{y},\bm{\mathcal{X}})\,Q_{\mathbf{y},-\nu^{\prime}}(\mathbf{x})\mathds{1}(-\nu^{\prime}=-\nu)$ $\displaystyle\qquad+(1/2)\,\pi(\mathbf{y})\,\mathds{1}(-\nu^{\prime}=\nu,\mathbf{y}=\mathbf{x})\left[(\rho_{-\nu^{\prime}}(\mathbf{y})+T_{-\nu^{\prime}}(\mathbf{y},\mathcal{X}))-T_{-\nu^{\prime}}(\mathbf{y},\mathcal{X})\right]$ $\displaystyle\qquad+(1/2)\,\pi(\mathbf{y})\,\mathds{1}(-\nu^{\prime}=-\nu,\mathbf{y}=\mathbf{x})\left[1-\rho_{-\nu^{\prime}}(\mathbf{y})-T_{-\nu^{\prime}}(\mathbf{y},\mathcal{X})\right],$ where we used the definition of $\alpha$ for the first term and that $\rho_{\nu}(\mathbf{x})-\rho_{-\nu}(\mathbf{x})=T_{-\nu}(\mathbf{x},\bm{\mathcal{X}})-T_{\nu}(\mathbf{x},\bm{\mathcal{X}})$ for the third term. Notice the sum on the right-hand side (RHS) is equal to the probability to reach some $(\mathbf{x},-\nu)$, starting from $(\mathbf{y},-\nu^{\prime})$: $(1/2)\,\pi(\mathbf{y})\,P((\mathbf{y},-\nu^{\prime}),(\mathbf{x},-\nu))$. Therefore, $\displaystyle\sum_{\mathbf{x},\nu}\pi(\mathbf{x})\,(1/2)\,P((\mathbf{x},\nu),(\mathbf{y},\nu^{\prime}))$ $\displaystyle=\sum_{\mathbf{x},\nu}(1/2)\,\pi(\mathbf{y})\,P((\mathbf{y},-\nu^{\prime}),(\mathbf{x},-\nu))$ $\displaystyle=(1/2)\,\pi(\mathbf{y}),$ which concludes the proof. ∎ We now present a lemma that will be useful in the next proofs. ###### Lemma 1. Let $Q$ be the Markov kernel of the Markov chain simulated by Algorithm 2, for any valid switching function $\rho_{\nu}$, $\nu\in\\{-1,1\\}$. Assume that $\bm{\mathcal{X}}$ is finite. Then, for any $f\in\mathcal{L}_{2}^{*}(\bar{\pi})$, $\lim_{\lambda\to 1}\sum_{k>0}\lambda^{k}\langle\,f,Q^{k}f\,\rangle=\sum_{k>0}\langle\,f,Q^{k}f\,\rangle\,.$ (17) ###### Proof. For each $f\in\mathcal{L}_{2}^{*}(\bar{\pi})$, define the sequence of functions $S_{n}:\lambda\mapsto\sum_{0<k\leq n}\lambda^{k}\langle\,f,Q^{k}f\,\rangle$ defined for $\lambda\in[0,1)$ and its limit $S(\lambda)=\sum_{k>0}\lambda^{k}\langle\,f,Q^{k}f\,\rangle$ (the dependance of $S_{n}$ and $S$ on $f$ and $Q$ is implicit). We now show that the partial sum $S_{n}$ converges uniformly to $S$ on $[0,1)$, and since for each $n\in\mathbb{N}$, the function $\lambda\to\lambda^{n}\langle\,f,Q^{n}f\,\rangle$ admits a limit when $\lambda\to 1$, we have that $S$ admits a limit when $\lambda\to 1$, given by $\lim_{\lambda\to 1}S(\lambda)=S(1)=\sum_{k>0}\langle\,f,Q^{k}f\,\rangle\,,$ which is Eq. (17). First, note that $\sup_{\lambda\in[0,1)}\left|S_{n}(\lambda)-S(\lambda)\right|=\sup_{\lambda\in[0,1)}\left|\sum_{k>n}\lambda^{k}\langle\,f,Q^{k}f\,\rangle\right|\leq\sup_{\lambda\in[0,1)}\sum_{k>n}\lambda^{k}\left|\langle\,f,Q^{k}f\,\rangle\right|=\sum_{k>n}\left|\langle\,f,Q^{k}f\,\rangle\right|\,.$ (18) Thus, to prove that $\sup_{\lambda\in[0,1)}\left|S_{n}(\lambda)-S(\lambda)\right|\to 0$, it is sufficient to prove that the series $\sum_{k>0}\left|\langle\,f,Q^{k}f\,\rangle\right|$ converges. By bilinearity of the inner product and by linearity of the iterated operators $Q,Q^{2},\ldots$, it can be checked that for any linear mapping $\phi$ on $\mathcal{L}_{2}^{\ast}(\bar{\pi})$ $\sum_{k=1}^{\infty}\left|\left\langle f,Q^{k}f\right\rangle\right|<\infty\Leftrightarrow\sum_{k=1}^{\infty}\left|\left\langle\phi(f),Q^{k}\phi(f)\right\rangle\right|<\infty\,.$ (19) Since $\bm{\mathcal{X}}$ is finite, if $f\in\mathcal{L}_{2}^{\ast}(\bar{\pi})$ then $\sup|f|<\infty$. As a consequence, we may use $\phi(f):=(f-\bar{\pi}f)/\sup|f|$ and $\bar{\pi}f:=\int f\mathrm{d}\bar{\pi}$. In the following we denote by $\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi})$ the subset of $\mathcal{L}_{2}^{\ast}(\bar{\pi})$ such that $\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi}):=\left\\{f\in\mathcal{L}_{2}^{\ast}(\bar{\pi})\,:\,\bar{\pi}f=0\,,\;\sup|f|\leq 1\right\\}\,.$ By Eq. (19), we only need to check that the series $\sum_{k>0}\left|\langle\,f,Q^{k}f\,\rangle\right|$ converges for each $f\in\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi})$. Since $\bm{\mathcal{X}}$ is finite, $Q$ is uniformly ergodic and there exists constants $\varrho\in(0,1)$ and $C\in(0,\infty)$ such that for any $t\in\mathbb{N}$, $\sup_{(x,\nu)\in\bm{\mathcal{X}}\times\\{-1,+1\\}}\|\delta_{x,\nu}Q^{t}-\bar{\pi}\|_{\mathrm{tv}}\leq C\varrho^{t}\,,\ $ (20) where for any signed measure $\mu$, $\|\mu\|_{\mathrm{tv}}$ denotes its total variation. On the one hand, note that for each $f\in\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi})$ $\langle\,f,Q^{k}f\,\rangle=\mathbb{E}f(X,\nu)Q^{k}f(X,\nu)\leq\mathbb{E}|f(X,\nu)||Q^{k}f(X,\nu)|\,,$ (21) and on the other hand, we have that for any $(x,\nu)\in\bm{\mathcal{X}}\times\\{-1,1\\}$, $|Q^{k}f(x,\nu)|=|Q^{k}f(x,\nu)-\bar{\pi}f|\leq\sup_{f\in\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi})}|Q^{k}f(x,\nu)-\bar{\pi}f|\,.$ (22) But $\|\mu\|_{\mathrm{tv}}=(1/2)\sup_{g:\bm{\mathcal{X}}\to[-1,1]}|\mu g|$, see for instance (Roberts and Rosenthal, 2004, Proposition 3). Since $f\in\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi})$, $|f|\leq 1$ and we have by inclusion that for all $(x,\nu)\in\bm{\mathcal{X}}\times\\{-1,1\\}$ $|Q^{k}f(x,\nu)|\leq\sup_{f\in\mathcal{L}_{2}^{\ast,0,1}(\bar{\pi})}|Q^{k}f(x,\nu)-\bar{\pi}f|\leq\sup_{g:\bm{\mathcal{X}}\times\\{-1,1\\}\to[-1,1]}|Q^{k}g(x,\nu)-\bar{\pi}g|\leq 2\|\delta_{x,\nu}Q^{k}-\bar{\pi}\|_{\mathrm{tv}}\,.$ (23) Taking the supremum over all $(x,\nu)\in\bm{\mathcal{X}}\times\\{-1,1\\}$ in Eq. (23), and combining with Eq. (20) yields $\sup_{(x,\nu)\in\bm{\mathcal{X}}\times\\{-1,1\\}}|Q^{k}f(x,\nu)|\leq 2C\varrho^{k}\,.$ Plugging this into Eq. (21), we have $\left|\langle\,f,Q^{k}f\,\rangle\right|\leq 2C\mathbb{E}|f(X,\nu)|\varrho^{k}\,.$ (24) which is clearly summable. As a consequence, $S_{n}$ converges uniformly to $S$ on $[0,1)$ which concludes the proof. ∎ ###### Proof of Corollary 1. The results of Theorem 3.15 in Andrieu and Livingstone (2019) holds in our framework, implying that $\mathrm{var}_{\lambda}(f,P_{\rho^{*}})\leq\mathrm{var}_{\lambda}(f,P_{\rho})\leq\mathrm{var}_{\lambda}(f,P_{\rho}^{\text{w}}),$ where $\mathrm{var}_{\lambda}(f,P_{\rho}):=\mathbb{V}\mathrm{ar}f(\mathbf{X},\nu)+2\sum_{k>0}\lambda^{k}\left\langle f,P_{\rho}^{k}f\right\rangle$ with $\lambda\in[0,1)$. Lemma 1 allows to conclude. ∎ ###### Proof of Corollary 2. The proof is an application of Theorem 3.17 in Andrieu and Livingstone (2019) which will allow to establish that $\mathrm{var}_{\lambda}(f,P_{\rho})\leq\mathrm{var}_{\lambda}(f,P_{\text{MH}}).$ We will thus be able to conclude using Lemma 1. In order to apply Theorem 3.17, we must verify that $q_{\mathbf{x}}(\mathbf{y})\,\alpha(\mathbf{x},\mathbf{y})=(1/2)\,q_{\mathbf{x},+1}(\mathbf{y})\,\alpha_{+1}(\mathbf{x},\mathbf{y})+(1/2)\,q_{\mathbf{x},-1}(\mathbf{y})\,\alpha_{-1}(\mathbf{x},\mathbf{y}),$ for all $\mathbf{x}$ and $\mathbf{y}$. This is straightforward to verify under the assumptions of Corollary 2: $\displaystyle(1/2)\,q_{\mathbf{x},+1}(\mathbf{y})\,\alpha_{+1}(\mathbf{x},\mathbf{y})+(1/2)\,q_{\mathbf{x},-1}(\mathbf{y})\,\alpha_{-1}(\mathbf{x},\mathbf{y})$ $\displaystyle\qquad=\frac{1}{2}\frac{1}{(|\mathcal{N}(\mathbf{x})|/2)}\left(1\wedge\frac{\pi(\mathbf{y})}{\pi(\mathbf{x})}\right)\mathds{1}_{\mathbf{y}\in\mathcal{N}_{+1}(\mathbf{x})}+\frac{1}{2}\frac{1}{(|\mathcal{N}(\mathbf{x})|/2)}\left(1\wedge\frac{\pi(\mathbf{y})}{\pi(\mathbf{x})}\right)\mathds{1}_{\mathbf{y}\in\mathcal{N}_{-1}(\mathbf{x})}$ $\displaystyle\qquad=\frac{1}{|\mathcal{N}(\mathbf{x})|}\left(1\wedge\frac{\pi(\mathbf{y})}{\pi(\mathbf{x})}\right)\left(\mathds{1}_{\mathbf{y}\in\mathcal{N}_{+1}(\mathbf{x})}+\mathds{1}_{\mathbf{y}\in\mathcal{N}_{-1}(\mathbf{x})}\right)=q_{\mathbf{x}}(\mathbf{y})\,\alpha(\mathbf{x},\mathbf{y}).$ ∎ ###### Proof of Theorem 1. We first prove that $\mathrm{var}(f,P_{\rho,n})\leq\mathrm{var}(f,P_{\text{rev.},n})$. This is done as in the proof of Corollary 2. We now analyse $\mathrm{var}(f,P_{\text{rev.},n})$: $\mathrm{var}(f,P_{\text{rev.},n})=\mathbb{V}\mathrm{ar}f(\mathbf{X})+2\sum_{k>0}\left\langle f,P_{\text{rev.},n}^{k}f\right\rangle=\mathbb{V}\mathrm{ar}f(\mathbf{X})+2\sum_{k>0}\text{Cov}_{P_{\text{rev.},n}}[f(\mathbf{X}(0)),f(\mathbf{X}(k))],$ where we omitted the dependence on $\nu$ because, as we mentioned in Section 3.1, it can be treated as a constant as a consequence of the restrictions on $f$. In the expression above, it is considered that the chain starts at stationarity and evolves using $P_{\text{rev.},n}$. We consider without loss of generality that $\mathbb{E}[f(\mathbf{X}(k))]=0$ (for any $k$). We first write $\displaystyle\mathbb{V}\mathrm{ar}f(\mathbf{X})$ $\displaystyle=\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}}]+\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}}]$ (25) $\displaystyle=\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})^{2}]-\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}+\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}+(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}}]$ (26) $\displaystyle\qquad+\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}}],$ (27) where $\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}$ denotes an expectation with respect to $\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$. Also, $\displaystyle\sum_{k>0}\text{Cov}_{P_{\text{rev.},n}}[f(\mathbf{X}(0)),f(\mathbf{X}(k))]=\sum_{k=1}^{\varrho(n)}\text{Cov}_{P_{\text{rev.},n}}[f(\mathbf{X}(0)),f(\mathbf{X}(k))]+\sum_{k=\varrho(n)+1}^{\infty}\text{Cov}_{P_{\text{rev.},n}}[f(\mathbf{X}(0)),f(\mathbf{X}(k))],$ where $\varrho(n)$ is chosen according to the statement of Theorem 1; therefore the second term on the RHS can be made as small as we want. For the first term, we have $\displaystyle\sum_{k=1}^{\varrho(n)}\text{Cov}_{P_{\text{rev.},n}}[f(\mathbf{X}(0)),f(\mathbf{X}(k))]$ $\displaystyle=\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))]$ $\displaystyle=\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cap_{m=0}^{k}A_{m}(\bm{\mathcal{X}}_{\varphi(n)-1})}]+\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cup_{m=0}^{k}A_{m}^{\mathsf{c}}(\bm{\mathcal{X}}_{\varphi(n)-1})}],$ where $A_{m}(\bm{\mathcal{X}}_{\varphi(n)-1}):=\\{\mathbf{X}(m)\in\bm{\mathcal{X}}_{\varphi(n)-1}\\}$. By assumption, the second term on the RHS can be made as small as we want. We have $\displaystyle\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cap_{m=0}^{k}A_{m}(\bm{\mathcal{X}}_{\varphi(n)-1})}]$ $\displaystyle=\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{P}_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))]$ $\displaystyle+(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\sum_{k=1}^{\varrho(n)}\mathbb{E}_{P_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))\mathds{1}_{\cap_{m=0}^{k}A_{m}(\bm{\mathcal{X}}_{\varphi(n)-1})}],$ where $\tilde{P}_{\text{rev.},n}$ is the Markov kernel whose stationary distribution is $\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$. This equality holds because the paths involving transition probabilities $P_{\text{rev.},n}(\mathbf{x},\mathbf{y})$ with $\mathbf{x},\mathbf{y}\in\bm{\mathcal{X}}_{\varphi(n)-1}$ have the same probabilities as those of the chain with stationary distribution $\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}$. We just have to renormalise the probabilities of the starting point by dividing by $\pi(\bm{\mathcal{X}}_{\varphi(n)})$ to complete the argument. Note that, by assumption, the second term on the RHS can be made as small as we want. Now, $\displaystyle\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{P}_{\text{rev.},n}}[f(\mathbf{X}(0))f(\mathbf{X}(k))]=\sum_{k=1}^{\varrho(n)}\text{Cov}_{\tilde{P}_{\text{rev.},n}}[f(\mathbf{X}(0)),f(\mathbf{X}(k))]+\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}.$ By assumption, the sum from $\varrho(n)+1$ to $\infty$ of the covariances is small as well. If we combine this with (25), we have $\displaystyle\mathrm{var}(f,P_{\text{rev.},n})$ $\displaystyle=\mathrm{var}(f,\tilde{P}_{\text{rev.},n})+\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}+(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}}]$ $\displaystyle\qquad+\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}}]+2\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}$ $\displaystyle\leq\mathrm{var}(f,\tilde{P}_{\text{MH},n})/\omega_{n}+((1-\omega_{n})/\omega_{n})\mathbb{V}\mathrm{ar}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}f(\mathbf{X})+\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}$ $\displaystyle\qquad+(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}}]+\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}}]+2\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}$ $\displaystyle=\mathrm{var}(f,\tilde{P}_{\text{MH},n})+\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}+(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}}]+\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}}]$ $\displaystyle\qquad+2\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}+(1/\omega_{n}-1)\mathrm{var}(f,\tilde{P}_{\text{MH},n})+((1-\omega_{n})/\omega_{n})\mathbb{V}\mathrm{ar}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}f(\mathbf{X}),$ using Theorem 2 of Zanella (2019) for the inequality and omitting the terms that can be made as small as we want. This theorem can be used as a result of the bound $\tilde{P}_{\text{rev.},n}(\mathbf{x},\mathbf{y})\geq\omega_{n}\tilde{P}_{\text{MH},n}(\mathbf{x},\mathbf{y})$ with $\omega_{n}$ defined in (9) (see Section 3.1). By assumption, the last two terms can be made as small as we want. Now we used the previous arguments in the reverse order to show that $\displaystyle\mathrm{var}(f,\tilde{P}_{\text{MH},n})+\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}+(1-1/\pi(\bm{\mathcal{X}}_{\varphi(n)}))\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}}]+\mathbb{E}[f(\mathbf{X})^{2}\mathds{1}_{\mathbf{X}\in\bm{\mathcal{X}}_{\varphi(n)}^{\mathsf{c}}}]$ $\displaystyle\qquad+2\sum_{k=1}^{\varrho(n)}\mathbb{E}_{\tilde{\pi}_{\bm{\mathcal{X}}_{\varphi(n)}}}[f(\mathbf{X})]^{2}=\mathrm{var}(f,P_{\text{MH},n})+\text{small error term},$ which yields the result. ∎ ###### Proof of Proposition 2. It suffices to prove that the probability to reach the state $\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime}\in A,\nu^{\prime}$ in one step is equal to the probability of this state under the target: $\displaystyle\sum_{\mathbf{x},\nu}\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime}))\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$ $\displaystyle=\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\,d\bm{\theta}_{\mathbf{y}}^{\prime},$ (28) where $P$ is the transition kernel. Note that we abuse notation here by denoting the measure $d\bm{\theta}_{\mathbf{y}}^{\prime}$ on the left-hand side (LHS) given that we may in fact use vectors of auxiliary variables to generate the proposal when switching models, which often do not have the same dimension as $\bm{\theta}_{\mathbf{y}}^{\prime}$. We consider two distinct events: a model switch is proposed, that we denote $S$, and a parameter update is proposed (therefore denoted $S^{\mathsf{c}}$). We know that the probabilities of these events are $1-q_{\mathbf{x},\nu}(\mathbf{x})$ and $q_{\mathbf{x},\nu}(\mathbf{x})$, respectively. We rewrite the LHS of (28) as $\displaystyle\sum_{\mathbf{x},\nu}\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime}))\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$ (29) $\displaystyle\quad=\sum_{\mathbf{x},\nu}\,(1-q_{\mathbf{x},\nu}(\mathbf{x}))\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime})\mid S)\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$ (30) $\displaystyle\qquad+\sum_{\mathbf{x},\nu}\,q_{\mathbf{x},\nu}(\mathbf{x})\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime})\mid S^{\mathsf{c}})\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}.$ (31) We analyse the two terms separately. We know that $P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime})\mid S^{\mathsf{c}})=\delta_{(\mathbf{x},\nu)}(\mathbf{y},\nu^{\prime})\,P_{S^{\mathsf{c}}}(\bm{\theta}_{\mathbf{x}},\bm{\theta}_{\mathbf{y}}^{\prime}),$ where $P_{S^{\mathsf{c}}}$ is the transition kernel associated with the method used to update the parameters. Therefore, the second term on the RHS of (29) is equal to $\displaystyle\sum_{k,\nu}\,q_{\mathbf{x},\nu}(\mathbf{x})\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime})\mid S^{\mathsf{c}})\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$ $\displaystyle\quad=q_{\mathbf{y},\nu^{\prime}}(\mathbf{y})\,\pi(\mathbf{y})\times(1/2)\int\pi(\bm{\theta}_{\mathbf{y}}\mid\mathbf{y})\left(\int_{A}P_{S^{\mathsf{c}}}(\bm{\theta}_{\mathbf{y}},\bm{\theta}_{\mathbf{y}}^{\prime})\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{y}}.$ We also know that $P_{S^{\mathsf{c}}}$ leaves the conditional distribution $\pi(\,\cdot\mid\mathbf{y})$ invariant, implying that $\displaystyle q_{\mathbf{y},\nu^{\prime}}(\mathbf{y})\,\pi(\mathbf{y})\times(1/2)\int\pi(\bm{\theta}_{\mathbf{y}}\mid\mathbf{y})\left(\int_{A}P_{S^{\mathsf{c}}}(\bm{\theta}_{\mathbf{y}},\bm{\theta}_{\mathbf{y}}^{\prime})\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{y}}$ (32) $\displaystyle\quad=q_{\mathbf{y},\nu^{\prime}}(\mathbf{y})\,\pi(\mathbf{y})\times(1/2)\int_{A}\pi(\bm{\theta}_{\mathbf{y}}^{\prime}\mid\mathbf{y})\,d\bm{\theta}_{\mathbf{y}}^{\prime}=q_{\mathbf{y},\nu^{\prime}}(\mathbf{y})\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\,d\bm{\theta}_{\mathbf{y}}^{\prime}.$ (33) For the model switching case (the first term on the RHS of (29)), we use the fact that there is a connection between $P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime})\mid S)$ and the kernel associated to a specific RJ. Consider that $q_{\mathbf{x}}(\mathbf{y})=(1/2)\,q_{\mathbf{x},-1}(\mathbf{y})+(1/2)\,q_{\mathbf{x},+1}(\mathbf{y})$ for all $\mathbf{x},\mathbf{y}$ and that all other proposal distributions in RJ are the same as in Algorithm 3 during model switches. In this case, $\alpha_{\text{RJ}}=\alpha_{\text{NRJ}}$ in the case where at the current iteration it is chosen to use $q_{\mathbf{x},\nu}$ (which happens with probability $1/2$) and in the reverse move it is chosen to use $q_{\mathbf{y},-\nu}$ (which also happens with probability $1/2$). Consider the case where Model $\mathbf{y}$ is reached from Model $\mathbf{x}\neq\mathbf{y}$ coming from direction $\nu^{\prime}=\nu$. Given the reversibility of RJ, the probability to go from Model $\mathbf{x}$ with parameters in $B$ to Model $\mathbf{y}\neq\mathbf{x}$ with parameters in $A$ is $\displaystyle\int_{B}\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\left(\int_{A}P_{\text{RJ}}((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime}))\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}=\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\left(\int_{B}P_{\text{RJ}}((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime}),(\mathbf{x},\bm{\theta}_{\mathbf{x}}))\,d\bm{\theta}_{\mathbf{x}}\right)\,d\bm{\theta}_{\mathbf{y}}^{\prime},$ (34) where $P_{\text{RJ}}$ is the transition kernel of the RJ. Note that $P_{\text{RJ}}((\mathbf{x},\bm{\theta}_{\mathbf{x}}),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime}))=(1/2)\,(1-q_{\mathbf{x},\nu}(\mathbf{x}))\,P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu)\mid S),$ given that the difference between both kernels is that in RJ, it is first decided to use $q_{\mathbf{x},\nu}$, there is thus an additional probability of $1/2$. Analogously, $P_{\text{RJ}}((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime}),(\mathbf{x},\bm{\theta}_{\mathbf{x}}))=(1/2)\,(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\,P((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},-\nu),(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)\mid S)$. Using this and taking $B$ equals the whole parameter (and auxiliary) space in (34), we have $\displaystyle(1-q_{\mathbf{x},\nu}(\mathbf{x}))\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu)\mid S)\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$ $\displaystyle\qquad=(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\left(\int P((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},-\nu),(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)\mid S)\,d\bm{\theta}_{\mathbf{x}}\right)\,d\bm{\theta}_{\mathbf{y}}^{\prime}.$ The only other case to consider for model switches is where Model $\mathbf{y}$ is reached from Model $\mathbf{y}$ (because the proposal is rejected) and the direction is $\nu^{\prime}=-\nu$. The probability of this transition is $(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\left(1-\sum_{\mathbf{x}\neq\mathbf{y}}\int P((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},-\nu),(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)\mid S)\,d\bm{\theta}_{\mathbf{x}}\right)\,d\bm{\theta}_{\mathbf{y}}^{\prime}.$ So, the total probability of reaching $\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime}\in A,\nu^{\prime}$ through a model switch is (recalling (29)): $\displaystyle\sum_{\mathbf{x},\nu}\,(1-q_{\mathbf{x},\nu}(\mathbf{x}))\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu^{\prime})\mid S)\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$ $\displaystyle\quad=\sum_{\mathbf{x}\neq\mathbf{y}}(1-q_{\mathbf{x},\nu}(\mathbf{x}))\int\pi(\mathbf{x},\bm{\theta}_{\mathbf{x}})\times(1/2)\left(\int_{A}P((\mathbf{x},\bm{\theta}_{\mathbf{x}},\nu),(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},\nu)\mid S)\,d\bm{\theta}_{\mathbf{y}}^{\prime}\right)\,d\bm{\theta}_{\mathbf{x}}$ $\displaystyle\qquad+(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\left(1-\sum_{\mathbf{x}\neq\mathbf{y}}\int P((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},-\nu),(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)\mid S)\,d\bm{\theta}_{\mathbf{x}}\right)\,d\bm{\theta}_{\mathbf{y}}^{\prime}$ $\displaystyle\quad=\sum_{\mathbf{x}\neq\mathbf{y}}(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\left(\int P((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},-\nu),(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)\mid S)\,d\bm{\theta}_{\mathbf{x}}\right)\,d\bm{\theta}_{\mathbf{y}}^{\prime}$ $\displaystyle\qquad+(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2)\left(1-\sum_{\mathbf{x}\neq\mathbf{y}}\int P((\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime},-\nu),(\mathbf{x},\bm{\theta}_{\mathbf{x}},-\nu)\mid S)\,d\bm{\theta}_{\mathbf{x}}\right)\,d\bm{\theta}_{\mathbf{y}}^{\prime}$ $\displaystyle\quad=(1-q_{\mathbf{y},-\nu}(\mathbf{y}))\int_{A}\pi(\mathbf{y},\bm{\theta}_{\mathbf{y}}^{\prime})\times(1/2).$ Combining this with (32) allows to conclude the proof. ∎

2024-09-04T02:54:59.374374

2003.05511

arxiv-papers

# Resource Allocation for Intelligent Reflecting Surface Aided Wireless Powered Mobile Edge Computing in OFDM Systems Tong Bai, , Cunhua Pan, , Hong Ren, , Yansha Deng, , Maged Elkashlan, , and Arumugam Nallanathan T. Bai, C. Pan, H. Ren, M. Elkashlan and A. Nallanathan are with the School of Electronic Engineering and Computer Science, Queen Mary University of London, London E1 4NS, U.K. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>[email protected]). Y. Deng is with the Department of Engineering, King’s College London, London, WC2R 2LS, U.K. (e-mail: [email protected]). ###### Abstract Wireless powered mobile edge computing (WP-MEC) has been recognized as a promising technique to provide both enhanced computational capability and sustainable energy supply to massive low-power wireless devices. However, its energy consumption becomes substantial, when the transmission link used for wireless energy transfer (WET) and for computation offloading is hostile. To mitigate this hindrance, we propose to employ the emerging technique of intelligent reflecting surface (IRS) in WP-MEC systems, which is capable of providing an additional link both for WET and for computation offloading. Specifically, we consider a multi-user scenario where both the WET and the computation offloading are based on orthogonal frequency-division multiplexing (OFDM) systems. Built on this model, an innovative framework is developed to minimize the energy consumption of the IRS-aided WP-MEC network, by optimizing the power allocation of the WET signals, the local computing frequencies of wireless devices, both the sub-band-device association and the power allocation used for computation offloading, as well as the IRS reflection coefficients. The major challenges of this optimization lie in the strong coupling between the settings of WET and of computing as well as the unit- modules constraint on IRS reflection coefficients. To tackle these issues, the technique of alternative optimization is invoked for decoupling the WET and computing designs, while two sets of locally optimal IRS reflection coefficients are provided for WET and for computation offloading separately relying on the successive convex approximation method. The numerical results demonstrate that our proposed scheme is capable of monumentally outperforming the conventional WP-MEC network without IRSs. Quantitatively, about $80\%$ energy consumption reduction is attained over the conventional MEC system in a single cell, where $3$ wireless devices are served via $16$ sub-bands, with the aid of an IRS comprising of $50$ elements. ## I Introduction ### I-A Motivation and Scope In the Internet-of-Things (IoT) era, a myriad of heterogeneous devices are envisioned to be interconnected [1]. However, due to the stringent constraints both on device sizes and on manufacturing cost, many of them have to be equipped with either life-limited batteries or low-performance processors. Consequently, if only relying on their local computing, these resource- constrained devices are incapable of accommodating the applications that require sustainable and low-latency computation, e.g. wireless body area networks [2] and environment monitoring [3]. Fortunately, wireless powered mobile edge computing (WP-MEC)[4, 5, 6, 7, 8, 9, 10, 11, 12, 13], which incorporates radio frequency (RF) based wireless energy transmission (WET) [14, 15, 16] and mobile edge computing (MEC) [17, 18], constitutes a promising solution of this issue. Specifically, at the time of writing, the commercial RF-based WET has already been capable of delivering $0.05\leavevmode\nobreak\ \rm{mW}$ to a distance of $12-14\leavevmode\nobreak\ \rm{m}$ [14], which is sufficient to charge many low-power devices, whilst the MEC technique may provide the cloud-like computing service at the edge of mobile networks [18]. In WP-MEC systems, hybrid access points (HAP) associated with edge computing nodes are deployed in the proximity of wireless devices, and the computation of these devices is typically realized in two phases, namely the WET phase and the computing phase. To elaborate, the batteries of the devices are replenished by harvesting WET signals from the HAP in the first phase, while in the computing phase, devices may decide whether to process their computational tasks locally or offload them to edge computing nodes via the HAP. Given that these wireless devices are fully powered by WET in WP-MEC systems, the power consumption at HAPs becomes substantial, which inevitably increases the expenditure on energy consumption and may potentially saturate power rectifiers. At the time of writing, the existing research contributions that focus on reducing the power consumption mainly rely on the joint optimization of the WET and of computing [5], as well as cooperative computation offloading [10, 11]. However, wireless devices are still suspicious to severe channel attenuation, which limits the performance of WP-MEC systems. To resolve this issue, we propose to deploy the emerging intelligent reflecting surfaces (IRS) [19, 20, 21] in the vicinity of devices, for providing an additional transmission link both for WET and for computation offloading. Then, the power consumption can be beneficially reduced both for the downlink and for the uplink. To elaborate, an IRS comprises of an IRS controller and a large number of low-cost passive reflection elements. Regulated by the IRS controller, each IRS reflection element may adapt both the amplitude and the phase of the incident signals reflected, for collaboratively modifying the signal propagation environment. The gain attained by IRSs is based on the combination of so-called the virtual array gain and the reflection-enabled beamforming gain [19]. More explicitly, the virtual array gain is achieved by combining the direct and IRS-reflected links, while the reflection-enabled beamforming gain is realized by proactively adjusting the reflection coefficients of the IRS elements. By combining these two types of gain together, IRSs are capable of reducing the power required both for WET and for computation offloading, thus improving the energy efficiency of WP-MEC systems. In this treatise, we aim for providing a holistic scheme to minimize the energy consumption of WP- MEC systems, relying on IRSs. ### I-B Related Works The current state-of-the-art contributions are reviewed from the perspectives of WP-MEC and of IRS-aided networks, as follows. #### I-B1 Wireless Powered Mobile Edge Computing This topic has attracted an increasing amount of research attention [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Specifically, You _et al._ firstly proposed the WP- MEC framework [4], where the probability of successfully computing was maximized subject to the constraints both on energy harvesting and on latency. The single-user system considered in this first trial limits its application in large-scale scenarios. For eliminating this shortage, an energy- minimization algorithm was proposed for the multi-user scenario [5], where the devices’ computation offloading was realized by the time division multiple access (TDMA) technique. Following this, Bi and Zhang maximized the weighted sum computation rate in a similar TDMA system [6], while an orthogonal frequency division multiple access (OFDMA) based multi-user WP-MEC system was investigated in [7]. A holistic online optimization algorithm was proposed for the WP-MEC in industrial IoT scenarios [8]. In the aforementioned works, the associated optimization is commonly realized with the aid of the alternative optimization (AO) method, because the pertinent optimization problems are usually not jointly convex. This inevitably imposes a delay on decision making. To mimic this issue, Huang _et al._ proposed a deep reinforcement learning based algorithm for maximizing the computation rate of WP-MEC systems [9], which may replace the aforementioned complicated optimization by a pre- trained look-up table. Furthermore, as for the system where both near and far devices have to be served, the energy consumption at the HAP has to be vastly increased, because the farther device harvests less energy while a higher transmit power is required for its computation offloading. Aimed for releasing this so-called “doubly near-far” issue, the technique of user cooperation was revisited [10, 11]. At the time of writing, the WET and computation offloading in WP-MEC systems in the face of hostile communication environments has not been well addressed. Against this background, we aim for tackling this issue by invoking IRSs. Let us now continue by reviewing the relevant research contributions on IRSs as follows. #### I-B2 IRS-Aided Networks In order to exploit the potential of IRSs, an upsurging number of research efforts have been devoted in its channel modeling [22, 23], analyzing the impact of limited-resolution phase shifts [24, 25], channel estimation [26, 27] as well as IRS reflection coefficient designs [28, 29, 30, 31]. Inspired by these impressive research contributions, the beneficial role of IRSs was evaluated in various application scenarios [32, 33, 34, 35, 36, 37, 38]. Specifically, an IRS was employed in multi-cell communications systems for mitigate the severe inter-cell interference [32], where an IRS comprising of $100$ reflection elements was shown to be capable of doubling the sum rate of the multi-cell system. Yang _et al._ investigated an IRS-enhanced OFDMA system [33], whose common rate was improved from around $2.75\leavevmode\nobreak\ \rm{bps/Hz}$ to $4.4\leavevmode\nobreak\ \rm{bps/Hz}$, with the aid of a $50$-element IRS. Apart from assisting the aforementioned throughput maximization in the conventional communications scenario, a sophisticated design of IRSs may also eminently upgrade the performance of diverse emerging wireless networks, e.g. protecting data transmission security [34, 35], assisting simultaneous wireless information and power transfer (SWIPT) [36], enhancing the user cooperation in wireless powered communications networks [37], as well as reducing the latency in IRS-aided MEC systems [38]. These impressive research contributions inspire us to exploit the beneficial role of IRSs in this momentous WP-MEC scenario. ### I-C Novelty and Contributions In this paper, an innovative IRS-aided WP-MEC framework is proposed, where we consider orthogonal frequency-division multiplexing (OFDM) systems for its WET and devices’ computation offloading. Under this framework, a joint WET and computing design is conceived for minimizing its energy consumption, by optimizing the power allocation of the WET signals over OFDM sub-bands, the local computing frequencies of wireless devices, both the sub-band-device association and the power allocation used for computation offloading, as well as the pertinent IRS reflection coefficient design. Let us now detail our contributions as follows. * • _Energy minimization problem formulation for the new IRS-aided WP-MEC design:_ In order to reduce the energy consumption of WP-MEC systems, we employ an IRS in WP-MEC systems and formulate a pertinent energy minimization problem. Owing to the coupling effects between the designs of WET and of computing, it is difficult to find its globally optimal solution. Alternatively, we provide an alternative optimization (AO) based solution to approach a locally optimal solution, by iteratively optimizing settings of WET and of computing. * • _WET design:_ The WET setting is realized by alternatively optimizing the power allocation of energy-carrying signals over OFDM sub-bands and the IRS reflection coefficients. Specifically, given a set of fixed IRS reflection coefficients, the power allocation problem can be simplified to be a linear programming problem, which can be efficiently solved by the existing optimization software. Given a fixed power allocation, the IRS reflection coefficient design becomes a feasibility-check problem, the solution of which is incapable of ensuring a rapid convergence. To tackle this issue, we reformulate the problem by introducing a number of auxiliary variables, and provide a locally optimal design of IRS reflection coefficients, with the aid of several steps of mathematical manipulations and of the successive convex approximation (SCA) method. * • _Computing design:_ The settings at the computing phase are specified by alternatively optimizing the joint sub-band-device association for and the power allocation for devices’ computation offloading, IRS reflection coefficients at the computing phase as well as the local computing frequencies. Specifically, as verified by [39], the duality gap vanishes when the number of sub-bands exceeds $8$. Hence, we provide a near-optimal solution for the joint sub-band-device association and power allocation problem, relying on the Lagrangian duality method. The IRS reflection coefficients are designed using the similar approach devised for that in the WET phase. Finally, our analysis reveals that the optimal local computing frequencies can be obtained by selecting their maximum allowable values. * • _Numerical validations:_ Our numerical results validates the benefits of employing IRSs in WP-MEC systems, and quantify the energy consumption of our proposed framework in diverse simulation environments, together with two benchmark schemes. The rest of the paper is organized as follows. In Section II, we describe the system model and formulate the pertinent problem. A solution of this problem is provided in Section III. The numerical results are presented in Section IV. Finally, our conclusions are drawn in Section V. Figure 1: An illustration of our IRS-aided WP-MEC system, where $K$ single- antenna devices are served by a mobile edge computing node via a single- antenna hybrid access point, with the aid of an $N$-element IRS. ## II System Model and Problem Formulation As illustrated in Fig. 1, we consider an OFDM-based WP-MEC system, where $K$ single-antenna devices are served by a single-antenna hybrid access point (HAP) associated with an edge computing node through $M$ equally-divided OFDM sub-bands. Similar to the assumption in [5, 6, 7], we assume that these devices do not have any embedded energy supply available, but are equipped with energy storage devices, e.g. rechargeable batteries or super-capacitors, for storing the energy harvested from RF signals. As shown in Fig. 2, relying on the so-called “harvest-then-computing” mechanism [5], the system operates in a two-phase manner in each time block. Specifically, during the WET phase, the HAP broadcasts energy-carrying signals to all $K$ devices for replenishing their batteries, while these $K$ devices process their computing tasks both by local computing and by computation offloading during the computing phase. We denote the duration of each time block by $T$, which is chosen to be no larger than the tolerant latency of MEC applications. The duration of the WET and of the computing phases are set as $\tau T$ and $(1-\tau)T$, respectively. Furthermore, to assist the WET and the devices’ computation offloading in this WP-MEC system, we place an IRS comprising of $N$ reflection elements in the proximity of devices. The reflection coefficients of these IRS reflection elements are controlled by an IRS controller in a real-time manner, based on the optimization results provided by the HAP. Figure 2: An illustration of the harvest-then-offloading protocol, where $\tau T$ and $(1-\tau)T$ refer to the duration of the WET and the computing phases, respectively. Let us continue by elaborating on the equivalent baseband time-domain channel as follows. We denote the equivalent baseband time-domain channel of the direct link between the $k$-th device and the HAP, the equivalent baseband time-domain channel between the $n$-th IRS element and the HAP, and the equivalent baseband time-domain channel between the $k$-th device and the $n$-th IRS element by $\hat{\boldsymbol{h}}^{d}_{k}\in\mathbb{C}^{L^{d}_{k}\times 1}$, $\hat{\boldsymbol{g}}_{n}\in\mathbb{C}^{L_{1}\times 1}$ and $\hat{\boldsymbol{r}}_{k,n}\in\mathbb{C}^{L_{2,k}\times 1}$, respectively, where $L^{d}_{k}$, $L_{1}$ and $L_{2,k}$ represent the respective number of delay spread taps. Without loss of generality, we assume that the above channels remain approximately constant over each time block. Furthermore, the channels are assumed to be reciprocal for the downlink and the uplink. As for the IRS, we denote the phase shift vector of and the amplitude response of the IRS reflection elements by $\boldsymbol{\theta}=[\theta_{1},\theta_{2},\ldots,\theta_{N}]^{T}$ and $\boldsymbol{\beta}=[\beta_{1},\beta_{2},\ldots,\beta_{N}]^{T}$, respectively, where we have $\theta_{n}\in[0,2\pi)$ and $\beta_{n}\in[0,1]$. Then, the corresponding reflection coefficients of the IRS are given by $\boldsymbol{\Theta}=[\Theta_{1},\Theta_{2},\ldots,\Theta_{N}]^{T}=[\beta_{1}e^{j\theta_{1}},\beta_{2}e^{j\theta_{2}},\ldots,\beta_{N}e^{j\theta_{N}}]^{T}$, where $j$ represents the imaginary unit and we have $|\Theta_{n}|\leq 1$ for $\forall n\in\mathcal{N}$. The baseband equivalent time-domain channel of the reflection link is the convolution of the device-IRS channel, of the IRS reflection response, and of the IRS-HAP channel. Specifically, the baseband equivalent time-domain channel reflected by the $n$-th IRS element is formulated as $\hat{\boldsymbol{h}}^{r}_{k,n}=\hat{\boldsymbol{r}}_{k,n}\ast\Theta_{n}\ast\hat{\boldsymbol{g}}_{n}=\Theta_{n}\hat{\boldsymbol{r}}_{k,n}\ast\hat{\boldsymbol{g}}_{n}$. Here, we have $\hat{\boldsymbol{h}}^{r}_{k,n}\in\mathbb{C}^{L^{r}_{k}\times 1}$ and $L^{r}_{k}=L_{1}+L_{2,k}-1$, which denotes the number of delay spread taps of the reflection channel. Furthermore, we denote the time-domain zero- padded concatenated device-IRS-HAP channel between the $k$-th device and the HAP via the $n$-th IRS element by $\boldsymbol{v}_{k,n}=[(\hat{\boldsymbol{r}}_{k,n}\ast\hat{\boldsymbol{g}}_{n})^{T},0,\ldots,0]^{T}\in\mathbb{C}^{M\times 1}$. Upon denoting $\boldsymbol{V}_{k}=[\boldsymbol{v}_{k,1},\ldots,\boldsymbol{v}_{k,N}]\in\mathbb{C}^{M\times N}$, we formulate the composite device-IRS-HAP channel between the $k$-th device and the HAP as $\boldsymbol{h}^{r}_{k}=\boldsymbol{V}_{k}\boldsymbol{\Theta}$. Similarly, we use $\boldsymbol{h}^{d}_{k}=[(\hat{\boldsymbol{h}}^{d}_{k})^{T},0,\ldots,0]^{T}\in\mathbb{C}^{M\times 1}$ to represent the zero-padded time-domain channel of the direct device-HAP link. To this end, we may readily arrive at the superposed channel impulse response (CIR) for the $k$-th device, formulated as $\displaystyle\boldsymbol{h}_{k}=\boldsymbol{h}^{d}_{k}+\boldsymbol{h}^{r}_{k}=\boldsymbol{h}^{d}_{k}+\boldsymbol{V}_{k}\boldsymbol{\Theta},\quad\forall k\in\mathcal{K},$ (1) whose number of delay spread taps is given by $L_{k}=\max(L^{d}_{k},L^{r}_{k})$. We assume that the number of cyclic prefixes (CP) is no smaller than the maximum number of delay spread taps for all devices, so that the inter-symbol interference (ISI) can be eliminated. Upon denoting the $m$-th row of the $M\times M$ discrete Fourier transform (DFT) matrix $\boldsymbol{F}_{M}$ by $\boldsymbol{f}^{H}_{m}$, we formulate the channel frequency response (CFR) for the $k$-th device at the $m$-th sub- band as $\displaystyle C_{k,m}(\boldsymbol{\Theta})=\boldsymbol{f}^{H}_{m}\boldsymbol{h}_{k}=\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta},\quad\forall k\in\mathcal{K},\forall m\in\mathcal{M}.$ (2) For ease of exposition, we assume that the knowledge of $\boldsymbol{h}^{d}_{k}$ and of $\boldsymbol{V}_{k}$ is perfectly known at the HAP. Naturally, this assumption is idealistic. Hence, the algorithm developed in this paper can be deemed to represent the best-case bound for the energy performance of realistic scenarios. Since different types of signals are transmitted in the WET and computing phases, the reflection coefficients of the IRS require separate designs in these two phases. The models of the WET and of computing phases are detailed as follows. ### II-A Model of the Wireless Energy Transfer Phase It is assumed that the capacity of devices’ battery is large enough so that all the harvest energy can be saved without energy overflow. Let us use $\boldsymbol{\Theta}^{E}=\big{\\{}\Theta^{E}_{1},\Theta^{E}_{2},\ldots,\Theta^{E}_{N}\big{\\}}$ to represent the IRS reflection-coefficient vector during the WET phase, where we have $|\Theta^{E}_{n}|\leq 1$ for $\forall n\in\mathcal{N}$. Then, the composite channel of the $m$-th sub-band for the $k$-th device during the WET phase $C_{k,m}(\boldsymbol{\Theta}^{E})$ can be obtained by (2). As a benefit of the broadcasting nature of WET, each device can harvest the energy from the RF signals transmitted over all $M$ sub-bands. Hence, upon denoting the power allocation for the energy-carrying RF signals at the $M$ sub-bands during the WET phase by $\boldsymbol{p}^{E}=[p^{E}_{1},p^{E}_{2},\ldots,p^{E}_{M}]$, we are readily to formulate the energy harvested by the $k$-th device as [5] $\displaystyle E_{k}(\tau,\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})=\sum^{M}_{m=1}\eta\tau Tp^{E}_{m}\big{|}C_{k,m}(\boldsymbol{\Theta}^{E})\big{|}^{2},$ (3) where $\eta\in[0,1]$ denotes the efficiency of the energy harvesting at the wireless devices. ### II-B Model of the Computing Phase We consider the data-partitioning based application [40], where a fraction of the data can be processed locally, while the other part can be offloaded to the edge node. For a specific time block, we use $L_{k}$ and $\ell_{k}$ to denote the number of bits to be processed by the $k$-th device and its computation offloading volume in terms of the number of bits, respectively. The models of local computing, of computation offloading and of edge computing are detailed as follows. #### II-B1 Local Computing We use $f_{k}$ and $c_{k}$ to represent its computing capability in terms of the number of central processing unit (CPU) cycles per second and the number of CPU cycles required to process a single bit, for the $k$-th device, respectively. The number of bits processed by local computing is readily calculated as $(1-\tau)Tf_{k}/c_{k}$, and the number of bits to be offloaded is given by $\ell_{k}=L_{k}-(1-\tau)Tf_{k}/c_{k}$. Furthermore, we assume that $f_{k}$ is controlled in the range of $[0,f_{max}]$ using the dynamic voltage scaling model [40]. Upon denoting the computation energy efficiency coefficient of the processor’s chip by $\kappa$, we formulate the power consumption of the local computing mode as $\kappa f_{k}^{2}$ for the $k$-th device [40]. #### II-B2 Computation offloading In order to mitigate the co-channel interference, the devices’ computation offloading is realized relying on the orthogonal frequency-division multiple access (OFDMA) scheme. In this case, each sub-band is allowed to be used by at most a single device. We use the binary vector $\boldsymbol{\alpha}_{k}=[\alpha_{k,1},\alpha_{k,2},\ldots,\alpha_{k,M}]^{T}$ and the non-negative vector $\boldsymbol{p}^{I}_{k}=[p^{I}_{k,1},p^{I}_{k,2},\ldots,p^{I}_{k,M}]^{T}$ to represent the association between the sub-band and devices as well as the power allocation of the $k$-th device to the $M$ sub-bands, respectively, where we have $\displaystyle\alpha_{k,m}$ $\displaystyle=\begin{cases}0,&\quad\text{if }p^{I}_{k,m}=0,\\\ 1,&\quad\text{if }p^{I}_{k,m}>0.\end{cases}$ (4) The power consumption of computation offloading is given by $\sum^{M}_{m=1}\alpha_{k,m}(p_{k,m}+p_{c})$, where $p_{c}$ represents a constant circuit power (caused by the digital-to-analog converter, filter, etc.) [5]. Let us denote the IRS reflection-coefficient vector during the computation offloading by $\boldsymbol{\Theta}^{I}=\big{\\{}\Theta^{I}_{1},\Theta^{I}_{2},\ldots,\Theta^{I}_{N}\big{\\}}$, where $|\Theta^{I}_{n}|\leq 1$ for $\forall n\in\mathcal{N}$. Then, the composite channel of the $k$-th device at the $m$-th sub-band denoted by $C_{k,m}(\boldsymbol{\Theta}^{I})$ can be obtained by (2). The corresponding achievable rate of computation offloading is formulated below for the $k$-th device $\displaystyle R_{k}(\boldsymbol{\alpha}_{k},\boldsymbol{p}^{I}_{k},\boldsymbol{\Theta}^{I})=\sum^{M}_{m=1}\alpha_{k,m}B\log_{2}\bigg{(}1+\frac{p_{k,m}|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\bigg{)},$ (5) where $\Gamma$ is the gap between the channel capacity and a specific modulation and coding scheme. Furthermore, in order to offload all the $\ell_{k}$ bits within the duration of the computation phase, the achievable offloading rate has to obey $R_{k}(\tau,\boldsymbol{\alpha}_{k},\boldsymbol{p}^{I}_{k},\boldsymbol{\Theta}^{I})\geq\frac{\ell_{k}}{(1-\tau)T}$. #### II-B3 Edge Computing Invoking the simplified linear model [5], we formulate the energy consumption at the edge node as $\vartheta\sum^{K}_{k=1}\ell_{k}=\vartheta\sum^{K}_{k=1}\big{[}L_{k}-(1-\tau)Tf_{k}/c_{k}\big{]}$. Furthermore, the latency imposed by edge computing comprises of two parts. The first part is caused by processing the computational tasks. Given that edge nodes typically possess high computational capabilities, this part can be negligible. The second part is induced by sending back the computational results, which are usually of a small volume. Hence, the duration of sending the feedback is also negligible. As such, we neglect the latency induced by edge computing. ### II-C Problem Formulation In this paper, we aim for minimizing the total energy consumption of the OFDM- based WP-MEC system, by optimizing the time allocation for WET and computing phases $\tau$, both the power allocation $\boldsymbol{p}^{E}$ and the IRS reflection coefficients $\boldsymbol{\Theta}^{E}$ at the WET phase, and the local CPU frequency at devices $\boldsymbol{f}$, the sub-band-device association $\\{\boldsymbol{\alpha}_{k}\\}$ and the power allocation $\\{\boldsymbol{p}_{k}\\}$ as well as the IRS reflection coefficients $\boldsymbol{\Theta}^{I}$ at the computing phase, subject to the energy constraint imposed by energy harvesting, the latency requirement of computation offloading and the sub-band-device association constraint in OFDMA systems as well as the constraint on IRS reflection coefficients. Since the batteries of all the wireless devices are replenished by the HAP, their energy consumption is covered by the energy consumption at the HAP during the WET phase. Hence, the total energy consumption of the system is formulated as the summation of the energy consumption both of the WET at the HAP and of the edge computing, i.e. $\tau T\sum^{M}_{m=1}p^{E}_{m}+\vartheta\sum^{K}_{k=1}\big{[}L_{k}-(1-\tau)Tf_{k}/c_{k}\big{]}$. To this end, the energy minimization problem is readily formulated for our OFDM-based WP-MEC system as $\displaystyle\mathcal{P}0\mathrel{\mathop{\mathchar 58\relax}}\mathop{\min}\limits_{\begin{subarray}{c}\tau,\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E},\boldsymbol{f},\\\ \\{\boldsymbol{\alpha}_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\Theta}^{I}\end{subarray}}\tau T\sum^{M}_{m=1}p^{E}_{m}+\vartheta\sum^{K}_{k=1}\bigg{[}L_{k}-\frac{(1-\tau)Tf_{k}}{c_{k}}\bigg{]}$ $\displaystyle\text{s.t.}\quad 0<\tau<1,$ (6a) $\displaystyle\quad\quad p^{E}_{m}\geq 0,\quad\forall m\in\mathcal{M},$ (6b) $\displaystyle\quad\quad|\Theta^{E}_{n}|\leq 1,\quad\forall n\in\mathcal{N},$ (6c) $\displaystyle\quad\quad 0\leq f_{k}\leq f_{max},\quad\forall k\in\mathcal{K},$ (6d) $\displaystyle\quad\quad\alpha_{k,m}\in\\{0,1\\},\quad\forall k\in\mathcal{K},\quad\forall m\in\mathcal{M},$ (6e) $\displaystyle\quad\quad\sum^{K}_{k=1}\alpha_{k,m}\leq 1,\quad\forall m\in\mathcal{M},$ (6f) $\displaystyle\quad\quad p^{I}_{k,m}\geq 0,\quad\forall k\in\mathcal{K},\quad\forall m\in\mathcal{M},$ (6g) $\displaystyle\quad\quad|\Theta^{I}_{n}|\leq 1,\quad\forall n\in\mathcal{N},$ (6h) $\displaystyle\quad\quad(1-\tau)T\bigg{[}\kappa f_{k}^{2}+\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})\bigg{]}\leq E_{k}(\tau,\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E}),\quad\forall k\in\mathcal{K},$ (6i) $\displaystyle\quad\quad(1-\tau)TR_{k}(\boldsymbol{\alpha}_{k},\boldsymbol{p}^{I}_{k},\boldsymbol{\Theta}^{I})\geq L_{k}-\frac{(1-\tau)Tf_{k}}{c_{k}},\quad\forall k\in\mathcal{K}.$ (6j) Constraint (6a) restricts the time allocation for the WET and for the computing phases. Constraint (6b) and (6c) represent the range of the power allocation and the IRS reflection coefficients at the WET phase, respectively. Constraint (6d) gives the range of tunable local computing frequencies. Constraint (6e) and (6f) detail the requirement of sub-band-device association in OFDMA systems. Constraint (6g) and (6h) restrict the range of the power allocation and the IRS reflection coefficient at the computing phase, respectively. Constraint (6i) indicates that the sum energy consumption of local computing and of computation offloading should not exceed the harvested energy for each device. Finally, Constraint (6j) implies that the communication link between the $k$-th device and the HAP is capable of offloading the corresponding computational tasks within the duration of the computing phase. ## III Joint Optimization of the Settings in the WET and the Computing Phases In this section, we propose to solve Problem $\mathcal{P}0$ in a two-step procedure. Firstly, given a fixed $\hat{\tau}\in(0,1)$, Problem $\mathcal{P}0$ can be simplified as follows $\displaystyle\mathcal{P}1\mathrel{\mathop{\mathchar 58\relax}}\mathop{\min}\limits_{\begin{subarray}{c}\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E},\boldsymbol{f},\\\ \\{\boldsymbol{\alpha}_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\Theta}^{I}\end{subarray}}\hat{\tau}T\sum^{M}_{m=1}p^{E}_{m}+\vartheta\sum^{K}_{k=1}\bigg{[}L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}\bigg{]}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_4},\eqref{eqn:P1_constraint_5},\eqref{eqn:P1_constraint_10},\eqref{eqn:P1_constraint_6},\eqref{eqn:P1_constraint_7},\eqref{eqn:P1_constraint_8},\eqref{eqn:P1_constraint_9},\eqref{eqn:P1_constraint_1},\eqref{eqn:P1_constraint_2}$ (7a) In the second step, we aim for finding the optimal $\hat{\tau}$ that is capable of minimizing the OF of Problem $\mathcal{P}0$ using the one- dimensional search method. In the rest of this section, we focus on solving Problem $\mathcal{P}1$. At a glance of Problem $\mathcal{P}1$, the optimization variables $\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$ and $\\{\boldsymbol{p}^{I}_{k}\\}$ are coupled with $\boldsymbol{p}^{E}$ and $\boldsymbol{\Theta}^{E}$ in Constraint (6i), which makes the problem difficult to solve. To tackle this issue, the AO technique is invoked. Specifically, upon initializing the setting of the computing phase, we may optimize the design of the WET phase while fixing the time allocation and the computing phase settings. Then, the computing phase settings could be optimized while fixing the time allocation and the design of the WET. A suboptimal solution can be obtained by iteratively optimizing the designs of the WET and of the computing phases. Let us detail the initialization as well as the designs of the WET and of the computing phases, as follows. ### III-A Initialization of the Time Allocation and the Computing Phase In order to ensure our WET design to be a feasible solution of Problem $\mathcal{P}1$, the initial settings of the computing phase denoted by $\boldsymbol{f}^{(0)},\big{\\{}\boldsymbol{\alpha}_{k}^{(0)}\big{\\}},\big{\\{}{\boldsymbol{p}^{I}_{k}}^{(0)}\big{\\}},{\boldsymbol{\Theta}^{I}}^{(0)}$ should satisfy Constraint (6d), (6e), (6f), (6g), (6h) and (6j). Without any loss of generality, their initialization is set as follows. * • Local computing frequency $\boldsymbol{f}^{(0)}$: Obeying the uniform distribution, each element of $\boldsymbol{f}^{(0)}$ is randomly set in the range of $[0,f_{max}]$. * • IRS reflection coefficient at the computing phase ${\boldsymbol{\Theta}^{I}}^{(0)}$: Obeying the uniform distribution, the amplitude response ${\beta^{I}_{n}}^{(0)}$ and the phase shift ${\theta^{I}_{n}}^{(0)}$ are randomly set in the range of $[0,1]$ and of $[0,2\pi)$, respectively. Then, ${\boldsymbol{\Theta}^{I}}^{(0)}=\\{{\beta^{I}_{1}}^{(0)}e^{j{\theta^{I}_{1}}^{(0)}},\ldots,{\beta^{I}_{N}}^{(0)}e^{j{\theta^{I}_{N}}^{(0)}}\\}$ can be obtained. * • Sub-band-device association at the computing phase $\big{\\{}\boldsymbol{\alpha}_{k}^{(0)}\big{\\}}$: We reserve a single sub- band for the devices associated with the index ranging from $k=1$ to $k=K$, sequentially. Specific to the $k$-th device, we use $k_{m}^{(0)}$ to denote the sub-band having the maximum $\big{|}C_{k,m}\big{(}{\boldsymbol{\Theta}^{I}}^{(0)}\big{)}\big{|}^{2}$ over the unassigned sub-bands, and assign this sub-band to the $k$-th device. * • Power allocation at the computing phase $\big{\\{}{\boldsymbol{p}^{I}_{k}}^{(0)}\big{\\}}$: For the $k$-th device, its power allocation at the computing phase should satisfy Constraint (6j). For minimizing the energy consumption, we assume the equivalence of two sides in Constraint (6j). Then, its initial power allocation is given by ${p^{I^{(0)}}_{k,k_{m}^{(0)}}}=\frac{\Gamma\sigma^{2}\Big{[}2^{\frac{L_{k}}{(1-\hat{\tau})TB}-\frac{f_{k}}{c_{k}B}}-1\Big{]}}{\big{|}c_{k,k^{(0)}_{m}}\big{(}{\boldsymbol{\Theta}^{I}}^{(0)}\big{)}\big{|}^{2}}$. For those sub-bands associated with the index $m\neq k^{(0)}_{m}$, we set ${p^{I^{(0)}}_{k,m}}=0$. ### III-B Design of the WET Phase While Fixing the Time Allocation and Computing Settings Given a fixed time allocation $\hat{\tau}$ and the settings of the computing phase $\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$, we may simplify Problem $\mathcal{P}1$ as $\displaystyle\mathcal{P}2\mathrel{\mathop{\mathchar 58\relax}}\mathop{\min}\limits_{\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E}}\hat{\tau}T\sum^{M}_{m=1}p^{E}_{m}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_4},\eqref{eqn:P1_constraint_5},$ $\displaystyle\quad\quad(1-\hat{\tau})T\bigg{[}\kappa f_{k}^{2}+\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})\bigg{]}\leq\sum^{M}_{m=1}\eta\hat{\tau}Tp^{E}_{m}\big{|}C_{k,m}(\boldsymbol{\Theta}^{E})\big{|}^{2},\quad\forall k\in\mathcal{K}.$ (8a) Since Constraint (8a) is not jointly convex regarding $\boldsymbol{p}^{E}$ and $\boldsymbol{\Theta}^{E}$, we optimize one of these two variables while fixing the other in an iterative manner, relying on the AO technique, as follows. #### III-B1 Optimizing the Power Allocation of the WET Phase While Fixing the Settings of the Time Allocation, the Computing Phase and the IRS Reflection Coefficient at the WET Phase Given an IRS phase shift design $\boldsymbol{\Theta}^{E}$, Problem $\mathcal{P}2$ is simplified as $\displaystyle\mathcal{P}2\text{-}1\mathrel{\mathop{\mathchar 58\relax}}\mathop{\min}\limits_{\boldsymbol{p}^{E}}\hat{\tau}T\sum^{M}_{m=1}p^{E}_{m}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_4},\eqref{eqn:P2_constraint_1}.$ (9a) It can be observed that Problem $\mathcal{P}2\text{-}1$ is a linear programming problem, which can be readily solved with the aid of the general implementation of interior-point methods, e.g. CVX [41]. The complexity is given by $\sqrt{M+KM}M[M+KM^{3}+M(M+KM^{2})+M^{2}]$ [42], i.e. $\mathcal{O}(K^{1.5}M^{4.5})$. #### III-B2 Optimizing the IRS Reflection Coefficient at the WET Phase While Fixing the Settings of the Time Allocation, the Computing Phase and the power Allocation at the WET Phase Given a power allocation at the WET phase $\boldsymbol{p}^{E}$, Problem $\mathcal{P}2$ becomes a feasibility-check problem, i.e. $\displaystyle\mathcal{P}2\text{-}2\mathrel{\mathop{\mathchar 58\relax}}\text{Find }\boldsymbol{\Theta}^{E}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_5},\eqref{eqn:P2_constraint_1}.$ (10a) As verified in [28], if one of the sub-problems is a feasibility-check problem, the iterative algorithm relying on the AO technique has a slow convergence. Specific to the problem considered, the operation of Find in Problem $\mathcal{P}2\text{-}2$ cannot guarantee the OF of Problem $\mathcal{P}2$ to be further reduced in each iteration. To address this issue, we reformulate Problem $\mathcal{P}2\text{-}2$ as follows, by introducing a set of auxiliary variables $\\{\xi_{k}\\}$ $\displaystyle\mathcal{P}2\text{-}2^{\prime}\mathrel{\mathop{\mathchar 58\relax}}\mathop{\max}\limits_{\boldsymbol{\Theta}^{E},\\{\xi_{k}\\}}\sum^{K}_{k=1}\xi_{k}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_5},$ $\displaystyle\quad\quad\xi_{k}+\kappa f_{k}^{2}+\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})\leq\frac{\sum^{M}_{m=1}\eta\hat{\tau}p^{E}_{m}\big{|}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta}^{E}\big{|}^{2}}{1-\hat{\tau}},\quad\forall k\in\mathcal{K},$ (11a) $\displaystyle\quad\quad\xi_{k}\geq 0,\quad\forall k\in\mathcal{K}.$ (11b) It is readily seen that the energy harvested by the wireless devices may increase after solving Problem $\mathcal{P}2\text{-}2^{\prime}$, which implies that the channel gain of the reflection link is enhanced. Then, a reduced power of energy signals can be guaranteed, when we turn back to solve Problem $\mathcal{P}2\text{-}1$. As such, a faster convergence can be obtained. However, at a glance of Problem $\mathcal{P}2\text{-}2^{\prime}$, Constraint (11a) is still non-convex regarding $\boldsymbol{\Theta}^{E}$. To tackle this issue, we manipulate the optimization problem in light of [33] as follows. Firstly, we transform Problem $\mathcal{P}2\text{-}2^{\prime}$ to its equivalent problem below, by introducing a set of auxiliary variables $\boldsymbol{y}^{E}$, $\boldsymbol{a}^{E}$ and $\boldsymbol{b}^{E}$ $\displaystyle\mathcal{P}2\text{-}2^{\prime}E1\mathrel{\mathop{\mathchar 58\relax}}\mathop{\max}\limits_{\boldsymbol{\Theta}^{E},\\{\xi_{k}\\},\boldsymbol{y}^{E},\boldsymbol{a}^{E},\boldsymbol{b}^{E}}\sum^{K}_{k=1}\xi_{k}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_5},\eqref{eqn:P2_2_constraint_11},$ $\displaystyle\quad\quad\xi_{k}+\kappa f_{k}^{2}+\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})\leq\frac{\sum^{M}_{m=1}\eta\hat{\tau}p^{E}_{m}y^{E}_{k,m}}{1-\hat{\tau}},\quad\forall k\in\mathcal{K},$ (12a) $\displaystyle\quad\quad a^{E}_{k,m}=\Re\big{\\{}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta}^{E}\big{\\}},\quad k\in\mathcal{K},\quad m\in\mathcal{M},$ (12b) $\displaystyle\quad\quad b^{E}_{k,m}=\Im\big{\\{}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta}^{E}\big{\\}},\quad k\in\mathcal{K},\quad m\in\mathcal{M},$ (12c) $\displaystyle\quad\quad y^{E}_{k,m}\leq(a^{E}_{k,m})^{2}+(b^{E}_{k,m})^{2},\quad k\in\mathcal{K},\quad m\in\mathcal{M},$ (12d) where $\Re\\{\bullet\\}$ and $\Im\\{\bullet\\}$ represent the real and imaginary part of $\bullet$, respectively. Following this, the successive convex approximation (SCA) method [43] is applied for tackling the non-convex constraint (12d). Specifically, the approximation function is constructed as follows. The right hand side of (12d) is lower-bounded by its first-order approximation at $(\tilde{a}^{E}_{k,m},\tilde{b}^{E}_{k,m})$, i.e. $(a^{E}_{k,m})^{2}+(b^{E}_{k,m})^{2}\geq\tilde{a}^{E}_{k,m}(2a^{E}_{k,m}-\tilde{a}^{E}_{k,m})+\tilde{b}^{E}_{k,m}(2b^{E}_{k,m}-\tilde{b}^{E}_{k,m})$, where the equality holds only when we have $\tilde{a}^{E}_{k,m}=a^{E}_{k,m}$ and $\tilde{b}^{E}_{k,m}=b^{E}_{k,m}$. Now we consider the following optimization problem $\displaystyle\mathcal{P}2\text{-}2^{\prime}E2\mathrel{\mathop{\mathchar 58\relax}}\mathop{\max}\limits_{\boldsymbol{\Theta}^{E},\\{\xi_{k}\\},\boldsymbol{y}^{E},\boldsymbol{a}^{E},\boldsymbol{b}^{E}}\sum^{K}_{k=1}\xi_{k}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_5},\eqref{eqn:P2_2_constraint_11},\eqref{eqn:P2_2'E_constraint_1},\eqref{eqn:P2_2'E_constraint_6},\eqref{eqn:P2_2'E_constraint_7},$ $\displaystyle\quad\quad y^{E}_{k,m}=\tilde{a}^{E}_{k,m}(2a^{E}_{k,m}-\tilde{a}^{E}_{k,m})+\tilde{b}^{E}_{k,m}(2b^{E}_{k,m}-\tilde{b}^{E}_{k,m}),\quad k\in\mathcal{K},\quad m\in\mathcal{M}.$ (13a) Both the OF and contraints in Problem $\mathcal{P}2\text{-}2^{\prime}E2$ are affine. Hence, Problem $\mathcal{P}2\text{-}2^{\prime}E2$ is a convex optimization problem, which can be solved by the implementation of interior- point methods, e.g. CVX [41]. Then, a locally optimal solution of $\mathcal{P}2\text{-}2^{\prime}$ can be approached by successively updating $\tilde{a}^{E}_{k,m}$ and $\tilde{b}^{E}_{k,m}$ based on the optimal solution of Problem $\mathcal{P}2\text{-}2^{\prime}E2$, whose procedure is detailed in Algorithm 1. The computation complexity of the SCA method is analyzed as follows. Problem $\mathcal{P}2\text{-}2^{\prime}E2$ involves $2KM$ linear equality constraints (equivalently $4KM$ inequality constraints) of size $2N+1$, $K$ linear inequality constraints of size $M+1$, $KM$ linear inequality constraints of size $3$, $K$ linear inequality constraints of size $1$, $N$ second-order cone inequality constraints of size $2$. Hence, the total complexity of Algorithm 1 is given by $\ln(1/\epsilon)\sqrt{4KM(2N+1)+K(M+1)+3KM+K+2N}(2N+3M+K)\\{4KM(2N+1)^{3}+K(M+1)^{3}+27KM+K+(2N+3M+K)[4KM(2N+1)^{2}+K(M+1)^{2}+9KM+K]+4N+(2N+3M+K)^{2}\\}$ [42], i.e. $\ln(1/\epsilon)\mathcal{O}(K^{1.5}M^{1.5}N^{4.5}+K^{1.5}M^{2.5}N^{3.5}+K^{1.5}M^{2.5}N^{1.5}+K^{2.5}M^{1.5}N^{3.5}+K^{1.5}M^{4.5}+K^{2.5}M^{2.5}N^{2.5}+K^{2.5}M^{3.5}+K^{3.5}M^{1.5}N^{2.5}+K^{3.5}M^{2.5})$. To this end, we summarize the procedure of solving Problem $\mathcal{P}2$ in Algorithm 2. Algorithm 1 SCA approach to design $\boldsymbol{\Theta}^{E}$, given the settings of the time allocation, the computing phase and the power allocation at the WET phase 0: $t_{max}$, $\epsilon$, $K$, $M$, $N$, $T$, $\eta$, $c_{k}$, $\kappa$, $f_{max}$, $p_{c}$, $\Gamma$, $L_{k}$, $\\{\boldsymbol{h}^{d}_{k}\\}$, $\\{\boldsymbol{V}_{k}\\}$, $\hat{\tau}$, $\boldsymbol{P}^{E}$, $\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$, $\boldsymbol{\Theta}^{I}$ and $\tilde{\boldsymbol{\Theta}}^{E}$ 0: $\boldsymbol{\Theta}^{E}$ 1\. Initialization Initialize $t_{1}=0$; $\epsilon_{1}=1$; $\xi_{k}=0,\forall k\in\mathcal{K}$ 2\. SCA approach to design $\boldsymbol{\Theta}^{E}$ while $t_{1}<t_{\text{max}}$ $\&\&$ $\epsilon_{1}>\epsilon$ do $\bullet$ Set $\tilde{a}^{E}_{k,m}=\Re\big{\\{}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\tilde{\boldsymbol{\Theta}}^{E}\big{\\}}$ and $\tilde{b}^{E}_{k,m}=\Im\big{\\{}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\tilde{\boldsymbol{\Theta}}^{E}\big{\\}},\forall k\in\mathcal{K},\forall m\in\mathcal{M}$ $\bullet$ Obtain ${\boldsymbol{\Theta}^{E}}$ and $\\{\xi_{k}\\}$ by solving Problem $\mathcal{P}2\text{-}2^{\prime}E2$ using CVX $\bullet$ Set $\epsilon_{1}=\frac{\big{|}\text{obj}\big{(}{\boldsymbol{\Theta}}^{E}\big{)}-\text{obj}\big{(}\tilde{\boldsymbol{\Theta}}^{E}\big{)}\big{|}}{\big{|}\text{obj}\big{(}\boldsymbol{\Theta}^{E}\big{)}\big{|}}$, $\tilde{\boldsymbol{\Theta}}^{E}\leftarrow\boldsymbol{\Theta}^{E}$, $t_{1}\leftarrow t_{1}+1$ end while3. Output optimal ${\boldsymbol{\Theta}^{E}}^{*}$ ${\boldsymbol{\Theta}^{E}}^{*}\leftarrow\tilde{\boldsymbol{\Theta}}^{E}$ Algorithm 2 Alternative optimization of $\boldsymbol{p}^{E}$ and $\boldsymbol{\Theta}^{E}$, given the settings of the time allocation and the computing phase 0: $t_{max}$, $\epsilon$, $K$, $M$, $N$, $T$, $\eta$, $c_{k}$, $\kappa$, $f_{max}$, $p_{c}$, $\Gamma$, $L_{k}$, $\\{\boldsymbol{h}^{d}_{k}\\}$, $\\{\boldsymbol{V}_{k}\\}$, $\hat{\tau}$, $\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$, $\boldsymbol{\Theta}^{I}$ and $\tilde{\boldsymbol{\Theta}}^{E}$ 0: $\boldsymbol{P}^{E}$ and $\boldsymbol{\Theta}^{E}$ 1\. Initialization $\bullet$ Initialize $t_{2}=0$; $\epsilon_{2}=1$; ${\boldsymbol{\Theta}^{E}}^{(0)}=\tilde{\boldsymbol{\Theta}}^{E}$ $\bullet$ Given ${\boldsymbol{\Theta}^{E}}^{(0)}$, find ${\boldsymbol{P}^{E}}^{(0)}$ by solving Problem $\mathcal{P}2\text{-}1$ via CVX 2\. Alternative optimization of $\boldsymbol{P}^{E}$ and $\boldsymbol{\Theta}^{E}$ while $t_{2}<t_{\text{max}}$ $\&\&$ $\epsilon_{2}>\epsilon$ do $\bullet$ Given ${\boldsymbol{P}^{E}}^{(t_{2})}$ and $\tilde{\boldsymbol{\Theta}}^{E}={\boldsymbol{\Theta}^{E}}^{(t_{2})}$, find ${\boldsymbol{\Theta}^{E}}^{(t_{2}+1)}$ by solving Problem $\mathcal{P}2\text{-}2^{\prime}E1$ using Algorithm 1 $\bullet$ Given ${\boldsymbol{\Theta}^{E}}^{(t_{2}+1)}$, find ${\boldsymbol{P}^{E}}^{(t_{2}+1)}$ by solving Problem $\mathcal{P}2\text{-}1$ via CVX $\bullet$ Set $\epsilon_{2}=\frac{\big{|}\text{obj}\big{(}{\boldsymbol{p}^{E}}^{(t_{2}+1)},{{\boldsymbol{\Theta}}^{E}}^{(t_{2}+1)}\big{)}-\text{obj}\big{(}{\boldsymbol{p}^{E}}^{(t_{2})},{{\boldsymbol{\Theta}}^{E}}^{(t_{2})}\big{)}\big{|}}{\big{|}\text{obj}\big{(}{\boldsymbol{p}^{E}}^{(t_{2}+1)},{{\boldsymbol{\Theta}}^{E}}^{(t_{2}+1)}\big{)}\big{|}}$, $t_{2}\leftarrow t_{2}+1$ end while3. Output optimal ${\boldsymbol{P}^{E}}^{*}$ and ${\boldsymbol{\Theta}^{E}}^{*}$ ${\boldsymbol{\Theta}^{E}}^{*}\leftarrow{\boldsymbol{\Theta}^{E}}^{(t_{2})}$ and ${\boldsymbol{P}^{E}}^{*}\leftarrow{\boldsymbol{P}^{E}}^{(t_{2})}$ ### III-C Design of the Computing Phase While Fixing the Time Allocation and WET Settings In this subsection, we aim for optimizing the design of the computing phase, while fixing the time allocation $\hat{\tau}$ and the WET settings $\boldsymbol{p}^{E}$ and $\boldsymbol{\Theta}^{E}$. In this case, we simplify Problem $\mathcal{P}1$ as $\displaystyle\mathcal{P}3\mathrel{\mathop{\mathchar 58\relax}}\mathop{\min}\limits_{\boldsymbol{f},\\{\boldsymbol{\alpha}_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\Theta}^{I}}\vartheta\sum^{K}_{k=1}\bigg{[}L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}\bigg{]}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_10},\eqref{eqn:P1_constraint_6},\eqref{eqn:P1_constraint_7},\eqref{eqn:P1_constraint_8},\eqref{eqn:P1_constraint_9},\eqref{eqn:P1_constraint_1},$ $\displaystyle\quad\quad\sum^{m}_{m=1}\alpha_{k,m}B\log_{2}\Bigg{[}1+\frac{p_{k,m}|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\Bigg{]}\geq\frac{L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}}{(1-\hat{\tau})T},\quad\forall k\in\mathcal{K}.$ (14a) Constraint (14a) is not jointly convex regarding $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$. Hence, it is difficult to find its globally optimal solution. Alternatively, its suboptimal solution is provided by iteratively optimizing the $\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$, again relying on the AO technique, as follows. #### III-C1 Alternative Optimization of the Sub-Band-Device Association and the Power Allocation as well as the IRS Reflection Coefficient at the Computing Phase Given a fixed local CPU frequency setting $\boldsymbol{f}$, the OF of Problem $\mathcal{P}3$ becomes deterministic. In other words, the optimization of $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$ seems not contributing to reducing the OF. However, this is not always true, because if a larger feasible set of $\boldsymbol{f}$ can be obtained by optimizing $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$, a reduced OF may be achieved when we turn back to optimize $\boldsymbol{f}$. Based on this observation, we formulate the problem below, by introducing a set of auxiliary variables $\\{\zeta_{k}\\}$ $\displaystyle\mathcal{P}3\text{-}1\mathrel{\mathop{\mathchar 58\relax}}\mathop{\max}\limits_{\\{\zeta_{k}\\},\\{\boldsymbol{\alpha}_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\Theta}^{I}}\sum^{K}_{k=1}\zeta_{k}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_6},\eqref{eqn:P1_constraint_7},\eqref{eqn:P1_constraint_8},\eqref{eqn:P1_constraint_9},\eqref{eqn:P3_constraint_2}$ $\displaystyle\quad\quad\zeta_{k}\geq 0,\quad\forall k\in\mathcal{K},$ (15a) $\displaystyle\quad\quad(1-\hat{\tau})T\bigg{[}\kappa f_{k}^{2}+\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})+\zeta_{k}\bigg{]}\leq\sum^{M}_{m=1}\eta\hat{\tau}Tp^{E}_{m}\big{|}C_{k,m}(\boldsymbol{\Theta}^{E})\big{|}^{2},\quad\forall k\in\mathcal{K}.$ (15b) As specified in (15a), the auxiliary variables $\\{\zeta_{k}\\}$ are non- negative, and thus a non-smaller set of $\boldsymbol{f}$ may be obtained after solving Problem $\mathcal{P}3\text{-}1$. Given that Constraint (14a) is not jointly convex regarding $\\{\boldsymbol{p}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$, we optimize $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$ in two steps iteratively. In the first step, we optimize $\\{\zeta_{k}\\},\\{\boldsymbol{\alpha}_{k}\\}$ and $\\{\boldsymbol{p}^{I}_{k}\\}$, while fixing the IRS reflection coefficient $\boldsymbol{\Theta}^{I}$. In this case, Problem $\mathcal{P}3\text{-}1$ can be simplified as $\displaystyle\mathcal{P}3\text{-}1a\mathrel{\mathop{\mathchar 58\relax}}\mathop{\max}\limits_{\\{\zeta_{k}\\},\\{\boldsymbol{\alpha}_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\}}\sum^{K}_{k=1}\zeta_{k}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_6},\eqref{eqn:P1_constraint_7},\eqref{eqn:P1_constraint_8},\eqref{eqn:P3_constraint_2},\eqref{eqn:P3_1_constraint_3},\eqref{eqn:P3_1_constraint_2}.$ (16a) Problem $\mathcal{P}3\text{-}1a$ is a combinatorial optimization problem, where the binary constraint (6e) is non-convex. The classic solution typically relies on the convex relaxation method [44], where the binary constraint imposed on $\\{\boldsymbol{\alpha}_{k}\\}$ is relaxed into a convex constraint by introducing time-sharing variables. However, the relaxed problem is different from the original problem, which might lead to a specific error. To address this issue, a near-optimal solution based on the Lagrangian duality was proposed [39], where it is verified that the duality gap vanishes in the system equipped with more than $8$ sub-bands. Hence, in this paper, the Lagrangian duality method [45] is invoked for solving Problem $\mathcal{P}3\text{-}1a$. Specifically, denoting the non-negative Lagrange multiplier vectors by $\boldsymbol{\lambda}=[\lambda_{1},\lambda_{2},\ldots,\lambda_{K}]^{T}$ and $\boldsymbol{\mu}=[\mu_{1},\mu_{2},\ldots,\mu_{K}]^{T}$, we formulate the Lagrangian function of Problem $\mathcal{P}3\text{-}1a$ over the domain $\mathcal{D}$ as $\displaystyle\mathcal{L}\big{(}\\{\zeta_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\lambda},\boldsymbol{\mu}\big{)}$ $\displaystyle=$ $\displaystyle\sum_{k=1}^{K}\zeta_{k}-\sum_{k=1}^{K}\lambda_{k}\bigg{[}\kappa f_{k}^{2}+\sum^{M}_{m=1}(p^{I}_{k,m}+p_{c})+\zeta_{k}-\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}\bigg{]}$ (17) $\displaystyle+\sum_{k=1}^{K}\mu_{k}\Bigg{[}\sum^{M}_{m=1}B\log_{2}\bigg{(}1+\frac{p^{I}_{k,m}|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\bigg{)}-\frac{L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}}{(1-\hat{\tau})T}\Bigg{]},$ where the domain $\mathcal{D}$ is defined as the set of all non-negative $p^{I}_{k,m}$ for $\forall k\in\mathcal{K}$ and for $\forall m\in\mathcal{M}$ such that for each $m$, only a single $p^{I}_{k,m}$ is positive for $k\in\mathcal{K}$. Then, the Lagrangian dual function of Problem $\mathcal{P}3\text{-}1a$ is given by $\displaystyle g(\boldsymbol{\lambda},\boldsymbol{\mu})=\mathop{\max}\limits_{\\{\zeta_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\}\in{\mathcal{D}}}\mathcal{L}\big{(}\\{\zeta_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\lambda},\boldsymbol{\mu}\big{)}.$ (18) (18) can be reformulated as $\displaystyle g(\boldsymbol{\lambda},\boldsymbol{\mu})$ $\displaystyle=$ $\displaystyle\sum_{m=1}^{M}\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})+\sum^{K}_{k=1}(1-\lambda_{k})\zeta_{k}-\sum^{K}_{k=1}\lambda_{k}\kappa f_{k}^{2}$ (19) $\displaystyle+\sum^{K}_{k=1}\lambda_{k}\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}-\sum^{K}_{k=1}\frac{\mu_{k}\Big{[}L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}\Big{]}}{(1-\hat{\tau})T},$ where we have $\displaystyle\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})\triangleq\mathop{\max}\limits_{\\{\boldsymbol{p}^{I}_{k}\\}\in{\mathcal{D}}}\Bigg{\\{}-\sum_{k=1}^{K}\lambda_{k}(p^{I}_{k,m}+p_{c})+\sum_{k=1}^{K}\mu_{k}B\log_{2}\Bigg{[}1+\frac{p^{I}_{k,m}|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\Bigg{]}\Bigg{\\}}.$ (20) It is readily seen that (20) is concave regarding $p^{I}_{k,m}$. Thus, upon letting its first-order derivative regarding $p^{I}_{k,m}$ be $0$, we may give the optimal power of the $m$-th sub-band when it is allocated to the $k$-th device as $\displaystyle\hat{p}^{I}_{k,m}(\lambda_{k},\mu_{k})=\bigg{[}\frac{\mu_{k}B}{\lambda_{k}\ln 2}-\frac{\Gamma\sigma^{2}}{|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}\bigg{]}^{+}.$ (21) Then, $\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})$ can be obtained, by searching over all possible assignments of the $m$-th sub-band for all the $K$ devices, as follows $\displaystyle\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})=\max_{k}\Bigg{\\{}-\lambda_{k}\Big{[}\hat{p}^{I}_{k,m}(\lambda_{k},\mu_{k})+p_{c}\Big{]}+\mu_{k}B\log_{2}\Bigg{[}1+\frac{\hat{p}^{I}_{k,m}(\lambda_{k},\mu_{k})|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\Bigg{]}\Bigg{\\}},$ (22) and the suitable device is given by $k^{*}=\arg\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})$. We set $\alpha_{k^{*},m}=1$ and $p^{I}_{k^{*},m}=\hat{p}^{I}_{k^{*},m}$ as well as $\alpha_{k,m}=0$ and $p^{I}_{k,m}=0$ for $\forall k\neq k^{*}$. We continue by calculating $\\{\zeta_{k}\\}$ as follows. At a glance of (21), it is observed that $\lambda_{k}$ has to yield $\lambda_{k}>0$, $\forall k\in\mathcal{K}$, which implies that Constraint (15b) is strictly binding for the optimal solution of Problem $\mathcal{P}3\text{-}1a$. Therefore, $\zeta_{k}$ can be set as $\displaystyle\zeta_{k}=\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}-\kappa f_{k}^{2}-\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c}).$ (23) Once all $\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})$ and $\zeta_{k}$ are obtained, $g(\boldsymbol{\lambda},\boldsymbol{\mu})$ can be calculated by (19). Bearing in mind that the obtained $g(\boldsymbol{\lambda},\boldsymbol{\mu})$ is not guaranteed to be optimal, we have to find a suitable set of $\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$ that minimize $g(\boldsymbol{\lambda},\boldsymbol{\mu})$, which can be realized by the ellipsoid method [45]. More explicitly, the Lagrange multipliers are iteratively updated following their sub-gradients towards their optimal settings. The corresponding sub-gradients are given as follows $\displaystyle s_{\lambda_{k}}=\kappa f_{k}^{2}+\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})-\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T},$ (24) $\displaystyle s_{\mu_{k}}=\frac{L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}}{(1-\hat{\tau})T}-\sum_{m=1}^{M}\alpha_{k,m}B\log_{2}\bigg{(}1+\frac{p^{I}_{k,m}|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\bigg{)}.$ (25) Upon denoting the iteration index by $t$, the Lagrange multipliers are updated obeying $\lambda_{k}(t+1)=[\lambda_{k}(t)+\delta_{\lambda}(t)s_{\lambda_{k}}]^{+}$ and $\mu_{k}(t+1)=[\mu_{k}(t)+\delta_{\mu}(t)s_{\mu_{k}}]^{+}$, where we set $\delta_{\lambda}(t)=\delta_{\lambda}(1)/t$ and $\delta_{\mu}(t)=\delta_{\mu}(1)/t$ for ensuring the convergence of the OF. In the problem considered, the ellipsoid method converges in $\mathcal{O}(K^{2})$ iterations [45, 39]. Within each iteration, the computational complexity is of $\mathcal{O}(KM)$. Hence, the total computational complexity is given by $\mathcal{O}(MK^{3})$. The procedure of this Lagrangian duality method is summarized in Algorithm 3. Algorithm 3 Design of $\\{\boldsymbol{\alpha}_{k}\\}$ and $\\{\boldsymbol{p}^{I}_{k}\\}$, given the settings of $\hat{\tau}$, $\boldsymbol{p}^{E}$, $\boldsymbol{\Theta}^{E}$, $\boldsymbol{f}$ and $\boldsymbol{\Theta}^{I}$ 0: $t_{max}$, $\epsilon$, $K$, $M$, $N$, $T$, $\eta$, $c_{k}$, $\kappa$, $f_{max}$, $p_{c}$, $\Gamma$, $L_{k}$, $\\{\boldsymbol{h}^{d}_{k}\\}$, $\\{\boldsymbol{V}_{k}\\}$, $\hat{\tau}$, $\boldsymbol{P}^{E}$, $\boldsymbol{\Theta}^{E}$, $\boldsymbol{\Theta}^{I}$, $\boldsymbol{f}$, $\boldsymbol{\Theta}^{I}$, $\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$ 0: $\\{\zeta_{k}\\}$, $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ 1\. Initialization Initialize $t_{3}=0$; $\epsilon_{3}=1$; Calculate $\mathcal{L}^{(0)}$ using (17) 2\. Optimization of $\\{\zeta_{k}\\}$, $\\{\boldsymbol{\alpha}_{k}\\}$ and $\\{\boldsymbol{p}^{I}_{k}\\}$ while $t_{3}<t_{\text{max}}$ $\&\&$ $\epsilon_{3}>\epsilon$ do for $m=1\mathrel{\mathop{\mathchar 58\relax}}M$ do $\bullet$ Calculate $\hat{p}^{I}_{k,m}$ using (21) for each $\forall k\in\mathcal{K}$ $\bullet$ Obtain the optimal device $k^{*}=\arg\hat{g}_{m}(\boldsymbol{\lambda},\boldsymbol{\mu})$ in (22) $\bullet$ Set $\alpha_{k^{*},m}=1$ and $p^{I}_{k^{*},m}=\hat{p}^{I}_{k^{*},m}$ as well as $\alpha_{k,m}=0$ and $p^{I}_{k,m}=0$ for $\forall k\neq k^{*}$ end for $\bullet$ Calculate $\zeta_{k}$ using (23) $\bullet$ Calculate $\mathcal{L}^{(t_{3}+1)}$ using (17) $\bullet$ Update $\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$ using the ellipsoid method $\bullet$ Set $\epsilon_{3}=\frac{\big{|}\mathcal{L}^{(t_{3}+1)}-\mathcal{L}^{(t_{3})}\big{|}}{\big{|}\mathcal{L}^{(t_{3}+1)}\big{|}}$, $t_{3}\leftarrow t_{3}+1$ end while3. Output optimal $\\{\zeta_{k}\\}^{*}$, $\\{\boldsymbol{\alpha}_{k}\\}^{*}$ and $\\{\boldsymbol{p}^{I}_{k}\\}^{*}$ $\\{\zeta_{k}\\}^{*}=\\{\zeta_{k}\\}$, $\\{\boldsymbol{\alpha}_{k}\\}^{*}=\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}^{*}=\\{\boldsymbol{p}^{I}_{k}\\}$ In the second step, we optimize the IRS reflection coefficient $\boldsymbol{\Theta}^{I}$, while fixing the settings of the resource allocation at the computing phase $\\{\boldsymbol{\alpha}_{k}\\}$ and $\\{\boldsymbol{p}^{I}\\}$. In this case, Problem $\mathcal{P}3\text{-}1$ becomes a feasibility-check problem below $\displaystyle\mathcal{P}3\text{-}1b\mathrel{\mathop{\mathchar 58\relax}}\text{Find }\boldsymbol{\Theta}^{I}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_9},\eqref{eqn:P3_constraint_2}.$ The problem can be solved using the approach devised in Section III-B2, detailed as follows. By introducing a set of auxiliary variables $\\{\chi_{k}\\}$, we transform $\mathcal{P}3\text{-}2$ to the problem below $\displaystyle\mathcal{P}3\text{-}1b\mathrel{\mathop{\mathchar 58\relax}}\mathop{\max}\limits_{\boldsymbol{\Theta}^{I},\\{\chi_{k}\\}}\sum^{K}_{k=1}\chi_{k}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_9},$ $\displaystyle\quad\quad\chi_{k}\geq 0,\quad\forall k\in\mathcal{K},$ (27a) $\displaystyle\quad\quad\sum^{m}_{m=1}\alpha_{k,m}B\log_{2}\bigg{[}1+\frac{p_{k,m}|C_{k,m}(\boldsymbol{\Theta}^{I})|^{2}}{\Gamma\sigma^{2}}\bigg{]}\geq\frac{L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}}{(1-\hat{\tau})T}+\chi_{k},\quad\forall k\in\mathcal{K}.$ (27b) Constraint (27b) is non-convex regarding $\boldsymbol{\Theta}^{I}$. To address this issue, firstly we transform Problem $\mathcal{P}3\text{-}1b$ to its equivalent form, by introducing a set of auxiliary variables $\boldsymbol{y}^{I}$, $\boldsymbol{a}^{I}$ and $\boldsymbol{b}^{I}$ $\displaystyle\mathcal{P}3\text{-}1bE1\mathrel{\mathop{\mathchar 58\relax}}\mathop{\max}\limits_{\boldsymbol{\Theta}^{I},\\{\chi_{k}\\},\boldsymbol{y}^{I},\boldsymbol{a}^{I},\boldsymbol{b}^{I}}\sum^{K}_{k=1}\chi_{k}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_9},\eqref{eqn:P3_2'_constraint_3},$ $\displaystyle\quad\quad\sum^{m}_{m=1}\alpha_{k,m}B\log_{2}\bigg{(}1+\frac{p_{k,m}y^{I}_{k,m}}{\Gamma\sigma^{2}}\bigg{)}\geq\frac{L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}}{(1-\hat{\tau})T}+\chi_{k},\quad\forall k\in\mathcal{K},$ (28a) $\displaystyle\quad\quad a^{I}_{k,m}=\Re\big{\\{}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta}^{I}\big{\\}},\quad k\in\mathcal{K},\quad m\in\mathcal{M},$ (28b) $\displaystyle\quad\quad b^{I}_{k,m}=\Im\big{\\{}\boldsymbol{f}^{H}_{m}\boldsymbol{h}^{d}_{k}+\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta}^{I}\big{\\}},\quad k\in\mathcal{K},\quad m\in\mathcal{M},$ (28c) $\displaystyle\quad\quad y^{I}_{k,m}=(a^{I}_{k,m})^{2}+(b^{I}_{k,m})^{2},\quad k\in\mathcal{K},\quad m\in\mathcal{M}.$ (28d) Then, upon invoking the so-called SCA method as detailed in Section III-B2, we approach the locally optimal solution by solving the problem below in a successive manner $\displaystyle\mathcal{P}3\text{-}1bE2\mathrel{\mathop{\mathchar 58\relax}}\mathop{\max}\limits_{\boldsymbol{\Theta}^{I},\\{\chi_{k}\\}}\sum^{K}_{k=1}\chi_{k}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_9},\eqref{eqn:P3_2'_constraint_3},\eqref{eqn:P3_2'E1_constraint_2},\eqref{eqn:P3_2'E1_constraint_6},\eqref{eqn:P3_2'E1_constraint_7},$ $\displaystyle\quad\quad y^{I}_{k,m}=\tilde{a}^{I}_{k,m}(2a^{I}_{k,m}-\tilde{a}^{I}_{k,m})+\tilde{b}^{I}_{k,m}(2b^{I}_{k,m}-\tilde{b}^{I}_{k,m}),\quad k\in\mathcal{K},\quad m\in\mathcal{M}.$ (29a) Problem $\mathcal{P}3\text{-}1bE2$ is a convex optimization problem, which can be readily solved with the aid of the software of CVX [41]. The computational complexity is the same as that given in Section III-B2. Note that the optimization of $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$ not only contributes to reducing the OF of Problem $\mathcal{P}2$, but also leads to a decreased OF of Problem $\mathcal{P}1$ by slacking its constraint (8a). Hence, we may still reduce the OF of Problem $\mathcal{P}1$ by iteratively optimizing the settings of the WET phase and the computing phase, even if $\boldsymbol{f}$ reaches its maximum value. #### III-C2 Design of CPU Frequencies Given the settings of the sub-band-device association $\\{\boldsymbol{\alpha}_{k}\\}$, the power allocation $\\{\boldsymbol{p}^{I}_{k}\\}$ and the IRS reflection coefficient $\boldsymbol{\Theta}^{I}$, Problem $\mathcal{P}3$ can be simplified as $\displaystyle\mathcal{P}3\text{-}2\mathrel{\mathop{\mathchar 58\relax}}\mathop{\min}\limits_{\boldsymbol{f},\\{\boldsymbol{\alpha}_{k}\\},\\{\boldsymbol{p}^{I}_{k}\\},\boldsymbol{\Theta}^{I}}\vartheta\sum^{K}_{k=1}\bigg{[}L_{k}-\frac{(1-\hat{\tau})Tf_{k}}{c_{k}}\bigg{]}$ $\displaystyle\text{s.t.}\quad\eqref{eqn:P1_constraint_10},\eqref{eqn:P3_1_constraint_2}.$ (30a) It can be seen that the OF of Problem $\mathcal{P}3\text{-}2$ decreases upon increasing $\boldsymbol{f}$. Hence, upon denoting $\displaystyle\hat{f}_{k}=\sqrt{\frac{\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}-\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})-\zeta_{k}}{\kappa}},$ (31) the optimal $\boldsymbol{f}$ can be obtained as: $\displaystyle f_{k}$ $\displaystyle=\begin{cases}0,&\quad\text{if }\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}-\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})-\zeta_{k}<0,\\\ \hat{f}_{k},&\quad\text{if }0\leq\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}-\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})-\zeta_{k}<\kappa f_{max}^{2},\\\ f_{max},&\quad\text{if }\frac{E_{k}(\hat{\tau},\boldsymbol{p}^{E},\boldsymbol{\Theta}^{E})}{(1-\hat{\tau})T}-\sum^{M}_{m=1}\alpha_{k,m}(p^{I}_{k,m}+p_{c})-\zeta_{k}\geq\kappa f_{max}^{2}.\end{cases}$ (32) The procedure of optimizing $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$, $\boldsymbol{\Theta}^{I}$ and $\boldsymbol{f}$ is summarized in Algorithm 4. To this end, it is readily to summarize the algorithm solving Problem $\mathcal{P}1$ under a given $\hat{\tau}$ in Algorithm 5, and an appropriate $\tau$ is found with the aid of numerical results, as detailed in Section IV-A. Algorithm 4 Alternative optimization of $\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$, given the settings of $\hat{\tau}$, $\boldsymbol{p}^{E}$ and $\boldsymbol{\Theta}^{E}$ 0: $t_{max}$, $\epsilon$, $K$, $M$, $N$, $T$, $\eta$, $c_{k}$, $\kappa$, $f_{max}$, $p_{c}$, $\Gamma$, $L_{k}$, $\\{\boldsymbol{h}^{d}_{k}\\}$, $\\{\boldsymbol{V}_{k}\\}$, $\hat{\tau}$, $\boldsymbol{P}^{E}$, $\boldsymbol{\Theta}^{E}$, $\boldsymbol{f}$, and $\tilde{\boldsymbol{\Theta}}^{I}$ 0: $\\{\boldsymbol{\alpha}_{k}\\}$ $\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$ 1\. Initialization $\bullet$ Initialize $t_{4}=0$; $\epsilon_{4}=1$; ${\boldsymbol{\Theta}^{I}}^{(0)}=\tilde{\boldsymbol{\Theta}}^{I}$ $\bullet$ Given ${\boldsymbol{\Theta}^{I}}^{(0)}$, find $\\{\boldsymbol{\alpha}_{k}\\}^{(0)}$ and $\\{\boldsymbol{p}^{I}_{k}\\}^{(0)}$ by solving Problem $\mathcal{P}3\text{-}1a$ via Algorithm 3 $\bullet$ Obtain $\boldsymbol{f}^{(0)}$ via (32) and calculate $\text{obj}\big{(}\boldsymbol{f}^{(0)}\big{)}$ 2\. Alternative optimization of $\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$, $\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$ while $t_{4}<t_{\text{max}}$ $\&\&$ $\epsilon_{4}>\epsilon$ do $\bullet$ Given $\\{\boldsymbol{\alpha}_{k}\\}^{(t_{4})}$, $\\{\boldsymbol{p}^{I}_{k}\\}^{(t_{4})}$ and $\tilde{\boldsymbol{\Theta}}^{I}={\boldsymbol{\Theta}^{I}}^{(t_{4})}$, find ${\boldsymbol{\Theta}^{I}}^{(t_{4}+1)}$ by solving Problem $\mathcal{P}3\text{-}1bE1$ via Algorithm 1 $\bullet$ Given ${\boldsymbol{\Theta}^{I}}^{(t_{4}+1)}$, find $\\{\boldsymbol{\alpha}_{k}\\}^{(t_{4}+1)}$ and $\\{\boldsymbol{p}^{I}_{k}\\}^{(t_{4}+1)}$ by solving Problem $\mathcal{P}3\text{-}1a$ via Algorithm 3 $\bullet$ Obtain $\boldsymbol{f}^{(t_{4}+1)}$ via (32) and calculate $\text{obj}\big{(}\boldsymbol{f}^{(t_{4}+1)}\big{)}$ $\bullet$ Set $\epsilon_{4}=\frac{\big{|}\text{obj}\big{(}\boldsymbol{f}^{(t_{4}+1)}\big{)}-\text{obj}\big{(}\boldsymbol{f}^{(t_{4})}\big{)}\big{|}}{\big{|}\text{obj}\big{(}\boldsymbol{f}^{(t_{4}+1)}\big{)}\big{|}}$, $t_{4}\leftarrow t_{4}+1$ end while3. Output optimal $\\{\boldsymbol{\alpha}_{k}\\}^{*}$ $\\{\boldsymbol{p}^{I}_{k}\\}^{*}$ and ${\boldsymbol{\Theta}^{I}}^{*}$ $\\{\boldsymbol{\alpha}_{k}\\}^{*}\leftarrow\\{\boldsymbol{\alpha}_{k}\\}^{(t_{4})}$, $\\{\boldsymbol{p}^{I}_{k}\\}^{*}\leftarrow\\{\boldsymbol{p}^{I}_{k}\\}^{(t_{4})}$ and ${\boldsymbol{\Theta}^{I}}^{*}\leftarrow{\boldsymbol{\Theta}^{I}}^{(t_{4})}$ Algorithm 5 Alternative optimization of the WET and computing phases, given the time allocation 0: $t_{max}$, $\epsilon$, $K$, $M$, $N$, $T$, $\eta$, $c_{k}$, $\kappa$, $f_{max}$, $p_{c}$, $\Gamma$, $L_{k}$, $\\{\boldsymbol{h}^{d}_{k}\\}$, $\\{\boldsymbol{V}_{k}\\}$ and $\hat{\tau}$ 0: $\boldsymbol{P}^{E}$, $\boldsymbol{\Theta}^{E}$, $\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$ $\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$ 1\. Initialization $\bullet$ Initialize $t_{5}=0$; $\epsilon_{5}=1$; $\tilde{\boldsymbol{\Theta}}^{E}$ $\bullet$ Initialize $\boldsymbol{f}^{(0)}$, $\\{\boldsymbol{\alpha}_{k}\\}^{(0)}$, $\\{\boldsymbol{p}^{I}_{k}\\}^{(0)}$ and ${\boldsymbol{\Theta}^{I}}^{(0)}$ following Section III-A $\bullet$ Given $\boldsymbol{f}^{(0)}$, $\\{\boldsymbol{\alpha}_{k}\\}^{(0)}$, $\\{\boldsymbol{p}^{I}_{k}\\}^{(0)}$ and ${\boldsymbol{\Theta}^{I}}^{(0)}$, find ${\boldsymbol{P}^{E}}^{(0)}$ and ${\boldsymbol{\Theta}^{E}}^{(0)}$ by solving Problem $\mathcal{P}2$ via Algorithm 2 2\. Alternative optimization of $\boldsymbol{P}^{E}$, $\boldsymbol{\Theta}^{E}$, $\boldsymbol{f}$, $\\{\boldsymbol{\alpha}_{k}\\}$ $\\{\boldsymbol{p}^{I}_{k}\\}$ and $\boldsymbol{\Theta}^{I}$ while $t_{5}<t_{\text{max}}$ $\&\&$ $\epsilon_{5}>\epsilon$ do $\bullet$ Given ${\boldsymbol{P}^{E}}^{(t_{5})}$, ${\boldsymbol{\Theta}^{E}}^{(t_{5})}$ and $\tilde{\boldsymbol{\Theta}}^{I}={\boldsymbol{\Theta}^{I}}^{(t_{5})}$, find $\boldsymbol{f}^{(t_{5}+1)}$, $\\{\boldsymbol{\alpha}_{k}\\}^{(t_{5}+1)}$, $\\{\boldsymbol{p}^{I}_{k}\\}^{(t_{5}+1)}$ and ${\boldsymbol{\Theta}^{I}}^{(t_{5}+1)}$ by solving Problem $\mathcal{P}3$ using Algorithm 4 $\bullet$ Given $\boldsymbol{f}^{(t_{5}+1)}$, $\\{\boldsymbol{\alpha}_{k}\\}^{(t_{5}+1)}$, $\\{\boldsymbol{p}^{I}_{k}\\}^{(t_{5}+1)}$, ${\boldsymbol{\Theta}^{I}}^{(t_{5}+1)}$ and $\tilde{\boldsymbol{\Theta}}^{E}={\boldsymbol{\Theta}^{E}}^{(t_{5})}$, find ${\boldsymbol{P}^{E}}^{(t_{5}+1)}$ and ${\boldsymbol{\Theta}^{E}}^{(t_{5}+1)}$ by solving Problem $\mathcal{P}2$ via Algorithm 2 $\bullet$ Set $\epsilon_{5}=\frac{\big{|}\text{obj}^{(t_{5}+1)}-\text{obj}^{(t_{5})}\big{|}}{\big{|}\text{obj}^{(t_{5}+1)}\big{|}}$, $t_{5}\leftarrow t_{5}+1$ end while3. Output optimal ${\boldsymbol{P}^{E}}^{*}$, ${\boldsymbol{\Theta}^{E}}^{*}$, $\boldsymbol{f}^{*}$, $\\{\boldsymbol{\alpha}_{k}\\}^{*}$ $\\{\boldsymbol{p}^{I}_{k}\\}^{*}$ and ${\boldsymbol{\Theta}^{I}}^{*}$ ${\boldsymbol{P}^{E}}^{*}\leftarrow{\boldsymbol{P}^{E}}^{(t_{5})}$, ${\boldsymbol{\Theta}^{E}}^{*}\leftarrow{\boldsymbol{\Theta}^{E}}^{(t_{5})}$, $\boldsymbol{f}^{*}\leftarrow\boldsymbol{f}^{(t_{5})}$, $\\{\boldsymbol{\alpha}_{k}\\}^{*}\leftarrow\\{\boldsymbol{\alpha}_{k}\\}^{(t_{5})}$, $\\{\boldsymbol{p}^{I}_{k}\\}^{*}\leftarrow\\{\boldsymbol{p}^{I}_{k}\\}^{(t_{5})}$ and ${\boldsymbol{\Theta}^{I}}^{*}\leftarrow{\boldsymbol{\Theta}^{I}}^{(t_{5})}$ ## IV Numerical Results In this section, we present the pertinent numerical results, for evaluating the performance of our proposed IRS-aided WP-MEC design. A top view of the HAP, of the wireless devices and of the IRS are shown in Fig. 3, where the HAP’s coverage radius is $R$ and the IRS is deployed at the cell edge. The locations of wireless devices are assumed to obey the uniform distribution within a circle, whose radius and locations are specified by $r$ as well as $d_{1}$ and $d_{2}$, respectively. Their default settings are specified in the block of “System model” in Table I. The efficiency of the energy harvesting $\eta$ is set as $0.5$. As for the communications channel, we consider both the small-scale fading and the large-scale path loss. More explicitly, the small-scale fading is assumed to be independent and identically distributed (i.i.d.) and obey the complex Gaussian distribution associated with zero mean and unit variance, while the path loss in $\rm{dB}$ is given by $\displaystyle\text{PL}=\text{PL}_{0}-10\beta\log_{10}\big{(}\frac{d}{d_{0}}\big{)},$ (33) where $\text{PL}_{0}$ is the path loss at the reference distance $d_{0}$; $\beta$ and $d$ denote the path loss exponent of and the distance of the communication link, respectively. Here we use $\beta_{ua}$, $\beta_{ui}$ and $\beta_{ia}$ to represent the path loss exponent of the links between the wireless devices and the HAP, between the wireless devices and the IRS, as well as between the IRS and the HAP, respectively111We assume that the channel of the direct link between the HAP and devices is hostile (due to an obstruction), while this obstruction can be partially avoided by the IRS- reflection link. Hence, we set a higher value for $\beta_{ua}$.. Furthermore, the additive while Gaussian noise associated with zero mean and the variable of $\sigma^{2}$ is imposed both on the energy signals and on the offloading signals. The default values of the parameters are set in the block of “Communications model” in Table I. As for the computing model, the variables of $L_{k}$ and $c_{k}$ are assumed to obey the uniform distribution. The offloaded tasks are assumed to be computed in parallel by a large number of CPUs at the edge computing node, where the computing capability of each CPU is $f_{e}=10^{9}\leavevmode\nobreak\ \rm{cycle/s}$. Then, the energy consumption at the edge for processing the offloaded computational tasks can be calculated as $\vartheta=c\kappa f_{e}^{2}=5\times 10^{-8}\leavevmode\nobreak\ \rm{Joule/bit}$. Figure 3: An illustration of the locations of the HAP, of devices and of the IRS in the IRS-aided WP-MEC system. Table I: Default simulation parameter setting Description | Parameter and Value ---|--- System model [27] | $M=16$, $N=30$, $K=3$, $T=10\leavevmode\nobreak\ \rm{ms}$ $R=12\leavevmode\nobreak\ \rm{m}$, $d_{1}=11\leavevmode\nobreak\ \rm{m}$, $d_{2}=1\leavevmode\nobreak\ \rm{m}$, $r=1\leavevmode\nobreak\ \rm{m}$ Wireless energy transfer model | $\eta=0.5$ Communication model [33] | $B=312.5\leavevmode\nobreak\ \rm{KHz}$ $\text{PL}_{0}=30\leavevmode\nobreak\ \rm{dB}$, $d_{0}=1\leavevmode\nobreak\ \rm{m}$, $\beta_{ua}=3.5$, $\beta_{ui}=2.2$, $\beta_{ia}=2.2$ $L^{d}_{k}=4$, $L_{1}=2$, $L_{2,k}=3$ $\sigma^{2}=1.24\times 10^{-12}\leavevmode\nobreak\ \rm{mW}$, $\Gamma=2$ Computing model [5] | $L_{k}=[15,20]\leavevmode\nobreak\ \rm{Kbit}$ $c_{k}=[400,500]\leavevmode\nobreak\ \rm{cycle/bit}$ $f_{max}=1\times 10^{8}\leavevmode\nobreak\ \rm{cycle/s}$ | $\kappa=10^{-28}$, $\vartheta=5\times 10^{-8}\leavevmode\nobreak\ \rm{Joule/bit}$ Convergence criterion | $\epsilon=0.001$ Apart from our algorithms developed in Section III, we also consider two benchmark schemes for comparison. Let us describe these three schemes as follows. * • _With IRS_ : In this scheme, we optimize both the power allocation $\boldsymbol{p}^{E}$ and the IRS reflection coefficients $\boldsymbol{\Theta}^{E}$ at the WET phase, as well as the local CPU frequency at devices $\boldsymbol{f}$, the sub-band-device association $\\{\boldsymbol{\alpha}_{k}\\}$, the power allocation $\\{\boldsymbol{p}_{k}\\}$ and the IRS reflection coefficients $\boldsymbol{\Theta}^{I}$ at the computing phase, relying on Algorithm 5. * • _RandPhase_ : The power allocation $\boldsymbol{p}^{E}$ at the WET phase, as well as the local CPU frequency at devices $\boldsymbol{f}$, the sub-band- device association $\\{\boldsymbol{\alpha}_{k}\\}$ and the power allocation $\\{\boldsymbol{p}_{k}\\}$ at the computing phase are optimized with the aid of Algorithm 5, while we skip the design of the IRS reflection coefficients $\boldsymbol{\Theta}^{E}$ and $\boldsymbol{\Theta}^{I}$, whose amplitude response is set to $1$ and phase shifts are randomly set in the range of $[0,2\pi)$ obeying the uniform distribution. * • _Without IRS_ : The composite channel $\boldsymbol{f}^{H}_{m}\boldsymbol{V}_{k}\boldsymbol{\Theta}$ is set to $0$ both for the WET and for the computation offloading. The power allocation $\boldsymbol{p}^{E}$ at the WET phase, as well as the local CPU frequency at devices $\boldsymbol{f}$, the sub-band-device association $\\{\boldsymbol{\alpha}_{k}\\}$ and the power allocation $\\{\boldsymbol{p}_{k}\\}$ at the computing phase are optimized with the aid of Algorithm 5, while we skip the optimization of the IRS reflection coefficient $\boldsymbol{\Theta}^{E}$ and $\boldsymbol{\Theta}^{I}$. Let us continue by presenting the selection of the time allocation, sub-band allocation in the WET and the computing phases, as well as the impact of diverse environment settings, as follows. ### IV-A Selection of the Time Allocation Figure 4: Simulation results of the total energy consumption versus the time allocation $\tau$. The parameter settings are specified in Table I. In order to find an appropriate time allocation for our WP-MEC system, we depict the total energy consumption (the OF of Problem $\mathcal{P}1$) versus the the time allocation $\tau$ in Fig. 4. It can be seen that the total energy consumption becomes higher upon increasing $\tau$ for all these three schemes considered. The reason behind it is explained as follows. For a given volume of the computational task to be offloaded within the time duration of $T$, an increase of $\tau$ implies a higher offloading rate required by computation offloading, while at a glance of (5), the computation offloading rate is formulated as a logarithmic function of the offloading power. Hence, we have to largely increase the transmit power of computation offloading for providing the extra offloading rate required by the increase of $\tau$, which results in a higher energy consumption at the wireless devices. Furthermore, since the energy required by WET is determined by the energy consumption at the wireless devices, the total energy consumption becomes higher upon increasing $\tau$. Based on this discussion, it seems that we should select the value of $\tau$ as small as possible. However, this may lead to an upsurge of the power consumption for WET, which might exceed the maximum allowable transmit power at the HAP. Therefore, as a compromise, for the environment associated with the default settings we select $\tau=0.1$, beyond which the total energy consumption becomes increasingly higher along with $\tau$. ### IV-B Joint Sub-Band and Power Allocation in the WET and Computing Phases (a) (b) (c) (d) Figure 5: Joint sub-band and of power allocation for the WET and the computing phases, relying on the Algorithm 5, where the number of bits to be processed is set the same as $20\leavevmode\nobreak\ \rm{Kbits}$ for the three wireless devices. (a) The channel gain at the WET phase; (b) The joint sub-band and power allocation at the WET phase; (3) The channel gain at the computing phase; (d) The joint sub-band and power allocation at the computing phase. The parameter settings are specified in Table I. Fig. 5 illustrates the channel gain as well as the joint sub-band and power allocation both for the WET and computing phases. Our observations are as follows. Firstly, as shown in Fig. 5b, only the $5$-th sub-band is activated for WET. This allocation is jointly determined by the power consumption of the computing phase and by the channel gain in the WET phase. Specifically, with the reference of Fig. 5d, Device 3 requires the highest power consumption for computation offloading. Given that the overall performance is dominated by the device having the highest energy consumption, we may reduce the energy consumption of WET, by activating the sub-band associated with the highest channel gain of Device 3, which is the $5$-th sub-band as shown in Fig. 5a. Secondly, with the reference of Fig. 5c, it can be observed that the power allocation in Fig. 5d obeys the water-filling principle for each device, i.e. allocating a higher power to the sub-band possessing a high channel gain. This corresponds to the power allocation obtained in (21). Thirdly, comparing Fig. 5a and Fig. 5c, we can see that the channel gains in the WET and computing phase are different for each device after we optimize the IRS reflection coefficients, which consolidates our motivation to conceive separate IRS designs for the WET and the computing phases. ### IV-C Performance of the Proposed Algorithms In order to evaluate the benefits of employing an IRS in WP-MEC systems, we compare the performance of our proposed algorithms with that of the benchmark schemes, under various settings of the number of IRS reflection elements, of the device location, of the path loss exponent of the IRS-related channel, and of the energy consumption per bit at the edge, as follows. #### IV-C1 Impact of the Number of IRS Reflection Elements Figure 6: Simulation results of the total energy consumption versus the number of IRS reflection elements $N$. The rest of parameters are specified in Table I. Figure 7: Simulation results of the total energy consumption versus the distance between the HAP and the wireless device circle $d_{1}$. Other parameters are set in Table I. Fig. 6 shows the simulation results of the total energy consumption versus the number of IRS reflection elements for the three schemes considered. We have the following observations. Firstly, the performance gap between the scheme “Without IRS” and the scheme “IRS RandPhase” increases along with $N$, which implies that the IRS is capable of assisting the energy consumption reduction in the WP-MEC system, even without carefully designing the IRS reflection coefficients. This is due to the so-called virtual array gain induced by the IRS, as mentioned in Section I. Secondly, the scheme “With IRS” outperforms the scheme “IRS RandPhase”, which indicates that our sophisticated design of IRS reflection coefficients may provide the so-called passive beamforming gain for computation offloading. Note that different from the conventional MEC systems [38] where WET is not employed, these two types of gain are exploited twice in WP-MEC systems (during the WET and computing phases, respectively). As such, IRSs are capable of efficiently reducing the energy consumption in WP-MEC systems. #### IV-C2 Impact of the Distance between the Device Circle and the IRS Fig. 7 presents the simulation results of the total energy consumption versus the distance between the HAP and the mobile wireless circles. Our observations are as follows. Firstly, the two IRS-aided schemes do not show any visible advantage over the scheme of “Without IRS” when we have $d_{1}<6\leavevmode\nobreak\ \rm{m}$, which indicates that each IRS has a limited coverage. Secondly, the benefit of deploying the IRS is becomes visible at $d_{1}>9\leavevmode\nobreak\ \rm{m}$ in the scheme of “IRS RandPhase”, while the advantage of the “With IRS” scheme is already notable at $d_{1}=7\leavevmode\nobreak\ \rm{m}$. This observation implies that our sophisticated design of IRS reflection coefficient is capable of extending the coverage of IRS. Figure 8: Simulation results of the total energy consumption versus the path loss exponent of the IRS reflection link $\beta$, where we set $\beta_{ui}=\beta_{ia}=\beta$. Other parameters are set in Table I. Figure 9: Simulation results of the total energy consumption versus the energy consumption per bit at the edge. Other parameters are set in Table I. #### IV-C3 Impact of Path Loss Exponent Fig. 8 depicts the simulation results of the total energy consumption versus the path loss exponent of the IRS related links. It can be seen that the total energy consumption decreases if a higher path loss exponent is encountered, which is because a higher $\beta$ leads to a lower channel gain of the IRS- reflected link. This observation provides an important engineering insight: the locations of IRSs should be carefully selected for avoiding obstacles. #### IV-C4 Impact of energy consumption at the edge Fig. 9 shows the simulation results of the total energy consumption versus the energy consumption per bit at the edge node. It can be observed that the advantage of deploying IRS is eminent when we have a small value of $\vartheta$, while the benefit becomes smaller upon increasing the value of $\vartheta$. The reason is explained as follows. The OF of Problem $\mathcal{P}1$ is the combination of the energy consumption of WET and of processing the offloaded computational tasks. If the energy consumption per bit at the edge node is of a small value, the energy consumption of WET plays a dominant role in the total energy consumption. In this case, the benefit of employing IRS is significant. By contrast, if $\vartheta$ becomes higher, the total energy consumption is dominated by that at the edge. 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2003.05514

arxiv-papers

# Projective plane graphs and 3-rigidity E. Kastis and S.C. Power Dept. Math. Stats. Lancaster University Lancaster LA1 4YF U.K<EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract. A ${\mathcal{P}}$-graph is a simple graph $G$ which is embeddable in the real projective plane ${\mathcal{P}}$. A $(3,6)$-tight ${\mathcal{P}}$-graph is shown to be constructible from one of 8 uncontractible ${\mathcal{P}}$-graphs by a sequence of vertex splitting moves. Also it is shown that a ${\mathcal{P}}$-graph is minimally generically 3-rigid if and only if it is $(3,6)$-tight. In particular this characterisation holds for graphs that are embeddable in the Möbius strip. 2010 Mathematics Subject Classification. 52C25 51E15 Key words and phrases: projective plane, embedded graphs, geometric rigidity This work was supported by the Engineering and Physical Sciences Research Council [EP/P01108X/1] ## 1\. Introduction Let $G$ be the graph of a triangulated sphere. Then an associated bar-joint framework $(G,p)$ in ${\mathbb{R}}^{3}$ is known to be minimally rigid if the placements $p(v)$ of the vertices $v$ is strictly convex (Cauchy [4]) or if the placement is generic. The latter case follows from Gluck’s result [12] that any generic placement is in fact infinitesimally rigid. An equivalent formulation of Gluck’s theorem asserts that if $G$ is a simple graph which is embeddable in the sphere then $G$ is minimally 3-rigid if and only if it satisfies a $(3,6)$-tight sparsity condition. We obtain here the exact analogue of this for simple graphs that are embeddable in the real projective plane ${\mathcal{P}}$. The proof rests on viewing these graphs as partial triangulations and deriving inductive arguments based on edge contractions for certain admissible edges. Accordingly we may state this result in the following form. An immediate corollary is that this combinatorial characterisation also holds for triangulated Möbius strips. A graph $G$ is _3-rigid_ if its generic bar-joint frameworks in ${\mathbb{R}}^{3}$ are infinitesimally rigid and is _minimally 3-rigid_ if no subgraph has this property. ###### Theorem 1.1. Let $G$ be a simple graph associated with a partial triangulation of the real projective plane. Then $G$ is minimally $3$-rigid if and only if $G$ is $(3,6)$-tight. Recall that a $(3,6)$-tight graph $G=(V,E)$ is one that satisfies the Maxwell count $|E|=3|V|-6$ and the sparsity condition $|E^{\prime}|\leq 3|V^{\prime}|-6$ for subgraphs $G^{\prime}$ with at least 3 vertices. In particular it follows from the Maxwell condition that such a graph falls 3 edges short of a full (possibly nonsimple) triangulation of ${\mathcal{P}}$. The proof of Theorem 1.1 depends heavily on our main result, Theorem 6.1, which is a purely combinatorial constructive characterisation of the ${\mathcal{P}}$-graphs which are (3,6)-tight. A key step is the identification of edge contraction moves, for certain edges that lie in two 3-cycle faces, such that the $(3,6)$-sparsity condition is preserved. This is done in Section 3 by exploiting the implicit topological structure of the graphs. The associated contraction sequences must terminate and the terminal graphs are said to in _irreducible_. They have the defining property that every contractible edge lies on a critical $4$-, $5$\- or $6$-cycle. For the remainder of the proof of Theorem 6.1 we show that an irreducible graph has no contractible edges (Section 5) and we determine the uncontractible graphs (Section 4). Determining the uncontractibles requires an extensive case-by- case analysis leading to the 8 “base” graphs given in Figures 3, 4, 5. The determination of construction schemes and their base graphs for various classes of graphs is of general interest, both for embedded graph theory and for the rigidity of bar-joint frameworks. We note, for example, that Barnette [1] employed vertex splitting moves for the construction of triangulations of 2-manifolds and showed that there are 2 (full) triangulations of ${\mathcal{P}}$ which are uncontractible. Also, Barnette and Edelson [2], [3] have shown that all 2-manifolds have finitely many minimal uncontractible triangulations. Our construction theorem is in a similar spirit to this and we expect our reduction methods, involving critical cycles and minimum hole incidence degree, for example, to be useful for more general surface graphs and for other sparsity classes. In particular, for $(3,6)$-tight ${\mathcal{P}}$-graphs we show that the irreducibles are the uncontractibles and it would be interesting to determine to what extent this phenomenon is true for other surfaces and sparsity classes. We define a _triangulated surface graph_ associated with a classical surface ${\mathcal{M}}$, with or without boundary and we represent embeddings of these graphs, and their connected subgraphs (${\mathcal{M}}$-graphs), in terms of _face graphs_. A face graph is a finite connected planar graph with a specified pairing of some or all of the edges in the outer boundary. Identifying the paired edges gives an identification graph $G=(V,E)$ together with a set $F$ of facial 3-cycles inherited from the finite planar graph. See Definitions 2.1, 2.2. In Section 3 we identify the obstacles, in terms of critical cycles of edges, which prevent edge contraction moves from preserving the sparsity condition. The determination in Section 4 of the 8 uncontractible ${\mathcal{P}}$-graphs is given in several stages, based on the nature of the “holes” in their partial triangulation. They may have one hole with 6-cycle boundary, two holes with boundary cycle lengths 5 and 4, or three holes, each with a 4-cycle boundary. Also we give a useful index for the successive determination of these uncontractible base graphs, namely the minimum hole incidence degree $h(G)$ (Definition 4.3). Since Whiteley’s demonstration [14] that vertex splitting preserves generic rigidity this construction move has become an important tool in combinatorial rigidity theory [11]. See for example the more recent studies of generic rigidity in the case of graphs for modified spheres [7], [8], [5], [13], and in the case of a partially triangulated torus [6]. The proof of Theorem 1.1 given in Section 6 follows quickly from Whiteley’s theorem, Theorem 6.1, and the 3-rigidity of the 8 base graphs. ## 2\. Graphs in Surfaces Let ${\mathcal{M}}$ be a classical surface, possibly with boundary. Then a _surface graph for ${\mathcal{M}}$_ is a triple $G=(V,E,F)$ where $(V,E)$ is a simple graph, $F$ is a set of $3$-cycles of edges, called facial 3-cycles, and where there exists a faithful embedding of $G$ in ${\mathcal{M}}$ for which the facial 3-cycles correspond to the 3-sided faces inthe embedding. A surface graph for ${\mathcal{M}}$, which we also refer to as an ${\mathcal{M}}$-graph, can thus be viewed as a simple graph obtained from a full triangulation of ${\mathcal{M}}$ by discarding vertices, edges and faces. Also, $G$ is a _triangulated surface graph for ${\mathcal{M}}$_ if the union of the embedded faces is equal to ${\mathcal{M}}$. The following equivalent definition, based on simplicial complexes rather than surfaces, is combinatorial and so more elementary. ###### Definition 2.1. A _triangulated surface graph_ is a graph $G=G(M)=(V,E,F)$ which is simple and is determined by the $1$-skeleton and the $2$-simplexes of a finite simplicial complex $M$ where $M$ has the following properties. 1. (i) $M$ consists of a finite set of $2$-simplexes $\sigma_{1},\sigma_{2},\dots$ together with their $1$-simplexes and $0$-simplexes. 2. (ii) Every $1$-simplex lies in at most two $2$-simplexes. Condition (i) implies that each 1-simplex lies in at least one 2-simplex. It follows that $M$ can be viewed as a _combinatorial surface_ and we define ${\mathcal{M}}={\mathcal{M}}(M)={\mathcal{M}}(G)$ to be the classical topological surface, possibly with boundary, which is determined by $M$, the simplicial complex [9]. Evidently, $G$ is a triangulated surface graph for ${\mathcal{M}}$. Classical compact surfaces are classified up to homeomorphism by combinatorial surfaces and, moreover, combinatorial surfaces arise from triangulated polygon graphs (also called triangulated discs) by means of an identification of certain pairs of boundary edges [9]. We now formally define such labelled triangulated discs which we refer to as _face graphs_. ###### Definition 2.2. A _face graph_ for a triangulated surface graph is a pair $(B,\lambda)$ where $B$ is the planar graph of a triangulated disc and $\lambda$ is a partition of the _boundary graph_ $\partial B$ of $B$, such that each set of the partition has $1$ or $2$ edges, and the paired edges of the partition are directed. A face graph $(B,\lambda)$ defines a simplicial complex $M$, with $1$-simplexes provided by edges and identified edge pairs, and 2-simplexes provided by the facial 3-cycles. Also, if the boundary graph of $B$ is a 3-cycle and $\lambda$ is trivial then this 3-cycle defines a 2-simplex of $M$. If the identification graph $G=B/\lambda$ is a simple graph then $M$ is of the type given in Definition 2.1, and so $G$ is a triangulated surface graph $G=(V,E,F)$. ### 2.1. ${\mathcal{M}}$-graphs We are concerned simple graphs that can be embedded in a connected classical surface ${\mathcal{M}}$. More precisely we shall be concerned with embedded graphs, which we refer to as ${\mathcal{M}}$-graphs, or surface graphs, and we can define them directly in terms of more general face graphs $(B_{0},\lambda)$, where $B_{0}\subseteq B$ with $\partial B\subset B_{0}$ and $(B,\lambda)$ is as in the previous definition. Thus a surface graph has the form $G=B_{0}/\lambda$ where $B_{0}$ is obtained from $B$ by the removal of the interior edges of $k$ interior-disjoint triangulated subdiscs of $B$. We refer to $k$ as the _number of holes_ of the embedded graph $G$. When $k=1$ we refer to $B_{0}$ as an _annular face graph_. $e$$f$$g$$e$$f$$g$$v_{1}$$v_{2}$$v_{3}$$v_{1}$$v_{2}$$v_{3}$ Figure 1. A face graph $(B_{0},\lambda)$ for a ${\mathcal{P}}$-graph. Figure 1 shows the annular face graph $(B_{0},\lambda)$ for a surface graph $G=B_{0}/\lambda$. The labelling of outer boundary edges and vertices determines how pairs of edges are identified. A planar triangulation of the interior of the inner 6-cycle gives a face graph $(B,\lambda)$ for the triangulated surface graph $S=B/\lambda$, if and only if $S$ is simple. In view of the identifications the topological surface ${\mathcal{M}}(S)$ is the real projective plane ${\mathcal{P}}$ and $G$ is a ${\mathcal{P}}$-graph. In this example the surface graph $G$ happens to be a (fully) triangulated surface graph for the Möbius strip. However, in general a surface graph may have “exposed” edges, that is, edges that belong to no facial 3-cycles, and so in this case the surface graph will not be a triangulated surface graph for any classical surface with boundary. ## 3\. Contraction moves and (3,6)-sparsity. Let $G=(V,E,F)$ be a surface graph. An edge of $G$ is of type $FF$ if it is contained in two facial $3$-cycles and an $FF$ edge is _contractible_ if it is not contained in any non-facial $3$-cycle. For such an edge $e=uv$ there is a natural contraction move $G\to G^{\prime}$ on the graph $G$, corresponding to a contraction of $e$, merging $u$ and $v$ to a single vertex, leading to a surface graph $G^{\prime}=(V^{\prime},E^{\prime},F^{\prime})$ where $|V^{\prime}|=|V|-1,|E^{\prime}|=|E|-3,|F^{\prime}|=|F|-2$. We also say that $G$ is _contractible_ if it has a contractible $FF$ edge. To define formally the contracted graph $G^{\prime}$, let $e=vw$ be a contractible $FF$ edge in $G$ and let $avw$ and $bvw$ be the two facial 3-cycles which contain $e$. Then $G^{\prime}$ is obtained from $G$ by an _edge contraction_ on $e=vw$ if $G^{\prime}$ is obtained by (i) deleting the edges $aw$ and $bw$, (ii) replacing all remaining edges of the form $xw$ with $xv$, (iii) deleting the edge $e$ and the vertex $w$. That $G^{\prime}$ is simple follows from the fact that a contractible $FF$ edge does not lie on a nonfacial 3-cycle. Given an edge contraction move $G\to G^{\prime}$ we note that the inverse move, recovering $G$ from $G^{\prime}$, is a _vertex splitting move_ at $v$ which in particular introduces a new vertex $w$ and the new $FF$ edge $vw$. Such vertex splitting move $G^{\prime}\to G$, which might be thought of as being locally planar, creates the new surface graph $G$ for the surface ${\mathcal{M}}$ from a given surface graph $G^{\prime}$ for ${\mathcal{M}}$. ### 3.1. (3,6)-sparse ${\mathcal{P}}$-graphs. If $G=(V,E)$ is a graph then its _freedom number_ is defined to be $f(G)=3|V|-|E|$. A graph $G$ is _$(3,6)$ -sparse_ if $f(G^{\prime})\geq 6$ for any subgraph $G^{\prime}$ with at least 3 vertices, and is _$(3,6)$ -tight_ if it is $(3,6)$-sparse and $f(G)=6$. In particular a $(3,6)$-sparse graph is a simple graph, with no loop edges and no parallel edges. Let $B$ be a triangulated disc such that the boundary cycle $\partial B$ is of even length $2r$. With the pairing partition $\lambda$ of opposite edges, directed in cyclic order, the pair $(B,\lambda)$ is a face graph. If $S=B/\lambda$ is simple then $S$ is a triangulated surface graph for the real projective plane ${\mathcal{P}}$. Also we observe that the freedom number $f(B)$ is equal to $6+(2r-3)$. This follows since $B$ may be viewed as a triangulated sphere (which has freedom number $6$) with $2r-3$ edges removed. Noting that $S$ is related to $B$ by the loss of $r$ vertices and $r$ edges it follows that $f(S)=(3+2r)-3r+r=3.$ Let $G$ be a surface graph for ${\mathcal{P}}$, the real projective plane, which is determined by the annular face $(B_{0},\lambda)$ where the inner boundary cycle of edges has length $s$. Then $f(G)=f(S)-(s-3)$ and in particular $G$ satisfies the so-called _Maxwell count_ $f(G)=6$ if and only if $s=6$. ###### Lemma 3.1. Let $G$ be a triangulated surface graph for the Möbius strip. Then $G$ is a surface graph for ${\mathcal{P}}$. Also $G$ satisfies the Maxwell count if and only if the boundary graph $\partial G$ is the graph of a simple 6-cycle. ###### Proof. Let $G(M)=(V,E,F)$ be a triangulated surface graph given by a finite simplicial complex $M$ for the Möbius strip, as in Definition 2.1. Then $G(M)$ is determined by a face graph $(B,\mu)$ where $\mu$ is obtained from an identification of two vertex-disjoint paths in the boundary of $B$, which have the same length and orientation and which have end vertices $w_{1},w_{2}$ and $w_{3},w_{4}$ respectively. In Figure 2 the boundary of $B$ is depicted as rectangular. $w_{1}$$w_{2}$$w_{3}$$w_{4}$$v_{1}$$v_{2}$$B$ Figure 2. A Möbius strip triangulated surface graph as a ${\mathcal{P}}$ with hole graph. Augment the planar graph $B$ to obtain a containing planar graph $B_{1}$ which has 2 additional vertices, $v_{1}$ and $v_{2}$ say, and additional edges $v_{1}w$ (resp. $v_{2}w$) which are incident to vertices on the boundary path from $w_{4}$ to $w_{1}$ (resp. $w_{2}$ to $w_{3}$). This defines a triangulated disc $B_{1}$ which is also indicated in Figure 2. Define a partition $\lambda$ for $B_{1}$ as the augmentation of $\mu$ by the two directed edge pairs $v_{1}w_{1},v_{2}w_{3}$ and $w_{2}v_{2},w_{3}v_{1}$ and let $(B_{1},\lambda)$ be the resulting face graph. Then $H=B_{1}/\lambda$ is a triangulated surface graph for ${\mathcal{P}}$. Moreover, the faces of $H$ that are incident to the vertex $v_{1}=v_{2}$ in $S$ are the faces of a triangulated disc and $G$ is a surface graph for ${\mathcal{P}}$. ∎ Let $G$ be a ${\mathcal{P}}$-graph, with $k$ holes. If $G$ satisfies the Maxwell count $f(G)=6$ then $k=1,2$ or $3$. For $k=1$ a representing face graph $(B_{0},\lambda)$ for $G$ is annular with a 6-cycle inner boundary. This inner boundary can intersect and even coincide with the outer boundary of $B$. For $k=2$ there are two inner boundaries of length $5$ and $4$ corresponding to the boundaries of the interior disjoint discs defining $G$, while for $k=3$ there are three inner boundaries which are 4-cycles. In particular, $3\leq|\partial G|\leq 12.$ ###### Definition 3.2. For $k=1,2,3$ the set ${\mathfrak{P}}_{k}$ is the set of $(3,6)$-tight ${\mathcal{P}}$-graphs which have $k$ holes. While a surface graph is a graph with extra structure we shall informally refer to the elements of ${\mathfrak{P}}_{k}$ as graphs. ### 3.2. When contracted graphs are $(3,6)$-tight A contraction move $G\to G^{\prime}$ on a contractible $FF$ edge $e$ of a surface graph preserves the Maxwell count but need not preserve $(3,6)$-tightness. We now examine this more closely in the case of a surface graph for the real projective plane ${\mathcal{P}}$. Suppose that $G_{1}\subseteq G$ and that both $G_{1}$ and $G$ are in ${\mathfrak{P}}_{1}$. If $e$ is a contractible $FF$ edge of $G$ which lies on the boundary graph of $G_{1}$ then, since $G_{1}$ contains only one of the facial 3-cycles incident to $e$, the contraction $G\to G^{\prime}$ for $e$ gives a contraction $G^{\prime}$ which is not $(3,6)$-sparse, since $f(G_{1}^{\prime})=5$. We shall show that the failure of any contraction to preserve $(3,6)$-sparsity is due to such a subgraph obstacle. The following general lemma, which we refer to as the filling in lemma, is useful for the identification of maximal $(3,6)$-tight subgraphs with specific properties. See also [6]. In particular this lemma plays a role in the identification of an obstacle subgraph. ###### Lemma 3.3. Let $G\in{\mathfrak{P}}_{1}$ and let $H$ be an embedded triangulated disc graph in G. (i) If $K$ is a $(3,6)$-tight subgraph of $G$ with $K\cap H=\partial H$ then $\partial H$ is a $3$-cycle graph. (ii) If $K$ is a $(3,6)$-sparse subgraph of $G$ with $f(K)=7$ and $K\cap H=\partial H$ then $\partial H$ is either a $3$-cycle or $4$-cycle graph. ###### Proof. (i) Write $H^{c}$ for the subgraph of $G$ which contains the edges of $\partial H$ and the edges of $G$ not contained in $H$. Since $G=H^{c}\cup H$ and $H^{c}\cap H=\partial H$ we have $6=f(G)=f(H^{c})+f(H)-f(\partial H).$ Since $f(H^{c})\geq 6$ we have $f(H)-f(\partial H)\leq 0$. On the other hand, $6\leq f(K\cup H)=f(K)+f(H)-f(\partial H)$ and $f(K)=6$ and so it follows that $f(H)-f(\partial H)=0$. It follows that $\partial H$ is a 3-cycle. (ii) The argument above leads to $-1\leq f(H)-f(\partial H)$ and hence to the inequality $-1\leq f(D)-f(\partial D)$. This implies that $\partial H$ is either a $3$-cycle or $4$-cycle graph. ∎ ###### Lemma 3.4. Let $G\in{\mathfrak{P}}_{1}$, let $e$ be a contractible $FF$ edge in $G$, and let $G^{\prime}$ be the simple graph arising from the contraction move $G\to G^{\prime}$ associated with $e$. Then either $G^{\prime}\in{\mathfrak{P}}_{1}$ or $e$ lies on the boundary of a subgraph $G_{1}$ of $G$ where $G_{1}\in{\mathfrak{P}}_{1}$. ###### Proof. Assume that $G^{\prime}\notin{\mathfrak{P}}_{1}$. It follows that $G^{\prime}$ must fail the $(3,6)$-sparsity count. Thus there exists a subgraph $K$ of $G$ containing $e$ for which the edge contraction results in a graph $K^{\prime}$ satisfying $f(K^{\prime})<6$. Let $e=vw$ and let $c$ and $d$ be the facial $3$-cycles which contain $e$. If both $c$ and $d$ are subgraphs of $K$ then $f(K)=f(K^{\prime})<6$, which contradicts the sparsity count for $G$. Thus $K$ must contain at most one of these facial $3$-cycles. _Case 1_. Suppose first that $K$ is a maximal subgraph among all subgraphs of $G$ which contain the cycle $c$, do not contain $d$, and for which contraction of $e$ results in a simple graph $K^{\prime}$ which fails the $(3,6)$-sparsity count. Note that $f(K)=f(K^{\prime})+1$ which implies $f(K)=6$ and $f(K^{\prime})=5$. In particular, $K$ is $(3,6)$-tight, and is a connected graph. Let $(B_{0},\lambda)$ be a face graph for $G$ with an associated face graph $(B,\lambda)$ for a triangulated surface graph for $S=(V,E,F)$ for ${\mathcal{P}}$ containing $G$. In particular $(B,\lambda)$ provides a faithful topological embedding $\pi:S\to{\mathcal{P}}$. Let $X(K)\subset{\mathcal{P}}$ be the closed set $\pi(E(K))$ and let $\tilde{X}(K)$ be the union of $X(K)$ and the embeddings of the faces for the facial $3$-cycles belonging $K$. Finally, let $U_{1},\dots,U_{n}$ be the maximal connected open sets of the complement of $\tilde{X}(K)$ in ${\mathcal{P}}$. Note that each such connected open set $U_{i}$ is determined by a set ${\mathcal{U}}_{i}$ of embedded faces of $S$ with the property: each pair of embedded faces of $U_{i}$ are the endpoints of a path of edge-sharing embedded faces in ${\mathcal{U}}_{i}$. From the topological nature of ${\mathcal{P}}$ it follows that $U_{i}$ has one of the following 3 properties. (i) $U_{i}$ is an open disc. (ii) $U_{i}$ is the interior of a Möbius strip. (iii) The complement of $U_{i}$ is not connected. The third property cannot hold since the embedding of $K$ is contained in the complement of $U_{i}$ and contains the boundary of $U_{i}$, and yet $K$ is a connected graph. From the second property it follows that $K$ is a planar graph, since it can be embedded in the complement of $U_{i}$ and this is a triangulated disc. This is also a contradiction, since the edge contraction of a contractible $FF$ edge in a planar triangulated graph preserves $(3,6)$-sparsity. Each set $U_{i}$ is therefore the interior of the closed set determined by an embedding of a triangulated disc graph in $B$, say $H(U_{i})$. (Indeed, the facial 3-cycles defining $H(U_{i})$ are those whose torus embedding have interior set contained in $U_{i}$.) We may assume that $U_{1}$ is the open set that contains the hole of $G$. (More precisely, $U_{1}$ contains the open set corresponding to the embedded faces for the triangulated disc in $B$ that determines $B_{1}$.) For $i>1$ by the filling in lemma, Lemma 3.3, it follows that $\partial H(U_{i})$ is a 3-cycle. By the maximality of $K$ we have $k=1$ (since adding the edges and vertices of $S$ interior to these nonfacial 3-cycles gives a subgraph of $G$ with the same freedom count). Thus, $K$ is a subgraph of $G$ and is equal to the surface graph for ${\mathcal{P}}$ defined by $B$ and the embedded triangulated disc $H(U_{1})$. Thus, with $G_{1}=K$, the proof is complete in this case. _Case 2._ It remains to consider the case for which $K$ contains neither of the facial $3$-cycles which contain $e$. Thus $f(K)=f(K^{\prime})+2$ and $f(K)\in\\{6,7\\}$. Once again we assume that $K$ is a maximal subgraph of $G$ with respect to these properties and consider the complementary components $U_{1},\dots,U_{k}$. As before each set $U_{i}$ is homeomorphic to a disc and determines an embedded triangulated disc graph $H(U_{i})$, one of which, say $H(U_{1})$, contains the triangulated disc which defines $G$. The filling in lemma and maximality now implies that each boundary of $H(U_{i})$, for $i>1$, is a $4$-cycle. By the maximality of $K$, we see once again that $k=1$ (since adding the missing edge for such a $4$-cycle gives a subgraph of $G$ with a lower freedom count) and the proof is completed as before. ∎ The filling in lemma holds for graphs in ${\mathfrak{P}}_{2},{\mathfrak{P}}_{3}$, with the same proof, and we may extend Lemma 3.4 to these families of graphs. ###### Lemma 3.5. Let $G\in{\mathfrak{P}}_{k}$, for $k=1,2$ or $3$, let $e$ be a contractible $FF$ edge in $G$, and let $G^{\prime}$ be the simple graph arising from the contraction move $G\to G^{\prime}$ associated with $e$. Then either $G^{\prime}\in{\mathfrak{P}}_{k}$ or $e$ lies on the boundary of a subgraph $G_{1}$ of $G$ where $G_{1}\in{\mathfrak{P}}_{l}$, for some $1\leq l\leq k$. ###### Proof. The proof follows the same pattern as in the case $k=1$. Thus we assume that $G^{\prime}\notin{\mathfrak{P}}_{k}$ and consider a subgraph $K$ of $G$ which is maximal amongst all subgraphs which do not contain one (or both, according to Cases 1 and 2) of the facial 3-cycles incident to $e$ and whose contraction $K^{\prime}$ has freedom number $f(K^{\prime})=5$ (or $4$). We consider the open set which is the complement of the embedding in ${\mathcal{P}}$ of $K$ and its facial 3-cycles. (The embedding here is denoted $\tilde{X}(K)$ in the case $k=1$.) This open set has components $U_{1},\dots,U_{n}$ and each is the interior of a union of an edge-connected set of ${\mathcal{P}}$-embedded facial 3-cycles of $S$. It follows as before, from the topological nature of ${\mathcal{P}}$, from $(3,6)$-sparsity and from the filling in lemma, that each $U_{j}$ is an open disc. Moreover, in the case $k=2$ each $U_{j}$ contains at least one of the 2 discs $D_{1},D_{2}$ which defines $G$ and so $n$ is 1 or 2 and it follows that $K$ belongs to ${\mathfrak{P}}_{n}$, as desired. Similarly, for $k=3$, each $U_{j}$ contains at least one of the 3 discs $D_{1},D_{2},D_{3}$ which defines $G$ and so $n$ is 1, 2 or 3 and $K$ belongs to ${\mathfrak{P}}_{l}$ for some $1\leq l\leq 3$. ∎ ## 4\. The uncontractibles Let $k=1,2$ or $3$. By the finiteness of a graph $G$ in ${\mathfrak{P}}_{k}$ it is evident that it admits a full reduction sequence $G=G_{1}\to G_{2}\to\dots\to G_{n}$ where (i) each graph is in ${\mathfrak{P}}_{k}$, (ii) each move $G_{k}\to G_{k+1}$ is an edge contraction for an $FF$ edge, as before, and (iii) $G_{n}$ is _irreducible_ in ${\mathfrak{P}}_{k}$ in the sense that it admits no edge contraction to a graph in ${\mathfrak{P}}_{k}$. Let us say that a surface graph is _uncontractible_ is every $FF$ edge lies on a nonfacial 3-cycle. An uncontractible graph $G\in{\mathfrak{P}}_{k}$ is certainly an irreducible graph in ${\mathfrak{P}}_{k}$ but we show in the next section that the two classes coincide. In Section 4.2 we shall prove that there are 8 uncontractible graphs but first we establish some useful properties of the irreducible graphs. ### 4.1. Some properties of irreducible graphs We say that a $k$-cycle of edges in $G$, $c$ say, is a _planar $k$-cycle_ in $G$ if there is a face graph $(B_{0},\lambda)$ for $G$, with containing face graph $(B,\lambda)$ for the triangulated surface graph $B/\lambda$ for ${\mathcal{P}}$, such that $c$ is determined by the boundary cycle $\hat{c}$ of a triangulated disc $D$ in $B$. Note that the holes of $G$ are defined by embedded triangulated discs $D_{i}$ in $B_{0}$, and so we may say that a planar cycle $c$ in $G$ _contains a hole of $G$_ if $D$ contains such an embedded disc $D_{i}$. Also we may say that $c$ _properly contains a hole_ if there is such an inclusion which is proper. The following lemma shows that an irreducible (3,6)-tight ${\mathcal{P}}$-graph contains no degree 3 vertex that is incident to an $FF$ edge or lies on a planar nonfacial triangle. ###### Lemma 4.1. Let $G$ be a graph in ${\mathfrak{P}}_{k}$, for $k=1,2$ or $3$. (i) If $e$ is an $FF$ edge incident to a degree 3 vertex then $G/e\in{\mathfrak{P}}_{k}$. (ii) If $v$ is an interior vertex of $G$ and $v$ lies on a planar nonfacial 3-cycle then there is a contractible edge $vw$ with $G/vw\in{\mathfrak{P}}_{k}$. ###### Proof. For (i) note that since $G$ is simple $e$ is a contractible edge. Write $e=uv$ with facial 3-cycles $uvx$ and $uvy$, with $\deg v=3$. Then $e$ cannot lie on a critical 4-, 5- or 6-cycle since one of the edges incident to $u$ would provide an interior chord for this cycle. Also $e$ does not lie on a nonfacial 3-cycle and so (i) follows. For (ii) let $H$ be the triangulated disc subgraph induced by the faces incident to $v$, with vertices $v_{1},\dots,v_{n}$ in cyclic order on the boundary of $H$. Considering the hypothesis, and relabelling, we may assume that there is an edge $f=v_{1}v_{j}$ with $3\leq j\leq n-2$ so that the edges $v_{3}v_{4},v_{4}v_{5},\dots,v_{j-1}v_{j},f$ are the boundary edges of a triangulated disc. It is straightforward to show that one of the vertices $v_{2},\dots,v_{j-1}$ has degree 3, and so (i) applies. ∎ The next lemma shows that if $G$ is irreducible then there is no critical $m$-cycle which properly contains an $m$-hole. ###### Lemma 4.2. Let $G$ be a graph in ${\mathfrak{P}}_{k}$, for $k=1,2$ or $3$, such that there is a critical $m$-cycle, for $m=1,2$ or 3, which properly contains an $m$-cycle hole, so that $G=G_{1}\cup A$ where $A$ is the annular graph determined by the two $m$-cycles. Then $G$ is constructible from $G_{1}$ by planar vertex splitting moves. ###### Proof. Fix $k$ and $m\leq k$. Suppose that $|V(G)|=|V(G_{1})|+1.$ Then there is a degree $3$ vertex on the boundary of the relevant hole of $G$. By Lemma 4.1(i) $G$ is constructible from $G_{1}$ by a single planar vertex splitting move. Assume next that the lemma is true whenever $|V(G)|=|V(G_{1})|+j$, for $j=1,2,\dots,N-1$, and suppose that $|V(G)|=|V(G_{1})|+N$. Let $e$ be an interior edge of the annular graph $A$. If the contraction $G/e$ is in ${\mathfrak{P}}_{m}$ then it follows from the induction step that $G$ is constructible from $G_{1}$ by planar vertex splitting moves. So, by Lemma 3.5 we may assume (i), that $e$ lies on a critical $m$-cycle, with associated graph $G^{\prime}$, or (ii), that $e$ lies on a nonfacial 3-cycle of $G$. In the former case we may take $G_{1}^{\prime\prime}$ to be the union of $G_{1}$ and $G_{1}^{\prime}$. Then $G_{1}^{\prime\prime}$ is also in ${\mathfrak{P}}_{k}$. Since $|V(G_{1}^{\prime\prime})|-|V(G_{1})|<N$ and $|V(G)|-|V(G_{1}^{\prime\prime})|<N$ it follows from the induction step that the lemma holds for $G$ and $G_{1}$. So we may assume that (ii) holds, and moreover, in view of Lemma 4.1(ii), that $e$ lies on a nonplanar nonfacial 3-cycle. To complete the proof we observe that this is not possible when $e$ is incident to a vertex on the hole which is not a vertex of the critical $m$-cycle. ∎ ### 4.2. The uncontractible graphs. We now identify 8 uncontractible $(3,6)$-tight ${\mathcal{P}}$-graphs. Figure 3 gives two uncontractibles specified by face graphs and Figure 4 gives three further uncontractibles as embedded graphs in ${\mathcal{P}}$. Here ${\mathcal{P}}$ is represented as a disc or a rectangle, with diagonally opposite points of the boundary identified. The 3 remaining irreducibles are given in Figure 5. The notation $G^{h}_{n}$ indicates that $n$ is the number of vertices and $h=h(G)$ is the minimum _hole incidence degree_ given in the following definition. Figure 3. The uncontractible graphs $G^{2}_{3}\in{\mathfrak{P}}_{1}$ and $G^{3}_{4}\in{\mathfrak{P}}_{3}$. Figure 4. The uncontractible graphs $G^{0}_{7},$ $G^{2}_{6,\alpha}$ and $G^{2}_{6,\beta}$ in ${\mathfrak{P}}_{3}$. Figure 5. The uncontractible graphs $G^{1}_{5}$ in ${\mathfrak{P}}_{2}$ and $G^{1}_{6,\alpha},$ $G^{1}_{6,\beta}$in ${\mathfrak{P}}_{3}$. ###### Definition 4.3. Let $v$ be a vertex of $G=(V,E,F)\in{\mathfrak{P}}_{k}$ for some $k=1,2,3$. Then (i) $\deg_{F}(v)$ is the number of facial 3-cycles incident to $v$, (ii) $\deg_{h}(v)=\deg(v)-\deg_{F}(v)$ is the _hole incidence degree_ for $v$, and (iii) $h(G)$ is the minimum hole incidence degree, $h(G)=\min_{v}\operatorname{deg}_{h}(v)$. In what follows, we shall usually consider graphs as ${\mathcal{P}}$-graphs, with facial structure. However, let us note that as graphs: $G^{2}_{3}$ is the triangle graph $K_{3}$; $G_{4}^{3}$ is $K_{4}$; $G_{7}^{0}$ is the cone graph over $K_{3,3}$; $G_{5}^{1}$ is $K_{5}-e$. Also the four remaining graphs, each with 6 vertices, are depletions of $K_{6}$ by 3 edges where these edges (i) form a copy of $K_{3}$, (ii) are disjoint, (iii) have one vertex shared by 2 edges, (iv) have 2 vertices of degree 1. These graphs account for all possible $(3,6)$-tight graphs on $n$ vertices for $n=3,4,5,6$, together with 1 of the 26 such graphs for $n=7$. (We remark that for $n=8,9,10$ the number of $(3,6)$-tight graphs rises steeply, with values 375, 11495, 613092 [10].) ###### Proposition 4.4. Let $G$ be an uncontractible graph in ${\mathfrak{P}}_{k}$, for $k=1,2$ or $3$, which has an interior vertex. Then $k=3$ and $G$ is the hexagon graph $G^{0}_{7}$. ###### Proof. Let $z$ be an interior vertex in $G$. Let $X(z)$ be the subgraph of $G$ induced by $z$ and its neighbours. Assume that $z$ has degree $n$ and label its neighbours, in cyclical order, as $v_{1},v_{2},\dots,v_{n}$. Then $X(z)$ has $n$ edges that are incident to $z$, plus $n$ perimeter edges $v_{1}v_{2},v_{2}v_{3},\dots,v_{n}v_{1}$, and additional edges between non- adjacent vertices $v_{1},\dots,v_{n}$. Since $G$ is uncontractible there exist at least $\left\lceil\frac{n}{2}\right\rceil$ additional edges. It follows now from the (3,)-sparsity that $\operatorname{deg}z=n\geq 6$. Also, since $G$ is uncontractible there can be no vertices $v_{i}$ of degree $3$, since otherwise the $FF$ edge $zv_{i}$ would be a contractible edge. For the same reason each vertex $v_{i}$ has at least one additional edge $v_{i}v_{j}$ for some $j$. Suppose that there is an additional edge $v_{i}v_{j}$ such that the (nonfacial) 3-cycle $zv_{i}v_{j}z$ is a planar 3-cycle. Then $G$ contains a triangulated disc $D$ with 3-cycle boundary with at least 4 vertices. Such a graph $D$ has a contractible $FF$ edge with an interior vertex and so this edge is also contractible in $G$, a contradiction. Consider one of the additional edges, $v_{i}v_{j}$ with $i<j$, and let $i^{\prime}\in\\{i+1,\dots,j-1\\}$. We claim that for every additional edge $v_{i^{\prime}}v_{j^{\prime}}$ we have $j^{\prime}\notin\\{i+1,\dots,j-1\\}$. Indeed, if this is not the case then there is a non-facial planar 3-cycle $c$ described by the edges $zv_{i^{\prime}},v_{i^{\prime}}v_{j^{\prime}},v_{j^{\prime}}z$ and by the previous paragraph this is a contradiction. Thus the additional edges have this _non-nested_ property. It follows by a simple inductive argument that the embedded graph $X(z)$ has faces with boundary cycles of length at most 4 since otherwise there must be perimeter vertices of degree 3. These 4-cycles are planar 4-cycles and so by Lemma 4.2 there are 3 holes. Thus $n=6$ and $G$ is the hexagon graph $G^{0}_{7}$. ∎ The next lemma is key to the determination of the uncontractible graphs in ${\mathfrak{P}}_{k}$ for $k=2$ or $3$. ###### Lemma 4.5. Let $G\in{\mathfrak{P}}_{k}$, for $k=2$ or 3, be an uncontractible (3,6)-tight graph with no interior vertex and let $v_{1}$ be a vertex with $\operatorname{deg}_{h}(v_{1})=1$ which lies on the boundary of a 4-cycle hole of $G$ with edges $v_{1}v_{2},v_{2}v_{3},v_{3}v_{4},v_{4}v_{1}$. Then $\operatorname{deg}(v_{1})=4$ if $v_{1}$ is not adjacent to $v_{3}$ and $\operatorname{deg}(v_{1})=5$ otherwise. ###### Proof. Let $v_{2}=w_{1},w_{2},\dots,w_{n}=v_{4}$ be the neighbours of $v_{1}$ in cyclical order. Since $\operatorname{deg}_{h}(v_{1})=1$, we also have the edges $w_{1}w_{2},\dots,w_{n-1}w_{n}$. Note that $\deg(v_{1})\geq 4$ since if the degree is 3 then the edge $v_{1}w_{2}$ is contractible. Case (a). $v_{3}\neq w_{i}$, for every $i\in\\{2,\dots,m-1\\}$. Suppose that $n\geq 5$. It follows from the uncontractibility that for each vertex $w_{i},2\leq i\leq n-1,$ there is an associated edge $w_{i}w_{r}$ for some $1\leq r\leq n$ and an associated edge $w_{i+1}w_{s}$ for some $s>r$. Since there are at most 3 holes there is an edge $w_{i}w_{i+1}$ for which the associated cycle through $w_{i},w_{i+1},w_{s},w_{r}$ is triangulated by faces. We claim that (i) it is a 4-cycle and (ii) it is triangulated by 2 faces. Note that at most one of the edges $w_{i}w_{s},w_{i+1}w_{r}$ exists. Indeed, although we can have $K_{4}\to{\mathcal{P}}$ with 3 faces this implies the existence of a degree 3 vertex and hence a contractible edge incident to it, a contradiction. If the face of the triangulation which contains $w_{i}w_{i+1}$ has third vertex $w$ not equal to $w_{s}$ or $w_{r}$, then at least one of the edges $w_{i}w$, $w_{i+1}w$ is contractible, a contradiction. Since an interior vertex $w$ does not exist the implied cycle is a 4-cycle and (i) and (ii) hold. Since $G$ is uncontractible $w_{i}w_{i+1}$ lies in a non-facial 3-cycle. Since $v_{1}w_{j}$ is also an $FF$ edge for every $j\in\\{2,\dots,n-1\\}$, it follows that there are just two candidate non-facial 3-cycles: $w_{i-1}w_{i}w_{i+1}w_{i-1}$ or $w_{i}w_{i+1}w_{i+2}w_{i}$. (i): If $w_{i}w_{i+1}$ lies on the cycle $w_{i-1}w_{i}w_{i+1}w_{i-1}$, then the 4-cycle $w_{i-1}v_{1}w_{r}w_{i+1}w_{i-1}$ contains strictly the hole boundary $v_{1}v_{2}v_{3}v_{4}v_{1}$, contradicting Lemma LABEL:l:4holelemma. Note that this 4-cycle does contain the hole in our sense since the shading in Figure 6 indicates a triangulated disc in ${\mathcal{P}}$ with boundary equal to this 4-cycle. $v_{1}$$w_{n}$$w_{r}$$w_{i+1}$$w_{i}$$w_{i-1}$$w_{1}$$v_{3}$ Figure 6. The 4-cycle $w_{i-1}v_{1}w_{r}w_{i+1}w_{i-1}$ contains strictly the 4-hole $v_{1}v_{2}v_{3}v_{4}v_{1}$. (ii): If $w_{i}w_{i+1}$ lies on the cycle $w_{i}w_{i+1}w_{i+2}w_{i}$, then, noting that $w_{i+2}w_{i}$ is an edge, we claim that the 5-cycle $w_{i}v_{1}w_{r}w_{i+1}w_{i+2}w_{i}$ contains all the holes, which is a contradiction. To see this note that by Lemma 4.2 the 4-cycle $v_{1}w_{r}w_{i}w_{i+2}v_{1}$ contains no holes. See Figure 7. $v_{1}$$w_{n}$$w_{r}$$w_{i+1}$$w_{i+2}$$w_{i}$$w_{1}$$v_{3}$ Figure 7. The 5-cycle $w_{i}v_{1}w_{r}w_{i+1}w_{i+2}w_{i}$ contains all the holes. Hence none of the edges $w_{2}w_{3},w_{3}w_{4},\dots,w_{n-2}w_{n-1}$ is an $FF$ edge. Also, the same holds for the edge $v_{2}w_{2}$, since it cannot lie in no non-facial 3-cycle. Thus every edge of the form $v_{2}w_{2},w_{2}w_{3},w_{3}w_{4},\dots,w_{n-1}w_{n}$ is on the boundary of a hole. Since every edge $v_{1}w_{j}$ is an $FF$ edge, $j=2,\dots,n-1$, it follows that $G$ contains at least $\left\lceil\frac{n}{2}+1\right\rceil$ holes. Thus, $n=4$. Case (b). $v_{4}=w_{i_{0}}$, for some $i_{0}\in\\{2,\dots,n-1\\}$. We have $\operatorname{deg}(v)\geq 5$ since $G$ is a simple graph. Suppose that $n\geq 6$. As in case (a) we may assume that there exists an $FF$ edge $w_{i}w_{i+1}$ with $i>1$ and $i+1<i_{0}$, and with vertex $w_{r}$ as before. (See Figure 8.) Then, the only possible non-facial 3-cycle for $w_{i}w_{i+1}$ is $v_{3}w_{i}w_{i+1}v_{3}$. However, this would lead to a contradiction since the 4-cycle $w_{i}v_{3}v_{4}v_{1}w_{i}$ strictly contains the hole $v_{1}v_{2}v_{3}v_{4}v_{1}$. . $v_{1}$$v_{4}$$v_{2}$$v_{3}$$w_{i}$$w_{r}$$w_{i+1}$$v_{3}$ Figure 8. The 4-cycle $v_{4}v_{3}v_{1}w_{i}v_{4}$ contains strictly the 4-hole $v_{1}v_{2}v_{3}v_{4}v_{1}$. Thus, each $w_{i}w_{i+1}$ is not an $FF$ edge. Similarly, we can argue that such a 4-cycle would be created if $w_{i_{0}-1}w_{i_{0}}$ was an $FF$ edge. Thus again we have that $w_{1}w_{2}$ is not an $FF$ edge, since it does not lie on a non-facial 3-cycle. Thus, the edges $w_{1}w_{2},w_{2}w_{3},w_{3}w_{4}$ should lie on the boundaries of different holes, which again contradicts the number of the holes of $G$. Thus $\operatorname{deg}(v_{1})=5$. ∎ ###### Proposition 4.6. Let $G\in{\mathfrak{P}}_{k}$, for $k=1,2$ or 3, be an uncontractible (3,6)-tight graph with no interior vertex. If there exists a vertex $v_{1}\in V(G)$ with $\operatorname{deg}_{h}(v_{1})=1$ then $G$ is one of the graphs $G^{1}_{6,\alpha},G^{1}_{6,\beta},G^{1}_{5}$. ###### Proof. Case (a). Assume first that $v_{1}$ lies on the 4-cycle boundary of the hole $H_{1}$, with vertices $v_{1},v_{2},v_{3},v_{4}$, and let $v_{2}=w_{1},w_{2},\dots,w_{n}=v_{4}$ be all the neighbours of $v_{1}$. Since $\operatorname{deg}_{h}(v_{1})=1$ the edges $w_{1}w_{2},\dots,w_{n-1}w_{n}$ exist. Also $\operatorname{deg}(v_{1})\geq 4$ since otherwise $v_{1}w_{2}$ is a contractible $FF$ edge. There are two subcases. (i): $v_{3}\neq w_{i}$, for every $i\in\\{1,2,\dots,m\\}$. By Lemma 4.5 we have $\operatorname{deg}(v_{1})=4$. By the uncontractibility of the edges $v_{1}w_{2}$ and $v_{1}w_{3}$ the edges $w_{2}w_{4}$ and $w_{1}w_{3}$ must exist. Thus $G$ contains the graph in Figure 9, except possibly for the edge $v_{3}w_{3}$. It follows that the 4-cycle $w_{1}w_{2}w_{4}w_{3}w_{1}$ must be the boundary of a 4-hole $H_{2}$, since otherwise the 5-cycle $v_{1}w_{1}w_{3}w_{2}w_{4}v_{1}$ contains all the holes, in the sense, as before, of being the boundary of an embedded disc, $B$ say, which contains the holes. This contradicts $(3,6)$-tightness. We claim now that the edge $v_{3}w_{2}$ or $v_{3}w_{3}$ must exist, for otherwise there is a contractible edge in $B$. To see this check that since $\operatorname{deg}(v_{3})\geq 3$, there exists a vertex $z$ in the interior of the 5-cycle $v_{3}w_{4}w_{2}w_{3}w_{1}v_{3}$, such that $v_{3}z\in E(G)$. Since $v_{3}z$ does not lie on a non facial 3-cycle, it follows that it lies on the boundary of the third 4-hole. Thus, if $v_{3}w_{3}$ is not allowed, we may assume by symmetry that $w_{1}z$ is an $FF$ edge in $E(G)$, so it lies on the non-facial 3 cycle $w_{1}zw_{2}w_{1}$. Hence the third hole is described by the 4-cycle $w_{4}v_{3}zw_{1}w_{4}$. However, this implies that $zw_{3}\in E(G)$, which is a contractible $FF$ edge, so we have proved the claim. Hence without loss of generality $G$ contains the subgraph $G^{1}_{6,\alpha}$ as indicated in Figure 9. Since $G$ is uncontractible it follows that $G=G^{1}_{6,\alpha}$. $v_{1}$$w_{4}$$w_{3}$$w_{2}$$w_{1}$$v_{3}$ Figure 9. The uncontractible graph $G^{1}_{6,\alpha}$. (ii): $v_{3}=w_{i_{0}}$ for some $i_{0}\in\\{3,\dots,n-2\\}$. By Lemma 4.5 $\operatorname{deg}(v_{1})=5$ and so $v_{3}=w_{3}$. Since $v_{1}w_{2}$ is an $FF$ edge, it follows that $w_{2}w_{4}\in E(G)$ and so $G$ contains the graph $G=G^{1}_{6,\beta}$ of Figure 10. Since $G$ is uncontractible it follows that this subgraph is equal to $G$. $v_{1}$$v_{4}$$w_{4}$$w_{2}$$v_{2}$$v_{3}$ Figure 10. The uncontractible graph $G^{1}_{6,\beta}$. Case (b). Let $v_{1}$ lie on the boundary of a 5-hole $H$ with boundary edges $v_{1}v_{2}$, $v_{2}v_{3}$, $v_{3}v_{4}$, $v_{4}v_{5}$, $v_{5}v_{1}$. We may assume that $\operatorname{deg}_{h}(v_{i})=2$, for every $i=2,3,4,5$, since otherwise there is a vertex $v$ on a 4-hole of $G$. Since $G$ has two holes it is straightforward to check that $\operatorname{deg}(v_{1})=4$ and that the second hole is described by the 4-cycle $v_{2}v_{3}v_{5}v_{4}v_{2}$. Thus we obtain that $G$ is the uncontractible (3,6)-tight given by Figure 11. $v_{1}$$v_{5}$$v_{4}$$v_{3}$$v_{2}$ Figure 11. The uncontractible graph $G^{1}_{5}$. ∎ Note that in the proof of the previous result we have determined the uncontractible graphs in 2-holed case and shown that there is a unique uncontractible graph, namely $G_{5}^{1}$. The next proposition completes the proof that there are 8 base graphs. ###### Proposition 4.7. Let $G\in{\mathcal{P}}$ be an uncontractible (3,6)-tight graph with $\operatorname{deg}_{h}(v)\geq 2$ for all $v\in V(G)$. Then $G$ is one of the four graphs $G^{2}_{6,\alpha},G^{2}_{6,\beta},G^{3}_{4},G_{3}^{2}$. ###### Proof. Suppose first that $G$ has 2 or 3 holes. Then the hole boundaries have length 4 or 5 and it follows from the simplicity of the graph that every vertex is common to at least 2 holes. Since there are either 2 or 3 holes it follows that $|V|\leq 6$. Case (a). Suppose that $G$ contains at least one $FF$ edge, say $v_{1}v_{2}$, with non facial 3-cycle $v_{1}v_{2}v_{3}$, and associated 3-cycle faces $v_{1}v_{2}v_{4}v_{1}$ and $v_{1}v_{2}v_{5}v_{1}$. We claim that one of the edges $v_{3}v_{4}$ or $v_{3}v_{5}$ lies in $E(G)$. Suppose, by way of contradiction, that neither edge exists. Then we show that the edge $v_{4}v_{5}$ is also absent. Indeed, if $v_{4}v_{5}\in E(G)$, then we have two planar 5-cycles; $v_{1}v_{4}v_{5}v_{2}v_{3}v_{1}$ and $v_{1}v_{5}v_{4}v_{2}v_{3}v_{1}$, as in Figure 12. $v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$ Figure 12. A subgraph with the 5-cycles $v_{1}v_{4}v_{5}v_{2}v_{3}v_{1}$ and $v_{1}v_{5}v_{4}v_{2}v_{3}v_{1}$. By the sparsity condition one of these has a vertex in the interior with 3 incident edges and the other has a single chordal edge in the interior and by symmetry we may assume that the planar 5-cycle $v_{1}v_{4}v_{5}v_{2}v_{3}v_{1}$ has the single chordal edge. However, of the 5 possibilities $v_{1}v_{2},v_{2}v_{4},v_{1}v_{5}$ are not available, by the simplicity of $G$, and the edges $v_{3}v_{4},v_{3}v_{5}$ are absent by assumption. This contradiction shows that $v_{4}v_{5}$ is indeed absent and so, since $v_{4},v_{5}$ have degree at least 2, the edges $v_{4}v_{6},v_{5}v_{6}$ must exist. Now the complement of the 2 3-cycle faces is bounded by two 6-cycles. By the sparsity condition there are now only 2 further edges to add and so there must be a 5-cycle hole, a contradiction, and so the claim holds. Without loss of generality we suppose that $v_{3}v_{4}\in E(G)$. Since $\operatorname{deg}_{h}(v_{2})\geq 2$, it follows that $v_{2}v_{6}\in E(G)$. Moreover, the edges $v_{6}v_{2}$,$v_{2}v_{3}$ should be on the boundary of a planar 4-hole $H_{1}$, and this implies that $v_{1}v_{6}\in E(G)$. Similarly we obtain that the two remaining holes are determined by the cycles $v_{1}v_{3}v_{4}v_{5}v_{1}$, and $v_{2}v_{5}v_{4}v_{6}v_{2}$. The resulting (3,6)-tight triangulated surface graph is given in Figure 13 and is the uncontractible graph $G_{6,\alpha}^{2}$. $v_{5}$$v_{2}$$v_{3}$$v_{1}$$v_{4}$$v_{6}$ Figure 13. The uncontractible graph with $h(G)=2$ and an $FF$ edge; $G_{6,\alpha}^{2}$. Case (b). Suppose now $G$ has at least one 3-cycle face, $v_{1}v_{2}v_{3}$, and no $FF$ edges. Then the edge $v_{1}v_{2}$ is on the boundary of a 4-hole $H_{1}$, that is determined by the edges $v_{1}v_{2}$, $v_{2}v_{4}$, $v_{4}v_{5}$ and $v_{5}v_{1}$. To see that $|V|\neq 5$ note that without loss of generality the edge $v_{3}v_{4}$ exists and $G$ contains the subgraph shown in Figure 14. Also, since $v_{5}$ cannot have degree 2 at least one of the edges $v_{5}v_{3},v_{5}v_{2}$ exists. $v_{1}$$v_{2}$$v_{4}$$v_{5}$$v_{4}$$v_{5}$$v_{3}$ Figure 14. A necessary subgraph. If $v_{5}v_{2}$ exists then the edge $v_{2}v_{3}$ is adjacent to a 4-cycle hole and $v_{5}v_{3}$ is absent. We note next that the planar 5-cycle $v_{3}v_{1}v_{5}v_{2}v_{4}v_{3}$ must contain a chord edge (and so provide the third 4-cycle hole). The only available edge (by simplicity) is $v_{3}v_{5}$. This however is inadmissible since it introduces a second 3-cycle face $v_{3}v_{5}v_{1}$ adjacent to $v_{1}v_{2}v_{3}$. Similarly, if $v_{5}v_{3}$ exists then we have the planar 6-cycle $v_{3}v_{1}v_{5}v_{3}v_{2}v_{4}v_{3}$ and there must exist a diameter edge to create the 2 additional 4-cycle holes. As there is no such edge we conclude that $|V|=6$. Introducing $v_{6}$ the fact that $v_{2}v_{3}$ and $v_{3}v_{1}$ lie on 4-cycle hole boundaries leads to the graph $G_{6,\beta}^{2}$ indicated in Figure 15. $v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$v_{6}$$v_{4}$$v_{5}$ Figure 15. The uncontractible graph $G_{6,\beta}^{2}$, with $h(G)=2$, no $FF$ edge and a 3-cycle face. Case (c). Let now $G$ be a graph with no 3-cycle faces. Since $\operatorname{deg}(v)\geq 3$ for each vertex it follows that $\operatorname{deg}_{h}(v)=3$ and $\deg(v)=3$, for all $v\in V(G)$. Thus $|V|=4$ and it follows that $G$ is the uncontractible (3,6)-tight graph $G_{4}^{3}$ given by Figure 16. $v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{2}$$v_{3}$$v_{4}$ Figure 16. The uncontractible graph $G_{4}^{3}$ with $h(G)=3$. Case (d). Finally, suppose that $G\in\mathfrak{P}_{1}$. We claim that the graph has no faces and the surface graph is given by Figure 19. Assume first that there exists an $FF$ edge, say $v_{1}v_{2}$, that lies on the faces $v_{1}v_{2}v_{3}v_{1}$ and $v_{1}v_{2}v_{4}v_{1}$. Since the graph is uncontractible, $v_{1}v_{2}$ lies on a non facial 3-cycle $v_{1}v_{2}v_{5}v_{1}$. Note that $v_{3}v_{4}\notin E(G)$, since otherwise the 6-hole would lie inside a 5-cycle, either $v_{1}v_{3}v_{4}v_{2}v_{5}v_{1}$ or $v_{1}v_{4}v_{3}v_{2}v_{5}v_{1}$, contradicting the sparsity of the graph. It follows that we cannot have $|V(G)|\leq 5$. Indeed, in this case (see Figure 17) $v_{3}v_{5}\in E(G)$, since $\operatorname{deg}(v_{3})\geq 3$, and so without loss of generality, in view of the symmetry, $v_{1}v_{3}$ is an $FF$ edge. But this edge does not lie on a non-facial 3-cycle, a contradiction. $v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{5}$ Figure 17. $|V(G)|\leq 5$ leads to a contradiction. Thus $|V(G)|=6$ and it remains to consider two subcases: 1. (i) $v_{3}v_{5}\in E(G)$. In this case $v_{1}v_{3}$ lies on the non-facial 3-cycle $v_{1}v_{3}v_{6}v_{1}$. However, this leads to a contradiction, since the 6-hole is contained either in the 5-cycle $v_{5}v_{3}v_{6}v_{1}v_{2}v_{5}$ or in the 5-cycle $v_{6}v_{3}v_{2}v_{5}v_{1}v_{6}$. Hence by symmetry neither of the edges $v_{3}v_{5},v_{4}v_{5}$ is allowed. 2. (ii) $v_{3}v_{6},v_{4}v_{6}\in E(G)$. In this case, indicated in Figure 18, we may assume that the hole is contained in the planar 6-cycle $v_{1}v_{5}v_{2}v_{4}v_{6}v_{3}v_{1}$ and that the planar 6-cycle $v_{1}v_{5}v_{2}v_{3}v_{6}v_{4}v_{1}$ is triangulated. This implies that $v_{2}v_{3}$ is an $FF$ edge and so lies on non-facial 3-cycle. However, the only candidate cycle is $v_{3}v_{2}v_{6}v_{3}$ and if $v_{2}v_{6}$ lies in $E(G)$ then the hole is contained in the 5-cycle $v_{1}v_{5}v_{2}v_{6}v_{3}v_{1}$, a contradiction. $v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{5}$$v_{6}$$v_{6}$ Figure 18. Edges $v_{3}v_{6},v_{4}v_{6}$ in $G$ leads to a contradiction. We have shown that no $FF$ edge is allowed. Suppose now that $G$ contains a face, described by the vertices $v_{1},v_{2}$ and $v_{3}$. Since there are no $FF$ edges, all edges $v_{1}v_{2},v_{2}v_{3}$ and $v_{1}v_{3}$ lie on the boundary of the hole. Moreover, since they form a face of the graph, they cannot form a 3-cycle path in the boundary of the hole. Only 3 edges of the boundary cycle are left to be determined, so we may assume that the path $v_{1}v_{2}v_{3}$ lies on the boundary. Therefore, without loss of generality, there exists a vertex $v_{4}$ on the boundary that connects the two paths, $v_{1}v_{2}v_{3}$and $v_{1}v_{3}$, so we obtain the 5-path $v_{1}v_{3}v_{4}v_{1}v_{2}v_{3}$. But this implies that the remaining edge of the 6-hole is $v_{1}v_{3}$, which would break the simplicity of the graph. Hence the graph contains no faces and the proof is complete. $v_{1}$$v_{2}$$v_{3}$ Figure 19. The uncontractible graph $G^{2}_{3}$. ∎ ## 5\. The irreducibles We show that an irreducible $(3,6)$-tight ${\mathcal{P}}$-graph is uncontractible. Thus, if a graph $G$ in ${\mathfrak{P}}_{k}$, for $k=1,2$ or $3$, has a contractible edge $e$ (so that $G/e$ is a simple graph) then there exists a contractible edge $f$, which need not be the edge $e$, such that the contracted graph is simple and satisfies the sparsity condition for membership in ${\mathfrak{P}}_{k}$. Recall that Lemma 3.5 identifies the obstacles to the preservation of $(3,6)$-sparsity when contracting a contractible edge of $G\in{\mathfrak{P}}_{k}$, namely that the edge lies on the boundary of a subgraph of $G$ which is in ${\mathfrak{P}}_{l}$ for some $l\leq k$. For $k=1$ this boundary corresponds to a directed 6-cycle $c$ and we also refer to it in subsequent proofs as a _critical 6-cycle_. Likewise for $k=2$ or $k=3$ the edge $e$ lies on the boundary of one of the holes of a subgraph $G\in{\mathfrak{P}}_{l}$ and we refer to the associated cycle as a _critical 5-cycle_ or _critical 4-cycle_. ###### Proposition 5.1. Let $G\in{\mathfrak{P}}_{1}$ be irreducible. Then $G$ is uncontractible. ###### Proof. Suppose that $G$ is irreducible with a contractible edge $e=xy$. By Lemma 3.5 there is a critical 6-cycle $c$, containing $e$, which is the boundary of a subgraph $G_{1}\in{\mathfrak{P}}_{1}$. Since $c$ properly contains the hole of $G$ this contradicts Lemma 4.2, completing the proof. ∎ ###### Proposition 5.2. Let $G$ be an irreducible graph in ${\mathfrak{P}}_{k}$, for $k=2$ or $3$. Then $G$ is uncontractible. ###### Proof. Suppose that $G$ is irreducible and $e$ is a contractible $FF$ edge in $G$. By Lemma 3.5 there is a decomposition $G=G_{1}\cup A$ with $e\in\partial G_{1},$ and $G_{1}\in{\mathfrak{P}}_{l},$ for some $l\leq k$. $e$$v$$v_{1}$$v_{2}$$w$$y$$x$$G_{1}$$v_{3}$$z$ Figure 20. A ${\mathcal{P}}$-diagram for a critical 6-cycle for $e$. _Case $k=2,l=1$._ Figure 20 illustrates the planar 6-cycle boundary $c$ of $G_{1}$ and we assume it includes the contractible edge $e$ and that it contains the planar 4- and 5-cycle boundaries of the two holes of $G$. Since $G$ is simple $c$ has 6 distinct vertices. For the first part of the proof we show that $G$ contains $vv_{1}$, perhaps after relabelling $v_{1},v_{2}$, that $yvv_{1}v_{2}v_{3}y$ is the boundary of the 5-cycle hole, and that $xvv_{1}wx$ is the boundary of the 4-cycle hole. Note that $G_{1}$ is a contractible graph, for otherwise, by the previous section, $G_{1}=G_{3}^{2}$, with 3 vertices. By Proposition 5.1 $G_{1}$ is reducible and so there is an $FF$ edge edge $h$ with $G_{1}/h\in{\mathfrak{P}}_{1}$. If $h$ lies on a critical 6-cycle $c^{\prime}$ in $G$ then it necessarily lies on a critical 6-cycle in $G_{1}$. This is because the subpath of $c^{\prime}$ which is interior to $c$ must have the same length as one of boundary paths of $c$ between the corresponding vertices. (Otherwise the 6-cycle hole is contained in a planar cycle of length at most 5.) Thus, since $G$ is irreducible, $h$ must lie on a nonfacial 3-cycle in $G$ with some edges that are internal to $c$. To avoid sparsity violation there must be 2 such edges, say $h_{1},h_{2}$. Moreover, since $h$ is a contractible edge in $G_{1}$ the edges $h_{1},h_{2}$ form a diameter of the 6-cycle $c$. This diameter together with subpaths of $c$, yields two planar 5-cycles which contain the holes of $G$. Considering the 5-cycle hole, Lemma 4.2 implies that, perhaps after relabelling, the pair $h_{1},h_{2}$ is equal to the pair $yv,vv_{1}$ or to a pair $wu,uv_{3}$ for some vertex $u\neq v$ interior $c$. In the first case $yvv_{1}v_{2}v_{3}y$ is the boundary of the 5-cycle hole and, by a further application of Lemma 4.2, $xvv_{1}wx$ is the boundary of the 4-cycle hole. The second case cannot occur, since one of the edges $xv,yv$ must be an $FF$ edge, and one can see that it does not lie on a nonfacial 3-cycle or a critical 4-, 5- or 6-cycle. For the next part of the proof we show that $G_{1}$ has no interior vertices. Let $u$ be an interior vertex of $G_{1}$ and let $f$ be one of its incident $FF$ edges. Then since $G$ is irreducible, by the hole inclusion lemma, Lemma 4.2, $f$ does not lie on a critical 6-cycle. Also if $f$ lies on a nonfacial 3-cycle then by Lemma 4.1 it lies on a nonplanar nonfacial 3-cycle. It follows from the $(3,6)$-sparsity of $G$ that $\deg v\geq 6$ and so there are at least 3 distinct nonplanar nonfacial 3-cycle through $u$. However this implies that every hole of $G$ is contained in a planar 4-cycle, a contradiction. $e$$v$$v_{1}$$v_{2}$$w$$y$$x$$G_{1}$$v_{3}$$z$$H$$H$ Figure 21. $G_{1}$ has no interior vertex and $z=v_{1}$. Since $z$ is not an interior vertex of $G_{1}$ it is equal to $v_{1}$ (see Figure 20). By Lemma 4.1(i) we have $\deg(v_{2})\geq 4$ and $\deg(v_{3})\geq 4$. Since $G_{1}$ is $(3,6)$-tight it follows that both vertices have degree 4 and that $G$ must have the structure indicated in Figure 21. In particular, $v_{3}w$ does not lie on a nonfacial 3-cycle or a critical cycle and so $G/v_{3}w$ is reducible, a contradiction. _Case $k=2,l=2$._ We argue by contradiction and assume that $G$ is irreducible and $e$ is a contractible $FF$ edge in $G$ which, by Lemma 3.5, lies on the boundary of the proper subgraph $G_{1}\in{\mathfrak{P}}_{2}$. Each of the two holes of $G_{1}$ must contain a hole of $G$, with the boundary cycles are of the same length. By Lemma 4.2 this is a contradiction. _Case $k=3,l=2$._ This case follows similarly. ∎ ## 6\. Constructibility and 3-rigidity Combining results of the previous sections we obtain the following construction theorem and the proof of Theorem 1.1. ###### Theorem 6.1. Let $G$ be a simple (3,6)-tight graph which is embeddable in the real projective plane ${\mathcal{P}}$. Then $G$ is constructible by a finite sequence of planar vertex splitting moves from at least one of the eight ${\mathcal{P}}$-graphs, $G_{3}^{2},G_{4}^{3},G_{5}^{1},G_{6,\alpha}^{1},G_{6,\beta}^{1},G_{6,\alpha}^{2},G_{6,\beta}^{2},G_{7}^{0}$. ###### Proof. As we have observed at the beginning of Section 4 it is evident that $G$ can be reduced to an irreducible (3,6)-tight ${\mathcal{P}}$-graph, $H$ say, by a sequence of planar edge contraction moves. By the results of Section 5 the irreducible graph $H$ is uncontractible, and so, by the results of Section 4, it is equal to one of the eight uncontractible ${\mathcal{P}}$-graphs. Since a planar edge-contraction move is the inverse of a planar vertex splitting move the proof is complete. ∎ ###### Proof of Theorem 1.1. Let $G$ be the graph of a partial triangulation of the real projective plane. If $G$ is minimally 3-rigid then it is well-known that $G$ is necessarily $(3,6)$-tight [11]. Suppose on the other hand that $G$ is $(3,6)$-tight. Then, by Theorem 6.1 the graph $G$ is constructible by planar vertex splitting moves from one of the eight uncontractible ${\mathcal{P}}$-graphs, each of which has fewer than $8$ vertices. It is well-known that all $(3,6)$-tight graphs with fewer than $8$ vertices are minimally 3-rigid. Since vertex splitting preserves minimal 3-rigidity (Whiteley [14]) it follows that $G$ is minimally 3-rigid. ∎ Acknowledgements. This research was supported by the EPSRC grant EP/P01108X/1 for the project _Infinite bond-node frameworks_ and by a visit to the Erwin Schroedinger Institute in September 2018 in connection with the workshop on _Rigidity and Flexibility of Geometric Structures_. ## References * [1] D. W. Barnette, Generating the triangulations of the projective plane, J. Comb. Theory33 (1982), 222-230. * [2] D. W. Barnette and A. Edelson, All orientable 2-manifolds have finitely many minimal triangulations, Israel. J. Math., 62 (1988), 90-98. * [3] D.W. Barnette, and A.L. Edelson, All 2-manifolds have finitely many minimal triangulations, Israel J. Math., 67 (1989), 123-128. * [4] A. Cauchy, Sur les polygones et polyèdres. Second Mémoir. J École Polytechn. 9 (1813) 87-99; Oeuvres. T. 1. Paris 1905, pp. 26-38. * [5] J. Cruickshank, D. Kitson and S.C. Power, The generic rigidity of triangulated spheres with blocks and holes, J. Combin. Theory Ser. B 122 (2017), 550-577. * [6] J. Cruickshank, D. Kitson and S.C. Power, The rigidity of a partially triangulated torus, Proc. London Math. Soc., 2018, https://doi.org/10.1112/plms.12215 * [7] W. Finbow-Singh and W. Whiteley, Isostatic block and hole frameworks, SIAM J. Discrete Math. 27 (2013) 991-1020. * [8] W. Finbow-Singh, E. Ross and W. Whiteley, The rigidity of spherical frameworks: Swapping blocks and holes in spherical frameworks, SIAM J. Discrete Math. 26 (2012), 280-304. * [9] N. D. Gilbert and T. Porter, Knots and Surfaces, Oxford University Press, 1994. * [10] G. Grassegar, personal communication. * [11] J. Graver, B. Servatius and H. Servatius, Combinatorial rigidity, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993. * [12] H. Gluck, Almost all simply connected closed surfaces are rigid, in Geometric Topology, Lecture Notes in Math., no. 438, Springer-Verlag, Berlin, 1975, pp. 225-239. * [13] T. Jordan and S. Tanigawa, Global rigidity of triangulations with braces, J. of Comb. Theory, Ser. B, 136 (2019), 249-288. * [14] W. Whiteley, Vertex splitting in isostatic frameworks, Structural Topology, 16 (1990), 23-30.

2024-09-04T02:54:59.425039

2003.05669

arxiv-papers

# ARAE: Adversarially Robust Training of Autoencoders Improves Novelty Detection Mohammadreza Salehi,1 Atrin Arya,1 Barbod Pajoum,1 Mohammad Otoofi,1 Amirreza Shaeiri,1 Mohammad Hossein Rohban,1 Hamid R. Rabiee1 ###### Abstract Autoencoders (AE) have recently been widely employed to approach the novelty detection problem. Trained only on the normal data, the AE is expected to reconstruct the normal data effectively while failing to regenerate the anomalous data. Based on this assumption, one could utilize the AE for novelty detection. However, it is known that this assumption does not always hold. More specifically, such an AE can often perfectly reconstruct the anomalous data as well, due to modeling of low-level and generic features in the input. To address this problem, we propose a novel training algorithm for the AE that facilitates learning of more semantically meaningful features. For this purpose, we exploit the fact that adversarial robustness promotes learning of meaningful features. Therefore, we force the AE to learn such features by making its bottleneck layer more stable against adversarial perturbations. This idea is general and can be applied to other autoencoder based approaches as well. We show that despite using a much simpler architecture in comparison to the prior methods, the proposed AE outperforms or is competitive to state- of-the-art on four benchmark datasets and two medical datasets. ## Introduction --- Figure 1: Unlike DAE, ARAE that is trained on the normal class, which is the digit $8$, reconstructs a normal instance when it is given an anomalous digit, from the class $1$. The first row shows the input images. The second and third rows show the DAE and ARAE reconstructions of the corresponding inputs, respectively. ARAE is trained based on bounded $\ell_{\infty}$, $\ell_{2}$, rotation, and translation perturbations. In many real-world problems, it is easy to gather normal data from the operating behavior of a system. However, collecting data from the same system in situations where it malfunctions or is being used clumsily may be difficult or even impossible. For instance, in a surveillance camera that captures daily activity in an environment, almost all frames are related to the normal behavior. This means that data associated with the anomalous behavior is difficult to obtain from such cameras. Anomaly/novelty detection refers to the set of solutions for such settings. The key point in the definition of anomaly detection is the outlier notion. In the literature, An outlier is defined as a data point that deviates from the bulk of the remaining data (Hawkins 1980; Chalapathy and Chawla 2019). Assuming that the normal data is generated by a distribution, the goal is to detect whether a new unseen observation is drawn from this distribution or not. In prior work, AE and Generative Adversarial Network (GAN) were extensively applied for novelty detection (Sabokrou et al. 2018; Perera, Nallapati, and Xiang 2019; Schlegl et al. 2017; Akcay, Atapour-Abarghouei, and Breckon 2018). In GAN-based approaches, one tries to train a model that could adversarially generate realistic images from the normal class. This means that if the model fails to generate a given input image, the input would probably be an anomalous one. However, GAN-based approaches face some challenges during the training. These include mode collapse that happens when the generator maps several inputs to a single image in the output space. In GAN, complete mode collapse is rare, while a partial collapse occurs more frequently (Goodfellow 2016; Kodali et al. 2017). Furthermore, high sensitivity of the training to the choices of hyperparameters, non-convergence problem, parameter oscillation, and non-reproducible results due to the unstable training are counted as the other challenges in training of the GAN (Martin and Lon 2017; Salimans et al. 2016). On the other hand, AE is more convenient to train and gives results that are easier to reproduce. Therefore, we propose our method based on AE-based approaches in this paper. An AE, which has learned features that are mostly unique to the normal class, could reconstruct the normal data perfectly, while when given an anomalous data, it either reconstructs a corrupted or a normal output; In the former case, the anomalous input is likely to have disjoint features compared to the normal class, while in the latter, the input may resemble a normal data in some aspects. Note that in both cases, unlike for the normal data, the reconstruction Mean Squared Error (MSE) is high for the anomalous data. This means that for such an AE, we could threshold the reconstruction loss to distinguish the normal vs. anomalous data. One could alternatively leverage a discriminator that is applied to the reconstructed image to distinguish between the anomalous and normal data (Sabokrou et al. 2018; Larsen et al. 2015). In any case, as mentioned, an important premise for the AE to work is that it learns mostly unique features to the normal class. We call such features “semantically meaningful” or “robust”, contrasted with generic low level features that are subject to change in presence of noise, in the rest of the paper. A common problem in using AE for novelty detection is its generalization ability to reconstruct some anomaly inputs, when they share common features with the normal class (Gong et al. 2019; Zong et al. 2018). Although this generalization property is useful in other contexts, such as restoration (Mao, Shen, and Yang 2016), it is considered as a drawback in novelty detection. In other papers (Hasan et al. 2016; Zhao et al. 2017; Sultani, Chen, and Shah 2018), the main underlying assumption behind the AE-based approaches is that the reconstruction error is high when the model is given an anomalous data, which as mentioned does not seem to be holding perfectly. There are two reasons why the main underlying assumption in these methods does not hold necessarily. First, the model behavior when facing the anomalous data is not observed and is not therefore predictable. Second, the learned latent space may capture mostly the features that are in common between the normal and anomalous data. When given the anomalous data, this would likely yield a perfectly reconstructed anomalous data. To address these issues, we aimed for a solution that learns an adversarially robust latent space, where the focus is on learning unique or semantically meaningful features of the normal inputs and their nuances. This could prevent the decoder from reconstructing the anomalies. It is shown in (Madry et al. 2017) that small imperceptible changes in the input can easily fool a deep neural network classifier. AE’s are subject to such attacks as well. This stems from the fact that a deep classifier or an AE would likely learn low level or brittle non-robust features (Ilyas et al. 2019). Low level features could be exploited to reconstruct any given image perfectly. Hence, the presence of such features seems to violate the main underlying assumption of the earlier work for novelty detection that is based on AE. Therefore, we propose to train an adversarially robust AE to overcome this issue. In Figure 1, reconstructions from DAE and the proposed method are shown. Here, the normal data is considered to be the number $8$ in the MNIST dataset and the models are trained only on the normal category. As opposed to the proposed ARAE, DAE generalizes and reconstructs the number $1$ perfectly. This is not desired in the novelty detection problem. This means that the latent space of DAE has learned features that are not necessarily meaningful. To train a robust AE for the novelty detection task, a new objective function based on adversarial attacks is proposed. The novel AE which is based on a simple architecture, is evaluated on MNIST, Fashion-MNIST, COIL-100, CIFAR-10, and two medical datasets. We will next review existing approaches in more details, and then describe our proposed idea along with its evaluation. We demonstrate that despite the simplicity of the underlying model, the proposed model outperforms or stays competitive with state-of-the-art in novelty detection. Moreover, we show that our method performs much better compared to another state-of-the-art method in presence of adversarial examples, which is more suitable for real-world applications. ## Related work Figure 2: The training procedure of our method. $L_{latent}$ and $L_{rec.}$ are obtained using the MSE distance and used to form $L_{AE}$. As explained earlier in the introduction, methods that are used in the literature are classified into two main categories: (1) modeling the normal behavior in the latent space; and (2) thresholding the AE reconstruction error. Of course, a hybrid of these two approaches was also considered in the field. DRAE (Zhou and Paffenroth 2017), takes the second approach, i.e. it is based on the MSE distance between the AE output and its input. An underlying assumption in this work is that the training data may contain abnormal samples. Therefore, the method tries to identify these samples throughout the training process. It finally uses only the reconstruction error in the test time. As an extension to the AE-based methods, in OCGAN (Perera, Nallapati, and Xiang 2019), a model is introduced in which the AE is trained by using 4 GANs, a classifier, and the “negative sample mining” technique. Here, both the encoder and decoder of the AE are considered as generators in the GAN. At the inference time, the method only uses MSE between the model output and input to make a prediction. The authors attempted to force the encoder output distribution to be approximately uniform. They also forced the decoder output distribution to resemble the normal input distribution in the whole latent domain. This is expected to result in a higher MSE distance between the decoder output and input for the abnormal data. This method achieved state-of- the-art results at the time of presentation. (Abati et al. 2019) and (Sabokrou et al. 2018) are the other examples in the AE-based approaches, except that in (Abati et al. 2019), additionally, the probability distribution over the latent space was obtained for the normal input data. Then, in the test time, the probability of a sample being normal, which is called the “surprise score”, is added to the reconstruction error before the thresholding happens. In (Sabokrou et al. 2018), there is a possibility of using the discriminator output, which is a real number between zero and one, as an alternative to the MSE distance in order to find the anomaly score. This is done by considering the AE as the generator in the GAN framework. In (Pidhorskyi, Almohsen, and Doretto 2018), a GAN is initially used to obtain the latent space, then the probability distribution of the normal class over the latent space is considered to be as the multiplication of two marginal distributions, which are learned empirically. (Ruff et al. 2018) (DSVDD) tries to model the normal latent space with the presumption that all normal data can be compressed into a hyper-sphere. This framework can be considered as a combination of Deep Learning and classical models such as (Chen, Zhou, and Huang 2001) (One-class SVM), that has the advantage of extracting more relevant features from the training data than the above-mentioned (Chen, Zhou, and Huang 2001) because the whole network training process is done in an end- to-end procedure. In (Schlegl et al. 2017), a GAN framework is used to model the latent space. It is assumed that if the test data is normal, then a sample could be found in a latent space such that the corresponding image that is made by the generator is classified as real by the GAN discriminator. ## Method Figure 3: Samples from the evaluation datasets. For the medical datasets, the top row samples are anomalous and the bottom row samples are normal. As we discussed earlier, the main problem of AE is its strong generalization ability. We observe that DAE does not necessarily learn distinctive features of the normal class. To remedy this problem, our approach is to force the AE latent space implicitly to model only unique features of the normal class. To make this happen, the framework for adversarial robustness, which is proposed in (Madry et al. 2017; Ilyas et al. 2019), is adopted. We propose to successively craft adversarial examples and then utilize them to train the AE. Adversarial examples are considered as those irrelevant small changes in the input that destabilize the latent encoding. We will next describe the details of the proposed adversarial training in the following sections. The training procedure is demonstrated in Figure 2. ### Adversarial Examples Crafting In a semantically meaningful latent space, two highly perceptually similar samples should share similar feature encodings. Therefore, searching for a sample $X^{*}$ that is perceptually similar to a sample $X$, but has a distant latent encoding from that of $X$, leads us to an adversarial sample. As opposed to the normal sample $X$, the adversarial sample $X^{*}$ is very likely to have a high reconstruction loss, thus it would be detected as abnormal by the AE, despite being perceptually similar to a normal sample. Therefore, based on this intuition, the following method is used to craft the adversarial samples. At the training epoch $i$, we craft a set of adversarial samples $S^{i}_{(adv)}$ based on the initial training dataset $S$. For this purpose, we slightly perturb each sample $X\in S$ to craft an adversarial sample $X^{*}$ that has two properties: (1) $X^{*}$ is perceptually similar to $X$, through controlling the $\ell_{\infty}$ distance of $X$ and $X^{*}$; (2) $X^{*}$ latent encoding is as far as possible from that of $X$. This is equivalent to solving the following optimization problem: $\max_{\delta_{X}}L_{\text{latent}}\mbox{ s.t. }{\|\delta_{X}\|}_{\infty}\leq\epsilon$ (1) $L_{\text{latent}}=\|\mbox{Enc}(X+\delta_{X})-\mbox{Enc}(X)\|^{2}_{2}\ $ (2) In this formulation, ${\|\ .\ \|}_{p}$ is the $\ell_{p}$-norm, $\epsilon$ is the attack magnitude, and $X^{*}=X+\delta_{X}$ is the adversarial sample. We solve this optimization problem for each sample $X\in S$ using the Projected Gradient Descent (PGD) (Madry et al. 2017) method, to obtain $S^{i}_{(adv)}$. ### Autoencoder Adversarial Training To train the AE using the crafted dataset $S^{i}_{(adv)}$ in the previous section, we propose the following loss function: $L_{\text{AE}}=L_{\text{rec.}}+\gamma L_{\text{latent}}$ (3) where $\gamma$ is a balancing hyperparameter, $L_{\text{latent}}$ refers to the loss function that is introduced in Eq. 2 and $L_{\text{rec.}}$ corresponds to the following loss function: $L_{\text{rec.}}=\|X-\mbox{Dec}(\mbox{Enc}(X^{*}))\|^{2}_{2}\ $ (4) At each step, the AE is trained one epoch on the adversarially crafted samples using this loss function. In the training procedure, the $L_{\text{rec.}}$ term forces the AE to reconstruct the adversarial samples properly, while the $L_{\text{latent}}$ term forces the adversarial samples to have closer representations to that of the corresponding normal samples in the latent space. We observe that the encoder decreases $L_{\text{latent}}$ to a limited extent by merely encoding the whole input space into a compact latent space. Too compact latent space results in a high $L_{\text{rec.}}$, which is not achievable when the network is trained using $L_{\text{AE}}$. A compact latent space causes the latent encodings of anomalous data to be close to that of normal data. Thus for any given input, the generated image is more likely to be a normal sample. To summarize, the whole training procedure is trying to solve the following saddle point problem (Wald 1945): $\begin{gathered}\delta^{*}_{X}:=\operatorname*{arg\,max}_{\|\delta_{X}\|_{\infty}\leq\epsilon}L_{\text{latent}}(X,\delta_{X},W)\\\ \min_{W}\operatorname{\mathbb{E}}_{X}\left[\gamma L_{\text{latent}}(X,\delta^{*}_{X},W)+L_{\text{rec.}}(X,\delta^{*}_{X},W)\right]\end{gathered}$ (5) where $W$ is denoted as the AE weights. Note that it was shown that the adversarial training could not be solved in a single shot by the Stochastic Gradient Descent (SGD), and one instead should try other optimization algorithms such as the PGD. This relies on Danskin theorem to solve the inner optimization followed by the outer optimization (Madry et al. 2017). ## Experiments In this section, we evaluate our method, which is denoted by ARAE, and compare it with state-of-the-art on common benchmark datasets that are used for the unsupervised novelty detection task. Moreover, we use two medical datasets to evaluate our method in real-world settings. We show that even though our method is based on a simple and efficient architecture, it performs competitively or superior compared to state-of-the-art approaches. Furthermore, we provide insights about the robustness of our method against adversarial attacks. The results are based on several evaluation strategies that are used in the literature. All results that are reported in this paper are reproducible by our publicly available implementation in the Keras framework (Chollet 2015)111https://github.com/rohban- lab/Salehi˙submitted˙2020. ### Experimental Setup #### Baselines Baseline and state-of-the-art approaches like VAE (Kingma and Welling 2013), OCSVM (Chen, Zhou, and Huang 2001), AnoGAN (Schlegl et al. 2017), DSVDD (Ruff et al. 2018), MTQM (Wang, Sun, and Yu 2019), OCGAN (Perera, Nallapati, and Xiang 2019), LSA (Abati et al. 2019), DAGMM (Zong et al. 2018), DSEBM (Zhai et al. 2016), GPND (Pidhorskyi, Almohsen, and Doretto 2018), $l_{1}$ thresholding (Soltanolkotabi, Candes et al. 2012), DPCP (Tsakiris and Vidal 2018), OutlierPursuit (Xu, Caramanis, and Sanghavi 2010), ALOCC (Sabokrou et al. 2018), LOF (Breunig et al. 2000), and DRAE (Xia et al. 2015) are selected to be compared with our method. Results of some of these methods were obtained from (Perera, Nallapati, and Xiang 2019; Wang, Sun, and Yu 2019; Pidhorskyi, Almohsen, and Doretto 2018). #### Datasets We evaluate our method on MNIST (LeCun, Cortes, and Burges 2010), Fashion- MNIST (Xiao, Rasul, and Vollgraf 2017), COIL-100 (Nene, Nayar, and Murase 1996), CIFAR-10 (Krizhevsky 2009), Head CT - hemorrhage (Kitamura 2018), and Brain MRI - Tumor (Chakrabarty 2019) datasets. Samples from each dataset are shown in Figure 3. These datasets differ in size, image shape, complexity and diversity. Next, we briefly introduce each of these datasets. * • MNIST: This dataset contains 70,000 $28\times 28$ grayscale handwritten digits from 0 to 9. * • Fashion-MNIST: A dataset similar to MNIST with 70,000 $28\times 28$ grayscale images of 10 fashion product categories. * • CIFAR-10: This dataset contains 60000 $32\times 32$ color images of 10 categories. * • COIL-100: A dataset of 7200 color images of 100 different object classes. Each class contains 72 images of one object captured in different poses. We downscale the images of this dataset to the size $32\times 32$. * • Head CT - Hemorrhage: A dataset with 100 normal head CT slices and 100 other with 4 different kinds of hemorrhage. Each slice comes from a different person and the image size is $128\times 128$. * • Brain MRI - Tumor: A dataset with 253 brain MRI images. 155 of them contain brain tumors and the rest 98 are normal. The image size is $256\times 256$. #### Protocols To carry out the training-testing procedure, we need to define the data partitions. For MNIST, Fashion-MNIST, and CIFAR-10, one class is considered as the normal class and samples from the other classes are assumed to be anomalous. For COIL-100, we randomly take $n$ classes as the normal classes, where $n\in\\{1,4,7\\}$, and use the samples from the remaining classes as the anomalous samples. For the mentioned dataset, this process is repeated 30 times and the results are averaged. For the medical datasets, the brain images with no damage are considered as the normal class and the rest form the anomalous class. To form the training and testing data, there are two protocols that are commonly used in the framework of unsupervised novelty detection(Pidhorskyi, Almohsen, and Doretto 2018; Perera, Nallapati, and Xiang 2019; Sabokrou et al. 2018), which are as follows: * • Protocol 1: The original training-testing splits of the dataset are merged, shuffled, and $80\%$ of the normal class samples are used to train the model. The remaining $20\%$ forms some specified portion (denoted as $\tau$) of the testing data. The other portion is formed by randomly sampling from the anomalous classes. * • Protocol 2: The original training-testing splits of the dataset are used to train and test the model. The training is carried out using the normal samples and the entire testing data is used for evaluation. We compare our method to other approaches using Area Under the Curve (AUC) of the Receiver Operating Characteristics (ROC) curve, the $F_{1}$ score, and the False Positive Rate (FPR) at $99.5\%$ True Positive Rate (TPR). Here, we let the positive class be the anomalous one unless otherwise specified. #### Architecture and Hyperparameters Our AE uses a 3-layer fully connected network with layer sizes of $(512,256,128)$, following the input-layer to encode the input. A decoder, whose architecture is mirroring that of the encoder, is used to reconstruct the output. Each layer of the network is followed by a sigmoid activation. This architecture is used for all the datasets except the medical ones and CIFAR-10. For the medical datasets and CIFAR-10, we use a convolutional AE which is explained in (Bergmann et al. 2019). For datasets with complex and detailed images like COIL-100, Fashion-MNIST, CIFAR-10, and the medical datasets, the hyperparameter $\epsilon$, which is the maximum perturbation $\ell_{\infty}$ norm as defined in Eq. 5, is set to $0.05$, while for MNIST it is set to $0.2$. The hyperparameter $\gamma$, defined in Eq. 5, is always set to $0.1$. ### Results Table 1: AUC values (in percentage) for the medical datasets. The standard deviation of the last 50 epochs’ AUCs are included for the Brain MRI - Tumor dataset. Dataset | OCGAN | LSA | ARAE ---|---|---|--- Head CT - Hemorrhage | 51.2 | 81.6 | 84.8 Brain MRI - Tumor | 91.7 | 95.6 | 97.0 $\pm 3$ | $\pm 1.4$ | $\pm 0.5$ Table 2: AUC values (in percentage) on MNIST and FMNIST (Fashion-MNIST). The standard deviation of the last 50 epochs’ AUCs are included for our method on MNIST. The values were obtained for each class using protocol 2. Dataset | Method | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Mean ---|---|---|---|---|---|---|---|---|---|---|---|--- MNIST | VAE | 98.5 | 99.7 | 94.3 | 91.6 | 94.5 | 92.9 | 97.7 | 97.5 | 86.4 | 96.7 | 95.0 OCSVM | 99.5 | 99.9 | 92.6 | 93.6 | 96.7 | 95.5 | 98.7 | 96.6 | 90.3 | 96.2 | 96.0 AnoGAN | 96.6 | 99.2 | 85.0 | 88.7 | 89.4 | 88.3 | 94.7 | 93.5 | 84.9 | 92.4 | 91.3 DSVDD | 98.0 | 99.7 | 91.7 | 91.9 | 94.9 | 88.5 | 98.3 | 94.6 | 93.9 | 96.5 | 94.8 MTQM | 99.5 | 99.8 | 95.3 | 96.3 | 96.6 | 96.2 | 99.2 | 96.9 | 95.5 | 97.7 | 97.3 OCGAN | 99.8 | 99.9 | 94.2 | 96.3 | 97.5 | 98.0 | 99.1 | 98.1 | 93.9 | 98.1 | 97.5 LSA | 99.3 | 99.9 | 95.9 | 96.6 | 95.6 | 96.4 | 99.4 | 98.0 | 95.3 | 98.1 | 97.5 ARAE | 99.8 | 99.9 | 96.0 | 97.2 | 97.0 | 97.4 | 99.5 | 96.9 | 92.4 | 98.5 | 97.5 $\pm 0.017$ | $\pm 0.003$ | $\pm 0.2$ | $\pm 0.17$ | $\pm 0.14$ | $\pm 0.1$ | $\pm 0.03$ | $\pm 0.1$ | $\pm 0.3$ | $\pm 0.04$ | $\pm 0.04$ FMNIST | VAE | 87.4 | 97.7 | 81.6 | 91.2 | 87.2 | 91.6 | 73.8 | 97.6 | 79.5 | 96.5 | 88.4 OCSVM | 91.9 | 99.0 | 89.4 | 94.2 | 90.7 | 91.8 | 83.4 | 98.8 | 90.3 | 98.2 | 92.8 DAGMM | 30.3 | 31.1 | 47.5 | 48.1 | 49.9 | 41.3 | 42.0 | 37.4 | 51.8 | 37.8 | 41.7 DSEBM | 89.1 | 56.0 | 86.1 | 90.3 | 88.4 | 85.9 | 78.2 | 98.1 | 86.5 | 96.7 | 85.5 MTQM | 92.2 | 95.8 | 89.9 | 93.0 | 92.2 | 89.4 | 84.4 | 98.0 | 94.5 | 98.3 | 92.8 LSA | 91.6 | 98.3 | 87.8 | 92.3 | 89.7 | 90.7 | 84.1 | 97.7 | 91.0 | 98.4 | 92.2 ARAE | 93.7 | 99.1 | 91.1 | 94.4 | 92.3 | 91.4 | 83.6 | 98.9 | 93.9 | 97.9 | 93.6 Table 3: AUC and $F_{1}$ values on the COIL-100 dataset. The values were obtained using protocol 1 for $n\in\\{1,4,7\\}$ and different $\tau$s, where n and $\tau$ represent the number of normal classes and the testing data portion of the normal samples, respectively. Parameters | Metric | OutlierPursuit | DPCP | $l_{1}$ thresholding | GPND | ARAE ---|---|---|---|---|---|--- $n=1$, $\tau=50\%$ | AUC | 0.908 | 0.900 | 0.991 | 0.968 | 0.998 $F_{1}$ | 0.902 | 0.882 | 0.978 | 0.979 | 0.993 $n=4$, $\tau=75\%$ | AUC | 0.837 | 0.859 | 0.992 | 0.945 | 0.997 $F_{1}$ | 0.686 | 0.684 | 0.941 | 0.960 | 0.973 $n=7$, $\tau=85\%$ | AUC | 0.822 | 0.804 | 0.991 | 0.919 | 0.993 $F_{1}$ | 0.528 | 0.511 | 0.897 | 0.941 | 0.941 Table 4: Mean AUC values (in percentage) on CIFAR-10 using protocol 2. Metric | OCSVM | OCGAN | LSA | ARAE ---|---|---|---|--- AUC | 67.8 | 73.3 | 73.1 | 71.7 We present our AUC results for MNIST and Fashion-MNIST in Table 2. The table contains AUC values for each class as the normal class, which were achieved using protocol 2. Moreover, we report our results on the COIL-100 dataset in Table 3. This table contains AUC and $F_{1}$ values for $n\in\\{1,4,7\\}$, where $n$ is the number of normal classes. We use protocol 1 for this dataset. For each $n\in\\{1,4,7\\}$, the percentage of the normal samples in the testing data ($\tau$) is defined in the table. The $F_{1}$ score is reported for the threshold value that is maximizing it. As shown in Tables 2 and 3, we achieve state-of-the-art results in all of these datasets while using a simpler architecture compared to other state-of-the-art methods, such as OCGAN, LSA, and GPND. Moreover, the results in Table 3 indicate that our method performs well when having multiple classes as normal. It also shows the low effect of the number of normal classes on our method performance. We also report our mean AUC results for the CIFAR-10 dataset using protocol 2, excluding the classes with AUC near 0.5 or below, in Table 4. Consider a classifier that labels each input as normal with probability $p$. By varying $p$ between 0 and 1, we can plot a ROC curve and compute its AUC. We observe that this method achieves an AUC of 0.5. So improvements below or near 0.5 aren’t valuable (see (Zhu, Zeng, and Wang 2010) for more details). Consequently, classes 1, 3, 5, 7, and 9 which contained AUC values below 0.6 were excluded. As shown in the table, we get competitive results compared to other state-of-the-art approaches. The AUC values of our method on the medical datasets are reported in Table 1. We used $90\%$ of the normal data for training and the rest in addition to the anomalous data were used to form the testing data. Our method clearly outperforms other state-of-the-art approaches, which shows the effectiveness of our method on medical real-world tasks, where the dataset might be small and complex. To show the stability of our training procedure, we compute the standard deviation of AUCs for the last 50 epochs of training. These values are reported for our method on MNIST in Table 2 and for all the methods on Brain MRI - Tumor in Table 1. From these tables, one can see the high stability of our training procedure. Moreover, It is apparent that our method is much more stable than other methods on the Brain MRI - Tumor dataset. We also evaluate our method using the $F_{1}$ score on the MNIST dataset. In this experiment, the normal class is the positive one. We use protocol 1 and vary $\tau$ between $50\%$ and $90\%$. We use $20\%$ of the training samples and sample from the anomalous classes to form a validation set with the same normal samples percentage as the testing data. This validation set is used to find the threshold that maximizes the $F_{1}$ score. As shown in Figure 4, we achieve slightly lower $F_{1}$ scores compared to that of GPND. However, this figure shows the low impact of the percentage of anomalous data on our method performance. Furthermore, FPR values at $99.5\%$ TPR on the MNIST dataset using protocol 2, for ARAE and LSA are compared in Figure 5. One can see that despite having equal AUCs, ARAE has lower FPR values compared to LSA and that it can reduce the FPR value more than $50\%$ in some cases. #### Adversarial Robustness To show the robustness of our model against adversarial attacks, we use PGD (Madry et al. 2017) with the $\epsilon$ parameter set to $0.05$ and $0.1$ on the reconstruction loss, to craft adversarial samples from the normal samples of the testing data. The normal samples of the testing data are replaced by the adversarial ones. The AUC results for this testing data are reported in Table 5 on the class 8 of the MNIST dataset, using protocol 2. As shown in the table, our method is significantly more robust against adversarial samples compared to LSA. ### Ablation Table 5: AUC values for the attacked models. The values are reported for class 8 of MNIST using protocol 2. Parameters | LSA | ARAE ---|---|--- $\epsilon=0.05$ | 0.56 | 0.86 $\epsilon=0.1$ | 0.17 | 0.76 Table 6: AUC values (in percentage) on MNIST using protocol 2. The results are reported for both one class and two classes as the normal data. Results for other variants of our method are reported. Method | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Mean ---|---|---|---|---|---|---|---|---|---|---|--- DAE | 99.6 | 99.9 | 93.9 | 93.5 | 96.4 | 94.3 | 99.0 | 95.8 | 89.1 | 97.5 | 95.9 ARAE | 99.8 | 99.9 | 96.0 | 97.2 | 97.0 | 97.4 | 99.5 | 96.9 | 92.4 | 98.5 | 97.5 ARAE-A | 99.1 | 99.7 | 95.2 | 96.7 | 97.7 | 98.3 | 99.2 | 97.1 | 95.6 | 96.8 | 97.5 ARAE-R | 99.3 | 99.9 | 93.2 | 92.5 | 96.2 | 96.6 | 99.3 | 97.3 | 91.2 | 98.2 | 96.4 Method | (4, 5) | (0, 7) | (1, 3) | (2, 6) | (8, 9) | (2, 9) | (0, 8) | (0, 1) | (2, 3) | (4, 9) | Mean DAE | 88.8 | 94.1 | 98.2 | 90.3 | 86.8 | 91.8 | 91.1 | 99.7 | 90.0 | 97.3 | 92.8 ARAE | 91.7 | 96.0 | 99.1 | 94.7 | 91.4 | 94.5 | 93.1 | 99.7 | 91.2 | 97.3 | 94.9 ARAE-A | 95.0 | 97.1 | 97.4 | 95.7 | 91.5 | 92.6 | 94.3 | 98.8 | 94.3 | 97.4 | 95.4 Figure 4: $F_{1}$ scores on the MNIST dataset using protocol 1, by taking the normal class as the positive one. Figure 5: FPR at $99.5\%$ TPR on the MNIST dataset using protocol 2. We train a DAE, as a baseline method, with a random uniform noise between 0 and $0.1$ using the same network as the one that is used in our approach. Furthermore, In addition to the $\ell_{\infty}$ perturbation set, we consider $\ell_{2}$, and also rotation and translation perturbation sets. We need to solve a similar optimization to the one in Eq. 5, with the only difference being the perturbation sets (Engstrom et al. 2017). Specifically, we solve this optimization problem on $\ell_{2}$-bounded perturbations for each sample $X\in S$ through PGD (Madry et al. 2017) again. We next solve this optimization on rotation and translation perturbation sets for each sample $X\in S$ by quantizing the parameter space, and performing a grid search on the quantized space and choosing the one with the highest latent loss. This is the most reliable approach for solving rotation and translation perturbations that is mentioned in (Engstrom et al. 2017). Following the approach in (Tramèr and Boneh 2019), we use the union of these perturbation sets to make the attack even stronger to avoid as much as brittle features that model might use (Ilyas et al. 2019). We present our results on MNIST using protocol 2, in Table 6. This variant of our method is denoted as ARAE-A. Notably, the AUC is improved further in this variant in the most challenging class $8$ in MNIST from $92.4$ based on $\ell_{\infty}$ attack to $95.6$ using the union of the mentioned attacks. Despite this improvement, the average AUC is still the same as in the original ARAE method. Instead of designing the attack based on the latent layer, one could directly use the reconstruction loss to do so. We denote this variant as ARAE-R. However, we observed that a model that is robust to the latter attack yields a lower improvement compared to ARAE (see Table 6). To justify this effect, we note that an AE model that is robust based on the latter attack does not necessarily have a stable latent layer. This stems from the fact that the encoder and decoder are almost inverse functions by construction, and a destabilization of the latent encoding by an attack could be repressed by the decoder. In summary, an attack based on the latent layer is stronger than an attack based on the reconstruction error, and hence the former promotes more robust features. We also report AUC values on MNIST by taking pairs of classes as the normal ones, in Table 6. These values show the improvement yield by both of the ARAE variants. Note that when having multiple classes as normal, one should tune the $\epsilon$ parameter based on diversity and complexity of the training data. ## Visualization In the experiments section, we showed that our method improves the AE performance and surpasses other state-of-the-art methods. In order to demonstrate the reasons behind this improvement, we show that ARAE learns more semantically meaningful features than DAE by interpreting these two approaches. MNIST | Fashion-MNIST ---|--- Input | DAE rec. | DAE map | ARAE rec. | ARAE map | Input | DAE rec. | DAE map | ARAE rec. | ARAE map | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Figure 6: ARAE and DAE reconstructions and saliency maps for ten random inputs from MNIST and Fashion-MNIST datasets. ARAE | | | | | | | | ---|---|---|---|---|---|---|---|--- DAE | | | | | | | | Figure 7: Local minima of inputs of ARAE and DAE, by initializing the input with random noise and optimizing the reconstruction loss with respect to the input. ARAE produces more realistic $8$ digits compared to DAE. ### Interpreting with Occlusion-1 In this method, we measure the effect of each part of the input on the output, by occluding it and observing the difference in the output. Finally, we visualize these differences as a saliency map (Zeiler and Fergus 2014; Ancona et al. 2017). In the occlusion-1 method, we iteratively set each pixel to black and then observe the reconstruction error. If it increases, we set the corresponding pixel in the saliency map to blue, and otherwise, we set it to red. The intensity of a pixel is determined by the amount that the reconstruction error has changed. We compare ARAE and DAE reconstructions and saliency maps on MNIST and Fashion-MNIST datasets, in Figure 6. For the MNIST dataset, the model has been trained on the class $8$ and noisy inputs are obtained by adding a uniform noise in the interval $[0,0.4]$. The outputs and saliency maps of ARAE and DAE are shown for five random inputs in the normal class. It is evident that DAE is focusing too much on the random noises and has a poorer reconstruction than our model. Similar to MNIST, we carry out the occlusion-1 method on the class dress of the Fashion-MNIST dataset. For Fashion-MNIST, it is also obvious that random noises have a larger effect on the output of DAE. Furthermore, DAE reconstructions are less accurate than those of ARAE. These observations are consistent with the known fact that adversarial robustness can increase the model interpretability (Tsipras et al. 2018) by avoiding the learning of brittle features (Ilyas et al. 2019). ### Local Minima Visualization We expect from an ideal model that is trained on the MNIST class $8$, to have a lower reconstruction error as the input gets more similar to a typical $8$. With this motivation, we start from random noise and iteratively modify it in order to minimize the reconstruction error using gradient descent. The results achieved by our model and DAE are shown in Figure 7. This figure demonstrates that inputs that lead to local minima in ARAE are much more similar to $8$, compared to DAE. ## Conclusions We introduced a variant of AE based on the robust adversarial training for novelty detection. This is motivated by the goal of learning representations of the input that are almost robust to small irrelevant adversarial changes in the input. A series of novelty detection experiments were performed to evaluate the proposed AE. Our experimental results of the proposed ARAE model show state-of-the-art performance on four publicly available benchmark datasets and two real-world medical datasets. This suggests that the benefits of adversarial robustness indeed go beyond security. Furthermore, by performing an ablation study, we discussed the effect of multiple perturbation sets on the model. Future work inspired by this observation could investigate the effect of other types of adversarial attacks in the proposed framework. ## References * Abati et al. 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Deep autoencoding gaussian mixture model for unsupervised anomaly detection . *[AE]: Autoencoders *[DAE]: Denoising Autoencoder *[ARAE]: Adversarially Robust trained Autoencoder *[GAN]: Generative Adversarial Network *[MSE]: Mean Squared Error *[PGD]: Projected Gradient Descent *[SGD]: Stochastic Gradient Descent